Compact Shape Trees: A Contribution to the Forest of Shape Correspondences and Matching Methods
This work addresses shape matching in computer vision, offering a novel method that is incremental in improving efficiency and invariance properties for single-boundary shapes.
The authors tackled the problem of computing correspondences for 2-D shapes by proposing compact shape trees, which achieve O(n^2) time complexity and demonstrate discriminatory power equivalent to O(n^2) collections of lines, with proven scale and rotation invariance in both spatial and Fourier domains.
We propose a novel technique, termed compact shape trees, for computing correspondences of single-boundary 2-D shapes in O(n2) time. Together with zero or 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 vectors emanating from any of the sample points on the shape's boundary to the rest of the sample points on the boundary. As it turns out, compact shape trees have a number of elegant properties both in the spatial and frequency domains. In particular, via a simple vector-algebraic argument, we show that the O(n) collection of vectors in a compact shape tree possesses at least the same discriminatory power as the O(n2) collection of lines emanating from each sample point to every other sample point on a shape's boundary. In addition, we describe neat approaches for achieving scale and rotation invariance with compact shape trees in the spatial domain; by viewing compact shape trees as aperiodic discrete signals, we also prove scale and rotation invariance properties for them in the Fourier domain. Towards these, along the way, using concepts from differential geometry and the Calculus, we propose a novel theory for sampling 2-D shape boundaries in a scale and rotation invariant manner. Finally, we propose a number of shape recognition experiments to test the efficacy of our concept.