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We propose a novel technique, termed compact shape trees, for computingcorrespondences of single-boundary 2-D shapes in O(n2) time. Together with zeroor more features defined at each of n sample points on the shape's boundary,the compact shape tree of a shape comprises the O(n) collection of vectorsemanating from any of the sample points on the shape's boundary to the rest ofthe sample points on the boundary. As it turns out, compact shape trees have anumber of elegant properties both in the spatial and frequency domains. Inparticular, via a simple vector-algebraic argument, we show that the O(n)collection of vectors in a compact shape tree possesses at least the samediscriminatory power as the O(n2) collection of lines emanating from eachsample point to every other sample point on a shape's boundary. In addition, wedescribe neat approaches for achieving scale and rotation invariance withcompact shape trees in the spatial domain; by viewing compact shape trees asaperiodic discrete signals, we also prove scale and rotation invarianceproperties for them in the Fourier domain. Towards these, along the way, usingconcepts from differential geometry and the Calculus, we propose a novel theoryfor sampling 2-D shape boundaries in a scale and rotation invariant manner.Finally, we propose a number of shape recognition experiments to test theefficacy of our concept.

Compact Shape Trees: A Contribution to the Forest of Shape Correspondences and Matching Methods