Bayesian Estimation of Deformation and Elastic Parameters in Non-rigid Registration

Elastic deformation models are frequently used when solving non-rigid registration problems that are associated with neurosurgical image guidance, however, establishing precise values for the material parameters of brain tissue remains challenging. In this work we include elastography in the registration process by formulating these parameters as unknown random variables with associated priors that may be broad or sharp, depending on the situation. A Bayesian registration model is introduced where the deformation probability is formulated by way of Boltzmann's equation and the linear elastic deformation and similarity energies. The full joint posterior on deformation and elastic random variables is characterized with a Markov Chain Monte Carlo method and can provide useful information beyond the usual "point estimates"; e.g. deformation uncertainty. Hard deformation constraints are easily accommodated in this framework which allows us to constrain the deformation of the brain to the intra-cranial space. We describe preliminary experiments with synthetic 3D brain images for which ground truth is known for the elastic and deformation parameters. We compare a model with separate elastic parameters for three compartments (white matter, gray matter, and CSF), to a single compartment model, and show convergence, improved deformation estimates for the three compartment model and that plausible posteriors on the elastic parameters are obtained from the elastography process.