Motion coherent image registration and demons: practical handling of deformation boundaries

Much effort has gone into the understanding of regularization in ill-posed problems in computer vision. Yuille and Grzywacz were among the first to propose use of the Gaussian Tikhonov regularizer for image registration, illustrating the ideal properties of this regularizer for preserving motion coherence. Nielsen et al. later described the intricate connection between the Gaussian Tikhonov regularizer and scale space theory; this work provided the basis with which Pennec et al. detailed the theoretical underpinnings of the Thirion’s Demons algorithm for deformable image registration. The Demons algorithm iteratively computes a force vector field to drive the deformation in the appropriate direction, and then smooths the force vector field by Gaussian convolution in order to update the deformation. The Gaussian convolution step, which can be performed in the Fourier domain or via recursive filters, explicitly incorporates motion-coherent regularization into the registration algorithm. However, these procedures do not allow for ideal treatment of the deformation at the image boundaries. Resolution in the Fourier domain forces the choice of periodic boundary conditions which do not mimic physical behavior. Recursive filters force the user to decide how to extrapolate image data, and they may yield deformations that are not endomorphic. In this article, we illustrate how to remove these limitations and define computationally efficient algorithms for motion-coherent registration under a wide variety of boundary conditions that enable endomorphic deformations and/or physically realistic behavior. The resulting algorithms enable a new degree of user control over the behavior of Demons-style motion coherent registration algorithms.