A Combinatorial Solution to Non-Rigid 3D Shape-to-Image Matching

TL;DR

This paper introduces a novel combinatorial approach for non-rigid 3D shape-to-image matching, modeling the shape as a mesh with rigid transformations per triangle, addressing a complex NP-hard problem with a graph-theoretic method.

Contribution

It presents the first combinatorial formulation for non-rigid 3D shape-to-image matching, including an efficient discretisation of SE(3) and solutions that do not need initialisation.

Findings

Effective non-rigid 3D shape-to-shape registration

Promising results in shape-to-image matching

Solutions within a bound of the optimal

Abstract

We propose a combinatorial solution for the problem of non-rigidly matching a 3D shape to 3D image data. To this end, we model the shape as a triangular mesh and allow each triangle of this mesh to be rigidly transformed to achieve a suitable matching to the image. By penalising the distance and the relative rotation between neighbouring triangles our matching compromises between image and shape information. In this paper, we resolve two major challenges: Firstly, we address the resulting large and NP-hard combinatorial problem with a suitable graph-theoretic approach. Secondly, we propose an efficient discretisation of the unbounded 6-dimensional Lie group SE(3). To our knowledge this is the first combinatorial formulation for non-rigid 3D shape-to-image matching. In contrast to existing local (gradient descent) optimisation methods, we obtain solutions that do not require a good initialisation and that are within a bound of the optimal solution. We evaluate the proposed method on the two problems of non-rigid 3D shape-to-shape and non-rigid 3D shape-to-image registration and demonstrate that it provides promising results.