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A Novel Truncated Norm Regularization Method for Multi-channel Color Image Denoising

Yiwen Shan, Dong Hu, Zhi Wang

TL;DR

The paper tackles color image denoising under both cross-channel noise differences and spatially varying noise by introducing DtNFM, a double-weighted regularization framework that combines a truncated nuclear norm minus truncated Frobenius norm $\Vert \mathbf{X} \Vert_{t,*-F}$ with two weight matrices $\mathbf{C}$ and $\mathbf{S}$ to model channel and patch-wise noise variations. By leveraging NSS to group similar patches and solving the resulting nonconvex optimization via ADMM, the authors derive closed-form proximal updates and establish convergence to a single critical point. Across spatially invariant, spatially variant, and real-world noisy datasets, DtNFM consistently outperforms state-of-the-art color denoising methods, with ablations confirming the complementary roles of $\mathbf{C}$ and $\mathbf{S}$ and a thorough exploration of hyperparameters $\lambda$ and $t$. The approach offers a robust, optimization-friendly route to high-quality color denoising without training data, with potential extensions to tensor formulations and adaptive parameter schemes for broader applicability.

Abstract

Due to the high flexibility and remarkable performance, low-rank approximation methods has been widely studied for color image denoising. However, those methods mostly ignore either the cross-channel difference or the spatial variation of noise, which limits their capacity in real world color image denoising. To overcome those drawbacks, this paper is proposed to denoise color images with a double-weighted truncated nuclear norm minus truncated Frobenius norm minimization (DtNFM) method. Through exploiting the nonlocal self-similarity of the noisy image, the similar structures are gathered and a series of similar patch matrices are constructed. For each group, the DtNFM model is conducted for estimating its denoised version. The denoised image would be obtained by concatenating all the denoised patch matrices. The proposed DtNFM model has two merits. First, it models and utilizes both the cross-channel difference and the spatial variation of noise. This provides sufficient flexibility for handling the complex distribution of noise in real world images. Second, the proposed DtNFM model provides a close approximation to the underlying clean matrix since it can treat different rank components flexibly. To solve the problem resulted from DtNFM model, an accurate and effective algorithm is proposed by exploiting the framework of the alternating direction method of multipliers (ADMM). The generated subproblems are discussed in detail. And their global optima can be easily obtained in closed-form. Rigorous mathematical derivation proves that the solution sequences generated by the algorithm converge to a single critical point. Extensive experiments on synthetic and real noise datasets demonstrate that the proposed method outperforms many state-of-the-art color image denoising methods.

A Novel Truncated Norm Regularization Method for Multi-channel Color Image Denoising

TL;DR

The paper tackles color image denoising under both cross-channel noise differences and spatially varying noise by introducing DtNFM, a double-weighted regularization framework that combines a truncated nuclear norm minus truncated Frobenius norm with two weight matrices and to model channel and patch-wise noise variations. By leveraging NSS to group similar patches and solving the resulting nonconvex optimization via ADMM, the authors derive closed-form proximal updates and establish convergence to a single critical point. Across spatially invariant, spatially variant, and real-world noisy datasets, DtNFM consistently outperforms state-of-the-art color denoising methods, with ablations confirming the complementary roles of and and a thorough exploration of hyperparameters and . The approach offers a robust, optimization-friendly route to high-quality color denoising without training data, with potential extensions to tensor formulations and adaptive parameter schemes for broader applicability.

Abstract

Due to the high flexibility and remarkable performance, low-rank approximation methods has been widely studied for color image denoising. However, those methods mostly ignore either the cross-channel difference or the spatial variation of noise, which limits their capacity in real world color image denoising. To overcome those drawbacks, this paper is proposed to denoise color images with a double-weighted truncated nuclear norm minus truncated Frobenius norm minimization (DtNFM) method. Through exploiting the nonlocal self-similarity of the noisy image, the similar structures are gathered and a series of similar patch matrices are constructed. For each group, the DtNFM model is conducted for estimating its denoised version. The denoised image would be obtained by concatenating all the denoised patch matrices. The proposed DtNFM model has two merits. First, it models and utilizes both the cross-channel difference and the spatial variation of noise. This provides sufficient flexibility for handling the complex distribution of noise in real world images. Second, the proposed DtNFM model provides a close approximation to the underlying clean matrix since it can treat different rank components flexibly. To solve the problem resulted from DtNFM model, an accurate and effective algorithm is proposed by exploiting the framework of the alternating direction method of multipliers (ADMM). The generated subproblems are discussed in detail. And their global optima can be easily obtained in closed-form. Rigorous mathematical derivation proves that the solution sequences generated by the algorithm converge to a single critical point. Extensive experiments on synthetic and real noise datasets demonstrate that the proposed method outperforms many state-of-the-art color image denoising methods.
Paper Structure (17 sections, 2 theorems, 43 equations, 16 figures, 10 tables, 2 algorithms)

This paper contains 17 sections, 2 theorems, 43 equations, 16 figures, 10 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notations and Related Works
  3. Notations
  4. Color Image Denoising Methods Based on Low-rank Approximation
  5. Alternating Direction Method of Multipliers (ADMM)
  6. The Proposed Model
  7. Problem Formulation
  8. Optimization
  9. Convergence and Complexity Analysis
  10. Differences From Existing Methods
  11. Experimental Results and Analysis
  12. Spatially Invariant Noise Removal
  13. Spatially Variant Noise Removal
  14. Real-World Noise Removal
  15. Ablation Studies
  16. ...and 2 more sections

Key Result

Theorem 1

Assume that $\tau>0$ and $\mathbf{B} \in \mathbb{R}^{m\times n}$ admits SVD as $\mathbf{U}_{\mathbf{B}} \mathrm{Diag}(\bm{\sigma}(\mathbf{B})) \mathbf{V}_{\mathbf{B}}^{\top}$, without loss of generality, let $m \ge n$. Then, the closed-form solution to is given by where where $\mathbf{r} = [0, \cdots, 0, \sigma_{t+1}(\mathbf{B}), \cdots, \sigma_{n}(\mathbf{B})]^{\top}$ and $\mathcal{S}_{\tau}(\

Figures (16)

  • Figure 1: An overview of our DtNFM method. (a) Group similar patches. For each key patch, the similar patches are found via the $k$NN algorithm. Those similar patch compose a patch matrix $\mathbf{Y}$. Using the data in $\mathbf{Y}$, the weight matrices $\mathbf{C}$ and $\mathbf{S}$ are constructed. (b) Denoise each patch matrix. The matrices $\mathbf{Y}, \mathbf{C}$ and $\mathbf{S}$ are used to formulate the DtNFM model in \ref{['eq_DtNFM']}. The generated optimization problem is solved by the proposed Algorithm \ref{['alg_admm']}. At each iteration, the variables $\mathbf{X}$, $\mathbf{Z}$, $\mathbf{A}$, and $\rho$ are updated in an alternating manner. (c) Generate the denoised image. The patch matrix $\mathbf{X}$ outputed from DtNFM model is decomposed to patches. Those denoised patches are settled to their original places. After processing all of the $P$ patch matrices, the denoised image is obtained.
  • Figure 2: Contours of the tNF and the truncated nuclear norm. Assume the matrix $\mathbf{X} \in \mathbb{R}^{n\times(t+2)}$ with $n \ge t+2$. As the $\Vert \mathbf{X} \Vert_{t,*-F}$ in (a) is minimized from 0.5 to 0.1, its curve approaches to the axes on which $\mathbf{Rank}(\mathbf{X}) = t+1$ holds. Thus the rank of matrix $\mathbf{X} \in \mathbb{R}^{n\times(t+2)}$ will be regularized down to $t+1$. And the low-rankness of $\mathbf{X}$ will be promoted by the tNF minimization. In contrast, the truncated nuclear norm behaves worse than the tNF on promoting the low-rankness.
  • Figure 3: An illustration of the definitions of $\mathbf{y}_{j}, \mathbf{x}_{j}, \mathbf{x}_{j}^{(c)}$, and $\mathbf{n}_{j}$.
  • Figure 4: Illustrations of the spatially invariant noise and the spatially variant noise. (a) the ground truth image "kodim20". (b) The surface of the standard deviation of spatially invariant noise. (c) The surface of the standard deviation of spatially variant noise, which is returned by MATLAB code "abs(peaks(512))". A point $(x,y,z)$ means the pixel at $x$th row and $y$th column has Gaussian noise with standard deviations being $z\times [\sigma_{r\_0};\sigma_{g\_0};\sigma_{b\_0}]$, where $z \in [0,1]$. Specifically, $z \equiv 1$ in the subfigure (b). (d) The image corrupted by the noise in subfigure (b) with $[\sigma_{r\_0};\sigma_{g\_0};\sigma_{b\_0}] = [30; 10; 50]$. (e) The image corrupted by the noise in subfigure (c) with $[\sigma_{r\_0};\sigma_{g\_0};\sigma_{b\_0}] = [30; 35; 40]$.
  • Figure 5: The ground truth images of the two dataset (enumerated from left-to-right and top--to-bottom).
  • ...and 11 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2