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Image registration is an ill-posed problem that has been studied widely in recent
years. The so-called curvature-based image registration method is one of the
most effective and well-known approaches, as it produces smooth solutions and
allows an automatic rigid alignment. An important outstanding issue is the
accurate and efficient numerical solution of the Euler-Lagrange system of two
coupled nonlinear biharmonic equations, addressed in this article. We propose a
fourth-order compact (FOC) finite difference scheme using a splitting operator
on a 9-point stencil, and discuss how the resulting nonlinear discrete system
can be solved efficiently by a nonlinear multi-grid (NMG) method. Thus after
measuring the h-ellipticity of the nonlinear discrete operator involved by a
local Fourier analysis (LFA), we show that our FOC finite difference method is
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 is shown to be effective by the LFA and numerical experiments. Real
medical images are used to compare the accuracy and efficiency of our approach
and the standard second-order central (SSOC) finite difference scheme in the
same NMG framework. As expected for a higher-order finite difference scheme, the
images generated by our FOC finite difference scheme prove significantly more
accurate than those computed using the SSOC finite difference scheme. Our
numerical results are consistent with the LFA analysis, and also demonstrate
that the NMG method converges within a few steps.
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