A Graph Matching Approach to Symmetry Detection and Analysis

Spectral relaxation was shown to provide an efficient approach for solving a gamut of computational problems, ranging from data mining to image registration. In this chapter we show that in the context of graph matching, spectral relaxation can be applied to the detection and analysis of symmetries in n-dimensions. First, we cast symmetry detection of a set of points in ℝ n as the self-alignment of the set to itself. Thus, by representing an object by a set of points S ∈ ℝ n , symmetry is manifested by multiple self-alignments. Secondly, we formulate the alignment problem as a quadratic binary optimization problem, solved efficiently via spectral relaxation. Thus, each eigenvalue corresponds to a potential self-alignment, and eigenvalues with multiplicity greater than one correspond to symmetric self-alignments. The corresponding eigenvectors reveal the point alignment and pave the way for further analysis of the recovered symmetry. We apply our approach to image analysis, by using local features to represent each image as a set of points. Last, we improve the scheme’s robustness by inducing geometrical constraints on the spectral analysis results. Our approach is verified by extensive experiments and was applied to two and three dimensional synthetic and real life images.