BREGMAN SPLIT ALGORITHM AND APPLICATION TO  IMAGE RECOVERY PROBLEM

Nguyễn Đình Dũng; Vũ Khắc Hưng

doi:10.34238/tnu-jst.11870

BREGMAN SPLIT ALGORITHM AND APPLICATION TO IMAGE RECOVERY PROBLEM

About this article

Received: 13/01/25 Revised: 19/03/25 Published: 21/03/25

Authors

1. Nguyen Dinh Dung, TNU - University of Information and Communication Technology
2. Vu Khac Hung , Thai Binh University

Abstract

The Split Bregman algorithm is a variation of the Bregman algorithm, which is an optimization method applied to non-smooth inverse problems in image reconstruction and restoration, particularly in total variation problems. Traditional methods often face challenges in handling non-differentiable problems and require significant computational effort. Therefore, this study aims to develop an improved algorithm based on Split Bregman to accelerate convergence and ensure the stability of the solution. The research method employs a splitting technique to separate non-smooth components, combined with a Bregman update step to solve the optimization problem for each component independently, thereby reducing computational complexity. The research results demonstrate that the improved algorithm achieves high performance in image reconstruction from noisy data, with significantly enhanced peak signal to noise ratio values and reduced mean squared error through iterations. Experimental computations illustrate that the improved Split Bregman method not only has high applicability but also opens new directions for research in optimizing parameters and processing more complex data in the future.

Keywords

Split Bregman Algorithm; Image reconstruction; Total variation; Inverse problems; Optimization methods

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References

[1] J.-F. Cai, S. Osher, and Z. Shen, "Convergence of the linearized Bregman iteration for ℓ₁-norm minimization," Mathematics of Computation, vol. 78, no. 268, pp. 2127-2136, 2009.

[2] T. Würfl, F. C. Ghesu, V. Christlein, and A. Maier, "Regularization of inverse problems in X-ray computed tomography with neural networks learned from imperfect data," Medical Image Analysis, vol. 54, pp. 68-82, 2019.

[3] M. Benning and M. Burger, "Modern regularization methods for inverse problems," Acta Numerica, vol. 27, pp. 1-111, 2018.

[4] A. Chambolle, V. Caselles, D. Cremers, M. Novaga, and T. Pock, "An introduction to total variation for image analysis," Theoretical Foundations and Numerical Methods for Sparse Recovery, vol. 9, pp. 263-340, 2010.

[5] W. Yin, S. Osher, D. Goldfarb, and J. Darbon, "Bregman iterative algorithms for L₁-minimization with applications to compressed sensing," SIAM Journal on Imaging Sciences, vol. 1, pp. 143-168, 2008.

[6] T. Goldstein and S. Osher, "The split Bregman method for L₁-regularized problems," SIAM Journal on Imaging Sciences, vol. 2, no. 2, pp. 323–343, 2009.

[7] L. Rudin, S. Osher, and E. Fatemi, "Nonlinear total variation based noise removal algorithms," Physica D: Nonlinear Phenomena, vol. 60, no. 1-4, pp. 259–268, 1992.

[8] Y. Wang, X. Liu, and Z. Li, "A modified non-convex Cauchy total variation regularization model for image restoration," Computational and Applied Mathematics, vol. 43, no. 5, pp. 1–20, 2024.

[9] J. Zhang, L. Chen, and H. Sun, "Total variation image reconstruction algorithm based on non-convex regularization," Signal, Image and Video Processing, vol. 18, no. 2, pp. 263–275, 2024.

[10] G. Pascal, "Rudin-Osher-Fatemi total variation denoising using split Bregman," Image Processing On Line, vol. 2, pp. 74–95, 2012.

[11] C. Chen and G. Xu, "A new linearized split Bregman iterative algorithm for image reconstruction in sparse-view X-ray computed tomography," Computers & Mathematics with Applications, vol. 71, no. 8, pp. 1537–1559, 2016.

[12] N. Parikh and S. Boyd, "Proximal algorithms," Foundations and Trends® in Optimization, vol. 1, no. 3, pp. 127–239, 2014.

[13] S. Hurault, U. Kamilov, A. Leclaire, and N. Papadakis, "Convergent Bregman Plug-and-Play image restoration for Poisson inverse problems," IEEE Transactions on Computational Imaging, vol. 7, pp. 123–136, 2021.

DOI: https://doi.org/10.34238/tnu-jst.11870

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