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In this paper, a new computational topological framework for hypergraph analysis and recognition is developed. "Topology provides scale" is the principle at the core of this set of algebraic topological tools, whose fundamental notion is that of a scale-space topological model (s 2 -model). The scale of this parameterized sequence of algebraic hypergraphs, all having the same Euler-Poincaré characteristic than the original hypergraph G , is provided by its relational topology in terms of evolution of incidence or adjacency connectivity maps. Its algebraic homological counterpart is again an s 2 -model, allowing the computation of new topological characteristics of G , which far exceeds current homological analytical techniques. Both scale-space algebraic dynamical systems are hypergraph isomorphic invariants. The hypergraph isomorphism problem is attacked here to demonstrate the power of the proposed framework, by proving the ability of s 2 -models to differentiate challenging cases that are difficult or even infeasible for state-of-the-art practical polynomial solvers. The processing, analysis, classification and learning power of the s 2 -model, at both combinatorial and algebraic levels, augurs positive prospects with respect to its application to physical, biological and social network analysis. • To develop the theoretical foundations of a new paradigm of computational analysis of hypergraphs based on s 2 -models, in which "Topology provides scale". • To prove that the homological information of an s 2 -model can be as well represented as an s 2 -model. • To demonstrate that the s 2 -model representation and its homology are hypergraph invariants (up to s 2 -model isomorphism). • To apply the proposed framework to the graph isomorphism problem, showing its potential use in network analysis. [ABSTRACT FROM AUTHOR]

• Foundational paper of the new homology boundary scale-space (hbss , for short) representation for geometric cell complexes. • Determination of a new invariant for cell complex isomorphisms in terms of Betti numbers. • Causality principle for connected subgraphs of the hbss -model. • Useful tool for distinguishing non-isomophic graphs and networks. In this paper we develop a new computational technique called boundary scale-space theory. This technique is based on the topol 1 1 comples -> complex ogical paradigm consisting of representing a geometric subdivided object K using a one-parameter family of geometric objects { K i } i ≥ 1 all of them having the same number of closed pieces than K. Each piece of K i (∀ i ≥ 1) presents the same interior part than the corresponding one in K , and a different boundary part depending on the scale i. Working with coefficients in a field, a scale is installed for the algebraic boundary of each piece and a new invariant for cell complex isomorphisms is given in terms of the Betti numbers of the generated boundary-scale-space cell complexes. Moreover, the so called homology boundary scale-space model of K (hbss -model for short) is introduced here. This model consists of a hierarchical graph whose nodes are the homology generators of the different boundary scale levels and whose edges are specified by homology generators of consecutive boundary scale indices linked by (hbss -transition maps) preserving homology classes. Various codes for each connected subgraph of an hbss -model are defined, which besides being fast and efficient similarity measures for cellular structures, they are as well relevant interpretive tools for the hbss -model. Finally, experimentations mainly aimed at clarifying and understanding the notion of hbss -model, as well as conjecturing about new graph isomorphism invariants (seeing graphs as a 1-dimensional cell complexes), are performed. [ABSTRACT FROM AUTHOR]

This paper presents a novel method for identifying vehicle components in a monocular traffic image sequence. In the proposed method, the vehicles are first divided into multiscale regions based on the center of gravity of the foreground vehicle mask and the calibrated-camera parameters. With these multiscale regions, textural couriers are generated based on the localized variances of the foreground vehicle image. A new scale-space model is subsequently created based on the textural couriers to provide a topological structure of the vehicle. In this model, key feature points of the vehicle can significantly be described based on the topological structure to determine the regions that are homogenous in texture from which vehicle components can be identified by segmenting the key feature points. Since no motion information is required in order to segment the vehicles prior to recognition, the proposed system can be used in situations where extensive observation time is not available or motion information is unreliable. This novel method can be used in real-world systems such as vehicle-shape reconstruction, vehicle classification, and vehicle recognition. This method was demonstrated and tested on 200 different vehicle samples captured in routine outdoor traffic images and achieved an average error rate of 6.8% with a variety of vehicles and traffic scenes. [ABSTRACT FROM PUBLISHER]