Shape matching, learning and classification
by
Anand Rangarajan
University of Florida

When we seek to compare shapes parameterized as a set of unlabeled points, we face the twin problems of i) shape correspondence and ii) shape deformation. Over the past few years, we have shown the efficacy of i) simultaneously solving for the correspondences and the deformation: TPS-RPM), ii) simultaneously clustering and matching the two shapes: JCM, iii) using the Jensen-Shannon (JS) divergence to solve for the deformation without parameterizing the correspondences, and iv) finding a deformation which minimizes a closed-form distance between two Gaussian mixture models for the shapes. We demonstrate both shape matching and atlas construction on 2D corpora callosa and 3D hippocampal datasets. When we turn to classification, the core new idea is to represent point-set shapes as the square root of probability densities expanded in a wavelet basis. We then use this representation to develop a natural similarity metric that respects the geometry of these probability distributions, i.e. under the wavelet expansion, densities are points on a unit hypersphere and the distance between densities is given by the separating arc length. The process uses a linear assignment solver for non-rigid alignment between densities prior to matching; this has the connotation of “sliding” wavelet coefficients akin to the sliding block puzzle L’Âne Rouge. We illustrate the utility of this framework by classifying shapes drawn from the MPEG-7 data set.

Date received: March 1, 2009