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Volume 5, Issue 4
A Fourth-Order Compact Finite Difference Scheme for Higher-Order PDE-Based Image Registration

Sopida Jewprasert, Noppadol Chumchob & Chantana Chantrapornchai

East Asian J. Appl. Math., 5 (2015), pp. 361-386.

Published online: 2018-02

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  • Abstract

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.

  • AMS Subject Headings

65F10, 65M55, 68U10

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{EAJAM-5-361, author = {}, title = {A Fourth-Order Compact Finite Difference Scheme for Higher-Order PDE-Based Image Registration}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {5}, number = {4}, pages = {361--386}, abstract = {

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.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.270415.280915a}, url = {http://global-sci.org/intro/article_detail/eajam/10821.html} }
TY - JOUR T1 - A Fourth-Order Compact Finite Difference Scheme for Higher-Order PDE-Based Image Registration JO - East Asian Journal on Applied Mathematics VL - 4 SP - 361 EP - 386 PY - 2018 DA - 2018/02 SN - 5 DO - http://doi.org/10.4208/eajam.270415.280915a UR - https://global-sci.org/intro/article_detail/eajam/10821.html KW - Curvature image registration, fourth-order compact finite difference scheme, local Fourier analysis, nonlinear multi-grid method, nonlinear biharmonic equation. AB -

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.

Sopida Jewprasert, Noppadol Chumchob & Chantana Chantrapornchai. (1970). A Fourth-Order Compact Finite Difference Scheme for Higher-Order PDE-Based Image Registration. East Asian Journal on Applied Mathematics. 5 (4). 361-386. doi:10.4208/eajam.270415.280915a
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