In this talk, we will present shape registration algorithms based on the implicit distance function representation. In its implicit representation, a shape is embedded in a higher-dimensional space as the zero level set of a distance function. In certain applications like shape registration, the implicit representation has advantages because it provides additional support to the registration process and requires matching of not only the shapes but also their clones that are positioned coherently in the embedding space. We propose an algorithm based on the mutual information criterion which supports global registration with any arbitrary transformation model. Furthermore, in order to establish dense correspondences between shape instances that undergo nonrigid deformations, we introduce B-spline based Incremental Free Form Deformations (IFFD) model to minimize a sum-of-squared differences (SSD) measure and recover a dense local registration field.
The use of distance functions for representing and matching shapes, however, suffers from the local minima problem because it assumes correspondence between “closest points” in each iteration. And it had been observed that longer “closest” distances under the L2 norm tend to be between false point correspondences, especially when outliers exist. We propose a mitigation scheme by replacing the squared-Euclidean distance model with a high-peak-fat-tail, two-component Gaussian Mixtures distance model, and seeking global optimum via a modified Particle Swarm Optimization (PSO) method. The shape registration methods above can also be used for matching “generalized shapes”, such as edge maps and gradient maps.
Time permitting, we will present our recent work on object matching using linear programming and a novel locally affine-invariant geometric constraint.