Application of trace-norm and low-rank matrix decomposition for computational anatomy

We propose a generative model to distinguish normal anatomical variations from abnormal deformations given a group of images with normal and abnormal subjects. We assume that abnormal subjects share common factors which characterize the abnormality. These factors are hard to discover due to large variance of normal anatomical differences. Assuming that the deformation fields are parametrized by their stationary velocity fields, these factors constitute a low-rank subspace (abnormal space) that is corrupted by high variance normal anatomical differences. We assume that these normal anatomical variations are not correlated. We form an optimization problem and propose an efficient iterative algorithm to recover the low-rank subspace. The algorithm iterates between image registration and the decomposition steps and hence can be seen as a group-wise registration algorithm. We apply our method on synthetic and real data and discover abnormality of the population that cannot be recovered by some of the well-known matrix decompositions (e.g. Singular Value Decomposition).