Cups products in Z2-cohomology of 3D polyhedral complexes

Let I=(Z3,26,6,B) be a 3D digital image, let Q(I) be the associated cubical complex and let ∂Q(I) be the subcomplex of Q(I) whose maximal cells are the quadrangles of Q(I) shared by a voxel of B in the foreground -- the object under study -- and by a voxel of Z3∖B in the background -- the ambient space. We show how to simplify the combinatorial structure of ∂Q(I) and obtain a 3D polyhedral complex P(I) homeomorphic to ∂Q(I) but with fewer cells. We introduce an algorithm that computes cup products on H∗(P(I);Z2) directly from the combinatorics. The computational method introduced here can be effectively applied to any polyhedral complex embedded in R3

Human gait recognition using topological information

This paper shows an image/video application using topological invariants in human gait recognition. The 3D volume of a gait cycle is built stacking silhouettes extracted using a background substraction approach. Ideally, the border cell complex is obtained from the 3D volume with one connected component and one cavity. Then, it is necessary to apply a topological enrichment strategy in order to obtain a robust and discriminative representation for person recognition. Using a sliding cutter plane normal to some direction of view it is possible to divide the border cell complex in different parts. The incremental algorithm is used to compute the homology on each part. A vectorial representation is built ordering the number of connected components and tunnels obtained for each cut. In order to evaluate the robustness of this representation the silhouettes were diminished to a quarter of the original size. At the same time, this is considered a simulation of a human gait captured at long distance. Even, under these difficult conditions it was possible to get a 74% of correct classification rates on CASIA-B database

Characterizing Configurations of critical points through LBP Extended Abstract

In this abstract we extend ideas and results submitted to [3] in which a new codification of Local Binary Patterns (LBP) is given using combinatorial maps and a method for obtaining a representative LBP image is developed based on merging regions and Minimum Contrast Algorithm. The LBP code characterizes the topological category (max, min, slope, saddle) of the 2D gray level landscape around the center region. We extend the result studying how to merge non-singular slopes with one of its neighbors and how to extend the results to nonwell formed images/maps. Some ideas related to robust LBP and isolines are also given in last section

Persistent-homology-based gait recognition

Gait recognition is an important biometric technique for video surveillance tasks, due to the advantage of using it at distance. In this paper, we present a persistent homology-based method to extract topological features (the so-called topological gait signature) from the the body silhouettes of a gait sequence. It has been used before in sev- eral conference papers of the same authors for human identi cation, gender classi cation, carried object detection and monitoring human activities at distance. The novelty of this paper is the study of the sta- bility of the topological gait signature under small perturbations and the number of gait cycles contained in a gait sequence. In other words, we show that the topological gait signature is robust to the presence of noise in the body silhouettes and to the number of gait cycles con- tained in a given gait sequence. We also show that computing our topological gait signature of only the lowest fourth part of the body silhouette, we avoid the upper body movements that are unrelated to the natural dynamic of the gait, caused for example by carrying a bag or wearing a coat.Ministerio de Economía y Competitividad MTM2015-67072-

An application for gait recognition using persistent homology

This Demo presents an application for gait recognition using persistent homology. Using a background subtraction approach, a silhouette sequence is obtained from a camera in a controlled environment. A border simplicial complex is built stacking silhouettes aligned by their gravity center. A multifiltration is applied on the border simplicial complex which captures relations among the parts of the human body when walking. Finally, the topological gait signature is extracted from the persistence barcode according to each filtration. The measure cosine is used to give a similarity value between topological signatures. The input of this Demo are videos with resolution 320x240 to 25f ps. The videos in CASIA-B database are used to prove the efficacy and efficiency. A computer with 2Gb of RAM memory and a DualCore processor was used to test the implementation of the proposed algorithm. In this Demo all related tasks have been programmed by the authors in the C++ programming language. OpenCV library has been used for the image processing part

Persistent homology-based gait recognition robust to upper body variations

Gait recognition is nowadays an important biometric technique for video surveillance tasks, due to the advantage of using it at distance. However, when the upper body movements are unrelated to the natural dynamic of the gait, caused for example by carrying a bag or wearing a coat, the reported results show low accuracy. With the goal of solving this problem, we apply persistent homology to extract topological features from the lowest fourth part of the body silhouettes. To obtain the features, we modify our previous algorithm for gait recognition, to improve its efficacy and robustness to variations in the amount of simplices of the gait complex. We evaluate our approach using the CASIA-B dataset, obtaining a considerable accuracy improvement of 93:8%, achieving at the same time invariance to upper body movements unrelated with the dynamic of the gait.Ministerio de Economía y Competitividad MTM2015-67072-

Topological signature for periodic motion recognition

In this paper, we present an algorithm that computes the topological signature for a given periodic motion sequence. Such signature consists of a vector obtained by persistent homology which captures the topological and geometric changes of the object that models the motion. Two topological signatures are compared simply by the angle between the corresponding vectors. With respect to gait recognition, we have tested our method using only the lowest fourth part of the body’s silhouette. In this way, the impact of variations in the upper part of the body, which are very frequent in real scenarios, decreases considerably. We have also tested our method using other periodic motions such as running or jumping. Finally, we formally prove that our method is robust to small perturbations in the input data and does not depend on the number of periods contained in the periodic motion sequence.Junta de Andalucía FQM-369Ministerio de Economía y Competitividad MTM2015-67072-

Application of algebraic topology to fingerprint recognitiony

In the present work, an algorithm for ngerprints verication based on an application of the algebraic topology is presented. Specically, we propose a representation of impressions as simplicial complexes and the denition of local structures based on local ltrations ordering from the complexes. This ltrations are determined by neighboring minutiae. It is also proposed the extraction of a set of features based on the analysis of the homology variation in these ltrations. The features combine information about the quantity and connectivity of papillary ridges in the local structures. In addition, a matching method based on the extracted topological information is presented. This paper shows that this information is discriminative and can be used in combination with classic geometric features to improve the description of local structures of the impressions and the accuracy in the comparison

Computing cup products in Z2\mathbb {Z}_2-cohomology of 3D polyhedral complexes

Let $I=(\mathbb {Z}^3,26,6,B)$ be a three-dimensional (3D) digital image, let $Q(I)$ be an associated cubical complex, and let $\partial Q(I)$ be a subcomplex of $Q(I)$ whose maximal cells are the quadrangles of $Q(I)$ shared by a voxel of $B$ in the foreground—the object under study—and by a voxel of $\mathbb {Z}^3\backslash B$ in the background—the ambient space. We show how to simplify the combinatorial structure of $\partial Q(I)$ and obtain a 3D polyhedral complex $P(I)$ homeomorphic to $\partial Q(I)$ but with fewer cells. We introduce an algorithm that computes cup products in $H^*(P(I);\mathbb {Z}_2)$ directly from the combinatorics. The computational method introduced here can be effectively applied to any polyhedral complex embedded in $\mathbb {R}^3$