A Fourth-Order Variational Image Registration Model and Its Fast Multigrid Algorithm

Abstract

Several partial differential equations (PDEs) based variational methods can be used for deformable image registration, mainly differing in how regularization for deformation fields is imposed [J. Modersitzki, Numerical Methods for Image Restoration, Oxford University Press, Oxford, 2004]. On one hand for smooth problems, models of elastic-, diffusion-, and fluid-image registration are known to generate globally smooth and satisfactory deformation fields. On the other hand for nonsmooth problems, models based on the total variation (TV) regularization are better for preserving discontinuities of the deformation fields. It is a challenge to design a deformation model suitable for both smooth and nonsmooth deformation problems. One promising model that is based on a curvature type regularizer and appears to deliver excellent results for both problems is proposed and studied in this paper. A related work due to B. Fischer and J. Modersitzki [J. Math. Imaging Vision, 18 (2003), pp. 81–85] and then refined by S. Henn and K. Witsch [Multiscale Model. Simul., 4 (2005), pp. 584–609] used an approximation of the mean curvature and obtained improved results over previous models. However, this paper investigates the full curvature model and finds that the new model is more robust than approximated curvature models and leads to further improvement. Associated with the new model is the apparent difficulty in developing a fast algorithm as the system of two coupled PDEs is highly nonlinear and of fourth order so standard application of multigrid methods does not work. In this paper, we first propose several fixed-point type smoothers. Then we use both local Fourier analysis and experiments to select the most effective smoother which turns out to be a primal-dual based method. Finally we use the recommended smoother to propose a nonlinear multigrid algorithm for the new model. Numerical tests using both synthetic and realistic images not only confirm that the proposed curvature model is more robust in registration quality for a wide range of applications than previous work [J. Modersitzki, Numerical Methods for Image Restoration, Oxford University Press, Oxford, 2004] and [S. Henn and K. Witsch, Multiscale Model. Simul., 4 (2005), pp. 584–609], but also that the proposed algorithm is fast and accurate in delivering visually pleasing registration results.