Deformable image registration by combining uncertainty estimates from supervoxel belief propagation

Discrete optimisation strategies have a number of advantages over their continuous counterparts for deformable registration of medical images. For example: it is not necessary to compute derivatives of the similarity term; dense sampling of the search space reduces the risk of becoming trapped in local optima; and (in principle) an optimum can be found without resorting to iterative coarse-to-fine warping strategies. However, the large complexity of high-dimensional medical data renders a direct voxel-wise estimation of deformation vectors impractical. For this reason, previous work on medical image registration using graphical models has largely relied on using a parameterised deformation model and on the use of iterative coarse-to-fine optimisation schemes. In this paper, we propose an approach that enables accurate voxel-wise deformable registration of high-resolution 3D images without the need for intermediate image warping or a multi-resolution scheme. This is achieved by representing the image domain as multiple comprehensive supervoxel layers and making use of the full marginal distribution of all probable displacement vectors after inferring regularity of the deformations using belief propagation. The optimisation acts on the coarse scale representation of supervoxels, which provides sufficient spatial context and is robust to noise in low contrast areas. Minimum spanning trees, which connect neighbouring supervoxels, are employed to model pair-wise deformation dependencies. The optimal displacement for each voxel is calculated by considering the probabilities for all displacements over all overlapping supervoxel graphs and subsequently seeking the mode of this distribution. We demonstrate the applicability of this concept for two challenging applications: first, for intra-patient motion estimation in lung CT scans; and second, for atlas-based segmentation propagation of MRI brain scans. For lung registration, the voxel-wise mode of displacements is found using the mean-shift algorithm, which enables us to determine continuous valued sub-voxel motion vectors. Finding the mode of brain segmentation labels is performed using a voxel-wise majority voting weighted by the displacement uncertainty estimates. Our experimental results show significant improvements in registration accuracy when using the additional information provided by the registration uncertainty estimates. The multi-layer approach enables fusion of multiple complementary proposals, extending the popular fusion approaches from multi-image registration to probabilistic one-to-one image registration.