A Fast Augmented Lagrangian Method for Euler's Elastica Models

A Fast Augmented Lagrangian Method for Euler's Elastica Models

Year:    2013

Numerical Mathematics: Theory, Methods and Applications, Vol. 6 (2013), Iss. 1 : pp. 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.

You do not have full access to this article.

Already a Subscriber? Sign in as an individual or via your institution

Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/nmtma.2013.mssvm03

Numerical Mathematics: Theory, Methods and Applications, Vol. 6 (2013), Iss. 1 : pp. 47–71

Published online:    2013-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    25

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

  1. A Fast Relaxed Normal Two Split Method and an Effective Weighted TV Approach for Euler's Elastica Image Inpainting

    Yashtini, Maryam | Kang, Sung Ha

    SIAM Journal on Imaging Sciences, Vol. 9 (2016), Iss. 4 P.1552

    https://doi.org/10.1137/16M1063757 [Citations: 34]
  2. Noise removal using an adaptive Euler’s elastica-based model

    Yang, Junci | Ma, Mingxi | Zhang, Jun | Wang, Chao

    The Visual Computer, Vol. 39 (2023), Iss. 11 P.5485

    https://doi.org/10.1007/s00371-022-02674-0 [Citations: 2]
  3. Euler’s elastica-based algorithm for Parallel MRI reconstruction using SENSitivity Encoding

    Yashtini, Maryam

    Optimization Letters, Vol. 14 (2020), Iss. 6 P.1435

    https://doi.org/10.1007/s11590-019-01451-8 [Citations: 4]
  4. Image Reconstruction by Minimizing Curvatures on Image Surface

    Zhong, Qiuxiang | Yin, Ke | Duan, Yuping

    Journal of Mathematical Imaging and Vision, Vol. 63 (2021), Iss. 1 P.30

    https://doi.org/10.1007/s10851-020-00992-3 [Citations: 14]
  5. Euler’s Elastica Strategy for Limited-angle Computed Tomography Image Reconstruction

    Zhang, Hanming | Wang, Linyuan | Duan, Yuping | Li, Lei | Hu, Guoen | Yan, Bin

    IEEE Transactions on Nuclear Science, Vol. (2017), Iss. P.1

    https://doi.org/10.1109/TNS.2017.2717864 [Citations: 3]
  6. Scale Space and Variational Methods in Computer Vision

    Alternating Direction Method of Multiplier for Euler’s Elastica-Based Denoising

    Yashtini, Maryam | Kang, Sung Ha

    2015

    https://doi.org/10.1007/978-3-319-18461-6_55 [Citations: 6]
  7. Consecutive Context Perceive Generative Adversarial Networks for Serial Sections Inpainting

    Zhang, Siqi | Wang, Lei | Zhang, Jie | Gu, Ling | Jiang, Xiran | Zhai, Xiaoyue | Sha, Xianzheng | Chang, Shijie

    IEEE Access, Vol. 8 (2020), Iss. P.190417

    https://doi.org/10.1109/ACCESS.2020.3031973 [Citations: 7]
  8. Fast Linearized Augmented Lagrangian Method for Euler's Elastica Model

    Zhang, Jun | Chen, Rongliang | Deng, Chengzhi | Wang, Shengqian

    Numerical Mathematics: Theory, Methods and Applications, Vol. 10 (2017), Iss. 1 P.98

    https://doi.org/10.4208/nmtma.2017.m1611 [Citations: 21]
  9. Elastica Models for Color Image Regularization

    Liu, Hao | Tai, Xue-Cheng | Kimmel, Ron | Glowinski, Roland

    SIAM Journal on Imaging Sciences, Vol. 16 (2023), Iss. 1 P.461

    https://doi.org/10.1137/22M147935X [Citations: 3]
  10. A Fast Minimization Algorithm for the Euler Elastica Model Based on a Bilinear Decomposition

    Liu, Zhifang | Sun, Baochen | Tai, Xue-Cheng | Wang, Qi | Chang, Huibin

    SIAM Journal on Scientific Computing, Vol. 46 (2024), Iss. 1 P.A290

    https://doi.org/10.1137/23M1552772 [Citations: 1]
  11. Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging

    On Variable Splitting and Augmented Lagrangian Method for Total Variation-Related Image Restoration Models

    Liu, Zhifang | Duan, Yuping | Wu, Chunlin | Tai, Xue-Cheng

    2023

    https://doi.org/10.1007/978-3-030-03009-4_84-2 [Citations: 0]
  12. An Efficient Method for Euler’s Elastica Based Image Deconvolution

    Wali, Samad | Shakoor, Abdul | Basit, Abdul | Xie, Lipeng | Huang, Chencheng | Li, Chunming

    IEEE Access, Vol. 7 (2019), Iss. P.61226

    https://doi.org/10.1109/ACCESS.2019.2912660 [Citations: 7]
  13. Poisson noise removal based on nonlocal total variation with Euler’s elastica pre-processing

    Liu, Hongyi | Zhang, Zhengrong | Xiao, Liang | Wei, Zhihui

    Journal of Shanghai Jiaotong University (Science), Vol. 22 (2017), Iss. 5 P.609

    https://doi.org/10.1007/s12204-017-1878-5 [Citations: 6]
  14. Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging

    On Variable Splitting and Augmented Lagrangian Method for Total Variation-Related Image Restoration Models

    Liu, Zhifang | Duan, Yuping | Wu, Chunlin | Tai, Xue-Cheng

    2021

    https://doi.org/10.1007/978-3-030-03009-4_84-1 [Citations: 1]
  15. An adaptive speckle removal algorithm for ultrasound image based on Euler's elastic model

    Zhang, Qiong | Shen, Minfen | Li, Bin

    2014 IEEE China Summit & International Conference on Signal and Information Processing (ChinaSIP), (2014), P.587

    https://doi.org/10.1109/ChinaSIP.2014.6889311 [Citations: 0]
  16. Adaptive Image Restoration via a Relaxed Regularization of Mean Curvature

    Ma, Mingxi | Zhang, Jun | Deng, Chengzhi | Liu, Zhaoyang | Wang, Yuanyun | Edalatpanah, S. A.

    Mathematical Problems in Engineering, Vol. 2020 (2020), Iss. P.1

    https://doi.org/10.1155/2020/3416907 [Citations: 0]
  17. A New Operator Splitting Method for the Euler Elastica Model for Image Smoothing

    Deng, Liang-Jian | Glowinski, Roland | Tai, Xue-Cheng

    SIAM Journal on Imaging Sciences, Vol. 12 (2019), Iss. 2 P.1190

    https://doi.org/10.1137/18M1226361 [Citations: 37]
  18. Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging

    On Variable Splitting and Augmented Lagrangian Method for Total Variation-Related Image Restoration Models

    Liu, Zhifang | Duan, Yuping | Wu, Chunlin | Tai, Xue-Cheng

    2023

    https://doi.org/10.1007/978-3-030-98661-2_84 [Citations: 0]