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Volume 6, Issue 1
A Fast Augmented Lagrangian Method for Euler's Elastica Models

Yuping Duan, Yu Wang & Jooyoung Hahn

DOI: 10.4208/nmtma.2013.mssvm03

Numer. Math. Theor. Meth. Appl., 6 (2013), pp. 47-71.

Published online: 2013-06

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  • Abstract

In this paper, a fast algorithm for Euler's elastica functional is proposed, in which the Euler's elastica functional is reformulated as a constrained minimization problem. Combining the augmented Lagrangian method and operator splitting techniques, the resulting saddle-point problem is solved by a serial of subproblems. To tackle the nonlinear constraints arising in the model, a novel fixed-point-based approach is proposed so that all the subproblems either is a linear problem or has a closed-form solution. We show the good performance of our approach in terms of speed and reliability using numerous numerical examples on synthetic, real-world and medical images for image denoising, image inpainting and image zooming problems.

  • Keywords

Euler's elastica, augmented Lagrangian method, image denoising, image inpainting, image zooming.

  • AMS Subject Headings

68U10, 65N21, 74S20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-6-47, author = {Yuping Duan, Yu Wang and Jooyoung Hahn}, title = {A Fast Augmented Lagrangian Method for Euler's Elastica Models}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2013}, volume = {6}, number = {1}, pages = {47--71}, abstract = {

In this paper, a fast algorithm for Euler's elastica functional is proposed, in which the Euler's elastica functional is reformulated as a constrained minimization problem. Combining the augmented Lagrangian method and operator splitting techniques, the resulting saddle-point problem is solved by a serial of subproblems. To tackle the nonlinear constraints arising in the model, a novel fixed-point-based approach is proposed so that all the subproblems either is a linear problem or has a closed-form solution. We show the good performance of our approach in terms of speed and reliability using numerous numerical examples on synthetic, real-world and medical images for image denoising, image inpainting and image zooming problems.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2013.mssvm03}, url = {http://global-sci.org/intro/article_detail/nmtma/5894.html} }
TY - JOUR T1 - A Fast Augmented Lagrangian Method for Euler's Elastica Models AU - Yuping Duan, Yu Wang & Jooyoung Hahn JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 47 EP - 71 PY - 2013 DA - 2013/06 SN - 6 DO - http://doi.org/10.4208/nmtma.2013.mssvm03 UR - https://global-sci.org/intro/article_detail/nmtma/5894.html KW - Euler's elastica, augmented Lagrangian method, image denoising, image inpainting, image zooming. AB -

In this paper, a fast algorithm for Euler's elastica functional is proposed, in which the Euler's elastica functional is reformulated as a constrained minimization problem. Combining the augmented Lagrangian method and operator splitting techniques, the resulting saddle-point problem is solved by a serial of subproblems. To tackle the nonlinear constraints arising in the model, a novel fixed-point-based approach is proposed so that all the subproblems either is a linear problem or has a closed-form solution. We show the good performance of our approach in terms of speed and reliability using numerous numerical examples on synthetic, real-world and medical images for image denoising, image inpainting and image zooming problems.

Yuping Duan, Yu Wang and Jooyoung Hahn. (2013). A Fast Augmented Lagrangian Method for Euler's Elastica Models. Numerical Mathematics: Theory, Methods and Applications. 6 (1). 47-71. doi:10.4208/nmtma.2013.mssvm03
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