Chapter 16 - Elastic Registration and Inference Using Oct-Tree Splines

Abstract

This chapter discusses a technique for registering three-dimensional sets of particles which are related by nonrigid deformations. These sets of particles are usually surfaces or sets of surface patches to which are assigned features such as gradient, curvature, etc. Two arguments which favored high-level feature-based approaches in the past were computational complexity and global correspondence search. The computational complexity argument assumed that if a small set of features was found and matched (say dozens of discrete features or hundreds of ridge points, as opposed to tens of thousands of surface particles), the overall complexity of the algorithm would be drastically lower. However, by using the oct-tree spline distance map, the complexity of each iterative adjustment step in our algorithm is linear in the number of sensed surface particles. The second argument in favor of discrete features is related to the first, and examines the combinatorial complexity of certain discrete matching (correspondence) algorithms. In its approach, the chapter uses a modified gradient descent which avoids combinatorial search but only finds locally optimal matches. In theory, volumetric deformation models are of a higher complexity than elastic surface models, that is, they represent the deformations over a dense region of three-dimensions, rather than over a two-dimensional manifold. However, the use of a multiresolution, coarse-to-fine strategy results in a very efficient matching algorithm. The estimation of elastic deformations in general is an ill-posed problem for which it is crucial to define some notion of minimal deformation that relates directly to the notion of smoothness.