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A Robust Low-Rank Prior Model for Structured Cartoon-Texture Image Decomposition with Heavy-Tailed Noise

Weihao Tang, Hongjin He

Abstract

Cartoon-texture image decomposition is a fundamental yet challenging problem in image processing. A significant hurdle in achieving accurate decomposition is the pervasive presence of noise in the observed images, which severely impedes robust results. To address the challenging problem of cartoon-texture decomposition in the presence of heavy-tailed noise, we in this paper propose a robust low-rank prior model. Our approach departs from conventional models by adopting the Huber loss function as the data-fidelity term, rather than the traditional $\ell_2$-norm, while retaining the total variation norm and nuclear norm to characterize the cartoon and texture components, respectively. Given the inherent structure, we employ two implementable operator splitting algorithms, tailored to different degradation operators. Extensive numerical experiments, particularly on image restoration tasks under high-intensity heavy-tailed noise, efficiently demonstrate the superior performance of our model.

A Robust Low-Rank Prior Model for Structured Cartoon-Texture Image Decomposition with Heavy-Tailed Noise

Abstract

Cartoon-texture image decomposition is a fundamental yet challenging problem in image processing. A significant hurdle in achieving accurate decomposition is the pervasive presence of noise in the observed images, which severely impedes robust results. To address the challenging problem of cartoon-texture decomposition in the presence of heavy-tailed noise, we in this paper propose a robust low-rank prior model. Our approach departs from conventional models by adopting the Huber loss function as the data-fidelity term, rather than the traditional -norm, while retaining the total variation norm and nuclear norm to characterize the cartoon and texture components, respectively. Given the inherent structure, we employ two implementable operator splitting algorithms, tailored to different degradation operators. Extensive numerical experiments, particularly on image restoration tasks under high-intensity heavy-tailed noise, efficiently demonstrate the superior performance of our model.

Paper Structure

This paper contains 11 sections, 2 theorems, 27 equations, 11 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notations
  3. The RLRP Model and Algorithms
  4. The RLRP model
  5. Solving model \ref{['RLRP']} with $\Phi=I$
  6. Solving model \ref{['RLRP']} with $\Phi\neq I$
  7. Experimental Results
  8. Pure image denoising
  9. Image inpainting
  10. Image deblurring
  11. Conclusion

Key Result

Theorem 1

Let $\Omega^*$ be the solution set of SD-one and $\Delta^*:=\left\{\zeta^*=(v^*,w^*,\lambda^*)\;|\;\xi^*\in\Omega^*\right\}$, where $\xi^*=(u^*, v^*, w^*, \lambda^*)$. Then, the sequences $\{\xi^k\}$, $\{\zeta^k\}$, and $\{\bar{\zeta}^k\}$ generated by Algorithm alg_I satisfy the following assertion

Figures (11)

  • Figure 1: A comparison between the CLRP model \ref{['CLRP']} and our RLRP model \ref{['RLRP']}, which provides a conceptual explanation of the robustness of our RLRP model to heavy-tailed noise.
  • Figure 2: Test images. (a) Boy synthetic image ($256 \times 256$); (b) Barbara natural image ($512 \times 512$); (c) Stone natural image ($512\times 512 \times 3$); (d) Towel natural image ($512\times 512 \times 3$); (e) Kodim_wall natural image ($512\times 512\times 3$); (f) Barbara_RGB natural image ( $512 \times 512 \times 3$ ); (g) TomJerry cartoon image ($512 \times 512$); (h) Sponge_Bob cartoon image ($512 \times 512 \times 3$); (i) Castle cartoon image ($512 \times 512 \times 3$); (j) Bole cartoon image ($512 \times 512 \times 3)$.
  • Figure 3: Results for clear image decomposition: $\Phi=\bm{I}$. From left to right: the noisy image, comparison of the cartoon part, comparison of the texture part, comparison of the restored images (left: CLRP, right: RLRP). From top to bottom: Boy, Barbara, Towel and Stone respectively.
  • Figure 4: Results for cartoon image decomposition: $\Phi=\bm{I}$. From left to right: the noisy image, comparison of the cartoon part, comparison of the texture part, comparison of the restored images (left: CLRP, right: RLRP). From top to bottom: TomJerry, Sponge_Bob, Castle and Bole respectively.
  • Figure 5: The decomposition results of Boy under different models (left: $u+v$, right: $v$.)
  • ...and 6 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2