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Towards a unified view of unsupervised non-local methods for image denoising: the NL-Ridge approach

Sébastien Herbreteau, Charles Kervrann

TL;DR

The paper addresses unsupervised non-local image denoising by reframing patch aggregation as a linear operator learned to minimize a quadratic risk. It introduces NL-Ridge, a two-step method where the first step uses Stein’s unbiased risk estimate (SURE) to obtain an initial weight matrix $\hat{\Theta}_1$, and the second step applies internal adaptation via multivariate Ridge regression to yield $\hat{\Theta}_2$, with a final reprojection weighting scheme for patch aggregation. A key contribution is showing that NL-Bayes and BM3D can be interpreted within the same NL-Ridge framework, yielding principled guidance on patch sizes and parameter choices while maintaining closed-form updates. Empirically, NL-Ridge achieves competitive or superior performance to state-of-the-art unsupervised denoisers (and even some unsupervised deep methods) on standard AWGN benchmarks, while offering a simpler, GPU-friendly implementation.

Abstract

We propose a unified view of unsupervised non-local methods for image denoising that linearily combine noisy image patches. The best methods, established in different modeling and estimation frameworks, are two-step algorithms. Leveraging Stein's unbiased risk estimate (SURE) for the first step and the "internal adaptation", a concept borrowed from deep learning theory, for the second one, we show that our NL-Ridge approach enables to reconcile several patch aggregation methods for image denoising. In the second step, our closed-form aggregation weights are computed through multivariate Ridge regressions. Experiments on artificially noisy images demonstrate that NL-Ridge may outperform well established state-of-the-art unsupervised denoisers such as BM3D and NL-Bayes, as well as recent unsupervised deep learning methods, while being simpler conceptually.

Towards a unified view of unsupervised non-local methods for image denoising: the NL-Ridge approach

TL;DR

The paper addresses unsupervised non-local image denoising by reframing patch aggregation as a linear operator learned to minimize a quadratic risk. It introduces NL-Ridge, a two-step method where the first step uses Stein’s unbiased risk estimate (SURE) to obtain an initial weight matrix , and the second step applies internal adaptation via multivariate Ridge regression to yield , with a final reprojection weighting scheme for patch aggregation. A key contribution is showing that NL-Bayes and BM3D can be interpreted within the same NL-Ridge framework, yielding principled guidance on patch sizes and parameter choices while maintaining closed-form updates. Empirically, NL-Ridge achieves competitive or superior performance to state-of-the-art unsupervised denoisers (and even some unsupervised deep methods) on standard AWGN benchmarks, while offering a simpler, GPU-friendly implementation.

Abstract

We propose a unified view of unsupervised non-local methods for image denoising that linearily combine noisy image patches. The best methods, established in different modeling and estimation frameworks, are two-step algorithms. Leveraging Stein's unbiased risk estimate (SURE) for the first step and the "internal adaptation", a concept borrowed from deep learning theory, for the second one, we show that our NL-Ridge approach enables to reconcile several patch aggregation methods for image denoising. In the second step, our closed-form aggregation weights are computed through multivariate Ridge regressions. Experiments on artificially noisy images demonstrate that NL-Ridge may outperform well established state-of-the-art unsupervised denoisers such as BM3D and NL-Bayes, as well as recent unsupervised deep learning methods, while being simpler conceptually.
Paper Structure (18 sections, 13 theorems, 49 equations, 1 figure, 2 tables)

This paper contains 18 sections, 13 theorems, 49 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. NL-Ridge for image denoising
  3. A two-step approach applied to patches
  4. Step 1: Stein's unbiased risk estimate
  5. Step 2: Internal adaptation
  6. Weighted average reprojection
  7. A unified view of non-local denoisers
  8. Analysis of NL-Bayes algorithm
  9. Analysis of BM3D algorithm
  10. Experimental results
  11. Setting of algorithm parameters
  12. Results on test datasets
  13. Conclusion
  14. Supplementary material
  15. Useful lemmas
  16. ...and 3 more sections

Key Result

Proposition 1

Let $Y = X + W$ where $Y, X, W \in \mathbb{R}^{n \times m}$ and $W_{i,j} \sim {\cal N}(0, \sigma^2)$ are independent along each row. An unbiased estimate of the risk $R_\Theta(X) = \mathbb{E}\|f_\Theta(Y) - X\|^2_F$ is Stein's unbiased risk estimate (SURE): where $\operatorname{tr}$ denotes the trace operator.

Figures (1)

  • Figure 1: Denoising results (in PSNR) on Barbara corrupted with additive white Gaussian noise ($\sigma = 20$).

Theorems & Definitions (23)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 13 more