Probabilistic matching of deformed images

The problem of determining the mapping between a pair of images is called image matching and is fundamental in image processing. We formalize a decision-theoretic framework for its solution by constructing a Bayesian model of the problem. Unlike traditional matching methods, the Bayesian formulation formally embodies the notions of uncertainty in the measurements and prior information that may be available about the problem. We illustrate the advantages of the approach and its development through the implementation of a volume warping system to ameliorate the difficult task of anatomical localization in tomographic scans of human anatomy. The likelihood of a mapping can in general be inferred from an observed image pair or their features by measuring the degree to which one image is made similar to the other through the mapping. A natural choice for our solution would be the mapping with the greatest likelihood. The problem of calculating the maximum likelihood estimate, however, is ill-posed because of the sparsity of informative features within the images. This motivates the introduction of prior information, in the form of constraints, with which to regularize the matching problem. We describe our prior expectations about the general form of the mapping, such as its smoothness, through Gibbs modeling. The most probable mapping given both the prior and sample information is the maximum a posteriori solution. We estimate a finite element representation of this value using a multi-level optimization scheme. In addition, an iterative Gibbs sampling algorithm is developed to stochastically estimate the minimum mean squared error solution. Its implementation capitalizes on our finite element approximation to the mapping, which allows the posterior distribution to be represented as a Markov random field. Beyond the estimation of our mappings, Bayesian analysis can also determine their uncertainty or reliability. We demonstrate these aspects of the approach on two- and three-dimensional images of the human brain