SNIPS: Solving Noisy Inverse Problems Stochastically

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.Preprint. Under review.

Bayesian Imaging Using Plug & Play Priors: When Langevin Meets Tweedie

“…Approaches based on score matching techniques [76,43] have also shown promising results recently [51,50]. These methods are linked with Plug & Play approaches as they also estimate a Stein score.…”

“…However, they do not rely on the asymptotic convergence of a diffusion, but instead aim at inverting a noising process stemming from an optimal transport problem [16]. The recent work [50] is particularly relevant in this context as it considers a range of imaging inverse problems, where it exploits the structure of the forward operator to perform posterior sampling in a coarse-to-fine manner. This also allows the use of multivariate step-sizes that are specific to each scale and ensure stability.…”

“…This also allows the use of multivariate step-sizes that are specific to each scale and ensure stability. However, to the best of our knowledge, the convergence properties of [50] have not been studied yet.…”

Solving Inverse Problems in Medical Imaging with Score-Based Generative Models

“…An unsupervised alternative is to approximate the conditional score function with an unconditionallytrained score model s θ ˚px t , tq « ∇ xt log p t px t q and the measurement distribution ppy | xq. Many existing works (Song et al, 2021;Kawar et al, 2021;Kadkhodaie & Simoncelli, 2020;Jalal et al, 2021) have implemented this idea in different ways. However, the methods in Kawar et al (2021) and Kadkhodaie & Simoncelli (2020) both require computing the singular value decomposition (SVD) of A P R mˆn , which can be difficult for many measurement processes in medical imaging.…”

“…Many existing works (Song et al, 2021;Kawar et al, 2021;Kadkhodaie & Simoncelli, 2020;Jalal et al, 2021) have implemented this idea in different ways. However, the methods in Kawar et al (2021) and Kadkhodaie & Simoncelli (2020) both require computing the singular value decomposition (SVD) of A P R mˆn , which can be difficult for many measurement processes in medical imaging. The method proposed in Jalal et al (2021) is only designed for a specific sampling method called annealed Langevin dynamics (ALD, Song & Ermon, 2019), which proves to be inferior to more advanced sampling algorithms such as Predictor-Corrector methods (Song et al, 2021).…”