🍩 Database of Original & Non-Theoretical Uses of Topology

(found 10 matches in 0.006569s)

  1. 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.

  2. 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.

  3. 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.

  4. 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.

  5. 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.

  6. 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.

  7. 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.

  8. 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.

  9. 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

  10. 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.