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Representation and Self-Similarity of Shapes
January 2003 (vol. 25 no. 1)
pp. 86-99

Abstract—Representing shapes in a compact and informative form is a significant problem for vision systems that must recognize or classify objects. We describe a compact representation model for two-dimensional (2D) shapes by investigating their self-similarities and constructing their shape axis trees (SA-trees). Our approach can be formulated as a variational one (or, equivalently, as MAP estimation of a Markov random field). We start with a 2D shape, its boundary contour, and two different parameterizations for the contour (one parameterization is oriented counterclockwise and the other clockwise). To measure its self-similarity, the two parameterizations are matched to derive the best set of one-to-one point-to-point correspondences along the contour. The cost functional used in the matching may vary and is determined by the adopted self-similarity criteria, e.g., cocircularity, distance variation, parallelism, and region homogeneity. The loci of middle points of the pairing contour points yield the shape axis and they can be grouped into a unique free tree structure, the SA-tree. By implicitly encoding the (local and global) shape information into an SA-tree, a variety of vision tasks, e.g., shape recognition, comparison, and retrieval, can be performed in a more robust and efficient way via various tree-based algorithms. A dynamic programming algorithm gives the optimal solution in O(N^4), where N is the size of the contour.

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Index Terms:
Shape representation, self-similarity, variational matching, dynamic programming, MRF.
Citation:
Davi Geiger, Tyng-Luh Liu, Robert V. Kohn, "Representation and Self-Similarity of Shapes," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 25, no. 1, pp. 86-99, Jan. 2003, doi:10.1109/TPAMI.2003.1159948
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