EM-LDDMM for 3D to 2D registration

Abstract

We examine the problem of mapping dense 3D atlases onto censored, sparsely sampled 2D target sections at micron and meso scales. We introduce a new class of large deformation diffeomorphic metric mapping (LD-DMM) algorithms for generating dense atlas correspondences onto sparse 2D samples by introducing a field of hidden variables which must be estimated representing a large class of target image uncertainties including (i) unknown parameters representing cross stain contrasts, (ii) censoring of tissue due to localized measurements of target subvolumes and (iii) sparse sampling of target tissue sections. For prediction of the hidden fields we introduce the generalized expectation-maximization algorithm (EM) for which the E-step calculates the conditional mean of the hidden variates simultaneously combined with the diffeomorphic correspondences between atlas and target coordinate systems. The algorithm is run to fixed points guaranteeing estimators satisfy the necessary maximizer conditions when interpreted as likelihood estimators. The dense mapping is an injective correspondence to the sparse targets implying all of the 3D variations are performed only on the atlas side with variation in the targets only 2D manipulations.