Topological Data Analysis with Bregman Divergences

Given a finite set in a metric space, the topological analysis generalizes hierarchical clustering using a 1-parameter family of homology groups to quantify connectivity in all dimensions. The connectivity is compactly described by the persistence diagram. One limitation of the current framework is the reliance on metric distances, whereas in many practical applications objects are compared by non-metric dissimilarity measures. Examples are the Kullback-Leibler divergence, which is commonly used for comparing text and images, and the Itakura-Saito divergence, popular for speech and sound. These are two members of the broad family of dissimilarities called Bregman divergences. We show that the framework of topological data analysis can be extended to general Bregman divergences, widening the scope of possible applications. In particular, we prove that appropriately generalized Cech and Delaunay (alpha) complexes capture the correct homotopy type, namely that of the corresponding union of Bregman balls. Consequently, their filtrations give the correct persistence diagram, namely the one generated by the uniformly growing Bregman balls. Moreover, we show that unlike the metric setting, the filtration of Vietoris-Rips complexes may fail to approximate the persistence diagram. We propose algorithms to compute the thus generalized Cech, Vietoris-Rips and Delaunay complexes and experimentally test their efficiency. Lastly, we explain their surprisingly good performance by making a connection with discrete Morse theory

Persistent Intersection Homology for the Analysis of Discrete Data

Topological data analysis is becoming increasingly relevant to support the analysis of unstructured data sets. A common assumption in data analysis is that the data set is a sample---not necessarily a uniform one---of some high-dimensional manifold. In such cases, persistent homology can be successfully employed to extract features, remove noise, and compare data sets. The underlying problems in some application domains, however, turn out to represent multiple manifolds with different dimensions. Algebraic topology typically analyzes such problems using intersection homology, an extension of homology that is capable of handling configurations with singularities. In this paper, we describe how the persistent variant of intersection homology can be used to assist data analysis in visualization. We point out potential pitfalls in approximating data sets with singularities and give strategies for resolving them.Comment: Topology-based Methods in Visualization 201

Topological data analysis to improve exemplar-based inpainting

Image inpainting is the process of filling in the missing region to preserve continuity of its overall content and semantic. In this paper, we present a novel approach to improve an existing scheme, called exemplar-based inpainting algorithm, using Topological Data Analysis (TDA). TDA is a mathematical approach concern studying shapes or objects to gain information about connectivity and closeness property of those objects. The challenge in using exemplar-based inpainting is that missing regions neighborhood area needs to have a relatively simple texture and structure. We studied the topological properties (e.g. number of connected components) of missing regions surrounding the missing area by building a sequence of simplicial complexes (known as persistent homology) based on a selected group of uniform Local binary Pattern LBP. Connected components of image regions generated by certain landmark pixels, at different thresholds, automatically quantify the texture nature of the missing regions surrounding areas. Such quantification help determine the appropriate size of patch propagation. We have modified the patch propagation priority function using geometrical properties of curvature of isophote and improved the matching criteria of patches by calculating the correlation coefficients from spatial, gradient and Laplacian domain. We use several image quality measures to illustrate the performance of our approach in comparison to similar inpainting algorithms. In particular, we shall illustrate that our proposed scheme outperforms the state-of-the-art exemplar-based inpainting algorithm

Topological Image Texture Analysis for Quality Assessment

Image quality is a major factor influencing pattern recognition accuracy and help detect image tampering for forensics. We are concerned with investigating topological image texture analysis techniques to assess different type of degradation. We use Local Binary Pattern (LBP) as a texture feature descriptor. For any image construct simplicial complexes for selected groups of uniform LBP bins and calculate persistent homology invariants (e.g. number of connected components). We investigated image quality discriminating characteristics of these simplicial complexes by computing these models for a large dataset of face images that are affected by the presence of shadows as a result of variation in illumination conditions. Our tests demonstrate that for specific uniform LBP patterns, the number of connected component not only distinguish between different levels of shadow effects but also help detect the infected regions as well

Mining topological structure in graphs through forest representations

We consider the problem of inferring simplified topological substructures—which we term backbones—in metric and non-metric graphs. Intuitively, these are subgraphs with ‘few’ nodes, multifurcations, and cycles, that model the topology of the original graph well. We present a multistep procedure for inferring these backbones. First, we encode local (geometric) information of each vertex in the original graph by means of the boundary coefficient (BC) to identify ‘core’ nodes in the graph. Next, we construct a forest representation of the graph, termed an f-pine, that connects every node of the graph to a local ‘core’ node. The final backbone is then inferred from the f-pine through CLOF (Constrained Leaves Optimal subForest), a novel graph optimization problem we introduce in this paper. On a theoretical level, we show that CLOF is NP-hard for general graphs. However, we prove that CLOF can be efficiently solved for forest graphs, a surprising fact given that CLOF induces a nontrivial monotone submodular set function maximization problem on tree graphs. This result is the basis of our method for mining backbones in graphs through forest representation. We qualitatively and quantitatively confirm the applicability, effectiveness, and scalability of our method for discovering backbones in a variety of graph-structured data, such as social networks, earthquake locations scattered across the Earth, and high-dimensional cell trajectory dat

Persistent Homology Tools for Image Analysis

Topological Data Analysis (TDA) is a new field of mathematics emerged rapidly since the first decade of the century from various works of algebraic topology and geometry. The goal of TDA and its main tool of persistent homology (PH) is to provide topological insight into complex and high dimensional datasets. We take this premise onboard to get more topological insight from digital image analysis and quantify tiny low-level distortion that are undetectable except possibly by highly trained persons. Such image distortion could be caused intentionally (e.g. by morphing and steganography) or naturally in abnormal human tissue/organ scan images as a result of onset of cancer or other diseases. The main objective of this thesis is to design new image analysis tools based on persistent homological invariants representing simplicial complexes on sets of pixel landmarks over a sequence of distance resolutions. We first start by proposing innovative automatic techniques to select image pixel landmarks to build a variety of simplicial topologies from a single image. Effectiveness of each image landmark selection demonstrated by testing on different image tampering problems such as morphed face detection, steganalysis and breast tumour detection. Vietoris-Rips simplicial complexes constructed based on the image landmarks at an increasing distance threshold and topological (homological) features computed at each threshold and summarized in a form known as persistent barcodes. We vectorise the space of persistent barcodes using a technique known as persistent binning where we demonstrated the strength of it for various image analysis purposes. Different machine learning approaches are adopted to develop automatic detection of tiny texture distortion in many image analysis applications. Homological invariants used in this thesis are the 0 and 1 dimensional Betti numbers. We developed an innovative approach to design persistent homology (PH) based algorithms for automatic detection of the above described types of image distortion. In particular, we developed the first PH-detector of morphing attacks on passport face biometric images. We shall demonstrate significant accuracy of 2 such morph detection algorithms with 4 types of automatically extracted image landmarks: Local Binary patterns (LBP), 8-neighbour super-pixels (8NSP), Radial-LBP (R-LBP) and centre-symmetric LBP (CS-LBP). Using any of these techniques yields several persistent barcodes that summarise persistent topological features that help gaining insights into complex hidden structures not amenable by other image analysis methods. We shall also demonstrate significant success of a similarly developed PH-based universal steganalysis tool capable for the detection of secret messages hidden inside digital images. We also argue through a pilot study that building PH records from digital images can differentiate breast malignant tumours from benign tumours using digital mammographic images. The research presented in this thesis creates new opportunities to build real applications based on TDA and demonstrate many research challenges in a variety of image processing/analysis tasks. For example, we describe a TDA-based exemplar image inpainting technique (TEBI), superior to existing exemplar algorithm, for the reconstruction of missing image regions