Joint Segmentation and Registration Through the Duality of Congealing and Maximum Likelihood Estimate

In this paper we consider the task of joint registration and segmentation. A popular method which aligns images and simultaneously estimates a simple statistical shape model was proposed by E. Learned-Miller and is known as congealing. It considers the entropy of a simple, pixel-wise independent distribution as the objective function for searching the unknown transformations. Besides being intuitive and appealing, this idea raises several theoretical and practical questions, which we try to answer in this paper. First, we analyse the approach theoretically and show that the original congealing is in fact the DC-dual task (difference of convex functions) for a properly formulated Maximum Likelihood estimation task. This interpretation immediately leads to a different choice for the algorithm which is substantially simpler than the known congealing algorithm. The second contribution is to show, how to generalise the task for models in which the shape prior is formulated in terms of segmentation labellings and is related to the signal domain via a parametric appearance model. We call this generalisation unsupervised congealing. The new approach is applied to the task of aligning and segmenting imaginal discs of Drosophila melanogaster larvae.