Simultaneous Nonrigid Registration of Multiple Point Sets and Atlas Construction

Abstract

Groupwise registration of a set of shapes represented by unlabeled point sets is a challenging problem since, usually, this involves solving for point correspondence in a nonrigid motion setting. In this paper, we propose a novel and robust algorithm that is capable of simultaneously computing the mean shape, represented by a probability density function, from multiple unlabeled point sets (represented by finite-mixture models), and registering them nonrigidly to this emerging mean shape. This algorithm avoids the correspondence problem by minimizing the Jensen-Shannon (JS) divergence between the point sets represented as finite mixtures of Gaussian densities. We motivate the use of the JS divergence by pointing out its close relationship to hypothesis testing. Essentially, minimizing the JS divergence is asymptotically equivalent to maximizing the likelihood ratio formed from a probability density of the pooled point sets and the product of the probability densities of the individual point sets. We derive the analytic gradient of the cost function, namely, the JS-divergence, in order to efficiently achieve the optimal solution. The cost function is fully symmetric, with no bias toward any of the given shapes to be registered and whose mean is being sought. A by-product of the registration process is a probabilistic atlas, which is defined as the convex combination of the probability densities of the input point sets being aligned. Our algorithm can be especially useful for creating atlases of various shapes present in images and for simultaneously (rigidly or nonrigidly) registering 3D range data sets (in vision and graphics applications), without having to establish any correspondence. We present experimental results on nonrigidly registering 2D and 3D real and synthetic data (point sets).

Generalized L2-Divergence and Its Application to Shape Alignment

“…Then, we will present the groupwise registration results achieved for the shapes from cardiac echocardiographic videos as well as human brain MRI. For the non-rigid registration experiments, we choose the thin-plate spline (TPS) to represent the deformation, which is similar to [2]. …”

“…Perhaps the methods that are closest in spirit to our approach are the recent work in [2,3]. They minimize the Jensen-Shannon divergence and the CDF-based Jensen-Shannon divergence respectively between the feature point-sets with respect to non-rigid deformation.…”

“…To motivate the derivation of generalized L2 divergence, we observe that an earlier divergence measure called Jensen-Shannon (JS) divergence had addressed the groupwise point-sets registration problem [2], where each point-set X i is modeled as probability density functions.…”

“…A recent approach models each point-set by a probability density function, and then quantify the distance between these probability densities using an information-theoretic measure[1,2,3]. Figure 1 illustrates this idea, wherein the right column of the figure depicts the density functions corresponding to the point-sets (representing the boundary and shape) drawn from corpus callosum shapes shown in the left column.…”

“…It is based on a novel information theoretic matching criterion called Generalized L 2 (GL2) divergence, which is used measure the similarity between the probability densities representing the shapes. Both the Jensen-Shannon (JS) divergence [2] and our Generalized L2-divergence are shown to belong the so-called Generalized Linear Divergence family. In this paper, we use Generalized L2-divergence for achieving non-rigid registration between multiple shapes represented using the points features.…”