Abstract. This paper shows an image/video application using topological invariants for human gait recognition. Using a background subtraction approach, a stack of silhouettes is extracted from a subsequence and glued through their gravity centers, forming a 3D digital image I. From this 3D representation, the border simplicial complex ∂K(I) is obtained. We order the triangles of ∂K(I) obtaining a sequence of subcomplexes of ∂K(I). The corresponding filtration F captures relations among the parts of the human body when walking. Finally, a topological gait signature is extracted from the persistence barcode according to F . In this work we obtain 98.5% correct classification rates on CASIA-B database 1 .

Abstract. There are surveillance scenarios where it is important to emit an alarm when a person carrying an object is detected. In order to detect when a person is carrying an object, we build models of naturally-walking and object-carrying persons using topological features. First, a stack of human silhouettes, extracted by background subtraction and thresholding, are glued through their gravity centers, forming a 3D digital image I. Second, different filters (i.e. orderings of the cells) are applied on ∂K(I) (cubical complex obtained from I) which capture relations among the parts of the human body when walking. Finally, a topological signature is extracted from the persistence diagrams according to each filter. We build some clusters of persons walking naturally, without carrying object and some clusters of persons carrying bags. We obtain vector prototypes for each cluster. Simple distances to the means are calculated for detecting the presence of carrying object. The measure cosine is used to give a similarity value between topological signatures. The accuracies obtained are 95.7% and 95.9% for naturally-walking and object-carrying respectively.

Let I = (Z 3 , 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 Z 3 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); Z 2 ) directly from the combinatorics. The computational method introduced here can be effectively applied to any polyhedral complex embedded in R 3 .

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 several conference papers of the same authors for human identification, gender classification, carried object detection and monitoring human activities at distance. The novelty of this paper is the study of the stability 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 contained 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.

Abstract. Let I be a 3D digital image, and let Q(I) be the associated cubical complex. In this paper we show how to simplify the combinatorial structure of Q(I) and obtain a homeomorphic cellular complex P (I) with fewer cells. We introduce formulas for a diagonal approximation on a general polygon and use it to compute cup products on the cohomology H * (P (I)). The cup product encodes important geometrical information not captured by the cohomology groups. Consequently, the ring structure of H * (P (I)) is a finer topological invariant. The algorithm proposed here can be applied to compute cup products on any polyhedral approximation of an object embedded in 3-space.