Deformable Density Matching for 3D Non-rigid Registration of Shapes

Abstract

Abstract. There exists a large body of literature on shape matching and registration in medical image analysis. However, most of the previous work is focused on matching particular sets of features-point-sets, lines, curves and surfaces. In this work, we forsake specific geometric shape representations and instead seek probabilistic representationsspecifically Gaussian mixture models-of shapes. We evaluate a closedform distance between two probabilistic shape representations for the general case where the mixture models differ in variance and the number of components. We then cast non-rigid registration as a deformable density matching problem. In our approach, we take one mixture density onto another by deforming the component centroids via a thin-plate spline (TPS) and also minimizing the distance with respect to the variance parameters. We validate our approach on synthetic and 3D arterial tree data and evaluate it on 3D hippocampal shapes.

PORTR: Pre-Operative and Post-Recurrence Brain Tumor Registration

“…We now define the pathology term E P in such a way that it penalizes mismatches between the aligned posteriors of the tumor in B and of the cavity in F , i.e., we measure the squared ℓ 2 -norm between the aligned p B , TU and p F , CA [31], [32] $$\begin{array}{c}{E}_{P}false({\mathit{f}}_{\mathit{CB}},{\mathit{f}}_{\mathit{CF}};p_B,p_{F}false)\\ \triangleq {italic\int }_{\mathbf{x}\in {\mathrm{normal\Omega }}_C}{\{p_{B,\mathrm{TU}}false({\mathit{f}}_{\mathit{CB}}false(\mathrm{boldx}false)false)-p_{F,\mathrm{CA}}false({\mathit{f}}_{\mathit{CF}}false(\mathrm{boldx}false)false)\}}^{2}d\mathrm{boldx}.\end{array}$$…”

Closed-Form Jensen-Renyi Divergence for Mixture of Gaussians and Applications to Group-Wise Shape Registration

“…In Jian et al [2] and Roy et al [6], nonrigid registration is between pairs of data sets using L2 distance on a mixture of Gaussians model of the data sets. Both methods, however, have not been extended to the problem of unbiased simultaneous matching of multiple point-sets being addressed in this paper.…”

Generalized L2-Divergence and Its Application to Shape Alignment

“…Consequently, their method is actually more similar to the image matching method of [5] discussed below, but with the advantage of not having to evaluate a cost function involving spatial integrals numerically, since a closed form expression is derived for the same. Roy et al [6] used a similar approach as in [1], except that they fit a density function to the data via maximum likelihood before the registration step. Both of the methods however have not been extended to the problem of unbiased simultaneous matching of multiple point-sets being addressed in this paper.…”

“…Using Gaussian mixtures keeps the complexity O ( L 2 ) and the estimation computationally simple. Since GL2 is a generalization of the popular L2 measure, our method is equivalent to the algorithms presented in Jian et al[1] and Roy et al [6] when applied to align pairwise pointsets.…”