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A Fourth-Order Compact Finite Difference Scheme for Higher-OrderPDE-Based Image Registration

Published online by Cambridge University Press:  10 November 2015

Sopida Jewprasert ,
Noppadol Chumchob  and
Chantana Chantrapornchai
Sopida Jewprasert
Affiliation:
Department of Mathematics, Faculty of Science, Silpakorn University, Nakorn Pathom 73000, Thailand
Noppadol Chumchob*
Affiliation:
Department of Mathematics, Faculty of Science, Silpakorn University, Nakorn Pathom 73000, Thailand Centre of Excellence in Mathematics, CHE, SiAyutthaya Rd., Bangkok 10400, Thailand
Chantana Chantrapornchai
Affiliation:
Department of Computer Engineering, Faculty of Engineering, Kasetsart University, Bangkok 10900, Thailand
*
*Corresponding author. Email addresses:[email protected](S. Jewprasert), [email protected](N.Chumchob), [email protected](C.Chantrapornchai)
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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 themost effective and well-known approaches, as it produces smooth solutions andallows an automatic rigid alignment. An important outstanding issue is theaccurate and efficient numerical solution of the Euler-Lagrange system of twocoupled nonlinear biharmonic equations, addressed in this article. We propose afourth-order compact (FOC) finite difference scheme using a splitting operatoron a 9-point stencil, and discuss how the resulting nonlinear discrete systemcan be solved efficiently by a nonlinear multi-grid (NMG) method. Thus aftermeasuring the h-ellipticity of the nonlinear discrete operator involved by alocal Fourier analysis (LFA), we show that our FOC finite difference method isamenable to multi-grid (MG) methods and an appropriate point-wise smoothingprocedure. A high potential point-wise smoother using an outer-inner iterationmethod is shown to be effective by the LFA and numerical experiments. Realmedical images are used to compare the accuracy and efficiency of our approachand the standard second-order central (SSOC) finite difference scheme in thesame NMG framework. As expected for a higher-order finite difference scheme, theimages generated by our FOC finite difference scheme prove significantly moreaccurate than those computed using the SSOC finite difference scheme. Ournumerical results are consistent with the LFA analysis, and also demonstratethat the NMG method converges within a few steps.

Keywords

65F1065M5568U10Curvature image registrationfourth-order compact finite difference schemelocal Fourier analysisnonlinear multi-grid methodnonlinear biharmonic equation
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 5 , Issue 4 , November 2015 , pp. 361 - 386
Copyright
Copyright © Global-Science Press 2015 

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