A Combinatorial Solution to Non-Rigid 3D Shape-to-Image Matching
This addresses the problem of non-rigid 3D shape-to-image matching for computer vision and medical imaging, offering a novel combinatorial approach that is incremental compared to existing local optimization methods.
The paper tackles the problem of non-rigidly matching 3D shapes to 3D image data by proposing a combinatorial solution that models shapes as triangular meshes with rigid transformations, penalizing distances and rotations between neighbors. It resolves NP-hard combinatorial challenges with graph-theoretic approaches and efficient discretization, achieving solutions within a bound of optimal without requiring good initialization.
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.