🍩 Database of Original & Non-Theoretical Uses of Topology

(found 7 matches in 0.005238s)

  1. 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.

  2. Equivariant Geometric Learning for Digital Rock Physics: Estimating Formation Factor and Effective Permeability Tensors From Morse Graph (2023)

    Chen Cai, Nikolaos Vlassis, Lucas Magee, Ran Ma, Zeyu Xiong, Bahador Bahmani, Teng-Fong Wong, Yusu Wang, WaiChing Sun
    Abstract We present a SE(3)-equivariant graph neural network (GNN) approach that directly predicts the formation factor and effective permeability from micro-CT images. ...

  3. Semantic Segmentation of Microscopic Neuroanatomical Data by Combining Topological Priors With Encoder–decoder Deep Networks (2020)

    Samik Banerjee, Lucas Magee, Dingkang Wang, Xu Li, Bing-Xing Huo, Jaikishan Jayakumar, Katherine Matho, Meng-Kuan Lin, Keerthi Ram, Mohanasankar Sivaprakasam, Josh Huang, Yusu Wang, Partha P. Mitra
    Abstract Understanding of neuronal circuitry at cellular resolution within the brain has relied on neuron tracing methods that involve careful observation and interpretation by experienced neuroscientists. With recent developments in imaging and digitization, this approach is no longer feasible with the large-scale (terabyte to petabyte range) images. Machine-learning-based techniques, using deep networks, provide an efficient alternative to the problem. However, these methods rely on very large volumes of annotated images for training and have error rates that are too high for scientific data analysis, and thus requires a substantial volume of human-in-the-loop proofreading. Here we introduce a hybrid architecture combining prior structure in the form of topological data analysis methods, based on discrete Morse theory, with the best-in-class deep-net architectures for the neuronal connectivity analysis. We show significant performance gains using our hybrid architecture on detection of topological structure (for example, connectivity of neuronal processes and local intensity maxima on axons corresponding to synaptic swellings) with precision and recall close to 90% compared with human observers. We have adapted our architecture to a high-performance pipeline capable of semantic segmentation of light-microscopic whole-brain image data into a hierarchy of neuronal compartments. We expect that the hybrid architecture incorporating discrete Morse techniques into deep nets will generalize to other data domains.

  4. 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.

  5. 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.

  6. 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.

  7. Efficient Map Reconstruction and Augmentation via Topological Methods (2015)

    Suyi Wang, Yusu Wang, Yanjie Li
    Abstract In recent years, with the rapid growth in the amount of publicly available Volunteered Geographic Information (VGI) data, automatic map generation from GPS trajectories has attracted great attention. Maps generated from these data can for example complement commercial maps in less developed areas. Two main challenges in the automatic generation of maps from volunteered GPS data are the handling of noise and of non-homogeneous sampling of road segments (for example, roads in downtown area can receive significantly more GPS traces than roads in residential areas). In this paper, we present a novel framework for map reconstruction based on a topological idea: the Morse theory. In particular, the use of Morse theory and topological simplification allows us to handle the issues of both noise and non-homogeneous sampling in an elegant unified framework. Our algorithm is significantly simpler than previous approaches, both conceptually and implementation speaking. Little pre- and post-processing is required, and yet the algorithm can reconstruct robust road-networks from challenging data sets (such as GPS traces for Berlin or Beijing cities) that are comparable or better than the output of previous state-of-the-art approaches. The new algorithm is also orders of magnitude faster than previous approaches on large data sets (for example, the entire processing of the Berlin city data with about 27189 trajectories takes less than one minute).Furthermore, our framework can be easily extended to handle the map integration problem, where one wishes to integrate multiple maps into a single one. Here, roads in different maps can have different confidence levels, and higher confident roads will have larger influence in the final integrated road. We also present an effective algorithm for a slightly different map augmentation problem, where one wishes to augment a map, say G2, using partial but more trust-worthy map G1, in the sense that in the final map, information in G1 needs to be completely preserved.