Special Issue "Computational Algebraic Topology and Neural Networks in Computer Vision"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematics and Computer Science".

Deadline for manuscript submissions: 30 September 2022 | Viewed by 7012

Special Issue Editors

Prof. Rocio Gonzalez Diaz
E-Mail Website
Guest Editor
Departmento de Matemática Aplicada I, Escuelta Técnica Superior de Ingeniería Informática, Universidad de Sevilla Campus, Reina Mercedes, 41012 Sevilla, Spain
Interests: computational algebraic topology, topological data analysis; computer vision; discrete geometry; combinatorial image analysis; neural networks
Prof. Dr. Matthias Zeppelzauer
E-Mail Website
Co-Guest Editor
Institute of Creative Media Technologies, University of Applied Sciences, St. Pölten, Matthias Corvinus-Strasse 15, 3100 St.Pölten, Austria
Interests: computer vision; pattern mining; topological data analysis; representation learning; active learning

Special Issue Information

Dear Colleagues,

Algebraic topology uses tools from abstract algebra to study topological spaces with the aim of defining algebraic invariants that classify topological spaces up to homeomorphism. Computational algebraic topology (CAT) provides methods to compute these invariants. The main topics in (Computational) Algebraic topology are simplicial and CW complexes, chain complexes, (co)homology and exact sequences. The recent field of Topological Data Analysis (TDA) is an approach to the analysis of datasets using techniques mainly from computational algebraic topology, being its leading tool persistent homology. In recent years, the development of methods based on neural networks is ubiquitous in computer vision. Recent research has shown that the integration of TDA into computer vision can improve performance. The integration of TDA into state-of-the-art computer vision approaches building heavily on deep neural networks is however still a challenging topic.
This Special Issue collects papers with the aim to develop novel topology-based approaches for computer vision and/or to apply topology-based approaches to improve the current state-of-the-art in computer vision. Of special interest are papers that deal with the challenge of integrating topology as the main tool into neural networks for computer vision applications.

This special issue is endorsed by the IAPR-TC3 (International Association for Pattern Recognition Technical Committee 03 - Neural Networks & Computational Intelligence), http://iapr-tc3.diism.unisi.it.

Prof. Rocio Gonzalez Diaz
Prof. Dr. Matthias Zeppelzauer
Guest Editors

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Keywords

  • Computational Algebraic Topology
  • Topological Data Analysis
  • Computer Vision
  • Neural Networks

Published Papers (7 papers)

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Research

Article
Image Inpainting for 3D Reconstruction Based on the Known Region Boundaries
Mathematics 2022, 10(15), 2761; https://doi.org/10.3390/math10152761 - 03 Aug 2022
Viewed by 221
Abstract
Pointcloud is a collection of 3D object coordinate systems in 3D scene. Generally, point data in pointclouds represent the outer surface of an object. It is widely used in 3D reconstruction applications in various fields. When obtaining pointcloud data from RGB-D images, if [...] Read more.
Pointcloud is a collection of 3D object coordinate systems in 3D scene. Generally, point data in pointclouds represent the outer surface of an object. It is widely used in 3D reconstruction applications in various fields. When obtaining pointcloud data from RGB-D images, if part of the information in the RGB-D images is lost or damaged, the pointcloud data will be hollow or too sparse. Moreover, it is not conducive to the subsequent application of pointcloud data. Based on the boundary of the region to be repaired, we proposes to repair the damaged image and synthesize the complete pointcloud data after a series of preprocessing steps related to the image. Experiments show that the our method can effectively improve the restoration of the lost details of the pixel in the target area and that it will have the fuller pointcloud data after synthesizing the restored image. Full article
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Article
TopoResNet: A Hybrid Deep Learning Architecture and Its Application to Skin Lesion Classification
Mathematics 2021, 9(22), 2924; https://doi.org/10.3390/math9222924 - 17 Nov 2021
Cited by 1 | Viewed by 509
Abstract
The application of artificial intelligence (AI) to various medical subfields has been a popular topic of research in recent years. In particular, deep learning has been widely used and has proven effective in many cases. Topological data analysis (TDA)—a rising field at the [...] Read more.
The application of artificial intelligence (AI) to various medical subfields has been a popular topic of research in recent years. In particular, deep learning has been widely used and has proven effective in many cases. Topological data analysis (TDA)—a rising field at the intersection of mathematics, statistics, and computer science—offers new insights into data. In this work, we develop a novel deep learning architecture that we call TopoResNet that integrates topological information into the residual neural network architecture. To demonstrate TopoResNet, we apply it to a skin lesion classification problem. We find that TopoResNet improves the accuracy and the stability of the training process. Full article
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Article
Isometry Invariant Shape Recognition of Projectively Perturbed Point Clouds by the Mergegram Extending 0D Persistence
Mathematics 2021, 9(17), 2121; https://doi.org/10.3390/math9172121 - 01 Sep 2021
Viewed by 761
Abstract
Rigid shapes should be naturally compared up to rigid motion or isometry, which preserves all inter-point distances. The same rigid shape can be often represented by noisy point clouds of different sizes. Hence, the isometry shape recognition problem requires methods that are independent [...] Read more.
Rigid shapes should be naturally compared up to rigid motion or isometry, which preserves all inter-point distances. The same rigid shape can be often represented by noisy point clouds of different sizes. Hence, the isometry shape recognition problem requires methods that are independent of a cloud size. This paper studies stable-under-noise isometry invariants for the recognition problem stated in the harder form when given clouds can be related by affine or projective transformations. The first contribution is the stability proof for the invariant mergegram, which completely determines a single-linkage dendrogram in general position. The second contribution is the experimental demonstration that the mergegram outperforms other invariants in recognizing isometry classes of point clouds extracted from perturbed shapes in images. Full article
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Article
Stable Topological Summaries for Analyzing the Organization of Cells in a Packed Tissue
Mathematics 2021, 9(15), 1723; https://doi.org/10.3390/math9151723 - 22 Jul 2021
Viewed by 619
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 [...] Read more.
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. Full article
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Article
Simplicial-Map Neural Networks Robust to Adversarial Examples
Mathematics 2021, 9(2), 169; https://doi.org/10.3390/math9020169 - 15 Jan 2021
Cited by 2 | Viewed by 997
Abstract
Broadly speaking, an adversarial example against a classification model occurs when a small perturbation on an input data point produces a change on the output label assigned by the model. Such adversarial examples represent a weakness for the safety of neural network applications, [...] Read more.
Broadly speaking, an adversarial example against a classification model occurs when a small perturbation on an input data point produces a change on the output label assigned by the model. Such adversarial examples represent a weakness for the safety of neural network applications, and many different solutions have been proposed for minimizing their effects. In this paper, we propose a new approach by means of a family of neural networks called simplicial-map neural networks constructed from an Algebraic Topology perspective. Our proposal is based on three main ideas. Firstly, given a classification problem, both the input dataset and its set of one-hot labels will be endowed with simplicial complex structures, and a simplicial map between such complexes will be defined. Secondly, a neural network characterizing the classification problem will be built from such a simplicial map. Finally, by considering barycentric subdivisions of the simplicial complexes, a decision boundary will be computed to make the neural network robust to adversarial attacks of a given size. Full article
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Article
Dynamic Graph Learning: A Structure-Driven Approach
Mathematics 2021, 9(2), 168; https://doi.org/10.3390/math9020168 - 15 Jan 2021
Cited by 1 | Viewed by 936
Abstract
The purpose of this paper is to infer a dynamic graph as a global (collective) model of time-varying measurements at a set of network nodes. This model captures both pairwise as well as higher order interactions (i.e., more than two nodes) among the [...] Read more.
The purpose of this paper is to infer a dynamic graph as a global (collective) model of time-varying measurements at a set of network nodes. This model captures both pairwise as well as higher order interactions (i.e., more than two nodes) among the nodes. The motivation of this work lies in the search for a connectome model which properly captures brain functionality across all regions of the brain, and possibly at individual neurons. We formulate it as an optimization problem, a quadratic objective functional and tensor information of observed node signals over short time intervals. The proper regularization constraints reflect the graph smoothness and other dynamics involving the underlying graph’s Laplacian, as well as the time evolution smoothness of the underlying graph. The resulting joint optimization is solved by a continuous relaxation of the weight parameters and an introduced novel gradient-projection scheme. While the work may be applicable to any time-evolving data set (e.g., fMRI), we apply our algorithm to a real-world dataset comprising recorded activities of individual brain cells. The resulting model is shown to be not only viable but also efficiently computable. Full article
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Article
Towards Personalized Diagnosis of Glioblastoma in Fluid-Attenuated Inversion Recovery (FLAIR) by Topological Interpretable Machine Learning
Mathematics 2020, 8(5), 770; https://doi.org/10.3390/math8050770 - 11 May 2020
Cited by 6 | Viewed by 1576
Abstract
Glioblastoma multiforme (GBM) is a fast-growing and highly invasive brain tumor, which tends to occur in adults between the ages of 45 and 70 and it accounts for 52 percent of all primary brain tumors. Usually, GBMs are detected by magnetic resonance images [...] Read more.
Glioblastoma multiforme (GBM) is a fast-growing and highly invasive brain tumor, which tends to occur in adults between the ages of 45 and 70 and it accounts for 52 percent of all primary brain tumors. Usually, GBMs are detected by magnetic resonance images (MRI). Among MRI, a fluid-attenuated inversion recovery (FLAIR) sequence produces high quality digital tumor representation. Fast computer-aided detection and segmentation techniques are needed for overcoming subjective medical doctors (MDs) judgment. This study has three main novelties for demonstrating the role of topological features as new set of radiomics features which can be used as pillars of a personalized diagnostic systems of GBM analysis from FLAIR. For the first time topological data analysis is used for analyzing GBM from three complementary perspectives—tumor growth at cell level, temporal evolution of GBM in follow-up period and eventually GBM detection. The second novelty is represented by the definition of a new Shannon-like topological entropy, the so-called Generator Entropy. The third novelty is the combination of topological and textural features for training automatic interpretable machine learning. These novelties are demonstrated by three numerical experiments. Topological Data Analysis of a simplified 2D tumor growth mathematical model had allowed to understand the bio-chemical conditions that facilitate tumor growth—the higher the concentration of chemical nutrients the more virulent the process. Topological data analysis was used for evaluating GBM temporal progression on FLAIR recorded within 90 days following treatment completion and at progression. The experiment had confirmed that persistent entropy is a viable statistics for monitoring GBM evolution during the follow-up period. In the third experiment we developed a novel methodology based on topological and textural features and automatic interpretable machine learning for automatic GBM classification on FLAIR. The algorithm reached a classification accuracy up to 97%. Full article
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