Volume 26, Issue 6
Multigrid Method for a Modified Curvature Driven Diffusion Model for Image Inpainting

Carlos Brito-Loeza & Ke Chen

DOI:

J. Comp. Math., 26 (2008), pp. 856-875

Published online: 2008-12

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  • Abstract

Digital inpainting is a fundamental problem in image processing and many variational models for this problem have appeared recently in the literature. Among them are the very successfully Total Variation (TV) model [11] designed for local inpainting and its improved version for large scale inpainting: the Curvature-Driven Diffusion (CDD) model [10]. For the above two models, their associated Euler Lagrange equations are highly nonlinear partial differential equations. For the TV model there exists a relatively fast and easy to implement fixed point method, so adapting the multigrid method of [24] to here is immediate. For the CDD model however, so far only the well known but usually very slow explicit time marching method has been reported and we explain why the implementation of a fixed point method for the CDD model is not straightforward. Consequently the multigrid method as in [Savage and Chen, Int. J. Comput. Math., 82 (2005), pp. 1001-1015] will not work here. This fact represents a strong limitation to the range of applications of this model since usually fast solutions are expected. In this paper, we introduce a modification designed to enable a fixed point method to work and to preserve the features of the original CDD model. As a result, a fast and efficient multigrid method is developed for the modified model. Numerical experiments are presented to show the very good performance of the fast algorithm.

  • Keywords

Image inpainting Variational models Regularization Multilevel methods

  • AMS Subject Headings

68U10 65F10 65K10

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COPYRIGHT: © Global Science Press

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@Article{JCM-26-856, author = {}, title = {Multigrid Method for a Modified Curvature Driven Diffusion Model for Image Inpainting}, journal = {Journal of Computational Mathematics}, year = {2008}, volume = {26}, number = {6}, pages = {856--875}, abstract = { Digital inpainting is a fundamental problem in image processing and many variational models for this problem have appeared recently in the literature. Among them are the very successfully Total Variation (TV) model [11] designed for local inpainting and its improved version for large scale inpainting: the Curvature-Driven Diffusion (CDD) model [10]. For the above two models, their associated Euler Lagrange equations are highly nonlinear partial differential equations. For the TV model there exists a relatively fast and easy to implement fixed point method, so adapting the multigrid method of [24] to here is immediate. For the CDD model however, so far only the well known but usually very slow explicit time marching method has been reported and we explain why the implementation of a fixed point method for the CDD model is not straightforward. Consequently the multigrid method as in [Savage and Chen, Int. J. Comput. Math., 82 (2005), pp. 1001-1015] will not work here. This fact represents a strong limitation to the range of applications of this model since usually fast solutions are expected. In this paper, we introduce a modification designed to enable a fixed point method to work and to preserve the features of the original CDD model. As a result, a fast and efficient multigrid method is developed for the modified model. Numerical experiments are presented to show the very good performance of the fast algorithm.}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8664.html} }
TY - JOUR T1 - Multigrid Method for a Modified Curvature Driven Diffusion Model for Image Inpainting JO - Journal of Computational Mathematics VL - 6 SP - 856 EP - 875 PY - 2008 DA - 2008/12 SN - 26 DO - http://dor.org/ UR - https://global-sci.org/intro/jcm/8664.html KW - Image inpainting KW - Variational models KW - Regularization KW - Multilevel methods AB - Digital inpainting is a fundamental problem in image processing and many variational models for this problem have appeared recently in the literature. Among them are the very successfully Total Variation (TV) model [11] designed for local inpainting and its improved version for large scale inpainting: the Curvature-Driven Diffusion (CDD) model [10]. For the above two models, their associated Euler Lagrange equations are highly nonlinear partial differential equations. For the TV model there exists a relatively fast and easy to implement fixed point method, so adapting the multigrid method of [24] to here is immediate. For the CDD model however, so far only the well known but usually very slow explicit time marching method has been reported and we explain why the implementation of a fixed point method for the CDD model is not straightforward. Consequently the multigrid method as in [Savage and Chen, Int. J. Comput. Math., 82 (2005), pp. 1001-1015] will not work here. This fact represents a strong limitation to the range of applications of this model since usually fast solutions are expected. In this paper, we introduce a modification designed to enable a fixed point method to work and to preserve the features of the original CDD model. As a result, a fast and efficient multigrid method is developed for the modified model. Numerical experiments are presented to show the very good performance of the fast algorithm.
Carlos Brito-Loeza & Ke Chen. (1970). Multigrid Method for a Modified Curvature Driven Diffusion Model for Image Inpainting. Journal of Computational Mathematics. 26 (6). 856-875. doi:
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