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SNIPS: Solving Noisy Inverse Problems Stochastically

Bahjat Kawar, Gregory Vaksman, Michael Elad

TL;DR

SNIPS introduces a novel stochastic algorithm to draw samples from the posterior of noisy linear inverse problems, addressing the perceptual-quality limitations of MMSE estimates by producing diverse, data-faithful reconstructions. The method combines annealed Langevin dynamics with a Newton-inspired diagonal preconditioner in the singular-value domain of the degradation operator and relies on a pre-trained Gaussian MMSE denoiser for the prior. The authors derive an explicit conditional score in the transformed domain, demonstrate SNIPS on image deblurring, super-resolution, and compressive sensing, and show that samples offer perceptual quality and uncertainty while remaining faithful to measurements. They also analyze measurement fidelity and discuss practical limitations and avenues for acceleration and scalability in future work.

Abstract

In this work we introduce a novel stochastic algorithm dubbed SNIPS, which draws samples from the posterior distribution of any linear inverse problem, where the observation is assumed to be contaminated by additive white Gaussian noise. Our solution incorporates ideas from Langevin dynamics and Newton's method, and exploits a pre-trained minimum mean squared error (MMSE) Gaussian denoiser. The proposed approach relies on an intricate derivation of the posterior score function that includes a singular value decomposition (SVD) of the degradation operator, in order to obtain a tractable iterative algorithm for the desired sampling. Due to its stochasticity, the algorithm can produce multiple high perceptual quality samples for the same noisy observation. We demonstrate the abilities of the proposed paradigm for image deblurring, super-resolution, and compressive sensing. We show that the samples produced are sharp, detailed and consistent with the given measurements, and their diversity exposes the inherent uncertainty in the inverse problem being solved.

SNIPS: Solving Noisy Inverse Problems Stochastically

TL;DR

SNIPS introduces a novel stochastic algorithm to draw samples from the posterior of noisy linear inverse problems, addressing the perceptual-quality limitations of MMSE estimates by producing diverse, data-faithful reconstructions. The method combines annealed Langevin dynamics with a Newton-inspired diagonal preconditioner in the singular-value domain of the degradation operator and relies on a pre-trained Gaussian MMSE denoiser for the prior. The authors derive an explicit conditional score in the transformed domain, demonstrate SNIPS on image deblurring, super-resolution, and compressive sensing, and show that samples offer perceptual quality and uncertainty while remaining faithful to measurements. They also analyze measurement fidelity and discuss practical limitations and avenues for acceleration and scalability in future work.

Abstract

In this work we introduce a novel stochastic algorithm dubbed SNIPS, which draws samples from the posterior distribution of any linear inverse problem, where the observation is assumed to be contaminated by additive white Gaussian noise. Our solution incorporates ideas from Langevin dynamics and Newton's method, and exploits a pre-trained minimum mean squared error (MMSE) Gaussian denoiser. The proposed approach relies on an intricate derivation of the posterior score function that includes a singular value decomposition (SVD) of the degradation operator, in order to obtain a tractable iterative algorithm for the desired sampling. Due to its stochasticity, the algorithm can produce multiple high perceptual quality samples for the same noisy observation. We demonstrate the abilities of the proposed paradigm for image deblurring, super-resolution, and compressive sensing. We show that the samples produced are sharp, detailed and consistent with the given measurements, and their diversity exposes the inherent uncertainty in the inverse problem being solved.

Paper Structure

This paper contains 15 sections, 2 theorems, 57 equations, 19 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. The Proposed Approach: Deriving the Conditional Score Function
  4. Problem Setting
  5. Derivation of the Conditional Score Function
  6. The Proposed Algorithm
  7. Experimental Results
  8. Assessing Faithfulness to the Measurements
  9. Conclusion and Future Work
  10. Conditional Score Derivation Proofs
  11. Step Size Derivation
  12. Alternative Definition of the Noise
  13. Implementation Details
  14. Comparison to RED
  15. Additional Results

Key Result

Theorem 1

Given $\mathbf{y} = \mathbf{Hx} + \mathbf{z}$, $\mathbf{z} \sim \mathcal{N}\left(0, \sigma_0^2 \mathbf{I}\right)$, $\mathbf{H} = \mathbf{U \Sigma V}^T$ is the SVD decomposition of $\mathbf{H}$, $\mathbf{y}_T = \mathbf{U}^T \mathbf{y}$, $\mathbf{n} = \mathbf{n}_i$ as constructed in sec:problem, $\mat

Figures (19)

  • Figure 1: Deblurring results on CelebA liu2015celeba images (uniform $5 \times 5$ blur and an additive noise with $\sigma_0=0.1$). Here and in all other shown figures, the standard deviation image is scaled by 4 for better visual inspection.
  • Figure 2: Super resolution results on LSUN bedroom yu2015lsun images (downscaling $4:1$ by plain averaging and adding noise with $\sigma_0 = 0.04$).
  • Figure 3: Compressive sensing results on a CelebA liu2015celeba image with an additive noise of $\sigma_0 = 0.1$.
  • Figure 4: Super resolution results on CelebA liu2015celeba images (downscaling $4:1$ by plain averaging and adding noise with $\sigma_0 = 0.1$).
  • Figure 5: Super resolution results on CelebA liu2015celeba images (downscaling $2:1$ by plain averaging and adding noise with $\sigma_0 = 0.1$).
  • ...and 14 more figures

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof