| Abstract: | Image registration is an ill-posed problem that has been studied widely in recentyears. The so-called curvature-based image registration method is one of the mosteffective and well-known approaches, as it produces smooth solutions and allows anautomatic rigid alignment. An important outstanding issue is the accurate and efficientnumerical solution of the Euler-Lagrange system of two coupled nonlinear biharmonicequations, addressed in this article. We propose a fourth-order compact (FOC) finitedifference scheme using a splitting operator on a 9-point stencil, and discuss how theresulting nonlinear discrete system can be solved efficiently by a nonlinear multi-grid(NMG) method. Thus after measuring the h-ellipticity of the nonlinear discrete operatorinvolved by a local Fourier analysis (LFA), we show that our FOC finite difference methodis amenable to multi-grid (MG) methods and an appropriate point-wise smoothing procedure.A high potential point-wise smoother using an outer-inner iteration method isshown to be effective by the LFA and numerical experiments. Real medical images areused to compare the accuracy and efficiency of our approach and the standard second-ordercentral (SSOC) finite difference scheme in the same NMG framework. As expectedfor a higher-order finite difference scheme, the images generated by our FOC finite differencescheme prove significantly more accurate than those computed using the SSOCfinite difference scheme. Our numerical results are consistent with the LFA analysis, andalso demonstrate that the NMG method converges within a few steps. |
