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Structural pattern recognition describes and classifies data based on the relationships of features and parts. Topological invariants, like the Euler number, characterize the structure of objects of any dimension. Cohomology can provide more refined algebraic invariants to a topological space than does homology. It assigns `quantities' to the chains used in homology to characterize holes of any dimension. Graph pyramids can be used to describe subdivisions of the same object at multiple levels of detail. This paper presents cohomology in the context of structural pattern recognition and introduces an algorithm to efficiently compute representative cocycles (the basic elements of cohomology) in 2D using a graph pyramid. An extension to obtain scanning and rotation invariant cocycles is given., Comment: Special issue on Graph-Based Representations in Computer Vision

We propose a mid-level image segmentation framework that combines multiple figure-ground hypothesis (FG) constrained at different locations and scales, into interpretations that tile the entire image. The problem is cast as optimization over sets of maximal cliques sampled from the graph connecting non-overlapping, putative figure-ground segment hypotheses. Potential functions over cliques combine unary Gestalt-based figure quality scores and pairwise compatibilities among spatially neighboring segments, constrained by T-junctions and the boundary interface statistics resulting from projections of real 3d scenes. Learning the model parameters is formulated as rank optimization, alternating between sampling image tilings and optimizing their potential function parameters. State of the art results are reported on both the Berkeley and the VOC2009 segmentation dataset, where a 28% improvement was achieved., Comment: 11 pages, 5 figures