A marginalized MAP approach and EM optimization for pair-wise registration

We formalize the pair-wise registration problem in a maximum a posteriori (MAP) framework that employs a multinomial model of joint intensities with parameters for which we only have a prior distribution. To obtain an MAP estimate of the aligning transformation alone, we treat the multinomial parameters as nuisance parameters, and marginalize them out. If the prior on those is uninformative, the marginalization leads to registration by minimization of joint entropy. With an informative prior, the marginalization leads to minimization of the entropy of the data pooled with pseudo observations from the prior. In addition, we show that the marginalized objective function can be optimized by the Expectation-Maximization (EM) algorithm, which yields a simple and effective iteration for solving entropy-based registration problems. Experimentally, we demonstrate the effectiveness of the resulting EM iteration for rapidly solving a challenging intra-operative registration problem.