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A unified framework of non-local parametric methods for image denoising

Sébastien Herbreteau, Charles Kervrann

TL;DR

This paper introduces NL-Ridge, a unified, risk-based framework for non-local image denoising that unifies established methods like NL-Bayes and BM3D under a quadratic-risk optimization. By processing groups of similar patches through linear patch combinations and optimizing weights with unbiased risk estimates (URE) and internal adaptation, NL-Ridge achieves competitive denoising performance across Gaussian, Poisson, and mixed noise, while remaining conceptually simple and computationally efficient. The authors show that NL-Bayes and BM3D can be recovered as special cases within this framework, enabling a cohesive understanding of non-local denoising strategies. Experimental results on standard benchmarks and real-world noisy images demonstrate NL-Ridge's strong performance, especially in texture-rich scenes, and its potential as a fast, unsupervised alternative to deep nets.

Abstract

We propose a unified view of non-local methods for single-image denoising, for which BM3D is the most popular representative, that operate by gathering noisy patches together according to their similarities in order to process them collaboratively. Our general estimation framework is based on the minimization of the quadratic risk, which is approximated in two steps, and adapts to photon and electronic noises. Relying on unbiased risk estimation (URE) for the first step and on ``internal adaptation'', a concept borrowed from deep learning theory, for the second, we show that our approach enables to reinterpret and reconcile previous state-of-the-art non-local methods. Within this framework, we propose a novel denoiser called NL-Ridge that exploits linear combinations of patches. While conceptually simpler, we show that NL-Ridge can outperform well-established state-of-the-art single-image denoisers.

A unified framework of non-local parametric methods for image denoising

TL;DR

This paper introduces NL-Ridge, a unified, risk-based framework for non-local image denoising that unifies established methods like NL-Bayes and BM3D under a quadratic-risk optimization. By processing groups of similar patches through linear patch combinations and optimizing weights with unbiased risk estimates (URE) and internal adaptation, NL-Ridge achieves competitive denoising performance across Gaussian, Poisson, and mixed noise, while remaining conceptually simple and computationally efficient. The authors show that NL-Bayes and BM3D can be recovered as special cases within this framework, enabling a cohesive understanding of non-local denoising strategies. Experimental results on standard benchmarks and real-world noisy images demonstrate NL-Ridge's strong performance, especially in texture-rich scenes, and its potential as a fast, unsupervised alternative to deep nets.

Abstract

We propose a unified view of non-local methods for single-image denoising, for which BM3D is the most popular representative, that operate by gathering noisy patches together according to their similarities in order to process them collaboratively. Our general estimation framework is based on the minimization of the quadratic risk, which is approximated in two steps, and adapts to photon and electronic noises. Relying on unbiased risk estimation (URE) for the first step and on ``internal adaptation'', a concept borrowed from deep learning theory, for the second, we show that our approach enables to reinterpret and reconcile previous state-of-the-art non-local methods. Within this framework, we propose a novel denoiser called NL-Ridge that exploits linear combinations of patches. While conceptually simpler, we show that NL-Ridge can outperform well-established state-of-the-art single-image denoisers.
Paper Structure (37 sections, 17 theorems, 105 equations, 5 figures, 1 table, 2 algorithms)

This paper contains 37 sections, 17 theorems, 105 equations, 5 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. NL-Ridge for image denoising
  3. Parametric linear patch combinations
  4. Parameter optimization
  5. Step 1: Unbiased risk estimate (URE)
  6. Minimization of the surrogate
  7. On the positive definiteness of Q1
  8. Particular case: homoscedastic Gaussian noise
  9. Step 2: Internal adaptation
  10. Minimization of the surrogate
  11. On the positive definiteness of Q2
  12. Particular case: homoscedastic Gaussian noise
  13. Weighted average reprojection
  14. A unified view of non-local denoisers
  15. Analysis of NL-Bayes algorithm
  16. ...and 22 more sections

Key Result

Lemma 1

Let $Q, C \in \mathbb{R}^{k \times k}$. If $Q$ is symmetric positive definite,

Figures (5)

  • Figure 1: Illustration of the grouping technique for image denoising.
  • Figure 2: Illustration of the parametric view of several popular non-local denoisers. Examples of parameterized functions unequivocally identifying the denoiser are given, whose optimal parameters are eventually selected for each group of patches by "internal adaptation" optimization.
  • Figure 3: Denoising results (in PSNR) on Barbara corrupted with additive white Gaussian noise ($\sigma = 20$).
  • Figure 4: Qualitative comparison of image denoising results on real-world noisy images from Darmstadt Noise Dataset DND. Zoom-in regions are indicated for each method.
  • Figure 5: The execution time on CPU for an image of size $512\times512$ v.s the average PSNR results on Set12 and BSD68 berkeley for synthetic Gaussian noise with $\sigma=25$ of the most effective popular methods drunetdncnnffdnetLIDIAscunetrestormernlridgenlbayesBM3DWNNMEPLL_unsupervisedTWSCrethinkingS2S. These results are calculated based on \ref{['nlridge_resultsPSNR00']}.

Theorems & Definitions (17)

  • Lemma 1: Multivariate quadratic programming
  • Lemma 2
  • Lemma 3: A closed-form expression for the quadratic risk
  • Proposition 1: Minimization of the quadratic risk
  • Proposition 2: Gaussian noise
  • Proposition 3: Poisson noise
  • Proposition 4: Mixed Poisson-Gaussian noise
  • Proposition 5: Minimization of the URE
  • Proposition 6: URE for a noisier risk and its minimization
  • Lemma 4: A closed-form expression for the quadratic risk
  • ...and 7 more