🍩 Database of Original & Non-Theoretical Uses of Topology

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  1. Delineation of a Conserved Arrestin-Biased Signaling Repertoire in Vivo (2015)

    Stuart Maudsley, Bronwen Martin, Diane Gesty-Palmer, Huey Cheung, Calvin Johnson, Shamit Patel, Kevin G. Becker, William H. Wood, Yongqing Zhang, Elin Lehrmann, Louis M. Luttrell
    Abstract Biased G protein–coupled receptor agonists engender a restricted repertoire of downstream events from their cognate receptors, permitting them to produce mixed agonist-antagonist effects in vivo. While this opens the possibility of novel therapeutics, it complicates rational drug design, since the in vivo response to a biased agonist cannot be reliably predicted from its in cellula efficacy. We have employed novel informatic approaches to characterize the in vivo transcriptomic signature of the arrestin pathway-selective parathyroid hormone analog [d-Trp12, Tyr34]bovine PTH(7-34) in six different murine tissues after chronic drug exposure. We find that [d-Trp12, Tyr34]bovine PTH(7-34) elicits a distinctive arrestin-signaling focused transcriptomic response that is more coherently regulated across tissues than that of the pluripotent agonist, human PTH(1-34). This arrestin-focused network is closely associated with transcriptional control of cell growth and development. Our demonstration of a conserved arrestin-dependent transcriptomic signature suggests a framework within which the in vivo outcomes of arrestin-biased signaling may be generalized.

  2. Capturing Dynamics of Time-Varying Data via Topology (2020)

    Lu Xian, Henry Adams, Chad M. Topaz, Lori Ziegelmeier
    Abstract One approach to understanding complex data is to study its shape through the lens of algebraic topology. While the early development of topological data analysis focused primarily on static data, in recent years, theoretical and applied studies have turned to data that varies in time. A time-varying collection of metric spaces as formed, for example, by a moving school of fish or flock of birds, can contain a vast amount of information. There is often a need to simplify or summarize the dynamic behavior. We provide an introduction to topological summaries of time-varying metric spaces including vineyards [17], crocker plots [52], and multiparameter rank functions [34]. We then introduce a new tool to summarize time-varying metric spaces: a crocker stack. Crocker stacks are convenient for visualization, amenable to machine learning, and satisfy a desirable stability property which we prove. We demonstrate the utility of crocker stacks for a parameter identification task involving an influential model of biological aggregations [54]. Altogether, we aim to bring the broader applied mathematics community up-to-date on topological summaries of time-varying metric spaces.

  4. Transfer Learning for Autonomous Chatter Detection in Machining (2022)

    Melih C. Yesilli, Firas A. Khasawneh, Brian P. Mann
    Abstract Large-amplitude chatter vibrations are one of the most important phenomena in machining processes. It is often detrimental in cutting operations causing a poor surface finish and decreased tool life. Therefore, chatter detection using machine learning has been an active research area over the last decade. Three challenges can be identified in applying machine learning for chatter detection at large in industry: an insufficient understanding of the universality of chatter features across different processes, the need for automating feature extraction, and the existence of limited data for each specific workpiece-machine tool combination, e.g., when machining one-off products. These three challenges can be grouped under the umbrella of transfer learning, which is concerned with studying how knowledge gained from one setting can be leveraged to obtain information in new settings. This paper studies automating chatter detection by evaluating transfer learning of prominent as well as novel chatter detection methods. We investigate chatter classification accuracy using a variety of features extracted from turning and milling experiments with different cutting configurations. The studied methods include Fast Fourier Transform (FFT), Power Spectral Density (PSD), the Auto-correlation Function (ACF), and decomposition based tools such as Wavelet Packet Transform (WPT) and Ensemble Empirical Mode Decomposition (EEMD). We also examine more recent approaches based on Topological Data Analysis (TDA) and similarity measures of time series based on Discrete Time Warping (DTW). We evaluate transfer learning potential of each approach by training and testing both within and across the turning and milling data sets. Four supervised classification algorithms are explored: support vector machine (SVM), logistic regression, random forest classification, and gradient boosting. In addition to accuracy, we also comment on the automation potential of feature extraction for each approach which is integral to creating autonomous manufacturing centers. Our results show that carefully chosen time-frequency features can lead to high classification accuracies albeit at the cost of requiring manual pre-processing and the tagging of an expert user. On the other hand, we found that the TDA and DTW approaches can provide accuracies and F1-scores on par with the time-frequency methods without the need for manual preprocessing via completely automatic pipelines. Further, we discovered that the DTW approach outperforms all other methods when trained using the milling data and tested on the turning data. Therefore, TDA and DTW approaches may be preferred over the time-frequency-based approaches for fully automated chatter detection schemes. DTW and TDA also can be more advantageous when pooling data from either limited workpiece-machine tool combinations, or from small data sets of one-off processes.

  5. RGB Image-Based Data Analysis via Discrete Morse Theory and Persistent Homology (2018)

    Chuan Du, Christopher Szul, Adarsh Manawa, Nima Rasekh, Rosemary Guzman, Ruth Davidson
    Abstract Understanding and comparing images for the purposes of data analysis is currently a very computationally demanding task. A group at Australian National University (ANU) recently developed open-source code that can detect fundamental topological features of a grayscale image in a computationally feasible manner. This is made possible by the fact that computers store grayscale images as cubical cellular complexes. These complexes can be studied using the techniques of discrete Morse theory. We expand the functionality of the ANU code by introducing methods and software for analyzing images encoded in red, green, and blue (RGB), because this image encoding is very popular for publicly available data. Our methods allow the extraction of key topological information from RGB images via informative persistence diagrams by introducing novel methods for transforming RGB-to-grayscale. This paradigm allows us to perform data analysis directly on RGB images representing water scarcity variability as well as crime variability. We introduce software enabling a a user to predict future image properties, towards the eventual aim of more rapid image-based data behavior prediction.

  6. TopoGAN: A Topology-Aware Generative Adversarial Network (2020)

    Fan Wang, Huidong Liu, Dimitris Samaras, Chao Chen
    Abstract Existing generative adversarial networks (GANs) focus on generating realistic images based on CNN-derived image features, but fail to preserve the structural properties of real images. This can be fatal in applications where the underlying structure (e.g.., neurons, vessels, membranes, and road networks) of the image carries crucial semantic meaning. In this paper, we propose a novel GAN model that learns the topology of real images, i.e., connectedness and loopy-ness. In particular, we introduce a new loss that bridges the gap between synthetic image distribution and real image distribution in the topological feature space. By optimizing this loss, the generator produces images with the same structural topology as real images. We also propose new GAN evaluation metrics that measure the topological realism of the synthetic images. We show in experiments that our method generates synthetic images with realistic topology. We also highlight the increased performance that our method brings to downstream tasks such as segmentation.

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  7. TDAExplore: Quantitative Analysis of Fluorescence Microscopy Images Through Topology-Based Machine Learning (2021)

    Parker Edwards, Kristen Skruber, Nikola Milićević, James B. Heidings, Tracy-Ann Read, Peter Bubenik, Eric A. Vitriol
    Abstract Recent advances in machine learning have greatly enhanced automatic methods to extract information from fluorescence microscopy data. However, current machine-learning-based models can require hundreds to thousands of images to train, and the most readily accessible models classify images without describing which parts of an image contributed to classification. Here, we introduce TDAExplore, a machine learning image analysis pipeline based on topological data analysis. It can classify different types of cellular perturbations after training with only 20–30 high-resolution images and performs robustly on images from multiple subjects and microscopy modes. Using only images and whole-image labels for training, TDAExplore provides quantitative, spatial information, characterizing which image regions contribute to classification. Computational requirements to train TDAExplore models are modest and a standard PC can perform training with minimal user input. TDAExplore is therefore an accessible, powerful option for obtaining quantitative information about imaging data in a wide variety of applications.

  8. Theory and Algorithms for Constructing Discrete Morse Complexes From Grayscale Digital Images (2011)

    V. Robins, P. J. Wood, A. P. Sheppard
    Abstract We present an algorithm for determining the Morse complex of a two or three-dimensional grayscale digital image. Each cell in the Morse complex corresponds to a topological change in the level sets (i.e., a critical point) of the grayscale image. Since more than one critical point may be associated with a single image voxel, we model digital images by cubical complexes. A new homotopic algorithm is used to construct a discrete Morse function on the cubical complex that agrees with the digital image and has exactly the number and type of critical cells necessary to characterize the topological changes in the level sets. We make use of discrete Morse theory and simple homotopy theory to prove correctness of this algorithm. The resulting Morse complex is considerably simpler than the cubical complex originally used to represent the image and may be used to compute persistent homology.

  9. Combining Geometric and Topological Information in Image Segmentation (2019)

    Hengrui Luo, Justin Strait
    Abstract A fundamental problem in computer vision is image segmentation, where the goal is to delineate the boundary of an object in the image. The focus of this work is on the segmentation of grayscale images and its purpose is two-fold. First, we conduct an in-depth study comparing active contour and topology-based methods in a statistical framework, two popular approaches for boundary detection of 2-dimensional images. Certain properties of the image dataset may favor one method over the other, both from an interpretability perspective as well as through evaluation of performance measures. Second, we propose the use of topological knowledge to assist an active contour method, which can potentially incorporate prior shape information. The latter is known to be extremely sensitive to algorithm initialization, and thus, we use a topological model to provide an automatic initialization. In addition, our proposed model can handle objects in images with more complex topological structures, including objects with holes and multiple objects within one image. We demonstrate this on artificially-constructed image datasets from computer vision, as well as real medical image data.

  10. Cubical Ripser: Software for Computing Persistent Homology of Image and Volume Data (2020)

    Shizuo Kaji, Takeki Sudo, Kazushi Ahara
    Abstract We introduce Cubical Ripser for computing persistent homology of image and volume data. To our best knowledge, Cubical Ripser is currently the fastest and the most memory-efficient program for computing persistent homology of image and volume data. We demonstrate our software with an example of image analysis in which persistent homology and convolutional neural networks are successfully combined. Our open source implementation is available at [14].

  11. Finite Topology as Applied to Image Analysis (1989)

    V. A Kovalevsky
    Abstract The notion of a cellular complex which is well known in the topology is applied to describe the structure of images. It is shown that the topology of cellular complexes is the only possible topology of finite sets. Under this topology no contradictions or paradoxes arise when defining connected subsets and their boundaries. Ways of encoding images as cellular complexes are discussed. The process of image segmentation is considered as splitting (in the topological sense) a cellular complex into blocks of cells. The notion of a cell list is introduced as a precise and compact data structure for encoding segmented images. Some applications of this data structure to the image analysis are demonstrated.

  12. A Multi-Parameter Persistence Framework for Mathematical Morphology (2021)

    Yu-Min Chung, Sarah Day, Chuan-Shen Hu
    Abstract The field of mathematical morphology offers well-studied techniques for image processing. In this work, we view morphological operations through the lens of persistent homology, a tool at the heart of the field of topological data analysis. We demonstrate that morphological operations naturally form a multiparameter filtration and that persistent homology can then be used to extract information about both topology and geometry in the images as well as to automate methods for optimizing the study and rendering of structure in images. For illustration, we apply this framework to analyze noisy binary, grayscale, and color images.

  13. TILT: Topological Interface Recovery in Limited-Angle Tomography (2024)

    Elli Karvonen, Matti Lassas, Pekka Pankka, Samuli Siltanen
    Abstract A wavelet-based sparsity-promoting reconstruction method is studied in the context of tomography with severely limited projection data. Such imaging problems are ill-posed inverse problems, or very sensitive to measurement and modeling errors. The reconstruction method is based on minimizing a sum of a data discrepancy term based on an \$\ell\textasciicircum2\$-norm and another term containing an \$\ell\textasciicircum1\$-norm of a wavelet coefficient vector. Depending on the viewpoint, the method can be considered (i) as finding the Bayesian maximum a posteriori (MAP) estimate using a Besov-space \$B_\11\\textasciicircum\1\(\\mathbb T\\textasciicircum\2\)\$ prior, or (ii) as deterministic regularization with a Besov-norm penalty. The minimization is performed using a tailored primal-dual path following interior-point method, which is applicable to problems larger in scale than commercially available general-purpose optimization package algorithms. The choice of “regularization parameter” is done by a novel technique called the S-curve method, which can be used to incorporate a priori information on the sparsity of the unknown target to the reconstruction process. Numerical results are presented, focusing on uniformly sampled sparse-angle data. Both simulated and measured data are considered, and noise-robust and edge-preserving multiresolution reconstructions are achieved. In sparse-angle cases with simulated data the proposed method offers a significant improvement in reconstruction quality (measured in relative square norm error) over filtered back-projection (FBP) and Tikhonov regularization.

  14. Hepatic Tumor Classification Using Texture and Topology Analysis of Non-Contrast-Enhanced Three-Dimensional T1-Weighted MR Images With a Radiomics Approach (2019)

    Asuka Oyama, Yasuaki Hiraoka, Ippei Obayashi, Yusuke Saikawa, Shigeru Furui, Kenshiro Shiraishi, Shinobu Kumagai, Tatsuya Hayashi, Jun’ichi Kotoku
    Abstract The purpose of this study is to evaluate the accuracy for classification of hepatic tumors by characterization of T1-weighted magnetic resonance (MR) images using two radiomics approaches with machine learning models: texture analysis and topological data analysis using persistent homology. This study assessed non-contrast-enhanced fat-suppressed three-dimensional (3D) T1-weighted images of 150 hepatic tumors. The lesions included 50 hepatocellular carcinomas (HCCs), 50 metastatic tumors (MTs), and 50 hepatic hemangiomas (HHs) found respectively in 37, 23, and 33 patients. For classification, texture features were calculated, and also persistence images of three types (degree 0, degree 1 and degree 2) were obtained for each lesion from the 3D MR imaging data. We used three classification models. In the classification of HCC and MT (resp. HCC and HH, HH and MT), we obtained accuracy of 92% (resp. 90%, 73%) by texture analysis, and the highest accuracy of 85% (resp. 84%, 74%) when degree 1 (resp. degree 1, degree 2) persistence images were used. Our methods using texture analysis or topological data analysis allow for classification of the three hepatic tumors with considerable accuracy, and thus might be useful when applied for computer-aided diagnosis with MR images.

  15. CCF-GNN: A Unified Model Aggregating Appearance, Microenvironment, and Topology for Pathology Image Classification (2023)

    Hongxiao Wang, Gang Huang, Zhuo Zhao, Liang Cheng, Anna Juncker-Jensen, Máté Levente Nagy, Xin Lu, Xiangliang Zhang, Danny Z. Chen
    Abstract Pathology images contain rich information of cell appearance, microenvironment, and topology features for cancer analysis and diagnosis. Among such features, topology becomes increasingly important in analysis for cancer immunotherapy. By analyzing geometric and hierarchically structured cell distribution topology, oncologists can identify densely-packed and cancer-relevant cell communities (CCs) for making decisions. Compared to commonly-used pixel-level Convolution Neural Network (CNN) features and cell-instance-level Graph Neural Network (GNN) features, CC topology features are at a higher level of granularity and geometry. However, topological features have not been well exploited by recent deep learning (DL) methods for pathology image classification due to lack of effective topological descriptors for cell distribution and gathering patterns. In this paper, inspired by clinical practice, we analyze and classify pathology images by comprehensively learning cell appearance, microenvironment, and topology in a fine-to-coarse manner. To describe and exploit topology, we design Cell Community Forest (CCF), a novel graph that represents the hierarchical formulation process of big-sparse CCs from small-dense CCs. Using CCF as a new geometric topological descriptor of tumor cells in pathology images, we propose CCF-GNN, a GNN model that successively aggregates heterogeneous features (e.g., appearance, microenvironment) from cell-instance-level, cell-community-level, into image-level for pathology image classification. Extensive cross-validation experiments show that our method significantly outperforms alternative methods on H&E-stained; immunofluorescence images for disease grading tasks with multiple cancer types. Our proposed CCF-GNN establishes a new topological data analysis (TDA) based method, which facilitates integrating multi-level heterogeneous features of point clouds (e.g., for cells) into a unified DL framework.

  16. Topological Data Analysis in Medical Imaging: Current State of the Art (2023)

    Yashbir Singh, Colleen M. Farrelly, Quincy A. Hathaway, Tim Leiner, Jaidip Jagtap, Gunnar E. Carlsson, Bradley J. Erickson
    Abstract Machine learning, and especially deep learning, is rapidly gaining acceptance and clinical usage in a wide range of image analysis applications and is regarded as providing high performance in detecting anatomical structures and identification and classification of patterns of disease in medical images. However, there are many roadblocks to the widespread implementation of machine learning in clinical image analysis, including differences in data capture leading to different measurements, high dimensionality of imaging and other medical data, and the black-box nature of machine learning, with a lack of insight into relevant features. Techniques such as radiomics have been used in traditional machine learning approaches to model the mathematical relationships between adjacent pixels in an image and provide an explainable framework for clinicians and researchers. Newer paradigms, such as topological data analysis (TDA), have recently been adopted to design and develop innovative image analysis schemes that go beyond the abilities of pixel-to-pixel comparisons. TDA can automatically construct filtrations of topological shapes of image texture through a technique known as persistent homology (PH); these features can then be fed into machine learning models that provide explainable outputs and can distinguish different image classes in a computationally more efficient way, when compared to other currently used methods. The aim of this review is to introduce PH and its variants and to review TDA’s recent successes in medical imaging studies.

  17. The Classification of Endoscopy Images With Persistent Homology (2016)

    Olga Dunaeva, Herbert Edelsbrunner, Anton Lukyanov, Michael Machin, Daria Malkova, Roman Kuvaev, Sergey Kashin
    Abstract Aiming at the automatic diagnosis of tumors using narrow band imaging (NBI) magnifying endoscopic (ME) images of the stomach, we combine methods from image processing, topology, geometry, and machine learning to classify patterns into three classes: oval, tubular and irregular. Training the algorithm on a small number of images of each type, we achieve a high rate of correct classifications. The analysis of the learning algorithm reveals that a handful of geometric and topological features are responsible for the overwhelming majority of decisions.

  18. Skeletonization and Partitioning of Digital Images Using Discrete Morse Theory (2015)

    Olaf Delgado-Friedrichs, Vanessa Robins, Adrian Sheppard
    Abstract We show how discrete Morse theory provides a rigorous and unifying foundation for defining skeletons and partitions of grayscale digital images. We model a grayscale image as a cubical complex with a real-valued function defined on its vertices (the voxel values). This function is extended to a discrete gradient vector field using the algorithm presented in Robins, Wood, Sheppard TPAMI 33:1646 (2011). In the current paper we define basins (the building blocks of a partition) and segments of the skeleton using the stable and unstable sets associated with critical cells. The natural connection between Morse theory and homology allows us to prove the topological validity of these constructions; for example, that the skeleton is homotopic to the initial object. We simplify the basins and skeletons via Morse-theoretic cancellation of critical cells in the discrete gradient vector field using a strategy informed by persistent homology. Simple working Python code for our algorithms for efficient vector field traversal is included. Example data are taken from micro-CT images of porous materials, an application area where accurate topological models of pore connectivity are vital for fluid-flow modelling.

  19. Stable Topological Summaries for Analyzing the Organization of Cells in a Packed Tissue (2021)

    Nieves Atienza, Maria-Jose Jimenez, Manuel Soriano-Trigueros
    Abstract We use topological data analysis tools for studying the inner organization of cells in segmented images of epithelial tissues. More specifically, for each segmented image, we compute different persistence barcodes, which codify the lifetime of homology classes (persistent homology) along different filtrations (increasing nested sequences of simplicial complexes) that are built from the regions representing the cells in the tissue. We use a complete and well-grounded set of numerical variables over those persistence barcodes, also known as topological summaries. A novel combination of normalization methods for both the set of input segmented images and the produced barcodes allows for the proven stability results for those variables with respect to small changes in the input, as well as invariance to image scale. Our study provides new insights to this problem, such as a possible novel indicator for the development of the drosophila wing disc tissue or the importance of centroids’ distribution to differentiate some tissues from their CVT-path counterpart (a mathematical model of epithelia based on Voronoi diagrams). We also show how the use of topological summaries may improve the classification accuracy of epithelial images using a Random Forest algorithm.

  21. Segmentation of Biomedical Images by a Computational Topology Framework (2017)

    Rodrigo Rojas Moraleda, Wei Xiong, Niels Halama, Katja Breitkopf-Heinlein, Steven Steven, Luis Salinas, Dieter W. Heermann, Nektarios A. Valous
    Abstract The segmentation of cell nuclei is an important step towards the automated analysis of histological images. The presence of a large number of nuclei in whole-slide images necessitates methods that are computationally tractable in addition to being effective. In this work, a method is developed for the robust segmentation of cell nuclei in histological images based on the principles of persistent homology. More specifically, an abstract simplicial homology approach for image segmentation is established. Essentially, the approach deals with the persistence of disconnected sets in the image, thus identifying salient regions that express patterns of persistence. By introducing an image representation based on topological features, the task of segmentation is less dependent on variations of color or texture. This results in a novel approach that generalizes well and provides stable performance. The method conceptualizes regions of interest (cell nuclei) pertinent to their topological features in a successful manner. The time cost of the proposed approach is lower-bounded by an almost linear behavior and upper-bounded by O(n2) in a worst-case scenario. Time complexity matches a quasilinear behavior which is O(n1+ɛ) for ε \textless 1. Images acquired from histological sections of liver tissue are used as a case study to demonstrate the effectiveness of the approach. The histological landscape consists of hepatocytes and non-parenchymal cells. The accuracy of the proposed methodology is verified against an automated workflow created by the output of a conventional filter bank (validated by experts) and the supervised training of a random forest classifier. The results are obtained on a per-object basis. The proposed workflow successfully detected both hepatocyte and non-parenchymal cell nuclei with an accuracy of 84.6%, and hepatocyte cell nuclei only with an accuracy of 86.2%. A public histological dataset with supplied ground-truth data is also used for evaluating the performance of the proposed approach (accuracy: 94.5%). Further validations are carried out with a publicly available dataset and ground-truth data from the Gland Segmentation in Colon Histology Images Challenge (GlaS) contest. The proposed method is useful for obtaining unsupervised robust initial segmentations that can be further integrated in image/data processing and management pipelines. The development of a fully automated system supporting a human expert provides tangible benefits in the context of clinical decision-making.

  22. Fast and Accurate Tumor Segmentation of Histology Images Using Persistent Homology and Deep Convolutional Features (2019)

    Talha Qaiser, Yee-Wah Tsang, Daiki Taniyama, Naoya Sakamoto, Kazuaki Nakane, David Epstein, Nasir Rajpoot
    Abstract Tumor segmentation in whole-slide images of histology slides is an important step towards computer-assisted diagnosis. In this work, we propose a tumor segmentation framework based on the novel concept of persistent homology profiles (PHPs). For a given image patch, the homology profiles are derived by efficient computation of persistent homology, which is an algebraic tool from homology theory. We propose an efficient way of computing topological persistence of an image, alternative to simplicial homology. The PHPs are devised to distinguish tumor regions from their normal counterparts by modeling the atypical characteristics of tumor nuclei. We propose two variants of our method for tumor segmentation: one that targets speed without compromising accuracy and the other that targets higher accuracy. The fast version is based on a selection of exemplar image patches from a convolution neural network (CNN) and patch classification by quantifying the divergence between the PHPs of exemplars and the input image patch. Detailed comparative evaluation shows that the proposed algorithm is significantly faster than competing algorithms while achieving comparable results. The accurate version combines the PHPs and high-level CNN features and employs a multi-stage ensemble strategy for image patch labeling. Experimental results demonstrate that the combination of PHPs and CNN features outperform competing algorithms. This study is performed on two independently collected colorectal datasets containing adenoma, adenocarcinoma, signet, and healthy cases. Collectively, the accurate tumor segmentation produces the highest average patch-level F1-score, as compared with competing algorithms, on malignant and healthy cases from both the datasets. Overall the proposed framework highlights the utility of persistent homology for histopathology image analysis.

  23. A Klein-Bottle-Based Dictionary for Texture Representation (2014)

    Jose A. Perea, Gunnar Carlsson
    Abstract A natural object of study in texture representation and material classification is the probability density function, in pixel-value space, underlying the set of small patches from the given image. Inspired by the fact that small \$\$n\times n\$\$n×nhigh-contrast patches from natural images in gray-scale accumulate with high density around a surface \$\$\fancyscript\K\\subset \\mathbb \R\\\textasciicircum\n\textasciicircum2\\$\$K⊂Rn2with the topology of a Klein bottle (Carlsson et al. International Journal of Computer Vision 76(1):1–12, 2008), we present in this paper a novel framework for the estimation and representation of distributions around \$\$\fancyscript\K\\$\$K, of patches from texture images. More specifically, we show that most \$\$n\times n\$\$n×npatches from a given image can be projected onto \$\$\fancyscript\K\\$\$Kyielding a finite sample \$\$S\subset \fancyscript\K\\$\$S⊂K, whose underlying probability density function can be represented in terms of Fourier-like coefficients, which in turn, can be estimated from \$\$S\$\$S. We show that image rotation acts as a linear transformation at the level of the estimated coefficients, and use this to define a multi-scale rotation-invariant descriptor. We test it by classifying the materials in three popular data sets: The CUReT, UIUCTex and KTH-TIPS texture databases.

  24. Feature Detection and Hypothesis Testing for Extremely Noisy Nanoparticle Images Using Topological Data Analysis (2023)

    Andrew M. Thomas, Peter A. Crozier, Yuchen Xu, David S. Matteson
    Abstract We propose a flexible algorithm for feature detection and hypothesis testing in images with ultra-low signal-to-noise ratio using cubical persistent homology. Our main application is in the identification of atomic columns and other features in Transmission Electron Microscopy (TEM). Cubical persistent homology is used to identify local minima and their size in subregions in the frames of nanoparticle videos, which are hypothesized to correspond to relevant atomic features. We compare the performance of our algorithm to other employed methods for the detection of columns and their intensity. Additionally, Monte Carlo goodness-of-fit testing using real-valued summaries of persistence diagrams derived from smoothed images (generated from pixels residing in the vacuum region of an image) is developed and employed to identify whether or not the proposed atomic features generated by our algorithm are due to noise. Using these summaries derived from the generated persistence diagrams, one can produce univariate time series for the nanoparticle videos, thus, providing a means for assessing fluxional behavior. A guarantee on the false discovery rate for multiple Monte Carlo testing of identical hypotheses is also established.

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  25. Persistent Betti Numbers for a Noise Tolerant Shape-Based Approach to Image Retrieval (2011)

    Patrizio Frosini, Claudia Landi
    Abstract In content-based image retrieval a major problem is the presence of noisy shapes. It is well known that persistent Betti numbers are a shape descriptor that admits a dissimilarity distance, the matching distance, stable under continuous shape deformations. In this paper we focus on the problem of dealing with noise that changes the topology of the studied objects. We present a general method to turn persistent Betti numbers into stable descriptors also in the presence of topological changes. Retrieval tests on the Kimia-99 database show the effectiveness of the method.

  26. A Proof-of-Concept Investigation Into Predicting Follicular Carcinoma on Ultrasound Using Topological Data Analysis and Radiomics (2025)

    Andrew M. Thomas, Ann C. Lin, Grace Deng, Yuchen Xu, Gustavo Fernandez Ranvier, Aida Taye, David S. Matteson, Denise Lee
    Abstract Background Sonographic risk patterns identified in established risk stratification systems (RSS) may not accurately stratify follicular carcinoma from adenoma, which share many similar US characteristics. The purpose of this study is to investigate the performance of a multimodal machine learning model utilizing radiomics and topological data analysis (TDA) to predict malignancy in follicular thyroid neoplasms on ultrasound. Patients & Methods This is a retrospective study of patients who underwent thyroidectomy with pathology confirmed follicular adenoma or carcinoma at a single academic medical center between 2010 and 2022. Features derived from radiomics and TDA were calculated from processed ultrasound images and high-dimensional features in each modality were projected onto their first two principal components. Logistic regression with L2 penalty was used to predict malignancy and performance was evaluated using leave-one-out cross-validation and area under the curve (AUC). Results Patients with follicular adenomas (n = 7) and follicular carcinomas (n = 11) with available imaging were included. The best multimodal model achieved an AUC of 0.88 (95% CI: [0.85, 1]), whereas the best radiomics model achieved an AUC of 0.68 (95% CI: [0.61, 0.84]). Conclusions We demonstrate that inclusion of topological features yields strong improvement over radiomics-based features alone in the prediction of follicular carcinoma on ultrasound. Despite low volume data, the TDA features explicitly capture shape information that likely augments performance of the multimodal machine learning model. This approach suggests that a quantitative based US RSS may contribute to the preoperative prediction of follicular carcinoma.

  27. PI-Net: A Deep Learning Approach to Extract Topological Persistence Images (2020)

    Anirudh Som, Hongjun Choi, Karthikeyan Natesan Ramamurthy, Matthew Buman, Pavan Turaga
    Abstract Topological features such as persistence diagrams and their functional approximations like persistence images (PIs) have been showing substantial promise for machine learning and computer vision applications. This is greatly attributed to the robustness topological representations provide against different types of physical nuisance variables seen in real-world data, such as view-point, illumination, and more. However, key bottlenecks to their large scale adoption are computational expenditure and difficulty incorporating them in a differentiable architecture. We take an important step in this paper to mitigate these bottlenecks by proposing a novel one-step approach to generate PIs directly from the input data. We design two separate convolutional neural network architectures, one designed to take in multi-variate time series signals as input and another that accepts multi-channel images as input. We call these networks Signal PI-Net and Image PINet respectively. To the best of our knowledge, we are the first to propose the use of deep learning for computing topological features directly from data. We explore the use of the proposed PI-Net architectures on two applications: human activity recognition using tri-axial accelerometer sensor data and image classification. We demonstrate the ease of fusion of PIs in supervised deep learning architectures and speed up of several orders of magnitude for extracting PIs from data. Our code is available at https://github.com/anirudhsom/PI-Net.

  28. Topological Descriptors of Histology Images (2014)

    Nikhil Singh, Heather D. Couture, J. S. Marron, Charles Perou, Marc Niethammer
    Abstract The purpose of this study is to investigate architectural characteristics of cell arrangements in breast cancer histology images. We propose the use of topological data analysis to summarize the geometric information inherent in tumor cell arrangements. Our goal is to use this information as signatures that encode robust summaries of cell arrangements in tumor tissue as captured through histology images. In particular, using ideas from algebraic topology we construct topological descriptors based on cell nucleus segmentations such as persistency charts and Betti sequences. We assess their performance on the task of discriminating the breast cancer subtypes Basal, Luminal A, Luminal B and HER2. We demonstrate that the topological features contain useful complementary information to image-appearance based features that can improve discriminatory performance of classifiers.

  30. Advancing Precision Medicine: Algebraic Topology and Differential Geometry in Radiology and Computational Pathology (2024)

    Richard M. Levenson, Yashbir Singh, Bastian Rieck, Quincy A. Hathaway, Colleen Farrelly, Jennifer Rozenblit, Prateek Prasanna, Bradley Erickson, Ashok Choudhary, Gunnar Carlsson, Deepa Sarkar

  31. Computing Robustness and Persistence for Images (2010)

    P. Bendich, H. Edelsbrunner, M. Kerber
    Abstract We are interested in 3-dimensional images given as arrays of voxels with intensity values. Extending these values to a continuous function, we study the robustness of homology classes in its level and interlevel sets, that is, the amount of perturbation needed to destroy these classes. The structure of the homology classes and their robustness, over all level and interlevel sets, can be visualized by a triangular diagram of dots obtained by computing the extended persistence of the function. We give a fast hierarchical algorithm using the dual complexes of oct-tree approximations of the function. In addition, we show that for balanced oct-trees, the dual complexes are geometrically realized in R3 and can thus be used to construct level and interlevel sets. We apply these tools to study 3-dimensional images of plant root systems.

  32. A Novel Multi-Task Machine Learning Classifier for Rare Disease Patterning Using Cardiac Strain Imaging Data (2024)

    Nanda K. Siva, Yashbir Singh, Quincy A. Hathaway, Partho P. Sengupta, Naveena Yanamala
    Abstract To provide accurate predictions, current machine learning-based solutions require large, manually labeled training datasets. We implement persistent homology (PH), a topological tool for studying the pattern of data, to analyze echocardiography-based strain data and differentiate between rare diseases like constrictive pericarditis (CP) and restrictive cardiomyopathy (RCM). Patient population (retrospectively registered) included those presenting with heart failure due to CP (n = 51), RCM (n = 47), and patients without heart failure symptoms (n = 53). Longitudinal, radial, and circumferential strains/strain rates for left ventricular segments were processed into topological feature vectors using Machine learning PH workflow. In differentiating CP and RCM, the PH workflow model had a ROC AUC of 0.94 (Sensitivity = 92%, Specificity = 81%), compared with the GLS model AUC of 0.69 (Sensitivity = 65%, Specificity = 66%). In differentiating between all three conditions, the PH workflow model had an AUC of 0.83 (Sensitivity = 68%, Specificity = 84%), compared with the GLS model AUC of 0.68 (Sensitivity = 52% and Specificity = 76%). By employing persistent homology to differentiate the “pattern” of cardiac deformations, our machine-learning approach provides reasonable accuracy when evaluating small datasets and aids in understanding and visualizing patterns of cardiac imaging data in clinically challenging disease states.

    Community Resources

  33. The Weighted Euler Curve Transform for Shape and Image Analysis (2020)

    Qitong Jiang, Sebastian Kurtek, Tom Needham
    Abstract The Euler Curve Transform (ECT) of Turner et al. is a complete invariant of an embedded simplicial complex, which is amenable to statistical analysis. We generalize the ECT to provide a similarly convenient representation for weighted simplicial complexes, objects which arise naturally, for example, in certain medical imaging applications. We leverage work of Ghrist et al. on Euler integral calculus to prove that this invariant—dubbed the Weighted Euler Curve Transform (WECT)—is also complete. We explain how to transform a segmented region of interest in a grayscale image into a weighted simplicial complex and then into a WECT representation. This WECT representation is applied to study Glioblastoma Multiforme brain tumor shape and texture data. We show that the WECT representation is effective at clustering tumors based on qualitative shape and texture features and that this clustering correlates with patient survival time.

  34. A Novel Quality Clustering Methodology on Fab-Wide Wafer Map Images in Semiconductor Manufacturing (2022)

    Yuan-Ming Hsu, Xiaodong Jia, Wenzhe Li, Jay Lee
    Abstract Abstract. In semiconductor manufacturing, clustering the fab-wide wafer map images is of critical importance for practitioners to understand the subclusters of wafer defects, recognize novel clusters or anomalies, and develop fast reactions to quality issues. However, due to the high-mix manufacturing of diversified wafer products of different sizes and technologies, it is difficult to cluster the wafer map images across the fab. This paper addresses this challenge by proposing a novel methodology for fab-wide wafer map data clustering. In the proposed methodology, a well-known deep learning technique, vision transformer with multi-head attention is first trained to convert binary wafer images of different sizes into condensed feature vectors for efficient clustering. Then, the Topological Data Analysis (TDA), which is widely used in biomedical applications, is employed to visualize the data clusters and identify the anomalies. The TDA yields a topological representation of high-dimensional big data as well as its local clusters by creating a graph that shows nodes corresponding to the clusters within the data. The effectiveness of the proposed methodology is demonstrated by clustering the public wafer map dataset WM-811k from the real application which has a total of 811,457 wafer map images. We further demonstrate the potential applicability of topology data analytics in the semiconductor area by visualization.

  35. Automatic Tree Ring Detection Using Jacobi Sets (2020)

    Kayla Makela, Tim Ophelders, Michelle Quigley, Elizabeth Munch, Daniel Chitwood, Asia Dowtin
    Abstract Tree ring widths are an important source of climatic and historical data, but measuring these widths typically requires extensive manual work. Computer vision techniques provide promising directions towards the automation of tree ring detection, but most automated methods still require a substantial amount of user interaction to obtain high accuracy. We perform analysis on 3D X-ray CT images of a cross-section of a tree trunk, known as a tree disk. We present novel automated methods for locating the pith (center) of a tree disk, and ring boundaries. Our methods use a combination of standard image processing techniques and tools from topological data analysis. We evaluate the efficacy of our method for two different CT scans by comparing its results to manually located rings and centers and show that it is better than current automatic methods in terms of correctly counting each ring and its location. Our methods have several parameters, which we optimize experimentally by minimizing edit distances to the manually obtained locations.

  36. Predicting Clinical Outcomes in Glioblastoma: An Application of Topological and Functional Data Analysis (2019)

    Lorin Crawford, Anthea Monod, Andrew X. Chen, Sayan Mukherjee, Raúl Rabadán
    Abstract Glioblastoma multiforme (GBM) is an aggressive form of human brain cancer that is under active study in the field of cancer biology. Its rapid progression and the relative time cost of obtaining molecular data make other readily available forms of data, such as images, an important resource for actionable measures in patients. Our goal is to use information given by medical images taken from GBM patients in statistical settings. To do this, we design a novel statistic—the smooth Euler characteristic transform (SECT)—that quantifies magnetic resonance images of tumors. Due to its well-defined inner product structure, the SECT can be used in a wider range of functional and nonparametric modeling approaches than other previously proposed topological summary statistics. When applied to a cohort of GBM patients, we find that the SECT is a better predictor of clinical outcomes than both existing tumor shape quantifications and common molecular assays. Specifically, we demonstrate that SECT features alone explain more of the variance in GBM patient survival than gene expression, volumetric features, and morphometric features. The main takeaways from our findings are thus 2-fold. First, they suggest that images contain valuable information that can play an important role in clinical prognosis and other medical decisions. Second, they show that the SECT is a viable tool for the broader study of medical imaging informatics. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.

  37. On the Local Behavior of Spaces of Natural Images (2008)

    Gunnar Carlsson, Tigran Ishkhanov, Vin de Silva, Afra Zomorodian
    Abstract In this study we concentrate on qualitative topological analysis of the local behavior of the space of natural images. To this end, we use a space of 3 by 3 high-contrast patches ℳ. We develop a theoretical model for the high-density 2-dimensional submanifold of ℳ showing that it has the topology of the Klein bottle. Using our topological software package PLEX we experimentally verify our theoretical conclusions. We use polynomial representation to give coordinatization to various subspaces of ℳ. We find the best-fitting embedding of the Klein bottle into the ambient space of ℳ. Our results are currently being used in developing a compression algorithm based on a Klein bottle dictionary.

  38. Algebraic Topology-Based Machine Learning Using MRI Predicts Outcomes in Primary Sclerosing Cholangitis (2022)

    Yashbir Singh, William A. Jons, John E. Eaton, Mette Vesterhus, Tom Karlsen, Ida Bjoerk, Andreas Abildgaard, Kristin Kaasen Jorgensen, Trine Folseraas, Derek Little, Aliya F. Gulamhusein, Kosta Petrovic, Anne Negard, Gian Marco Conte, Joseph D. Sobek, Jaidip Jagtap, Sudhakar K. Venkatesh, Gregory J. Gores, Nicholas F. LaRusso, Konstantinos N. Lazaridis, Bradley J. Erickson
    Abstract Primary sclerosing cholangitis (PSC) is a chronic cholestatic liver disease that can lead to cirrhosis and hepatic decompensation. However, predicting future outcomes in patients with PSC is challenging. Our aim was to extract magnetic resonance imaging (MRI) features that predict the development of hepatic decompensation by applying algebraic topology-based machine learning (ML).

  39. Imaging-Based Representation and Stratification of Intra-Tumor Heterogeneity via Tree-Edit Distance (2022)

    Lara Cavinato, Matteo Pegoraro, Alessandra Ragni, Francesca Ieva
    Abstract Personalized medicine is the future of medical practice. In oncology, tumor heterogeneity assessment represents a pivotal step for effective treatment planning and prognosis prediction. Despite new procedures for DNA sequencing and analysis, non-invasive methods for tumor characterization are needed to impact on daily routine. On purpose, imaging texture analysis is rapidly scaling, holding the promise to surrogate histopathological assessment of tumor lesions. In this work, we propose a tree-based representation strategy for describing intra-tumor heterogeneity of patients affected by metastatic cancer. We leverage radiomics information extracted from PET/CT imaging and we provide an exhaustive and easily readable summary of the disease spreading. We exploit this novel patient representation to perform cancer subtyping according to hierarchical clustering technique. To this purpose, a new heterogeneity-based distance between trees is defined and applied to a case study of prostate cancer. Clusters interpretation is explored in terms of concordance with severity status, tumor burden and biological characteristics. Results are promising, as the proposed method outperforms current literature approaches. Ultimately, the proposed method draws a general analysis framework that would allow to extract knowledge from daily acquired imaging data of patients and provide insights for effective treatment planning.

  40. A Sheaf and Topology Approach to Generating Local Branch Numbers in Digital Images (2020)

    Chuan-Shen Hu, Yu-Min Chung
    Abstract This paper concerns a theoretical approach that combines topological data analysis (TDA) and sheaf theory. Topological data analysis, a rising field in mathematics and computer science, concerns the shape of the data and has been proven effective in many scientific disciplines. Sheaf theory, a mathematics subject in algebraic geometry, provides a framework for describing the local consistency in geometric objects. Persistent homology (PH) is one of the main driving forces in TDA, and the idea is to track changes of geometric objects at different scales. The persistence diagram (PD) summarizes the information of PH in the form of a multi-set. While PD provides useful information about the underlying objects, it lacks fine relations about the local consistency of specific pairs of generators in PD, such as the merging relation between two connected components in the PH. The sheaf structure provides a novel point of view for describing the merging relation of local objects in PH. It is the goal of this paper to establish a theoretic framework that utilizes the sheaf theory to uncover finer information from the PH. We also show that the proposed theory can be applied to identify the branch numbers of local objects in digital images.

  41. Multiphase Mixing Quantification by Computational Homology and Imaging Analysis (2011)

    Jianxin Xu, Hua Wang, Hui Fang
    Abstract The purpose of this study is to introduce a new technique for quantifying the efficiency of multiphase mixing. This technique based on algebraic topology is illustrated by using the hydraulic modeling of gas agitated reactors stirred by top lance gas injection and image analysis. The zeroth Betti numbers are used to estimate the numbers of pieces in the patterns, leading to a useful parameter to characterize the mixture homogeneity. The first Betti numbers are introduced to characterize the nonhomogeneity of the mixture. The mixing efficiency can be characterized by the Betti numbers for binary images of the patterns. This novel method may be applied for studying a variety of multiphase mixing problems in which multiphase components or tracers are visually distinguishable.

  42. Hypothesis Testing for Shapes Using Vectorized Persistence Diagrams (2020)

    Chul Moon, Nicole A. Lazar
    Abstract Topological data analysis involves the statistical characterization of the shape of data. Persistent homology is a primary tool of topological data analysis, which can be used to analyze those topological features and perform statistical inference. In this paper, we present a two-stage hypothesis test for vectorized persistence diagrams. The first stage filters elements in the vectorized persistence diagrams to reduce false positives. The second stage consists of multiple hypothesis tests, with false positives controlled by false discovery rates. We demonstrate applications of the proposed procedure on simulated point clouds and three-dimensional rock image data. Our results show that the proposed hypothesis tests can provide flexible and informative inferences on the shape of data with lower computational cost compared to the permutation test.

  43. A Primer on Topological Data Analysis to Support Image Analysis Tasks in Environmental Science (2023)

    Lander Ver Hoef, Henry Adams, Emily J. King, Imme Ebert-Uphoff
    Abstract Abstract Topological data analysis (TDA) is a tool from data science and mathematics that is beginning to make waves in environmental science. In this work, we seek to provide an intuitive and understandable introduction to a tool from TDA that is particularly useful for the analysis of imagery, namely, persistent homology. We briefly discuss the theoretical background but focus primarily on understanding the output of this tool and discussing what information it can glean. To this end, we frame our discussion around a guiding example of classifying satellite images from the sugar, fish, flower, and gravel dataset produced for the study of mesoscale organization of clouds by Rasp et al. We demonstrate how persistent homology and its vectorization, persistence landscapes, can be used in a workflow with a simple machine learning algorithm to obtain good results, and we explore in detail how we can explain this behavior in terms of image-level features. One of the core strengths of persistent homology is how interpretable it can be, so throughout this paper we discuss not just the patterns we find but why those results are to be expected given what we know about the theory of persistent homology. Our goal is that readers of this paper will leave with a better understanding of TDA and persistent homology, will be able to identify problems and datasets of their own for which persistent homology could be helpful, and will gain an understanding of the results they obtain from applying the included GitHub example code. Significance Statement Information such as the geometric structure and texture of image data can greatly support the inference of the physical state of an observed Earth system, for example, in remote sensing to determine whether wildfires are active or to identify local climate zones. Persistent homology is a branch of topological data analysis that allows one to extract such information in an interpretable way—unlike black-box methods like deep neural networks. The purpose of this paper is to explain in an intuitive manner what persistent homology is and how researchers in environmental science can use it to create interpretable models. We demonstrate the approach to identify certain cloud patterns from satellite imagery and find that the resulting model is indeed interpretable.

  44. Image-Based Phenotyping for Identification of QTL Determining Fruit Shape and Size in American Cranberry (Vaccinium Macrocarpon L.) (2018)

    Luis Diaz-Garcia, Giovanny Covarrubias-Pazaran, Brandon Schlautman, Edward Grygleski, Juan Zalapa
    Abstract Image-based phenotyping methodologies are powerful tools to determine quality parameters for fruit breeders and processors. The fruit size and shape of American cranberry (Vaccinium macrocarpon L.) are particularly important characteristics that determine the harvests’ processing value and potential end-use products (e.g., juice vs. sweetened dried cranberries). However, cranberry fruit size and shape attributes can be difficult and time consuming for breeders and processors to measure, especially when relying on manual measurements and visual ratings. Therefore, in this study, we implemented image-based phenotyping techniques for gathering data regarding basic cranberry fruit parameters such as length, width, length-to-width ratio, and eccentricity. Additionally, we applied a persistent homology algorithm to better characterize complex shape parameters. Using this high-throughput artificial vision approach, we characterized fruit from 351 progeny from a full-sib cranberry population over three field seasons. Using a covariate analysis to maximize the identification of well-supported quantitative trait loci (QTL), we found 252 single QTL in a 3-year period for cranberry fruit size and shape descriptors from which 20% were consistently found in all years. The present study highlights the potential for the identified QTL and the image-based methods to serve as a basis for future explorations of the genetic architecture of fruit size and shape in cranberry and other fruit crops.

  49. Persistent Homology for Breast Tumor Classification Using Mammogram Scans (2022)

    Aras Asaad, Dashti Ali, Taban Majeed, Rasber Rashid
    Abstract An Important tool in the field topological data analysis is known as persistent Homology (PH) which is used to encode abstract representation of the homology of data at different resolutions in the form of persistence diagram (PD). In this work we build more than one PD representation of a single image based on a landmark selection method, known as local binary patterns, that encode different types of local textures from images. We employed different PD vectorizations using persistence landscapes, persistence images, persistence binning (Betti Curve) and statistics. We tested the effectiveness of proposed landmark based PH on two publicly available breast abnormality detection datasets using mammogram scans. Sensitivity of landmark based PH obtained is over 90% in both datasets for the detection of abnormal breast scans. Finally, experimental results give new insights on using different types of PD vectorizations which help in utilising PH in conjunction with machine learning classifiers.

  50. Advancing Precision Medicine: Algebraic Topology and Differential Geometry in Radiology and Computational Pathology (2024)

    Richard M. Levenson, Yashbir Singh, Bastian Rieck, Ashok Choudhary, Gunnar Carlsson, Deepa Sarkar, Quincy A. Hathaway, Colleen Farrelly, Jennifer Rozenblit, Prateek Prasanna, Bradley Erickson
    Abstract Precision medicine aims to provide personalized care based on individual patient characteristics, rather than guideline-directed therapies for groups of diseases or patient demographics. Images—both radiology- and pathology-derived—are a major source of information on presence, type, and status of disease. Exploring the mathematical relationship of pixels in medical imaging (“radiomics”) and cellular-scale structures in digital pathology slides (“pathomics”) offers powerful tools for extracting both qualitative and, increasingly, quantitative data. These analytical approaches, however, may be significantly enhanced by applying additional methods arising from fields of mathematics such as differential geometry and algebraic topology that remain underexplored in this context. Geometry’s strength lies in its ability to provide precise local measurements, such as curvature, that can be crucial for identifying abnormalities at multiple spatial levels. These measurements can augment the quantitative features extracted in conventional radiomics, leading to more nuanced diagnostics. By contrast, topology serves as a robust shape descriptor, capturing essential features such as connected components and holes. The field of topological data analysis was initially founded to explore the shape of data, with functional network connectivity in the brain being a prominent example. Increasingly, its tools are now being used to explore organizational patterns of physical structures in medical images and digitized pathology slides. By leveraging tools from both differential geometry and algebraic topology, researchers and clinicians may be able to obtain a more comprehensive, multi-layered understanding of medical images and contribute to precision medicine’s armamentarium

  51. TDA-Net: Fusion of Persistent Homology and Deep Learning Features for COVID-19 Detection From Chest X-Ray Images (2021)

    Mustafa Hajij, Ghada Zamzmi, Fawwaz Batayneh
    Abstract Topological Data Analysis (TDA) has emerged recently as a robust tool to extract and compare the structure of datasets. TDA identifies features in data (e.g., connected components and holes) and assigns a quantitative measure to these features. Several studies reported that topological features extracted by TDA tools provide unique information about the data, discover new insights, and determine which feature is more related to the outcome. On the other hand, the overwhelming success of deep neural networks in learning patterns and relationships has been proven on various data applications including images. To capture the characteristics of both worlds, we propose TDA-Net, a novel ensemble network that fuses topological and deep features for the purpose of enhancing model generalizability and accuracy. We apply the proposed TDA-Net to a critical application, which is the automated detection of COVID-19 from CXR images. Experimental results showed that the proposed network achieved excellent performance and suggested the applicability of our method in practice.

  52. Capturing Shape Information With Multi-Scale Topological Loss Terms For 3D Reconstruction (2022)

    Dominik J. E. Waibel, Scott Atwell, Matthias Meier, Carsten Marr, Bastian Rieck
    Abstract Reconstructing 3D objects from 2D images is both challenging for our brains and machine learning algorithms. To support this spatial reasoning task, contextual information about the overall shape of an object is critical. However, such information is not captured by established loss terms (e.g. Dice loss). We propose to complement geometrical shape information by including multi-scale topological features, such as connected components, cycles, and voids, in the reconstruction loss. Our method uses cubical complexes to calculate topological features of 3D volume data and employs an optimal transport distance to guide the reconstruction process. This topology-aware loss is fully differentiable, computationally efficient, and can be added to any neural network. We demonstrate the utility of our loss by incorporating it into SHAPR, a model for predicting the 3D cell shape of individual cells based on 2D microscopy images. Using a hybrid loss that leverages both geometrical and topological information of single objects to assess their shape, we find that topological information substantially improves the quality of reconstructions, thus highlighting its ability to extract more relevant features from image datasets.

  53. Histopathological Cancer Detection With Topological Signatures (2023)

    Ankur Yadav, Faisal Ahmed, Ovidiu Daescu, Reyhan Gedik, Baris Coskunuzer
    Abstract We present a transformative approach to histopathological cancer detection and grading by introducing a very powerful feature extraction method based on the latest topological data analysis tools. By analyzing the evolution of topological patterns in different color channels, we discovered that every tumor class leaves its own topological footprint in histopathological images, allowing to extract feature vectors that can be used to reliably identify tumor classes.Our topological signatures, even when combined with traditional machine learning methods, provide very fast and highly accurate results in various settings. While most DL models work well for one type of cancer, our model easily adapts to different scenarios, and consistently gives highly competitive results with the state-of-the-art models on benchmark datasets across multiple cancer types including bone, colon, breast, cervical (cytopathology), and prostate cancer. Unlike most DL models, our proposed Topo-ML model does not need any data augmentation or pre-processing steps and works perfectly on small datasets. The model is computationally very efficient, with end-to-end processing taking only a few hours for datasets consisting of thousands of images.

  54. Morse Theory and Persistent Homology for Topological Analysis of 3D Images of Complex Materials (2014)

    O. Delgado-Friedrichs, V. Robins, A. Sheppard
    Abstract We develop topologically accurate and compatible definitions for the skeleton and watershed segmentation of a 3D digital object that are computed by a single algorithm. These definitions are based on a discrete gradient vector field derived from a signed distance transform. This gradient vector field is amenable to topological analysis and simplification via For-man's discrete Morse theory and provides a filtration that can be used as input to persistent homology algorithms. Efficient implementations allow us to process large-scale x-ray micro-CT data of rock cores and other materials.

  55. Topological Data Analysis of Spatial Patterning in Heterogeneous Cell Populations: Clustering and Sorting With Varying Cell-Cell Adhesion (2023)

    Dhananjay Bhaskar, William Y. Zhang, Alexandria Volkening, Björn Sandstede, Ian Y. Wong
    Abstract Different cell types aggregate and sort into hierarchical architectures during the formation of animal tissues. The resulting spatial organization depends (in part) on the strength of adhesion of one cell type to itself relative to other cell types. However, automated and unsupervised classification of these multicellular spatial patterns remains challenging, particularly given their structural diversity and biological variability. Recent developments based on topological data analysis are intriguing to reveal similarities in tissue architecture, but these methods remain computationally expensive. In this article, we show that multicellular patterns organized from two interacting cell types can be efficiently represented through persistence images. Our optimized combination of dimensionality reduction via autoencoders, combined with hierarchical clustering, achieved high classification accuracy for simulations with constant cell numbers. We further demonstrate that persistence images can be normalized to improve classification for simulations with varying cell numbers due to proliferation. Finally, we systematically consider the importance of incorporating different topological features as well as information about each cell type to improve classification accuracy. We envision that topological machine learning based on persistence images will enable versatile and robust classification of complex tissue architectures that occur in development and disease.

  56. A Morphometric Analysis of Vegetation Patterns in Dryland Ecosystems (2017)

    Luke Mander, Stefan C. Dekker, Mao Li, Washington Mio, Surangi W. Punyasena, Timothy M. Lenton
    Abstract Vegetation in dryland ecosystems often forms remarkable spatial patterns. These range from regular bands of vegetation alternating with bare ground, to vegetated spots and labyrinths, to regular gaps of bare ground within an otherwise continuous expanse of vegetation. It has been suggested that spotted vegetation patterns could indicate that collapse into a bare ground state is imminent, and the morphology of spatial vegetation patterns, therefore, represents a potentially valuable source of information on the proximity of regime shifts in dryland ecosystems. In this paper, we have developed quantitative methods to characterize the morphology of spatial patterns in dryland vegetation. Our approach is based on algorithmic techniques that have been used to classify pollen grains on the basis of textural patterning, and involves constructing feature vectors to quantify the shapes formed by vegetation patterns. We have analysed images of patterned vegetation produced by a computational model and a small set of satellite images from South Kordofan (South Sudan), which illustrates that our methods are applicable to both simulated and real-world data. Our approach provides a means of quantifying patterns that are frequently described using qualitative terminology, and could be used to classify vegetation patterns in large-scale satellite surveys of dryland ecosystems.

  57. Data-Driven and Automatic Surface Texture Analysis Using Persistent Homology (2021)

    Melih C. Yesilli, Firas A. Khasawneh
    Abstract Surface roughness plays an important role in analyzing engineering surfaces. It quantifies the surface topography and can be used to determine whether the resulting surface finish is acceptable or not. Nevertheless, while several existing tools and standards are available for computing surface roughness, these methods rely heavily on user input thus slowing down the analysis and increasing manufacturing costs. Therefore, fast and automatic determination of the roughness level is essential to avoid costs resulting from surfaces with unacceptable finish, and user-intensive analysis. In this study, we propose a Topological Data Analysis (TDA) based approach to classify the roughness level of synthetic surfaces using both their areal images and profiles. We utilize persistent homology from TDA to generate persistence diagrams that encapsulate information on the shape of the surface. We then obtain feature matrices for each surface or profile using Carlsson coordinates, persistence images, and template functions. We compare our results to two widely used methods in the literature: Fast Fourier Transform (FFT) and Gaussian filtering. The results show that our approach yields mean accuracies as high as 97%. We also show that, in contrast to existing surface analysis tools, our TDA-based approach is fully automatable and provides adaptive feature extraction.

  58. Barcodes Distinguish Morphology of Neuronal Tauopathy (2022)

    David Beers, Despoina Goniotaki, Diane P. Hanger, Alain Goriely, Heather A. Harrington
    Abstract The geometry of neurons is known to be important for their functions. Hence, neurons are often classified by their morphology. Two recent methods, persistent homology and the topological morphology descriptor, assign a morphology descriptor called a barcode to a neuron equipped with a given function, such as the Euclidean distance from the root of the neuron. These barcodes can be converted into matrices called persistence images, which can then be averaged across groups. We show that when the defining function is the path length from the root, both the topological morphology descriptor and persistent homology are equivalent. We further show that persistence images arising from the path length procedure provide an interpretable summary of neuronal morphology. We introduce \topological morphology functions\, a class of functions similar to Sholl functions, that can be recovered from the associated topological morphology descriptor. To demonstrate this topological approach, we compare healthy cortical and hippocampal mouse neurons to those affected by progressive tauopathy. We find a significant difference in the morphology of healthy neurons and those with a tauopathy at a postsymptomatic age. We use persistence images to conclude that the diseased group tends to have neurons with shorter branches as well as fewer branches far from the soma.

  59. Exploring Surface Texture Quantification in Piezo Vibration Striking Treatment (PVST) Using Topological Measures (2022)

    Melih C. Yesilli, Max M. Chumley, Jisheng Chen, Firas A. Khasawneh, Yang Guo
    Abstract Abstract. Surface texture influences wear and tribological properties of manufactured parts, and it plays a critical role in end-user products. Therefore, quantifying the order or structure of a manufactured surface provides important information on the quality and life expectancy of the product. Although texture can be intentionally introduced to enhance aesthetics or to satisfy a design function, sometimes it is an inevitable byproduct of surface treatment processes such as Piezo Vibration Striking Treatment (PVST). Measures of order for surfaces have been characterized using statistical, spectral, and geometric approaches. For nearly hexagonal lattices, topological tools have also been used to measure the surface order. This paper explores utilizing tools from Topological Data Analysis for measuring surface texture. We compute measures of order based on optical digital microscope images of surfaces treated using PVST. These measures are applied to the grid obtained from estimating the centers of tool impacts, and they quantify the grid’s deviations from the nominal one. Our results show that TDA provides a convenient framework for characterization of pattern type that bypasses some limitations of existing tools such as difficult manual processing of the data and the need for an expert user to analyze and interpret the surface images.

  60. Topology-Aware Segmentation Using Discrete Morse Theory (2021)

    Xiaoling Hu, Yusu Wang, Li Fuxin, Dimitris Samaras, Chao Chen
    Abstract In the segmentation of fine-scale structures from natural and biomedical images, per-pixel accuracy is not the only metric of concern. Topological correctness, such as vessel connectivity and membrane closure, is crucial for downstream analysis tasks. In this paper, we propose a new approach to train deep image segmentation networks for better topological accuracy. In particular, leveraging the power of discrete Morse theory (DMT), we identify global structures, including 1D skeletons and 2D patches, which are important for topological accuracy. Trained with a novel loss based on these global structures, the network performance is significantly improved especially near topologically challenging locations (such as weak spots of connections and membranes). On diverse datasets, our method achieves superior performance on both the DICE score and topological metrics.

  61. Euler Characteristic Surfaces (2021)

    Gabriele Beltramo, Rayna Andreeva, Ylenia Giarratano, Miguel O. Bernabeu, Rik Sarkar, Primoz Skraba
    Abstract We study the use of the Euler characteristic for multiparameter topological data analysis. Euler characteristic is a classical, well-understood topological invariant that has appeared in numerous applications, including in the context of random fields. The goal of this paper is to present the extension of using the Euler characteristic in higher-dimensional parameter spaces. While topological data analysis of higher-dimensional parameter spaces using stronger invariants such as homology continues to be the subject of intense research, Euler characteristic is more manageable theoretically and computationally, and this analysis can be seen as an important intermediary step in multi-parameter topological data analysis. We show the usefulness of the techniques using artificially generated examples, and a real-world application of detecting diabetic retinopathy in retinal images.

  62. Topological Regularization for Dense Prediction (2021)

    Deqing Fu, Bradley J. Nelson
    Abstract Dense prediction tasks such as depth perception and semantic segmentation are important applications in computer vision that have a concrete topological description in terms of partitioning an image into connected components or estimating a function with a small number of local extrema corresponding to objects in the image. We develop a form of topological regularization based on persistent homology that can be used in dense prediction tasks with these topological descriptions. Experimental results show that the output topology can also appear in the internal activations of trained neural networks which allows for a novel use of topological regularization to the internal states of neural networks during training, reducing the computational cost of the regularization. We demonstrate that this topological regularization of internal activations leads to improved convergence and test benchmarks on several problems and architectures.

  63. Lung Topology Characteristics in Patients With Chronic Obstructive Pulmonary Disease (2018)

    Francisco Belchi, Mariam Pirashvili, Joy Conway, Michael Bennett, Ratko Djukanovic, Jacek Brodzki
    Abstract Quantitative features that can currently be obtained from medical imaging do not provide a complete picture of Chronic Obstructive Pulmonary Disease (COPD). In this paper, we introduce a novel analytical tool based on persistent homology that extracts quantitative features from chest CT scans to describe the geometric structure of the airways inside the lungs. We show that these new radiomic features stratify COPD patients in agreement with the GOLD guidelines for COPD and can distinguish between inspiratory and expiratory scans. These CT measurements are very different to those currently in use and we demonstrate that they convey significant medical information. The results of this study are a proof of concept that topological methods can enhance the standard methodology to create a finer classification of COPD and increase the possibilities of more personalized treatment.

  64. Persistent Homology Machine Learning for Fingerprint Classification (2019)

    N. Giansiracusa, R. Giansiracusa, C. Moon
    Abstract The fingerprint classification problem is to sort fingerprints into predetermined groups, such as arch, loop, and whorl. It was asserted in the literature that minutiae points, which are commonly used for fingerprint matching, are not useful for classification. We show that, to the contrary, near state-of-the-art classification accuracy rates can be achieved when applying topological data analysis (TDA) to 3-dimensional point clouds of oriented minutiae points. We also apply TDA to fingerprint ink-roll images, which yields a lower accuracy rate but still shows promise; moreover, combining the two approaches outperforms each one individually. These methods use supervised learning applied to persistent homology and allow us to explore feature selection on barcodes, an important topic at the interface between TDA and machine learning. We test our classification algorithms on the NIST fingerprint database SD-27.

  65. Topological Data Analysis for the Characterization of Atomic Scale Morphology From Atom Probe Tomography Images (2018)

    Tianmu Zhang, Scott R. Broderick, Krishna Rajan
    Abstract Atom probe tomography (APT) represents a revolutionary characterization tool for materials that combine atomic imaging with a time-of-flight (TOF) mass spectrometer to provide direct space three-dimensional, atomic scale resolution images of materials with the chemical identities of hundreds of millions of atoms. It involves the controlled removal of atoms from a specimen’s surface by field evaporation and then sequentially analyzing them with a position sensitive detector and TOF mass spectrometer. A paradox in APT is that while on the one hand, it provides an unprecedented level of imaging resolution in three dimensions, it is very difficult to obtain an accurate perspective of morphology or shape outlined by atoms of similar chemistry and microstructure. The origins of this problem are numerous, including incomplete detection of atoms and the complexity of the evaporation fields of atoms at or near interfaces. Hence, unlike scattering techniques such as electron microscopy, interfaces appear diffused, not sharp. This, in turn, makes it challenging to visualize and quantitatively interpret the microstructure at the “meso” scale, where one is interested in the shape and form of the interfaces and their associated chemical gradients. It is here that the application of informatics at the nanoscale and statistical learning methods plays a critical role in both defining the level of uncertainty and helping to make quantitative, statistically objective interpretations where heuristics often dominate. In this chapter, we show how the tools of Topological Data Analysis provide a new and powerful tool in the field of nanoinformatics for materials characterization.

  66. Revisiting Abnormalities in Brain Network Architecture Underlying Autism Using Topology-Inspired Statistical Inference (2018)

    Sourabh Palande, Vipin Jose, Brandon Zielinski, Jeffrey Anderson, P. Thomas Fletcher, Bei Wang
    Abstract A large body of evidence relates autism with abnormal structural and functional brain connectivity. Structural covariance magnetic resonance imaging (scMRI) is a technique that maps brain regions with covarying gray matter densities across subjects. It provides a way to probe the anatomical structure underlying intrinsic connectivity networks (ICNs) through analysis of gray matter signal covariance. In this article, we apply topological data analysis in conjunction with scMRI to explore network-specific differences in the gray matter structure in subjects with autism versus age-, gender-, and IQ-matched controls. Specifically, we investigate topological differences in gray matter structure captured by structural correlation graphs derived from three ICNs strongly implicated in autism, namely the salience network, default mode network, and executive control network. By combining topological data analysis with statistical inference, our results provide evidence of statistically significant network-specific structural abnormalities in autism.

  67. Classification of COVID-19 via Homology of CT-SCAN (2021)

    Sohail Iqbal, H. Fareed Ahmed, Talha Qaiser, Muhammad Imran Qureshi, Nasir Rajpoot
    Abstract In this worldwide spread of SARS-CoV-2 (COVID-19) infection, it is of utmost importance to detect the disease at an early stage especially in the hot spots of this epidemic. There are more than 110 Million infected cases on the globe, sofar. Due to its promptness and effective results computed tomography (CT)-scan image is preferred to the reverse-transcription polymerase chain reaction (RT-PCR). Early detection and isolation of the patient is the only possible way of controlling the spread of the disease. Automated analysis of CT-Scans can provide enormous support in this process. In this article, We propose a novel approach to detect SARS-CoV-2 using CT-scan images. Our method is based on a very intuitive and natural idea of analyzing shapes, an attempt to mimic a professional medic. We mainly trace SARS-CoV-2 features by quantifying their topological properties. We primarily use a tool called persistent homology, from Topological Data Analysis (TDA), to compute these topological properties. We train and test our model on the "SARS-CoV-2 CT-scan dataset" i̧tep\soares2020sars\, an open-source dataset, containing 2,481 CT-scans of normal and COVID-19 patients. Our model yielded an overall benchmark F1 score of \$99.42\% \$, accuracy \$99.416\%\$, precision \$99.41\%\$, and recall \$99.42\%\$. The TDA techniques have great potential that can be utilized for efficient and prompt detection of COVID-19. The immense potential of TDA may be exploited in clinics for rapid and safe detection of COVID-19 globally, in particular in the low and middle-income countries where RT-PCR labs and/or kits are in a serious crisis.

  68. Constructing Shape Spaces From a Topological Perspective (2017)

    Christoph Hofer, Roland Kwitt, Marc Niethammer, Yvonne Höller, Eugen Trinka, Andreas Uhl
    Abstract We consider the task of constructing (metric) shape space(s) from a topological perspective. In particular, we present a generic construction scheme and demonstrate how to apply this scheme when shape is interpreted as the differences that remain after factoring out translation, scaling and rotation. This is achieved by leveraging a recently proposed injective functional transform of 2D/3D (binary) objects, based on persistent homology. The resulting shape space is then equipped with a similarity measure that is (1) by design robust to noise and (2) fulfills all metric axioms. From a practical point of view, analyses of object shape can then be carried out directly on segmented objects obtained from some imaging modality without any preprocessing, such as alignment, smoothing, or landmark selection. We demonstrate the utility of the approach on the problem of distinguishing segmented hippocampi from normal controls vs. patients with Alzheimer’s disease in a challenging setup where volume changes are no longer discriminative.

  69. Quantitative and Interpretable Order Parameters for Phase Transitions From Persistent Homology (2020)

    Alex Cole, Gregory J. Loges, Gary Shiu
    Abstract We apply modern methods in computational topology to the task of discovering and characterizing phase transitions. As illustrations, we apply our method to four two-dimensional lattice spin models: the Ising, square ice, XY, and fully-frustrated XY models. In particular, we use persistent homology, which computes the births and deaths of individual topological features as a coarse-graining scale or sublevel threshold is increased, to summarize multiscale and high-point correlations in a spin configuration. We employ vector representations of this information called persistence images to formulate and perform the statistical task of distinguishing phases. For the models we consider, a simple logistic regression on these images is sufficient to identify the phase transition. Interpretable order parameters are then read from the weights of the regression. This method suffices to identify magnetization, frustration, and vortex-antivortex structure as relevant features for phase transitions in our models. We also define "persistence" critical exponents and study how they are related to those critical exponents usually considered.

  70. Algebraic Topology-Based Machine Learning Using MRI Predicts Outcomes in Primary Sclerosing Cholangitis (2022)

    Yashbir Singh, William A. Jons, John E. Eaton, Mette Vesterhus, Tom Karlsen, Ida Bjoerk, Andreas Abildgaard, Kristin Kaasen Jorgensen, Folseraas Trine, Derek Little, Aliya F. Gulamhusein, Kosta Petrovic, Anne Negard, Gian Marco Conte, Joseph D. Sobek, Jaidip Jagtap, Sudhakar K. Venkatesh, Gregory J. Gores, Nicholas F. LaRusso, Konstantinos N. Lazaridis, Bradley J. Erickson
    Abstract Background: Primary sclerosing cholangitis (PSC) is a chronic cholestatic liver disease that can lead to cirrhosis and hepatic decompensation. However, predicting future outcomes in patients with PSC is challenging. Our aim was to extract magnetic resonance imaging (MRI) features that predict the development of hepatic decompensation by applying algebraic topology-based machine learning (ML). Methods: We conducted a retrospective multicenter study among adults with large duct PSC who underwent MRI. A topological data analysis-inspired nonlinear framework was used to predict the risk of hepatic decompensation, which was motivated by algebraic topology theory-based ML. The topological representations (persistence images) were employed as input for classifcation to predict who developed early hepatic decompensation within one year after their baseline MRI. Results: We reviewed 590 patients; 298 were excluded due to poor image quality or inadequate liver coverage, leaving 292 potentially eligible subjects, of which 169 subjects were included in the study. We trained our model using contrast-enhanced delayed phase T1-weighted images on a single center derivation cohort consisting of 54 patients (hepatic decompensation, n = 21; no hepatic decompensation, n = 33) and a multicenter independent validation cohort of 115 individuals (hepatic decompensation, n = 31; no hepatic decompensation, n = 84). When our model was applied in the independent validation cohort, it remained predictive of early hepatic decompensation (area under the receiver operating characteristic curve = 0.84). Conclusions: Algebraic topology-based ML is a methodological approach that can predict outcomes in patients with PSC and has the potential for application in other chronic liver diseases

  71. Mapping Geometric and Electromagnetic Feature Spaces With Machine Learning for Additively Manufactured RF Devices (2022)

    Deanna Sessions, Venkatesh Meenakshisundaram, Andrew Gillman, Alexander Cook, Kazuko Fuchi, Philip R. Buskohl, Gregory H. Huff
    Abstract Multi-material additive manufacturing enables transformative capabilities in customized, low-cost, and multi-functional electromagnetic devices. However, process-specific fabrication anomalies can result in non-intuitive effects on performance; we propose a framework for identifying defect mechanisms and their performance impact by mapping geometric variances to electromagnetic performance metrics. This method can accelerate additive fabrication feedback while avoiding the high computational cost of in-line electromagnetic simulation. We first used dimension reduction to explore the population of geometric manufacturing anomalies and electromagnetic performance. Convolutional neural networks are then trained to predict the electromagnetic performance of the printed geometries. In generating the networks, we explored two inputs: one image-derived geometric description and one using the same description with additional simulated electromagnetic information. Network latent space analysis shows the networks learned both geometric and electromagnetic values even without electromagnetic input. This result demonstrates it is possible to create accelerated additive feedback systems predicting electromagnetic performance without in-line simulation.

  72. Topological Data Analysis Distinguishes Parameter Regimes in the Anderson-Chaplain Model of Angiogenesis (2021)

    John T. Nardini, Bernadette J. Stolz, Kevin B. Flores, Heather A. Harrington, Helen M. Byrne
    Abstract Angiogenesis is the process by which blood vessels form from pre-existing vessels. It plays a key role in many biological processes, including embryonic development and wound healing, and contributes to many diseases including cancer and rheumatoid arthritis. The structure of the resulting vessel networks determines their ability to deliver nutrients and remove waste products from biological tissues. Here we simulate the Anderson-Chaplain model of angiogenesis at different parameter values and quantify the vessel architectures of the resulting synthetic data. Specifically, we propose a topological data analysis (TDA) pipeline for systematic analysis of the model. TDA is a vibrant and relatively new field of computational mathematics for studying the shape of data. We compute topological and standard descriptors of model simulations generated by different parameter values. We show that TDA of model simulation data stratifies parameter space into regions with similar vessel morphology. The methodologies proposed here are widely applicable to other synthetic and experimental data including wound healing, development, and plant biology.

  73. A Topological Machine Learning Pipeline for Classification (2022)

    Francesco Conti, Davide Moroni, Maria Antonietta Pascali
    Abstract In this work, we develop a pipeline that associates Persistence Diagrams to digital data via the most appropriate filtration for the type of data considered. Using a grid search approach, this pipeline determines optimal representation methods and parameters. The development of such a topological pipeline for Machine Learning involves two crucial steps that strongly affect its performance: firstly, digital data must be represented as an algebraic object with a proper associated filtration in order to compute its topological summary, the Persistence Diagram. Secondly, the persistence diagram must be transformed with suitable representation methods in order to be introduced in a Machine Learning algorithm. We assess the performance of our pipeline, and in parallel, we compare the different representation methods on popular benchmark datasets. This work is a first step toward both an easy and ready-to-use pipeline for data classification using persistent homology and Machine Learning, and to understand the theoretical reasons why, given a dataset and a task to be performed, a pair (filtration, topological representation) is better than another.

  74. Topological Persistence for Relating Microstructure and Capillary Fluid Trapping in Sandstones (2019)

    A. L. Herring, V. Robins, A. P. Sheppard
    Abstract Results from a series of two-phase fluid flow experiments in Leopard, Berea, and Bentheimer sandstones are presented. Fluid configurations are characterized using laboratory-based and synchrotron based 3-D X-ray computed tomography. All flow experiments are conducted under capillary-dominated conditions. We conduct geometry-topology analysis via persistent homology and compare this to standard topological and watershed-partition-based pore-network statistics. Metrics identified as predictors of nonwetting fluid trapping are calculated from the different analytical methods and are compared to levels of trapping measured during drainage-imbibition cycles in the experiments. Metrics calculated from pore networks (i.e., pore body-throat aspect ratio and coordination number) and topological analysis (Euler characteristic) do not correlate well with trapping in these samples. In contrast, a new metric derived from the persistent homology analysis, which incorporates counts of topological features as well as their length scale and spatial distribution, correlates very well (R2 = 0.97) to trapping for all systems. This correlation encompasses a wide range of porous media and initial fluid configurations, and also applies to data sets of different imaging and image processing protocols.

  75. Dissecting Glial Scar Formation by Spatial Point Pattern and Topological Data Analysis (2024)

    Daniel Manrique-Castano, Dhananjay Bhaskar, Ayman ElAli
    Abstract Glial scar formation represents a fundamental response to central nervous system (CNS) injuries. It is mainly characterized by a well-defined spatial rearrangement of reactive astrocytes and microglia. The mechanisms underlying glial scar formation have been extensively studied, yet quantitative descriptors of the spatial arrangement of reactive glial cells remain limited. Here, we present a novel approach using point pattern analysis (PPA) and topological data analysis (TDA) to quantify spatial patterns of reactive glial cells after experimental ischemic stroke in mice. We provide open and reproducible tools using R and Julia to quantify spatial intensity, cell covariance and conditional distribution, cell-to-cell interactions, and short/long-scale arrangement, which collectively disentangle the arrangement patterns of the glial scar. This approach unravels a substantial divergence in the distribution of GFAP+ and IBA1+ cells after injury that conventional analysis methods cannot fully characterize. PPA and TDA are valuable tools for studying the complex spatial arrangement of reactive glia and other nervous cells following CNS injuries and have potential applications for evaluating glial-targeted restorative therapies.

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  76. Characterising Epithelial Tissues Using Persistent Entropy (2019)

    N. Atienza, L. M. Escudero, M. J. Jimenez, M. Soriano-Trigueros
    Abstract In this paper, we apply persistent entropy, a novel topological statistic, for characterization of images of epithelial tissues. We have found out that persistent entropy is able to summarize topological and geometric information encoded by \$\$\alpha \$\$α-complexes and persistent homology. After using some statistical tests, we can guarantee the existence of significant differences in the studied tissues.

  82. Diverse 3D Cellular Patterns Underlie the Development of Cardamine Hirsuta and Arabidopsis Thaliana Ovules (2023)

    Tejasvinee Atul Mody, Alexander Rolle, Nico Stucki, Fabian Roll, Ulrich Bauer, Kay Schneitz
    Abstract A fundamental question in biology is how organ morphogenesis comes about. The ovules of Arabidopsis thaliana have been established as a successful model to study numerous aspects of tissue morphogenesis; however, little is known regarding the relative contributions and dynamics of differential tissue and cellular growth and architecture in establishing ovule morphogenesis in different species. To address this issue, we generated a 3D digital atlas of Cardamine hirsuta ovule development with full cellular resolution. We combined quantitative comparative morphometrics and topological analysis to explore similarities and differences in the 3D cellular architectures underlying ovule development of the two species. We discovered that they show diversity in the way the three radial cell layers of the primordium contribute to its growth, in the formation of a new cell layer in the inner integument and, in certain cases, in the topological properties of the 3D cell architectures of homologous tissues despite their similar shape. Our work demonstrates the power of comparative 3D cellular morphometry and the importance of internal tissues and their cellular architecture in organ morphogenesis. Summary Statement Quantitative morphometric comparison of 3D digital ovules at full cellular resolution reveals diversity in internal 3D cellular architectures between similarly shaped ovules of Cardamine hirsuta and Arabidopsis thaliana.

  83. Specimen-Based Analysis of Morphology and the Environment in Ecologically Dominant Grasses: The Power of the Herbarium (2019)

    Christine A. McAllister, Michael R. McKain, Mao Li, Bess Bookout, Elizabeth A. Kellogg
    Abstract Herbaria contain a cumulative sample of the world's flora, assembled by thousands of people over centuries. To capitalize on this resource, we conducted a specimen-based analysis of a major clade in the grass tribe Andropogoneae, including the dominant species of the world's grasslands in the genera Andropogon, Schizachyrium, Hyparrhenia and several others. We imaged 186 of the 250 named species of the clade, georeferenced the specimens and extracted climatic variables for each. Using semi- and fully automated image analysis techniques, we extracted spikelet morphological characters and correlated these with environmental variables. We generated chloroplast genome sequences to correct for phylogenetic covariance and here present a new phylogeny for 81 of the species. We confirm and extend earlier studies to show that Andropogon and Schizachyrium are not monophyletic. In addition, we find all morphological and ecological characters are homoplasious but variable among clades. For example, sessile spikelet length is positively correlated with awn length when all accessions are considered, but when separated by clade, the relationship is positive for three sub-clades and negative for three others. Climate variables showed no correlation with morphological variation in the spikelet pair; only very weak effects of temperature and precipitation were detected on macrohair density. This article is part of the theme issue ‘Biological collections for understanding biodiversity in the Anthropocene'.

  84. Interpretable Phase Detection and Classification With Persistent Homology (2020)

    Alex Cole, Gregory J. Loges, Gary Shiu
    Abstract We apply persistent homology to the task of discovering and characterizing phase transitions, using lattice spin models from statistical physics for working examples. Persistence images provide a useful representation of the homological data for conducting statistical tasks. To identify the phase transitions, a simple logistic regression on these images is sufficient for the models we consider, and interpretable order parameters are then read from the weights of the regression. Magnetization, frustration and vortex-antivortex structure are identified as relevant features for characterizing phase transitions.

  85. Can Neural Networks Learn Persistent Homology Features? (2020)

    Guido Montúfar, Nina Otter, Yuguang Wang
    Abstract Topological data analysis uses tools from topology -- the mathematical area that studies shapes -- to create representations of data. In particular, in persistent homology, one studies one-parameter families of spaces associated with data, and persistence diagrams describe the lifetime of topological invariants, such as connected components or holes, across the one-parameter family. In many applications, one is interested in working with features associated with persistence diagrams rather than the diagrams themselves. In our work, we explore the possibility of learning several types of features extracted from persistence diagrams using neural networks.

  86. Topology-Based Kernels With Application to Inference Problems in Alzheimer’s Disease (2011)

    Deepti Pachauri, Chris Hinrichs, Moo K. Chung, Sterling C. Johnson, Vikas Singh
    Abstract Alzheimer’s disease (AD) research has recently witnessed a great deal of activity focused on developing new statistical learning tools for automated inference using imaging data. The workhorse for many of these techniques is the Support Vector Machine (SVM) framework (or more generally kernel based methods). Most of these require, as a first step, specification of a kernel matrix between input examples (i.e., images). The inner product between images Ii and Ij in a feature space can generally be written in closed form, and so it is convenient to treat as “given”. However, in certain neuroimaging applications such an assumption becomes problematic. As an example, it is rather challenging to provide a scalar measure of similarity between two instances of highly attributed data such as cortical thickness measures on cortical surfaces. Note that cortical thickness is known to be discriminative for neurological disorders, so leveraging such information in an inference framework, especially within a multi-modal method, is potentially advantageous. But despite being clinically meaningful, relatively few works have successfully exploited this measure for classification or regression. Motivated by these applications, our paper presents novel techniques to compute similarity matrices for such topologically-based attributed data. Our ideas leverage recent developments to characterize signals (e.g., cortical thickness) motivated by the persistence of their topological features, leading to a scheme for simple constructions of kernel matrices. As a proof of principle, on a dataset of 356 subjects from the ADNI study, we report good performance on several statistical inference tasks without any feature selection, dimensionality reduction, or parameter tuning.

  87. Optimal Topological Cycles and Their Application in Cardiac Trabeculae Restoration (2017)

    Pengxiang Wu, Chao Chen, Yusu Wang, Shaoting Zhang, Changhe Yuan, Zhen Qian, Dimitris Metaxas, Leon Axel
    Abstract In cardiac image analysis, it is important yet challenging to reconstruct the trabeculae, namely, fine muscle columns whose ends are attached to the ventricular walls. To extract these fine structures, traditional image segmentation methods are insufficient. In this paper, we propose a novel method to jointly detect salient topological handles and compute the optimal representations of them. The detected handles are considered hypothetical trabeculae structures. They are further screened using a classifier and are then included in the final segmentation. We show in experiments the significance of our contribution compared with previous standard segmentation methods without topological priors, as well as with previous topological method in which non-optimal representations of topological handles are used.

  88. MRI and Biomechanics Multidimensional Data Analysis Reveals R2 -R1ρ as an Early Predictor of Cartilage Lesion Progression in Knee Osteoarthritis (2017)

    Valentina Pedoia, Jenny Haefeli, Kazuhito Morioka, Hsiang-Ling Teng, Lorenzo Nardo, Richard B. Souza, Adam R. Ferguson, Sharmila Majumdar
    Abstract PURPOSE: To couple quantitative compositional MRI, gait analysis, and machine learning multidimensional data analysis to study osteoarthritis (OA). OA is a multifactorial disorder accompanied by biochemical and morphological changes in the articular cartilage, modulated by skeletal biomechanics and gait. While we can now acquire detailed information about the knee joint structure and function, we are not yet able to leverage the multifactorial factors for diagnosis and disease management of knee OA. MATERIALS AND METHODS: We mapped 178 subjects in a multidimensional space integrating: demographic, clinical information, gait kinematics and kinetics, cartilage compositional T1ρ and T2 and R2 -R1ρ (1/T2 -1/T1ρ ) acquired at 3T and whole-organ magnetic resonance imaging score morphological grading. Topological data analysis (TDA) and Kolmogorov-Smirnov test were adopted for data integration, analysis, and hypothesis generation. Regression models were used for hypothesis testing. RESULTS: The results of the TDA showed a network composed of three main patient subpopulations, thus potentially identifying new phenotypes. T2 and T1ρ values (T2 lateral femur P = 1.45*10-8 , T1ρ medial tibia P = 1.05*10-5 ), the presence of femoral cartilage defects (P = 0.0013), lesions in the meniscus body (P = 0.0035), and race (P = 2.44*10-4 ) were key markers in the subpopulation classification. Within one of the subpopulations we observed an association between the composite metric R2 -R1ρ and the longitudinal progression of cartilage lesions. CONCLUSION: The analysis presented demonstrates some of the complex multitissue biochemical and biomechanical interactions that define joint degeneration and OA using a multidimensional approach, and potentially indicates that R2 -R1ρ may be an imaging biomarker for early OA. LEVEL OF EVIDENCE: 3 Technical Efficacy: Stage 2 J. Magn. Reson. Imaging 2018;47:78-90.

  89. Persistence Diagrams for Exploring the Shape Variability of Abdominal Aortic Aneurysms (2024)

    Dario Arnaldo Domanin, Matteo Pegoraro, Santi Trimarchi, Maurizio Domanin, Piercesare Secchi
    Abstract Abdominal aortic aneurysm consists of a permanent dilation in the abdominal portion of the aorta and, along with its associated pathologies like calcifications and intraluminal thrombi, is one of the most important pathologies of the circulatory system. The shape of the aorta is among the primary drivers for these health issues, with particular reference to all the characteristics which affect the hemodynamics. Starting from the computed tomography angiography of a patient, we propose to summarize such information using tools derived from Topological Data Analysis, obtaining persistence diagrams which describe the irregularities of the lumen of the aorta. We showcase the effectiveness of such shape-related descriptors with a series of supervised and unsupervised case studies.

  90. Fast Estimation of Recombination Rates Using Topological Data Analysis (2019)

    Devon P. Humphreys, Melissa R. McGuirl, Michael Miyagi, Andrew J. Blumberg
    Abstract Accurate estimation of recombination rates is critical for studying the origins and maintenance of genetic diversity. Because the inference of recombination rates under a full evolutionary model is computationally expensive, we developed an alternative approach using topological data analysis (TDA) on genome sequences. We find that this method can analyze datasets larger than what can be handled by any existing recombination inference software, and has accuracy comparable to commonly used model-based methods with significantly less processing time. Previous TDA methods used information contained solely in the first Betti number (\textlessimg class="highwire-embed" alt="Embedded Image" src="http://www.genetics.org/sites/default/files/highwire/genetics/211/4/1191/embed/mml-math-1.gif"/\textgreater) of a set of genomes, which aims to capture the number of loops that can be detected within a genealogy. These explorations have proven difficult to connect to the theory of the underlying biological process of recombination, and, consequently, have unpredictable behavior under perturbations of the data. We introduce a new topological feature, which we call ψ, with a natural connection to coalescent models, and present novel arguments relating \textlessimg class="highwire-embed" alt="Embedded Image" src="http://www.genetics.org/sites/default/files/highwire/genetics/211/4/1191/embed/mml-math-2.gif"/\textgreater to population genetic models. Using simulations, we show that ψ and \textlessimg class="highwire-embed" alt="Embedded Image" src="http://www.genetics.org/sites/default/files/highwire/genetics/211/4/1191/embed/mml-math-3.gif"/\textgreater are differentially affected by missing data, and package our approach as TREE (Topological Recombination Estimator). TREE’s efficiency and accuracy make it well suited as a first-pass estimator of recombination rate heterogeneity or hotspots throughout the genome. Our work empirically and theoretically justifies the use of topological statistics as summaries of genome sequences and describes a new, unintuitive relationship between topological features of the distribution of sequence data and the footprint of recombination on genomes.

  91. Quantification of the Immune Content in Neuroblastoma: Deep Learning and Topological Data Analysis in Digital Pathology (2021)

    Nicole Bussola, Bruno Papa, Ombretta Melaiu, Aurora Castellano, Doriana Fruci, Giuseppe Jurman
    Abstract We introduce here a novel machine learning (ML) framework to address the issue of the quantitative assessment of the immune content in neuroblastoma (NB) specimens. First, the EUNet, a U-Net with an EfficientNet encoder, is trained to detect lymphocytes on tissue digital slides stained with the CD3 T-cell marker. The training set consists of 3782 images extracted from an original collection of 54 whole slide images (WSIs), manually annotated for a total of 73,751 lymphocytes. Resampling strategies, data augmentation, and transfer learning approaches are adopted to warrant reproducibility and to reduce the risk of overfitting and selection bias. Topological data analysis (TDA) is then used to define activation maps from different layers of the neural network at different stages of the training process, described by persistence diagrams (PD) and Betti curves. TDA is further integrated with the uniform manifold approximation and projection (UMAP) dimensionality reduction and the hierarchical density-based spatial clustering of applications with noise (HDBSCAN) algorithm for clustering, by the deep features, the relevant subgroups and structures, across different levels of the neural network. Finally, the recent TwoNN approach is leveraged to study the variation of the intrinsic dimensionality of the U-Net model. As the main task, the proposed pipeline is employed to evaluate the density of lymphocytes over the whole tissue area of the WSIs. The model achieves good results with mean absolute error 3.1 on test set, showing significant agreement between densities estimated by our EUNet model and by trained pathologists, thus indicating the potentialities of a promising new strategy in the quantification of the immune content in NB specimens. Moreover, the UMAP algorithm unveiled interesting patterns compatible with pathological characteristics, also highlighting novel insights into the dynamics of the intrinsic dataset dimensionality at different stages of the training process. All the experiments were run on the Microsoft Azure cloud platform.

  92. Localization in the Crowd With Topological Constraints (2020)

    Shahira Abousamra, Minh Hoai, Dimitris Samaras, Chao Chen
    Abstract We address the problem of crowd localization, i.e., the prediction of dots corresponding to people in a crowded scene. Due to various challenges, a localization method is prone to spatial semantic errors, i.e., predicting multiple dots within a same person or collapsing multiple dots in a cluttered region. We propose a topological approach targeting these semantic errors. We introduce a topological constraint that teaches the model to reason about the spatial arrangement of dots. To enforce this constraint, we define a persistence loss based on the theory of persistent homology. The loss compares the topographic landscape of the likelihood map and the topology of the ground truth. Topological reasoning improves the quality of the localization algorithm especially near cluttered regions. On multiple public benchmarks, our method outperforms previous localization methods. Additionally, we demonstrate the potential of our method in improving the performance in the crowd counting task.

  93. Pore Geometry Characterization by Persistent Homology Theory (2018)

    Fei Jiang, Takeshi Tsuji, Tomoyuki Shirai
    Abstract Rock pore geometry has heterogeneous characteristics and is scale dependent. This feature in a geological formation differs significantly from artificial materials and makes it difficult to predict hydrologic and elastic properties. To characterize pore heterogeneity, we propose an evaluation method that exploits the recently developed persistent homology theory. In the proposed method, complex pore geometry is first represented as sphere cloud data using a pore-network extraction method. Then, a persistence diagram (PD) is calculated from the point cloud, which represents the spatial distribution of pore bodies. A new parameter (distance index H) derived from the PD is proposed to characterize the degree of rock heterogeneity. Low H value indicates high heterogeneity. A new empirical equation using this index H is proposed to predict the effective elastic modulus of porous media. The results indicate that the proposed PD analysis is very efficient for extracting topological feature of pore geometry.

  94. Quantitative Analysis of Phase Transitions in Two-Dimensional XY Models Using Persistent Homology (2022)

    Nicholas Sale, Jeffrey Giansiracusa, Biagio Lucini
    Abstract We use persistent homology and persistence images as an observable of three different variants of the two-dimensional XY model in order to identify and study their phase transitions. We examine models with the classical XY action, a topological lattice action, and an action with an additional nematic term. In particular, we introduce a new way of computing the persistent homology of lattice spin model configurations and, by considering the fluctuations in the output of logistic regression and k-nearest neighbours models trained on persistence images, we develop a methodology to extract estimates of the critical temperature and the critical exponent of the correlation length. We put particular emphasis on finite-size scaling behaviour and producing estimates with quantifiable error. For each model we successfully identify its phase transition(s) and are able to get an accurate determination of the critical temperatures and critical exponents of the correlation length.

  95. A Mayer–Vietoris Formula for Persistent Homology With an Application to Shape Recognition in the Presence of Occlusions (2011)

    Barbara Di Fabio, Claudia Landi
    Abstract In algebraic topology it is well known that, using the Mayer–Vietoris sequence, the homology of a space X can be studied by splitting X into subspaces A and B and computing the homology of A, B, and A∩B. A natural question is: To what extent does persistent homology benefit from a similar property? In this paper we show that persistent homology has a Mayer–Vietoris sequence that is generally not exact but only of order 2. However, we obtain a Mayer–Vietoris formula involving the ranks of the persistent homology groups of X, A, B, and A∩B plus three extra terms. This implies that persistent homological features of A and B can be found either as persistent homological features of X or of A∩B. As an application of this result, we show that persistence diagrams are able to recognize an occluded shape by showing a common subset of points.

  96. The Euler Characteristic: A General Topological Descriptor for Complex Data (2021)

    Alexander Smith, Victor Zavala
    Abstract Datasets are mathematical objects (e.g., point clouds, matrices, graphs, images, fields/functions) that have shape. This shape encodes important knowledge about the system under study. Topology is an area of mathematics that provides diverse tools to characterize the shape of data objects. In this work, we study a specific tool known as the Euler characteristic (EC). The EC is a general, low-dimensional, and interpretable descriptor of topological spaces defined by data objects. We revise the mathematical foundations of the EC and highlight its connections with statistics, linear algebra, field theory, and graph theory. We discuss advantages offered by the use of the EC in the characterization of complex datasets; to do so, we illustrate its use in different applications of interest in chemical engineering such as process monitoring, flow cytometry, and microscopy. We show that the EC provides a descriptor that effectively reduces complex datasets and that this reduction facilitates tasks such as visualization, regression, classification, and clustering.

  97. Reviews: Topological Distances and Losses for Brain Networks (2021)

    Moo K. Chung, Alexander Smith, Gary Shiu
    Abstract Almost all statistical and machine learning methods in analyzing brain networks rely on distances and loss functions, which are mostly Euclidean or matrix norms. The Euclidean or matrix distances may fail to capture underlying subtle topological differences in brain networks. Further, Euclidean distances are sensitive to outliers. A few extreme edge weights may severely affect the distance. Thus it is necessary to use distances and loss functions that recognize topology of data. In this review paper, we survey various topological distance and loss functions from topological data analysis (TDA) and persistent homology that can be used in brain network analysis more effectively. Although there are many recent brain imaging studies that are based on TDA methods, possibly due to the lack of method awareness, TDA has not taken as the mainstream tool in brain imaging field yet. The main purpose of this paper is provide the relevant technical survey of these powerful tools that are immediately applicable to brain network data.

  98. Acute Lymphoblastic Leukemia Classification Using Persistent Homology (2024)

    Waqar Hussain Shah, Abdullah Baloch, Rider Jaimes-Reátegui, Sohail Iqbal, Syeda Rafia Fatima, Alexander N. Pisarchik
    Abstract Acute Lymphoblastic Leukemia (ALL) is a prevalent form of childhood blood cancer characterized by the proliferation of immature white blood cells that rapidly replace normal cells in the bone marrow. The exponential growth of these leukemic cells can be fatal if not treated promptly. Classifying lymphoblasts and healthy cells poses a significant challenge, even for domain experts, due to their morphological similarities. Automated computer analysis of ALL can provide substantial support in this domain and potentially save numerous lives. In this paper, we propose a novel classification approach that involves analyzing shapes and extracting topological features of ALL cells. We employ persistent homology to capture these topological features. Our technique accurately and efficiently detects and classifies leukemia blast cells, achieving a recall of 98.2% and an F1-score of 94.6%. This approach has the potential to significantly enhance leukemia diagnosis and therapy.

  99. Topological Detection of Phenomenological Bifurcations With Unreliable Kernel Density Estimates (2024)

    Sunia Tanweer, Firas A. Khasawneh
    Abstract Phenomenological (P-type) bifurcations are qualitative changes in stochastic dynamical systems whereby the stationary probability density function (PDF) changes its topology. The current state of the art for detecting these bifurcations requires reliable kernel density estimates computed from an ensemble of system realizations. However, in several real world signals such as Big Data, only a single system realization is available—making it impossible to estimate a reliable kernel density. This study presents an approach for detecting P-type bifurcations using unreliable density estimates. The approach creates an ensemble of objects from Topological Data Analysis (TDA) called persistence diagrams from the system’s sole realization and statistically analyzes the resulting set. We compare several methods for replicating the original persistence diagram including Gibbs point process modelling, Pairwise Interaction Point Modelling, and subsampling. We show that for the purpose of predicting a bifurcation, the simple method of subsampling exceeds the other two methods of point process modelling in performance.

  100. Analysis of Kolmogorov Flow and Rayleigh–Bénard Convection Using Persistent Homology (2016)

    Miroslav Kramár, Rachel Levanger, Jeffrey Tithof, Balachandra Suri, Mu Xu, Mark Paul, Michael F. Schatz, Konstantin Mischaikow
    Abstract We use persistent homology to build a quantitative understanding of large complex systems that are driven far-from-equilibrium. In particular, we analyze image time series of flow field patterns from numerical simulations of two important problems in fluid dynamics: Kolmogorov flow and Rayleigh–Bénard convection. For each image we compute a persistence diagram to yield a reduced description of the flow field; by applying different metrics to the space of persistence diagrams, we relate characteristic features in persistence diagrams to the geometry of the corresponding flow patterns. We also examine the dynamics of the flow patterns by a second application of persistent homology to the time series of persistence diagrams. We demonstrate that persistent homology provides an effective method both for quotienting out symmetries in families of solutions and for identifying multiscale recurrent dynamics. Our approach is quite general and it is anticipated to be applicable to a broad range of open problems exhibiting complex spatio-temporal behavior.

  101. When Remote Sensing Meets Topological Data Analysis (2018)

    Ludovic Duponchel
    Abstract Author Summary: Hyperspectral remote sensing plays an increasingly important role in many scientific domains and everyday life problems. Indeed, this imaging concept ends up in applications as varied as catching tax-evaders red-handed by locating new construction and building alterations, searching for aircraft and saving lives after fatal crashes, detecting oil spills for marine life and environmental preservation, spying on enemies with reconnaissance satellites, watching algae grow as an indicator of environmental health, forecasting weather to warn about natural disasters and much more. From an instrumental point of view, we can say that the actual spectrometers have rather good characteristics, even if we can always increase spatial resolution and spectral range. In order to extract ever more information from such experiments and develop new applications, we must, therefore, propose multivariate data analysis tools able to capture the shape of data sets and their specific features. Nevertheless, actual methods often impose a data model which implicitly defines the geometry of the data set. The aim of the paper is thus to introduce the concept of topological data analysis in the framework of remote sensing, making no assumptions about the global shape of the data set, but also allowing the capture of its local features.

  102. Persistence Images: A Stable Vector Representation of Persistent Homology (2017)

    Henry Adams, Tegan Emerson, Michael Kirby, Rachel Neville, Chris Peterson, Patrick Shipman, Sofya Chepushtanova, Eric Hanson, Francis Motta, Lori Ziegelmeier
    Abstract Many data sets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a multiscale description of the homological features within a data set. A useful representation of this homological information is a persistence diagram (PD). Efforts have been made to map PDs into spaces with additional structure valuable to machine learning tasks. We convert a PD to a finite-dimensional vector representation which we call a persistence image (PI), and prove the stability of this transformation with respect to small perturbations in the inputs. The discriminatory power of PIs is compared against existing methods, showing significant performance gains. We explore the use of PIs with vector-based machine learning tools, such as linear sparse support vector machines, which identify features containing discriminating topological information. Finally, high accuracy inference of parameter values from the dynamic output of a discrete dynamical system (the linked twist map) and a partial differential equation (the anisotropic Kuramoto-Sivashinsky equation) provide a novel application of the discriminatory power of PIs.

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  103. A Novel Multi-Task Machine Learning Classifier for Rare Disease Patterning Using Cardiac Strain Imaging Data (2024)

    Nanda K. Siva, Yashbir Singh, Quincy A. Hathaway, Partho P. Sengupta, Naveena Yanamala
    Abstract To provide accurate predictions, current machine learning-based solutions require large, manually labeled training datasets. We implement persistent homology (PH), a topological tool for studying the pattern of data, to analyze echocardiography-based strain data and differentiate between rare diseases like constrictive pericarditis (CP) and restrictive cardiomyopathy (RCM). Patient population (retrospectively registered) included those presenting with heart failure due to CP (n = 51), RCM (n = 47), and patients without heart failure symptoms (n = 53). Longitudinal, radial, and circumferential strains/strain rates for left ventricular segments were processed into topological feature vectors using Machine learning PH workflow. In differentiating CP and RCM, the PH workflow model had a ROC AUC of 0.94 (Sensitivity = 92%, Specificity = 81%), compared with the GLS model AUC of 0.69 (Sensitivity = 65%, Specificity = 66%). In differentiating between all three conditions, the PH workflow model had an AUC of 0.83 (Sensitivity = 68%, Specificity = 84%), compared with the GLS model AUC of 0.68 (Sensitivity = 52% and Specificity = 76%). By employing persistent homology to differentiate the “pattern” of cardiac deformations, our machine-learning approach provides reasonable accuracy when evaluating small datasets and aids in understanding and visualizing patterns of cardiac imaging data in clinically challenging disease states.

  104. Topological Data Analysis and Diagnostics of Compressible Magnetohydrodynamic Turbulence (2018)

    Irina Makarenko, Paul Bushby, Andrew Fletcher, Robin Henderson, Nikolay Makarenko, Anvar Shukurov
    Abstract The predictions of mean-field electrodynamics can now be probed using direct numerical simulations of random flows and magnetic fields. When modelling astrophysical magnetohydrodynamics, it is important to verify that such simulations are in agreement with observations. One of the main challenges in this area is to identify robust quantitative measures to compare structures found in simulations with those inferred from astrophysical observations. A similar challenge is to compare quantitatively results from different simulations. Topological data analysis offers a range of techniques, including the Betti numbers and persistence diagrams, that can be used to facilitate such a comparison. After describing these tools, we first apply them to synthetic random fields and demonstrate that, when the data are standardized in a straightforward manner, some topological measures are insensitive to either large-scale trends or the resolution of the data. Focusing upon one particular astrophysical example, we apply topological data analysis to H i observations of the turbulent interstellar medium (ISM) in the Milky Way and to recent magnetohydrodynamic simulations of the random, strongly compressible ISM. We stress that these topological techniques are generic and could be applied to any complex, multi-dimensional random field.

  105. Topological Early Warning Signals: Quantifying Varying Routes to Extinction in a Spatially Distributed Population Model (2022)

    Laura S. Storch, Sarah L. Day
    Abstract Understanding and predicting critical transitions in spatially explicit ecological systems is particularly challenging due to their complex spatial and temporal dynamics and high dimensionality. Here, we explore changes in population distribution patterns during a critical transition (an extinction event) using computational topology. Computational topology allows us to quantify certain features of a population distribution pattern, such as the level of fragmentation. We create population distribution patterns via a simple coupled patch model with Ricker map growth and nearest neighbors dispersal on a two dimensional lattice. We observe two dominant paths to extinction within the explored parameter space that depend critically on the dispersal rate d and the rate of parameter drift, Δϵ. These paths to extinction are easily topologically distinguishable, so categorization can be automated. We use this population model as a theoretical proof-of-concept for the methodology, and argue that computational topology is a powerful tool for analyzing dynamical changes in systems with noisy data that are coarsely resolved in space and/or time. In addition, computational topology can provide early warning signals for chaotic dynamical systems where traditional statistical early warning signals would fail. For these reasons, we envision this work as a helpful addition to the critical transitions prediction toolbox.

  106. Relational Persistent Homology for Multispecies Data With Application to the Tumor Microenvironment (2023)

    Bernadette J. Stolz, Jagdeep Dhesi, Joshua A. Bull, Heather A. Harrington, Helen M. Byrne, Iris H. R. Yoon
    Abstract Topological data analysis (TDA) is an active field of mathematics for quantifying shape in complex data. Standard methods in TDA such as persistent homology (PH) are typically focused on the analysis of data consisting of a single entity (e.g., cells or molecular species). However, state-of-the-art data collection techniques now generate exquisitely detailed multispecies data, prompting a need for methods that can examine and quantify the relations among them. Such heterogeneous data types arise in many contexts, ranging from biomedical imaging, geospatial analysis, to species ecology. Here, we propose two methods for encoding spatial relations among different data types that are based on Dowker complexes and Witness complexes. We apply the methods to synthetic multispecies data of a tumor microenvironment and analyze topological features that capture relations between different cell types, e.g., blood vessels, macrophages, tumor cells, and necrotic cells. We demonstrate that relational topological features can extract biological insight, including the dominant immune cell phenotype (an important predictor of patient prognosis) and the parameter regimes of a data-generating model. The methods provide a quantitative perspective on the relational analysis of multispecies spatial data, overcome the limits of traditional PH, and are readily computable.

  107. The Persistent Homology Mathematical Framework Provides Enhanced Genotype-to-Phenotype Associations for Plant Morphology (2018)

    Mao Li, Margaret H. Frank, Viktoriya Coneva, Washington Mio, Daniel H. Chitwood, Christopher N. Topp
    Abstract Efforts to understand the genetic and environmental conditioning of plant morphology are hindered by the lack of flexible and effective tools for quantifying morphology. Here, we demonstrate that persistent-homology-based topological methods can improve measurement of variation in leaf shape, serrations, and root architecture. We apply these methods to 2D images of leaves and root systems in field-grown plants of a domesticated introgression line population of tomato (Solanum pennellii). We find that compared with some commonly used conventional traits, (1) persistent-homology-based methods can more comprehensively capture morphological variation; (2) these techniques discriminate between genotypes with a larger normalized effect size and detect a greater number of unique quantitative trait loci (QTLs); (3) multivariate traits, whether statistically derived from univariate or persistent-homology-based traits, improve our ability to understand the genetic basis of phenotype; and (4) persistent-homology-based techniques detect unique QTLs compared to conventional traits or their multivariate derivatives, indicating that previously unmeasured aspects of morphology are now detectable. The QTL results further imply that genetic contributions to morphology can affect both the shoot and root, revealing a pleiotropic basis to natural variation in tomato. Persistent homology is a versatile framework to quantify plant morphology and developmental processes that complements and extends existing methods.

  108. A Topological Framework for Identifying Phenomenological Bifurcations in Stochastic Dynamical Systems (2024)

    Sunia Tanweer, Firas A. Khasawneh, Elizabeth Munch, Joshua R. Tempelman
    Abstract Changes in the parameters of dynamical systems can cause the state of the system to shift between different qualitative regimes. These shifts, known as bifurcations, are critical to study as they can indicate when the system is about to undergo harmful changes in its behavior. In stochastic dynamical systems, there is particular interest in P-type (phenomenological) bifurcations, which can include transitions from a monostable state to multi-stable states, the appearance of stochastic limit cycles and other features in the probability density function (PDF) of the system’s state. Current practices are limited to systems with small state spaces, cannot detect all possible behaviors of the PDFs and mandate human intervention for visually identifying the change in the PDF. In contrast, this study presents a new approach based on Topological Data Analysis that uses superlevel persistence to mathematically quantify P-type bifurcations in stochastic systems through a “homological bifurcation plot”—which shows the changing ranks of 0th and 1st homology groups, through Betti vectors. Using these plots, we demonstrate the successful detection of P-bifurcations on the stochastic Duffing, Raleigh-Vander Pol and Quintic Oscillators given their analytical PDFs, and elaborate on how to generate an estimated homological bifurcation plot given a kernel density estimate (KDE) of these systems by employing a tool for finding topological consistency between PDFs and KDEs.

  109. Manifold Learning for Coherent Design Interpolation Based on Geometrical and Topological Descriptors (2023)

    D. Muñoz, O. Allix, F. Chinesta, J. J. Ródenas, E. Nadal
    Abstract In the context of intellectual property in the manufacturing industry, know-how is referred to practical knowledge on how to accomplish a specific task. This know-how is often difficult to be synthesised in a set of rules or steps as it remains in the intuition and expertise of engineers, designers, and other professionals. Today, a new research line in this concern spot-up thanks to the explosion of Artificial Intelligence and Machine Learning algorithms and its alliance with Computational Mechanics and Optimisation tools. However, a key aspect with industrial design is the scarcity of available data, making it problematic to rely on deep-learning approaches. Assuming that the existing designs live in a manifold, in this paper, we propose a synergistic use of existing Machine Learning tools to infer a reduced manifold from the existing limited set of designs and, then, to use it to interpolate between the individuals, working as a generator basis, to create new and coherent designs. For this, a key aspect is to be able to properly interpolate in the reduced manifold, which requires a proper clustering of the individuals. From our experience, due to the scarcity of data, adding topological descriptors to geometrical ones considerably improves the quality of the clustering. Thus, a distance, mixing topology and geometry is proposed. This distance is used both, for the clustering and for the interpolation. For the interpolation, relying on optimal transport appear to be mandatory. Examples of growing complexity are proposed to illustrate the goodness of the method.

  110. Pattern Characterization Using Topological Data Analysis: Application to Piezo Vibration Striking Treatment (2023)

    Max M. Chumley, Melih C. Yesilli, Jisheng Chen, Firas A. Khasawneh, Yang Guo
    Abstract Quantifying patterns in visual or tactile textures provides important information about the process or phenomena that generated these patterns. In manufacturing, these patterns can be intentionally introduced as a design feature, or they can be a byproduct of a specific process. Since surface texture has significant impact on the mechanical properties and the longevity of the workpiece, it is important to develop tools for quantifying surface patterns and, when applicable, comparing them to their nominal counterparts. While existing tools may be able to indicate the existence of a pattern, they typically do not provide more information about the pattern structure, or how much it deviates from a nominal pattern. Further, prior works do not provide automatic or algorithmic approaches for quantifying other pattern characteristics such as depths’ consistency, and variations in the pattern motifs at different level sets. This paper leverages persistent homology from Topological Data Analysis (TDA) to derive noise-robust scores for quantifying motifs’ depth and roundness in a pattern. Specifically, sublevel persistence is used to derive scores that quantify the consistency of indentation depths at any level set in Piezo Vibration Striking Treatment (PVST) surfaces. Moreover, we combine sublevel persistence with the distance transform to quantify the consistency of the indentation radii, and to compare them with the nominal ones. Although the tool in our PVST experiments had a semi-spherical profile, we present a generalization of our approach to tools/motifs of arbitrary shapes thus making our method applicable to other pattern-generating manufacturing processes.

  111. Steinhaus Filtration and Stable Paths in the Mapper (2020)

    Dustin L. Arendt, Matthew Broussard, Bala Krishnamoorthy, Nathaniel Saul
    Abstract Two central concepts from topological data analysis are persistence and the Mapper construction. Persistence employs a sequence of objects built on data called a filtration. A Mapper produces insightful summaries of data, and has found widespread applications in diverse areas. We define a new filtration called the cover filtration built from a single cover based on a generalized Steinhaus distance, which is a generalization of Jaccard distance. We prove a stability result: the cover filtrations of two covers are \$\alpha/m\$ interleaved, where \$\alpha\$ is a bound on bottleneck distance between covers and \$m\$ is the size of smallest set in either cover. We also show our construction is equivalent to the Cech filtration under certain settings, and the Vietoris-Rips filtration completely determines the cover filtration in all cases. We then develop a theory for stable paths within this filtration. Unlike standard results on stability in topological persistence, our definition of path stability aligns exactly with the above result on stability of cover filtration. We demonstrate how our framework can be employed in a variety of applications where a metric is not obvious but a cover is readily available. First we present a new model for recommendation systems using cover filtration. For an explicit example, stable paths identified on a movies data set represent sequences of movies constituting gentle transitions from one genre to another. As a second application in explainable machine learning, we apply the Mapper for model induction, providing explanations in the form of paths between subpopulations. Stable paths in the Mapper from a supervised machine learning model trained on the FashionMNIST data set provide improved explanations of relationships between subpopulations of images.

  112. Using Multidimensional Topological Data Analysis to Identify Traits of Hip Osteoarthritis (2018)

    Jasmine Rossi‐deVries, Valentina Pedoia, Michael A. Samaan, Adam R. Ferguson, Richard B. Souza, Sharmila Majumdar
    Abstract Background Osteoarthritis (OA) is a multifaceted disease with many variables affecting diagnosis and progression. Topological data analysis (TDA) is a state-of-the-art big data analytics tool that can combine all variables into multidimensional space. TDA is used to simultaneously analyze imaging and gait analysis techniques. Purpose To identify biochemical and biomechanical biomarkers able to classify different disease progression phenotypes in subjects with and without radiographic signs of hip OA. Study Type Longitudinal study for comparison of progressive and nonprogressive subjects. Population In all, 102 subjects with and without radiographic signs of hip osteoarthritis. Field Strength/Sequence 3T, SPGR 3D MAPSS T1ρ/T2, intermediate-weighted fat-suppressed fast spin-echo (FSE). Assessment Multidimensional data analysis including cartilage composition, bone shape, Kellgren–Lawrence (KL) classification of osteoarthritis, scoring hip osteoarthritis with MRI (SHOMRI), hip disability and osteoarthritis outcome score (HOOS). Statistical Tests Analysis done using TDA, Kolmogorov–Smirnov (KS) testing, and Benjamini-Hochberg to rank P-value results to correct for multiple comparisons. Results Subjects in the later stages of the disease had an increased SHOMRI score (P \textless 0.0001), increased KL (P = 0.0012), and older age (P \textless 0.0001). Subjects in the healthier group showed intact cartilage and less pain. Subjects found between these two groups had a range of symptoms. Analysis of this subgroup identified knee biomechanics (P \textless 0.0001) as an initial marker of the disease that is noticeable before the morphological progression and degeneration. Further analysis of an OA subgroup with femoroacetabular impingement (FAI) showed anterior labral tears to be the most significant marker (P = 0.0017) between those FAI subjects with and without OA symptoms. Data Conclusion The data-driven analysis obtained with TDA proposes new phenotypes of these subjects that partially overlap with the radiographic-based classical disease status classification and also shows the potential for further examination of an early onset biomechanical intervention. Level of Evidence: 2 Technical Efficacy: Stage 2 J. Magn. Reson. Imaging 2018;48:1046–1058.

  116. Topological Data Analysis Quantifies Biological Nano-Structure From Single Molecule Localization Microscopy (2020)

    Jeremy A. Pike, Abdullah O. Khan, Chiara Pallini, Steven G. Thomas, Markus Mund, Jonas Ries, Natalie S. Poulter, Iain B. Styles
    Abstract AbstractMotivation. Localization microscopy data is represented by a set of spatial coordinates, each corresponding to a single detection, that form a point cl

  121. Uncovering Precision Phenotype-Biomarker Associations in Traumatic Brain Injury Using Topological Data Analysis (2017)

    Jessica L. Nielson, Shelly R. Cooper, John K. Yue, Marco D. Sorani, Tomoo Inoue, Esther L. Yuh, Pratik Mukherjee, Tanya C. Petrossian, Jesse Paquette, Pek Y. Lum, Gunnar E. Carlsson, Mary J. Vassar, Hester F. Lingsma, Wayne A. Gordon, Alex B. Valadka, David O. Okonkwo, Geoffrey T. Manley, Adam R. Ferguson, Track-Tbi Investigators
    Abstract Background Traumatic brain injury (TBI) is a complex disorder that is traditionally stratified based on clinical signs and symptoms. Recent imaging and molecular biomarker innovations provide unprecedented opportunities for improved TBI precision medicine, incorporating patho-anatomical and molecular mechanisms. Complete integration of these diverse data for TBI diagnosis and patient stratification remains an unmet challenge. Methods and findings The Transforming Research and Clinical Knowledge in Traumatic Brain Injury (TRACK-TBI) Pilot multicenter study enrolled 586 acute TBI patients and collected diverse common data elements (TBI-CDEs) across the study population, including imaging, genetics, and clinical outcomes. We then applied topology-based data-driven discovery to identify natural subgroups of patients, based on the TBI-CDEs collected. Our hypothesis was two-fold: 1) A machine learning tool known as topological data analysis (TDA) would reveal data-driven patterns in patient outcomes to identify candidate biomarkers of recovery, and 2) TDA-identified biomarkers would significantly predict patient outcome recovery after TBI using more traditional methods of univariate statistical tests. TDA algorithms organized and mapped the data of TBI patients in multidimensional space, identifying a subset of mild TBI patients with a specific multivariate phenotype associated with unfavorable outcome at 3 and 6 months after injury. Further analyses revealed that this patient subset had high rates of post-traumatic stress disorder (PTSD), and enrichment in several distinct genetic polymorphisms associated with cellular responses to stress and DNA damage (PARP1), and in striatal dopamine processing (ANKK1, COMT, DRD2). Conclusions TDA identified a unique diagnostic subgroup of patients with unfavorable outcome after mild TBI that were significantly predicted by the presence of specific genetic polymorphisms. Machine learning methods such as TDA may provide a robust method for patient stratification and treatment planning targeting identified biomarkers in future clinical trials in TBI patients. Trial Registration ClinicalTrials.gov Identifier NCT01565551

  122. Sheaves Are the Canonical Data Structure for Sensor Integration (2017)

    Michael Robinson
    Abstract A sensor integration framework should be sufficiently general to accurately represent many sensor modalities, and also be able to summarize information in a faithful way that emphasizes important, actionable information. Few approaches adequately address these two discordant requirements. The purpose of this expository paper is to explain why sheaves are the canonical data structure for sensor integration and how the mathematics of sheaves satisfies our two requirements. We outline some of the powerful inferential tools that are not available to other representational frameworks.

  123. Continuous Indexing of Fibrosis (CIF): Improving the Assessment and Classification of MPN Patients (2022)

    Hosuk Ryou, Korsuk Sirinukunwattana, Alan Aberdeen, Gillian Grindstaff, Bernadette Stolz, Helen Byrne, Heather A. Harrington, Nikolaos Sousos, Anna L. Godfrey, Claire N. Harrison, Bethan Psaila, Adam J. Mead, Gabrielle Rees, Gareth D. H. Turner, Jens Rittscher, Daniel Royston
    Abstract The detection and grading of fibrosis in myeloproliferative neoplasms (MPN) is an important component of disease classification, prognostication and disease monitoring. However, current fibrosis grading systems are only semi-quantitative and fail to capture sample heterogeneity. To improve the detection, quantitation and representation of reticulin fibrosis, we developed a machine learning (ML) approach using bone marrow trephine (BMT) samples (n = 107) from patients diagnosed with MPN or a reactive / nonneoplastic marrow. The resulting Continuous Indexing of Fibrosis (CIF) enhances the detection and monitoring of fibrosis within BMTs, and aids the discrimination of MPN subtypes. When combined with megakaryocyte feature analysis, CIF discriminates between the frequently challenging differential diagnosis of essential thrombocythemia (ET) and pre-fibrotic myelofibrosis (pre-PMF) with high predictive accuracy [area under the curve = 0.94]. CIF also shows significant promise in the identification of MPN patients at risk of disease progression; analysis of samples from 35 patients diagnosed with ET and enrolled in the Primary Thrombocythemia-1 (PT-1) trial identified features predictive of post-ET myelofibrosis (area under the curve = 0.77). In addition to these clinical applications, automated analysis of fibrosis has clear potential to further refine disease classification boundaries and inform future studies of the micro-environmental factors driving disease initiation and progression in MPN and other stem cell disorders. The image analysis methods used to generate CIF can be readily integrated with those of other key morphological features in MPNs, including megakaryocyte morphology, that lie beyond the scope of conventional histological assessment. Key PointsMachine learning enables an objective and quantitative description of reticulin fibrosis within the bone marrow of patients with myeloproliferative neoplasms (MPN),Automated analysis and Continuous Indexing of Fibrosis (CIF) captures heterogeneity within MPN samples and has utility in refined classification and disease monitoringQuantitative fibrosis assessment combined with topological data analysis may help to predict patients at increased risk of progression to post-ET myelofibrosis, and assist in the discrimination of ET and pre-fibrotic PMF (pre-PMF)

  124. Possible Clinical Use of Big Data: Personal Brain Connectomics (2018)

    Dong Soo Lee
    Abstract The biggest data is brain imaging data, which waited for clinical use during the last three decades. Topographic data interpretation prevailed for the first two decades, and only during the last decade, connectivity or connectomics data began to be analyzed properly. Owing to topological data interpretation and timely introduction of likelihood method based on hierarchical generalized linear model, we now foresee the clinical use of personal connectomics for classification and prediction of disease prognosis for brain diseases without any clue by currently available diagnostic methods.

  125. Persistent Homology to Quantify the Quality of Surface-Supported Covalent Networks (2019)

    Abraham Gutierrez, Mickaël Buchet, Sylvain Clair
    Abstract Covalent networks formed by on-surface synthesis usually suffer from the presence of a large number of defects. We report on a methodology to characterize such two-dimensional networks from their experimental images obtained by scanning probe microscopy. The computation is based on a persistent homology approach and provides a quantitative score indicative of the network homogeneity. We compare our scoring method with results previously obtained using minimal spanning tree analyses and we apply it to some molecular systems appearing in the existing literature.

  126. Topological Autoencoders (2020)

    Michael Moor, Max Horn, Bastian Rieck, Karsten Borgwardt
    Abstract We propose a novel approach for preserving topological structures of the input space in latent representations of autoencoders. Using persistent homology, a technique from topological data analysis, we calculate topological signatures of both the input and latent space to derive a topological loss term. Under weak theoretical assumptions, we construct this loss in a differentiable manner, such that the encoding learns to retain multi-scale connectivity information. We show that our approach is theoretically well-founded and that it exhibits favourable latent representations on a synthetic manifold as well as on real-world image data sets, while preserving low reconstruction errors.

  127. Investigation of Flash Crash via Topological Data Analysis (2020)

    Wonse Kim, Younng-Jin Kim, Gihyun Lee, Woong Kook
    Abstract Topological data analysis has been acknowledged as one of the most successful mathematical data analytic methodologies in various fields including medicine, genetics, and image analysis. In this paper, we explore the potential of this methodology in finance by applying persistence landscape and dynamic time series analysis to analyze an extreme event in the stock market, known as Flash Crash. We will provide results of our empirical investigation to confirm the effectiveness of our new method not only for the characterization of this extreme event but also for its prediction purposes.

  128. Classification of Skin Lesions by Topological Data Analysis Alongside With Neural Network (2020)

    Naiereh Elyasi, Mehdi Hosseini Moghadam
    Abstract In this paper we use TDA mapper alongside with deep convolutional neural networks in the classification of 7 major skin diseases. First we apply kepler mapper with neural network as one of its filter steps to classify the dataset HAM10000. Mapper visualizes the classification result by a simplicial complex, where neural network can not do this alone, but as a filter step neural network helps to classify data better. Furthermore we apply TDA mapper and persistent homology to understand the weights of layers of mobilenet network in different training epochs of HAM10000. Also we use persistent diagrams to visualize the results of analysis of layers of mobilenet network.

  129. Fruit Flies and Moduli: Interactions Between Biology and Mathematics (2015)

    Ezra Miller
    Abstract Possibilities for using geometry and topology to analyze statistical problems in biology raise a host of novel questions in geometry, probability, algebra, and combinatorics that demonstrate the power of biology to influence the future of pure mathematics. This expository article is a tour through some biological explorations and their mathematical ramifications. The article starts with evolution of novel topological features in wing veins of fruit flies, which are quantified using the algebraic structure of multiparameter persistent homology. The statistical issues involved highlight mathematical implications of sampling from moduli spaces. These lead to geometric probability on stratified spaces, including the sticky phenomenon for Frechet means and the origin of this mathematical area in the reconstruction of phylogenetic trees.

  130. Topological Singularity Detection at Multiple Scales (2023)

    Julius von Rohrscheidt, Bastian Rieck
    Abstract The manifold hypothesis, which assumes that data lies on or close to an unknown manifold of low intrinsic dimension, is a staple of modern machine learning research. However, recent work has shown that real-world data exhibits distinct non-manifold structures, i.e. singularities, that can lead to erroneous findings. Detecting such singularities is therefore crucial as a precursor to interpolation and inference tasks. We address this issue by developing a topological framework that (i) quantifies the local intrinsic dimension, and (ii) yields a Euclidicity score for assessing the ’manifoldness’ of a point along multiple scales. Our approach identifies singularities of complex spaces, while also capturing singular structures and local geometric complexity in image data.

  131. Topological Data Analysis of C. Elegans Locomotion and Behavior (2021)

    Ashleigh Thomas, Kathleen Bates, Alex Elchesen, Iryna Hartsock, Hang Lu, Peter Bubenik
    Abstract Video of nematodes/roundworms was analyzed using persistent homology to study locomotion and behavior. In each frame, an organism's body posture was represented by a high-dimensional vector. By concatenating points in fixed-duration segments of this time series, we created a sliding window embedding (sometimes called a time delay embedding) where each point corresponds to a sequence of postures of an organism. Persistent homology on the points in this time series detected behaviors and comparisons of these persistent homology computations detected variation in their corresponding behaviors. We used average persistence landscapes and machine learning techniques to study changes in locomotion and behavior in varying environments.

  132. Phase-Field Investigation of the Coarsening of Porous Structures by Surface Diffusion (2019)

    Pierre-Antoine Geslin, Mickaël Buchet, Takeshi Wada, Hidemi Kato
    Abstract Nano and microporous connected structures have attracted increasing attention in the past decades due to their high surface area, presenting interesting properties for a number of applications. These structures generally coarsen by surface diffusion, leading to an enlargement of the structure characteristic length scale. We propose to study this coarsening behavior using a phase-field model for surface diffusion. In addition to reproducing the expected scaling law, our simulations enable to investigate precisely the evolution of the topological and morphological characteristics along the coarsening process. In particular, we show that after a transient regime, the coarsening is self-similar as exhibited by the evolution of both morphological and topological features. In addition, the influence of surface anisotropy is discussed and comparisons with experimental tomographic observations are presented.

  133. Contagion Dynamics for Manifold Learning (2020)

    Barbara I. Mahler
    Abstract Contagion maps exploit activation times in threshold contagions to assign vectors in high-dimensional Euclidean space to the nodes of a network. A point cloud that is the image of a contagion map reflects both the structure underlying the network and the spreading behaviour of the contagion on it. Intuitively, such a point cloud exhibits features of the network's underlying structure if the contagion spreads along that structure, an observation which suggests contagion maps as a viable manifold-learning technique. We test contagion maps as a manifold-learning tool on a number of different real-world and synthetic data sets, and we compare their performance to that of Isomap, one of the most well-known manifold-learning algorithms. We find that, under certain conditions, contagion maps are able to reliably detect underlying manifold structure in noisy data, while Isomap fails due to noise-induced error. This consolidates contagion maps as a technique for manifold learning.

  134. Topological Extraction and Tracking of Defects in Crystal Structures (2011)

    Sebastian Grottel, Carlos A. Dietrich, João L. D. Comba, Thomas Ertl
    Abstract Interfaces between materials with different mechanical properties play an important role in technical applications. Nowadays molecular dynamics simulations are used to observe the behavior of such compound materials at the atomic level. Due to different atom crystal sizes, dislocations in the atom crystal structure occur once external forces are applied, and it has been observed that studying the change of thesedislocations can provide further understanding of macroscopic attributes like elasticity and plasticity. Standard visualization techniques such as the rendering of individual atoms work for 2D data or sectional views; however, visualizingdislocations in 3D using such methods usually fail due to occlusion and clutter. In this work we propose to extract and visualize the structure ofdislocations, which summarizes the commonly employed filtered atomistic renderings into a concise representation. The benefits of our approach are clearer images while retaining relevant data and easier visual tracking of topological changes over time.

  135. Topological Analysis of Population Activity in Visual Cortex (2008)

    Gurjeet Singh, Facundo Memoli, Tigran Ishkhanov, Guillermo Sapiro, Gunnar Carlsson, Dario L. Ringach
    Abstract Information in the cortex is thought to be represented by the joint activity of neurons. Here we describe how fundamental questions about neural representation can be cast in terms of the topological structure of population activity. A new method, based on the concept of persistent homology, is introduced and applied to the study of population activity in primary visual cortex (V1). We found that the topological structure of activity patterns when the cortex is spontaneously active is similar to those evoked by natural image stimulation and consistent with the topology of a two sphere. We discuss how this structure could emerge from the functional organization of orientation and spatial frequency maps and their mutual relationship. Our findings extend prior results on the relationship between spontaneous and evoked activity in V1 and illustrates how computational topology can help tackle elementary questions about the representation of information in the nervous system.

  136. The Extended Persistent Homology Transform of Manifolds With Boundary (2022)

    Katharine Turner, Vanessa Robins, James Morgan
    Abstract The Extended Persistent Homology Transform (XPHT) is a topological transform which takes as input a shape embedded in Euclidean space, and to each unit vector assigns the extended persistence module of the height function over that shape with respect to that direction. We can define a distance between two shapes by integrating over the sphere the distance between their respective extended persistence modules. By using extended persistence we get finite distances between shapes even when they have different Betti numbers. We use Morse theory to show that the extended persistence of a height function over a manifold with boundary can be deduced from the extended persistence for that height function restricted to the boundary, alongside labels on the critical points as positive or negative critical. We study the application of the XPHT to binary images; outlining an algorithm for efficient calculation of the XPHT exploiting relationships between the PHT of the boundary curves to the extended persistence of the foreground.

  137. A Novel Approach for Wafer Defect Pattern Classification Based on Topological Data Analysis (2023)

    Seungchan Ko, Dowan Koo
    Abstract In semiconductor manufacturing, wafer map defect pattern provides critical information for facility maintenance and yield management, so the classification of defect patterns is one of the most important tasks in the manufacturing process. In this paper, we propose a novel way to represent the shape of the defect pattern as a finite-dimensional vector, which will be used as an input for a neural network algorithm for classification. The main idea is to extract the topological features of each pattern by using the theory of persistent homology from topological data analysis (TDA). Through some experiments with a simulated dataset, we show that the proposed method is faster and much more efficient in training with higher accuracy, compared with the method using convolutional neural networks (CNN) which is the most common approach for wafer map defect pattern classification. Moreover, it was shown that our method outperforms the CNN-based method when the number of training data is not enough and is imbalanced.

  138. Persistent Homology for Path Planning in Uncertain Environments (2015)

    S. Bhattacharya, R. Ghrist, V. Kumar
    Abstract We address the fundamental problem of goal-directed path planning in an uncertain environment represented as a probability (of occupancy) map. Most methods generally use a threshold to reduce the grayscale map to a binary map before applying off-the-shelf techniques to find the best path. This raises the somewhat ill-posed question, what is the right (optimal) value to threshold the map? We instead suggest a persistent homology approach to the problem-a topological approach in which we seek the homology class of trajectories that is most persistent for the given probability map. In other words, we want the class of trajectories that is free of obstacles over the largest range of threshold values. In order to make this problem tractable, we use homology in ℤ2 coefficients (instead of the standard ℤ coefficients), and describe how graph search-based algorithms can be used to find trajectories in different homology classes. Our simulation results demonstrate the efficiency and practical applicability of the algorithm proposed in this paper.paper.

  139. Emotion Recognition in Talking-Face Videos Using Persistent Entropy and Neural Networks (2022)

    Eduardo Paluzo-Hidalgo, Rocio Gonzalez-Diaz, Guillermo Aguirre-Carrazana, Eduardo Paluzo-Hidalgo, Rocio Gonzalez-Diaz, Guillermo Aguirre-Carrazana
    Abstract \textlessabstract\textgreater\textlessp\textgreaterThe automatic recognition of a person's emotional state has become a very active research field that involves scientists specialized in different areas such as artificial intelligence, computer vision, or psychology, among others. Our main objective in this work is to develop a novel approach, using persistent entropy and neural networks as main tools, to recognise and classify emotions from talking-face videos. Specifically, we combine audio-signal and image-sequence information to compute a \textlessitalic\textgreatertopology signature\textless/italic\textgreater (a 9-dimensional vector) for each video. We prove that small changes in the video produce small changes in the signature, ensuring the stability of the method. These topological signatures are used to feed a neural network to distinguish between the following emotions: calm, happy, sad, angry, fearful, disgust, and surprised. The results reached are promising and competitive, beating the performances achieved in other state-of-the-art works found in the literature.\textless/p\textgreater\textless/abstract\textgreater

  140. Fibers of Failure: Classifying Errors in Predictive Processes (2020)

    Leo S. Carlsson, Mikael Vejdemo-Johansson, Gunnar Carlsson, Pär G. Jönsson
    Abstract Predictive models are used in many different fields of science and engineering and are always prone to make faulty predictions. These faulty predictions can be more or less malignant depending on the model application. We describe fibers of failure (FiFa), a method to classify failure modes of predictive processes. Our method uses Mapper, an algorithm from topological data analysis (TDA), to build a graphical model of input data stratified by prediction errors. We demonstrate two ways to use the failure mode groupings: either to produce a correction layer that adjusts predictions by similarity to the failure modes; or to inspect members of the failure modes to illustrate and investigate what characterizes each failure mode. We demonstrate FiFa on two scenarios: a convolutional neural network (CNN) predicting MNIST images with added noise, and an artificial neural network (ANN) predicting the electrical energy consumption of an electric arc furnace (EAF). The correction layer on the CNN model improved its prediction accuracy significantly while the inspection of failure modes for the EAF model provided guiding insights into the domain-specific reasons behind several high-error regions.

  141. Identification of Key Features Using Topological Data Analysis for Accurate Prediction of Manufacturing System Outputs (2017)

    Wei Guo, Ashis G. Banerjee
    Abstract Topological data analysis (TDA) has emerged as one of the most promising approaches to extract insights from high-dimensional data of varying types such as images, point clouds, and meshes, in an unsupervised manner. To the best of our knowledge, here, we provide the first successful application of TDA in the manufacturing systems domain. We apply a widely used TDA method, known as the Mapper algorithm, on two benchmark data sets for chemical process yield prediction and semiconductor wafer fault detection, respectively. The algorithm yields topological networks that capture the intrinsic clusters and connections among the clusters present in the data sets, which are difficult to detect using traditional methods. We select key process variables or features that impact the system outcomes by analyzing the network shapes. We then use predictive models to evaluate the impact of the selected features. Results show that the models achieve at least the same level of high prediction accuracy as with all the process variables, thereby, providing a way to carry out process monitoring and control in a more cost-effective manner.

  142. From Trees to Barcodes and Back Again: Theoretical and Statistical Perspectives (2020)

    Lida Kanari, Adélie Garin, Kathryn Hess
    Abstract Methods of topological data analysis have been successfully applied in a wide range of fields to provide useful summaries of the structure of complex data sets in terms of topological descriptors, such as persistence diagrams. While there are many powerful techniques for computing topological descriptors, the inverse problem, i.e., recovering the input data from topological descriptors, has proved to be challenging. In this article we study in detail the Topological Morphology Descriptor (TMD), which assigns a persistence diagram to any tree embedded in Euclidean space, and a sort of stochastic inverse to the TMD, the Topological Neuron Synthesis (TNS) algorithm, gaining both theoretical and computational insights into the relation between the two. We propose a new approach to classify barcodes using symmetric groups, which provides a concrete language to formulate our results. We investigate to what extent the TNS recovers a geometric tree from its TMD and describe the effect of different types of noise on the process of tree generation from persistence diagrams. We prove moreover that the TNS algorithm is stable with respect to specific types of noise.

  143. Persistent Homology Analysis of Brain Transcriptome Data in Autism (2019)

    Daniel Shnier, Mircea A. Voineagu, Irina Voineagu
    Abstract Persistent homology methods have found applications in the analysis of multiple types of biological data, particularly imaging data or data with a spatial and/or temporal component. However, few studies have assessed the use of persistent homology for the analysis of gene expression data. Here we apply persistent homology methods to investigate the global properties of gene expression in post-mortem brain tissue (cerebral cortex) of individuals with autism spectrum disorders (ASD) and matched controls. We observe a significant difference in the geometry of inter-sample relationships between autism and healthy controls as measured by the sum of the death times of zero-dimensional components and the Euler characteristic. This observation is replicated across two distinct datasets, and we interpret it as evidence for an increased heterogeneity of gene expression in autism. We also assessed the topology of gene-level point clouds and did not observe significant differences between ASD and control transcriptomes, suggesting that the overall transcriptome organization is similar in ASD and healthy cerebral cortex. Overall, our study provides a novel framework for persistent homology analyses of gene expression data for genetically complex disorders.

  144. Statistical Inference for Persistent Homology Applied to Simulated fMRI Time Series Data (2023)

    Hassan Abdallah, Adam Regalski, Mohammad Behzad Kang, Maria Berishaj, Nkechi Nnadi, Asadur Chowdury, Vaibhav A. Diwadkar, Andrew Salch
    Abstract Time-series data are amongst the most widely-used in biomedical sciences, including domains such as functional Magnetic Resonance Imaging (fMRI). Structure within time series data can be captured by the tools of topological data analysis (TDA). Persistent homology is the mostly commonly used data-analytic tool in TDA, and can effectively summarize complex high-dimensional data into an interpretable 2-dimensional representation called a persistence diagram. Existing methods for statistical inference for persistent homology of data depend on an independence assumption being satisfied. While persistent homology can be computed for each time index in a time-series, time-series data often fail to satisfy the independence assumption. This paper develops a statistical test that obviates the independence assumption by implementing a multi-level block sampled Monte Carlo test with sets of persistence diagrams. Its efficacy for detecting task-dependent topological organization is then demonstrated on simulated fMRI data. This new statistical test is therefore suitable for analyzing persistent homology of fMRI data, and of non-independent data in general.

  145. Determining Structural Properties of Artificial Neural Networks Using Algebraic Topology (2021)

    David Pérez Fernández, Asier Gutiérrez-Fandiño, Jordi Armengol-Estapé, Marta Villegas
    Abstract Artificial Neural Networks (ANNs) are widely used for approximating complex functions. The process that is usually followed to define the most appropriate architecture for an ANN given a specific function is mostly empirical. Once this architecture has been defined, weights are usually optimized according to the error function. On the other hand, we observe that ANNs can be represented as graphs and their topological 'fingerprints' can be obtained using Persistent Homology (PH). In this paper, we describe a proposal focused on designing more principled architecture search procedures. To do this, different architectures for solving problems related to a heterogeneous set of datasets have been analyzed. The results of the evaluation corroborate that PH effectively characterizes the ANN invariants: when ANN density (layers and neurons) or sample feeding order is the only difference, PH topological invariants appear; in the opposite direction in different sub-problems (i.e. different labels), PH varies. This approach based on topological analysis helps towards the goal of designing more principled architecture search procedures and having a better understanding of ANNs.

  146. Uncovering the Topology of Time-Varying fMRI Data Using Cubical Persistence (2020)

    Bastian Rieck, Tristan Yates, Christian Bock, Karsten Borgwardt, Guy Wolf, Nicholas Turk-Browne, Smita Krishnaswamy
    Abstract Functional magnetic resonance imaging (fMRI) is a crucial technology for gaining insights into cognitive processes in humans. Data amassed from fMRI measurements result in volumetric data sets that vary over time. However, analysing such data presents a challenge due to the large degree of noise and person-to-person variation in how information is represented in the brain. To address this challenge, we present a novel topological approach that encodes each time point in an fMRI data set as a persistence diagram of topological features, i.e. high-dimensional voids present in the data. This representation naturally does not rely on voxel-by-voxel correspondence and is robust to noise. We show that these time-varying persistence diagrams can be clustered to find meaningful groupings between participants, and that they are also useful in studying within-subject brain state trajectories of subjects performing a particular task. Here, we apply both clustering and trajectory analysis techniques to a group of participants watching the movie 'Partly Cloudy'. We observe significant differences in both brain state trajectories and overall topological activity between adults and children watching the same movie.

  147. Topological Eulerian Synthesis of Slow Motion Periodic Videos (2018)

    Christopher Tralie, Matthew Berger
    Abstract We consider the problem of taking a video that is comprised of multiple periods of repetitive motion, and reordering the frames of the video into a single period, producing a detailed, single cycle video of motion. This problem is challenging, as such videos often contain noise, drift due to camera motion and from cycle to cycle, and irrelevant background motion/occlusions, and these factors can confound the relevant periodic motion we seek in the video. To address these issues in a simple and efficient manner, we introduce a tracking free Eulerian approach for synthesizing a single cycle of motion. Our approach is geometric: we treat each frame as a point in high-dimensional Euclidean space, and analyze the sliding window embedding formed by this sequence of points, which yields samples along a topological loop regardless of the type of periodic motion. We combine tools from topological data analysis and spectral geometric analysis to estimate the phase of each window, and we exploit the sliding window structure to robustly reorder frames. We show quantitative results that highlight the robustness of our technique to camera shake, noise, and occlusions, and qualitative results of single-cycle motion synthesis across a variety of scenarios.

  148. Persistent Homology Index as a Robust Quantitative Measure of Immunohistochemical Scoring (2017)

    Akihiro Takiyama, Takashi Teramoto, Hiroaki Suzuki, Katsushige Yamashiro, Shinya Tanaka
    Abstract Immunohistochemical data (IHC) plays an important role in clinical practice, and is typically gathered in a semi-quantitative fashion that relies on some degree of visual scoring. However, visual scoring by a pathologist is inherently subjective and manifests both intra-observer and inter-observer variability. In this study, we introduce a novel computer-aided quantification methodology for immunohistochemical scoring that uses the algebraic concept of persistent homology. Using 8 bit grayscale image data derived from 90 specimens of invasive ductal carcinoma of the breast, stained for the replicative marker Ki-67, we computed homology classes. These were then compared to nuclear grades and the Ki-67 labeling indices obtained by visual scoring. Three metrics for IHC staining were newly defined: Persistent Homology Index (PHI), center coordinates of positive and negative groups, and the sum of squares within groups (WSS). This study demonstrates that PHI, a novel index for immunohistochemical labeling using persistent homology, can produce highly similar data to that generated by a pathologist using visual evaluation. The potential benefits associated with our novel technology include both improved quantification and reproducibility. Since our method reflects cellularity and nuclear atypia, it carries a greater quantity of biologic data compared to conventional evaluation using Ki-67.

  149. Topological Methods Reveal High and Low Functioning Neuro-Phenotypes Within Fragile X Syndrome (2014)

    David Romano, Monica Nicolau, Eve-Marie Quintin, Paul K. Mazaika, Amy A. Lightbody, Heather Cody Hazlett, Joseph Piven, Gunnar Carlsson, Allan L. Reiss
    Abstract Fragile X syndrome (FXS), due to mutations of the FMR1 gene, is the most common known inherited cause of developmental disability as well as the most common single-gene risk factor for autism. Our goal was to examine variation in brain structure in FXS with topological data analysis (TDA), and to assess how such variation is associated with measures of IQ and autism-related behaviors. To this end, we analyzed imaging and behavioral data from young boys (n = 52; aged 1.57–4.15 years) diagnosed with FXS. Application of topological methods to structural MRI data revealed two large subgroups within the study population. Comparison of these subgroups showed significant between-subgroup neuroanatomical differences similar to those previously reported to distinguish children with FXS from typically developing controls (e.g., enlarged caudate). In addition to neuroanatomy, the groups showed significant differences in IQ and autism severity scores. These results suggest that despite arising from a single gene mutation, FXS may encompass two biologically, and clinically separable phenotypes. In addition, these findings underscore the potential of TDA as a powerful tool in the search for biological phenotypes of neuropsychiatric disorders. Hum Brain Mapp 35:4904–4915, 2014. © 2014 Wiley Periodicals, Inc.

  150. Persistence-Based Pooling for Shape Pose Recognition (2016)

    Thomas Bonis, Maks Ovsjanikov, Steve Oudot, Frédéric Chazal
    Abstract In this paper, we propose a novel pooling approach for shape classification and recognition using the bag-of-words pipeline, based on topological persistence, a recent tool from Topological Data Analysis. Our technique extends the standard max-pooling, which summarizes the distribution of a visual feature with a single number, thereby losing any notion of spatiality. Instead, we propose to use topological persistence, and the derived persistence diagrams, to provide significantly more informative and spatially sensitive characterizations of the feature functions, which can lead to better recognition performance. Unfortunately, despite their conceptual appeal, persistence diagrams are difficult to handle, since they are not naturally represented as vectors in Euclidean space and even the standard metric, the bottleneck distance is not easy to compute. Furthermore, classical distances between diagrams, such as the bottleneck and Wasserstein distances, do not allow to build positive definite kernels that can be used for learning. To handle this issue, we provide a novel way to transform persistence diagrams into vectors, in which comparisons are trivial. Finally, we demonstrate the performance of our construction on the Non-Rigid 3D Human Models SHREC 2014 dataset, where we show that topological pooling can provide significant improvements over the standard pooling methods for the shape pose recognition within the bag-of-words pipeline.

  151. Optimizing Porosity Detection in Wire Laser Metal Deposition Processes Through Data-Driven AI Classification Techniques (2023)

    Meritxell Gomez-Omella, Jon Flores, Basilio Sierra, Susana Ferreiro, Nicolas Hascoët, Francisco Chinesta
    Abstract Additive manufacturing (AM) is an attractive solution for many companies that produce geometrically complex parts. This process consists of depositing material layer by layer following a sliced CAD geometry. It brings several benefits to manufacturing capabilities, such as design freedom, reduced material waste, and short-run customization. However, one of the current challenges faced by users of the process, mainly in wire laser metal deposition (wLMD), is to avoid defects in the manufactured part, especially the porosity. This defect is caused by extreme conditions and metallurgical transformations of the process. And not only does it directly affect the mechanical performance of the parts, especially the fatigue properties, but it also means an increase in costs due to the inspection tasks to which the manufactured parts must be subjected. This work compares three operational solution approaches, product-centric, based on signal-based feature extraction and Topological Data Analysis together with statistical and Machine Learning (ML) techniques, for the early detection and prediction of porosity failure in a wLMD process. The different forecasting and validation strategies demonstrate the variety of conclusions that can be drawn with different objectives in the analysis of the monitored data in AM problems.

  152. Multiresolution Persistent Homology for Excessively Large Biomolecular Datasets (2015)

    Kelin Xia, Zhixiong Zhao, Guo-Wei Wei
    Abstract Although persistent homology has emerged as a promising tool for the topological simplification of complex data, it is computationally intractable for large datasets. We introduce multiresolution persistent homology to handle excessively large datasets. We match the resolution with the scale of interest so as to represent large scale datasets with appropriate resolution. We utilize flexibility-rigidity index to access the topological connectivity of the data set and define a rigidity density for the filtration analysis. By appropriately tuning the resolution of the rigidity density, we are able to focus the topological lens on the scale of interest. The proposed multiresolution topological analysis is validated by a hexagonal fractal image which has three distinct scales. We further demonstrate the proposed method for extracting topological fingerprints from DNA molecules. In particular, the topological persistence of a virus capsid with 273 780 atoms is successfully analyzed which would otherwise be inaccessible to the normal point cloud method and unreliable by using coarse-grained multiscale persistent homology. The proposed method has also been successfully applied to the protein domain classification, which is the first time that persistent homology is used for practical protein domain analysis, to our knowledge. The proposed multiresolution topological method has potential applications in arbitrary data sets, such as social networks, biological networks, and graphs.

  153. Branching and Circular Features in High Dimensional Data (2011)

    B. Wang, B. Summa, V. Pascucci, M. Vejdemo-Johansson
    Abstract Large observations and simulations in scientific research give rise to high-dimensional data sets that present many challenges and opportunities in data analysis and visualization. Researchers in application domains such as engineering, computational biology, climate study, imaging and motion capture are faced with the problem of how to discover compact representations of highdimensional data while preserving their intrinsic structure. In many applications, the original data is projected onto low-dimensional space via dimensionality reduction techniques prior to modeling. One problem with this approach is that the projection step in the process can fail to preserve structure in the data that is only apparent in high dimensions. Conversely, such techniques may create structural illusions in the projection, implying structure not present in the original high-dimensional data. Our solution is to utilize topological techniques to recover important structures in high-dimensional data that contains non-trivial topology. Specifically, we are interested in high-dimensional branching structures. We construct local circle-valued coordinate functions to represent such features. Subsequently, we perform dimensionality reduction on the data while ensuring such structures are visually preserved. Additionally, we study the effects of global circular structures on visualizations. Our results reveal never-before-seen structures on real-world data sets from a variety of applications.

  154. WDR76 Co-Localizes With Heterochromatin Related Proteins and Rapidly Responds to DNA Damage (2016)

    Joshua M. Gilmore, Mihaela E. Sardiu, Brad D. Groppe, Janet L. Thornton, Xingyu Liu, Gerald Dayebgadoh, Charles A. Banks, Brian D. Slaughter, Jay R. Unruh, Jerry L. Workman, Laurence Florens, Michael P. Washburn
    Abstract Proteins that respond to DNA damage play critical roles in normal and diseased states in human biology. Studies have suggested that the S. cerevisiae protein CMR1/YDL156w is associated with histones and is possibly associated with DNA repair and replication processes. Through a quantitative proteomic analysis of affinity purifications here we show that the human homologue of this protein, WDR76, shares multiple protein associations with the histones H2A, H2B, and H4. Furthermore, our quantitative proteomic analysis of WDR76 associated proteins demonstrated links to proteins in the DNA damage response like PARP1 and XRCC5 and heterochromatin related proteins like CBX1, CBX3, and CBX5. Co-immunoprecipitation studies validated these interactions. Next, quantitative imaging studies demonstrated that WDR76 was recruited to laser induced DNA damage immediately after induction, and we compared the recruitment of WDR76 to laser induced DNA damage to known DNA damage proteins like PARP1, XRCC5, and RPA1. In addition, WDR76 co-localizes to puncta with the heterochromatin proteins CBX1 and CBX5, which are also recruited to DNA damage but much less intensely than WDR76. This work demonstrates the chromatin and DNA damage protein associations of WDR76 and demonstrates the rapid response of WDR76 to laser induced DNA damage.

  155. Topology Highlights Mesoscopic Functional Equivalence Between Imagery and Perception: The Case of Hypnotizability (2019)

    Esther Ibáñez-Marcelo, Lisa Campioni, Angkoon Phinyomark, Giovanni Petri, Enrica L. Santarcangelo
    Abstract The functional equivalence (FE) between imagery and perception or motion has been proposed on the basis of neuroimaging evidence of large spatially overlapping activations between real and imagined sensori-motor conditions. However, similar local activation patterns do not imply the same mesoscopic integration of brain regions, which can be described by tools from Topological Data Analysis (TDA). On the basis of behavioral findings, stronger FE has been hypothesized in the individuals with high scores of hypnotizability scores (highs) with respect to low hypnotizable participants (lows) who differ between each other in the proneness to modify memory, perception and behavior according to specific imaginative suggestions. Here we present the first EEG evidence of stronger FE in highs. In fact, persistent homology shows that the highs EEG topological asset during real and imagined sensory conditions is significantly more similar than the lows. As a corollary finding, persistent homology shows lower restructuring of the EEG asset in highs than in lows during both sensory and imagery tasks with respect to basal conditions. Present findings support the view that greater embodiment of mental images may be responsible for the highs greater proneness to respond to sensori-motor suggestions and to report involuntariness in action. In addition, findings indicate hypnotizability-related sensory and cognitive information processing and suggest that the psycho-physiological trait of hypnotizability may modulate more than one aspect of the everyday life.

  156. Topological Data Analysis of Zebrafish Patterns (2020)

    Melissa R. McGuirl, Alexandria Volkening, Björn Sandstede
    Abstract Self-organized pattern behavior is ubiquitous throughout nature, from fish schooling to collective cell dynamics during organism development. Qualitatively these patterns display impressive consistency, yet variability inevitably exists within pattern-forming systems on both microscopic and macroscopic scales. Quantifying variability and measuring pattern features can inform the underlying agent interactions and allow for predictive analyses. Nevertheless, current methods for analyzing patterns that arise from collective behavior capture only macroscopic features or rely on either manual inspection or smoothing algorithms that lose the underlying agent-based nature of the data. Here we introduce methods based on topological data analysis and interpretable machine learning for quantifying both agent-level features and global pattern attributes on a large scale. Because the zebrafish is a model organism for skin pattern formation, we focus specifically on analyzing its skin patterns as a means of illustrating our approach. Using a recent agent-based model, we simulate thousands of wild-type and mutant zebrafish patterns and apply our methodology to better understand pattern variability in zebrafish. Our methodology is able to quantify the differential impact of stochasticity in cell interactions on wild-type and mutant patterns, and we use our methods to predict stripe and spot statistics as a function of varying cellular communication. Our work provides an approach to automatically quantifying biological patterns and analyzing agent-based dynamics so that we can now answer critical questions in pattern formation at a much larger scale.

  157. Topological Data Analysis of Collective and Individual Epithelial Cells Using Persistent Homology of Loops (2021)

    Dhananjay Bhaskar, William Y. Zhang, Ian Y. Wong
    Abstract Interacting, self-propelled particles such as epithelial cells can dynamically self-organize into complex multicellular patterns, which are challenging to classify without a priori information. Classically, different phases and phase transitions have been described based on local ordering, which may not capture structural features at larger length scales. Instead, topological data analysis (TDA) determines the stability of spatial connectivity at varying length scales (i.e. persistent homology), and can compare different particle configurations based on the “cost” of reorganizing one configuration into another. Here, we demonstrate a topology-based machine learning approach for unsupervised profiling of individual and collective phases based on large-scale loops. We show that these topological loops (i.e. dimension 1 homology) are robust to variations in particle number and density, particularly in comparison to connected components (i.e. dimension 0 homology). We use TDA to map out phase diagrams for simulated particles with varying adhesion and propulsion, at constant population size as well as when proliferation is permitted. Next, we use this approach to profile our recent experiments on the clustering of epithelial cells in varying growth factor conditions, which are compared to our simulations. Finally, we characterize the robustness of this approach at varying length scales, with sparse sampling, and over time. Overall, we envision TDA will be broadly applicable as a model-agnostic approach to analyze active systems with varying population size, from cytoskeletal motors to motile cells to flocking or swarming animals.

  158. Persistent Homology of Time-Dependent Functional Networks Constructed From Coupled Time Series (2017)

    Bernadette J. Stolz, Heather A. Harrington, Mason A. Porter
    Abstract We use topological data analysis to study “functional networks” that we construct from time-series data from both experimental and synthetic sources. We use persistent homology with a weight rank clique filtration to gain insights into these functional networks, and we use persistence landscapes to interpret our results. Our first example uses time-series output from networks of coupled Kuramoto oscillators. Our second example consists of biological data in the form of functional magnetic resonance imaging data that were acquired from human subjects during a simple motor-learning task in which subjects were monitored for three days during a five-day period. With these examples, we demonstrate that (1) using persistent homology to study functional networks provides fascinating insights into their properties and (2) the position of the features in a filtration can sometimes play a more vital role than persistence in the interpretation of topological features, even though conventionally the latter is used to distinguish between signal and noise. We find that persistent homology can detect differences in synchronization patterns in our data sets over time, giving insight both on changes in community structure in the networks and on increased synchronization between brain regions that form loops in a functional network during motor learning. For the motor-learning data, persistence landscapes also reveal that on average the majority of changes in the network loops take place on the second of the three days of the learning process.

  159. The Shape of Cancer Relapse: Topological Data Analysis Predicts Recurrence in Paediatric Acute Lymphoblastic Leukaemia (2021)

    Salvador Chulián, Bernadette J. Stolz, Álvaro Martínez-Rubio, Cristina Blázquez Goñi, Juan F. Rodríguez Gutiérrez, Teresa Caballero Velázquez, Águeda Molinos Quintana, Manuel Ramírez Orellana, Ana Castillo Robleda, José Luis Fuster Soler, Alfredo Minguela Puras, María Victoria Martínez Sánchez, María Rosa, Víctor M. Pérez-García, Helen Byrne
    Abstract Acute Lymphoblastic Leukaemia (ALL) is the most frequent paediatric cancer. Modern therapies have improved survival rates, but approximately 15-20 % of patients relapse. At present, patients’ risk of relapse are assessed by projecting high-dimensional flow cytometry data onto a subset of biomarkers and manually estimating the shape of this reduced data. Here, we apply methods from topological data analysis (TDA), which quantify shape in data via features such as connected components and loops, to pre-treatment ALL datasets with known outcomes. We combine these fully unsupervised analyses with machine learning to identify features in the pre-treatment data that are prognostic for risk of relapse. We find significant topological differences between relapsing and non-relapsing patients and confirm the predictive power of CD10, CD20, CD38, and CD45. Further, we are able to use the TDA descriptors to predict patients who relapsed. We propose three prognostic pipelines that readily extend to other haematological malignancies. Teaser Topology reveals features in flow cytometry data which predict relapse of patients with acute lymphoblastic leukemia

  160. Measuring Hidden Phenotype: Quantifying the Shape of Barley Seeds Using the Euler Characteristic Transform (2021)

    Erik J. Amézquita, Michelle Y. Quigley, Tim Ophelders, Jacob B. Landis, Daniel Koenig, Elizabeth Munch, Daniel H. Chitwood
    Abstract Shape plays a fundamental role in biology. Traditional phenotypic analysis methods measure some features but fail to measure the information embedded in shape comprehensively. To extract, compare, and analyze this information embedded in a robust and concise way, we turn to Topological Data Analysis (TDA), specifically the Euler Characteristic Transform. TDA measures shape comprehensively using mathematical representations based on algebraic topology features. To study its use, we compute both traditional and topological shape descriptors to quantify the morphology of 3121 barley seeds scanned with X-ray Computed Tomography (CT) technology at 127 micron resolution. The Euler Characteristic Transform measures shape by analyzing topological features of an object at thresholds across a number of directional axes. A Kruskal-Wallis analysis of the information encoded by the topological signature reveals that the Euler Characteristic Transform picks up successfully the shape of the crease and bottom of the seeds. Moreover, while traditional shape descriptors can cluster the seeds based on their accession, topological shape descriptors can cluster them further based on their panicle. We then successfully train a support vector machine (SVM) to classify 28 different accessions of barley based exclusively on the shape of their grains. We observe that combining both traditional and topological descriptors classifies barley seeds better than using just traditional descriptors alone. This improvement suggests that TDA is thus a powerful complement to traditional morphometrics to comprehensively describe a multitude of “hidden” shape nuances which are otherwise not detected.

  161. Topological Descriptors Help Predict Guest Adsorption in Nanoporous Materials (2020)

    Aditi S. Krishnapriyan, Maciej Haranczyk, Dmitriy Morozov
    Abstract Machine learning has emerged as an attractive alternative to experiments and simulations for predicting material properties. Usually, such an approach relies on specific domain knowledge for feature design: each learning target requires careful selection of features that an expert recognizes as important for the specific task. The major drawback of this approach is that computation of only a few structural features has been implemented so far, and it is difficult to tell a priori which features are important for a particular application. The latter problem has been empirically observed for predictors of guest uptake in nanoporous materials: local and global porosity features become dominant descriptors at low and high pressures, respectively. We investigate a feature representation of materials using tools from topological data analysis. Specifically, we use persistent homology to describe the geometry of nanoporous materials at various scales. We combine our topological descriptor with traditional structural features and investigate the relative importance of each to the prediction tasks. We demonstrate an application of this feature representation by predicting methane adsorption in zeolites, for pressures in the range of 1-200 bar. Our results not only show a considerable improvement compared to the baseline, but they also highlight that topological features capture information complementary to the structural features: this is especially important for the adsorption at low pressure, a task particularly difficult for the traditional features. Furthermore, by investigation of the importance of individual topological features in the adsorption model, we are able to pinpoint the location of the pores that correlate best to adsorption at different pressure, contributing to our atom-level understanding of structure-property relationships.

  162. Topological Data Analysis Reveals Robust Alterations in the Whole-Brain and Frontal Lobe Functional Connectomes in Attention-Deficit/Hyperactivity Disorder (2020)

    Zeus Gracia-Tabuenca, Juan Carlos Díaz-Patiño, Isaac Arelio, Sarael Alcauter
    Abstract Visual Abstract \textlessimg class="highwire-fragment fragment-image" alt="Figure" src="https://www.eneuro.org/content/eneuro/7/3/ENEURO.0543-19.2020/F1.medium.gif" width="369" height="440"/\textgreaterDownload figureOpen in new tabDownload powerpoint Attention-deficit/hyperactivity disorder (ADHD) is a developmental disorder characterized by difficulty to control the own behavior. Neuroimaging studies have related ADHD with the interplay of fronto-parietal attention systems with the default mode network (DMN; Castellanos and Aoki, 2016). However, some results have been inconsistent, potentially due to methodological differences in the analytical strategies when defining the brain functional network, i.e., the functional connectivity threshold and/or the brain parcellation scheme. Here, we make use of topological data analysis (TDA) to explore the brain connectome as a function of the filtration value (i.e., the connectivity threshold), instead of using a static connectivity threshold. Specifically, we characterized the transition from all nodes being isolated to being connected into a single component as a function of the filtration value. We explored the utility of such a method to identify differences between 81 children with ADHD (45 male, age: 7.26–17.61 years old) and 96 typically developing children (TDC; 59 male, age: 7.17–17.96 years old), using a public dataset of resting state (rs)fMRI in human subjects. Results were highly congruent when using four different brain segmentations (atlases), and exhibited significant differences for the brain topology of children with ADHD, both at the whole-brain network and the functional subnetwork levels, particularly involving the frontal lobe and the DMN. Therefore, this is a solid approach that complements connectomics-related methods and may contribute to identify the neurophysio-pathology of ADHD.

  163. Applications of Persistent Homology to Time Varying Systems (2013)

    Elizabeth Munch
    Abstract \textlessp\textgreaterThis dissertation extends the theory of persistent homology to time varying systems. Most of the previous work has been dedicated to using this powerful tool in topological data analysis to study static point clouds. In particular, given a point cloud, we can construct its persistence diagram. Since the diagram varies continuously as the point cloud varies continuously, we study the space of time varying persistence diagrams, called vineyards when they were introduced by Cohen-Steiner, Edelsbrunner, and Morozov.\textless/p\textgreater\textlessp\textgreaterWe will first show that with a good choice of metric, these vineyards are stable for small perturbations of their associated point clouds. We will also define a new mean for a set of persistence diagrams based on the work of Mileyko et al. which, unlike the previously defined mean, is continuous for geodesic vineyards. \textless/p\textgreater\textlessp\textgreaterNext, we study the sensor network problem posed by Ghrist and de Silva, and their application of persistent homology to understand when a set of sensors covers a given region. Giving each of these sensors a probability of failure over time, we show that an exact computation of the probability of failure of the whole system is NP-hard, but give an algorithm which can predict failure in the case of a monitored system.\textless/p\textgreater\textlessp\textgreaterFinally, we apply these methods to an automated system which can cluster agents moving in aerial images by their behaviors. We build a data structure for storing and querying the information in real-time, and define behavior vectors which quantify behaviors of interest. This clustering by behavior can be used to find groups of interest, for which we can also quantify behaviors in order to determine whether the group is working together to achieve a common goal, and we speculate that this work can be extended to improving tracking algorithms as well as behavioral predictors.\textless/p\textgreater