[Paper Review] Diffusion Model Based Posterior Sampling for Noisy Linear Inverse Problems

Xiangming Meng, Yoshiyuki Kabashima|arXiv (Cornell University)|Nov 20, 2022

Photoacoustic and Ultrasonic ImagingEngineering20 citations

TL;DR

The paper introduces DMPS, an unsupervised diffusion-model-based posterior sampling method for noisy linear inverse problems, using a noise-perturbed pseudo-likelihood score with an uninformative prior to enable fast, competitive reconstruction.

ABSTRACT

With the rapid development of diffusion models and flow-based generative models, there has been a surge of interests in solving noisy linear inverse problems, e.g., super-resolution, deblurring, denoising, colorization, etc, with generative models. However, while remarkable reconstruction performances have been achieved, their inference time is typically too slow since most of them rely on the seminal diffusion posterior sampling (DPS) framework and thus to approximate the intractable likelihood score, time-consuming gradient calculation through back-propagation is needed. To address this issue, this paper provides a fast and effective solution by proposing a simple closed-form approximation to the likelihood score. For both diffusion and flow-based models, extensive experiments are conducted on various noisy linear inverse problems such as noisy super-resolution, denoising, deblurring, and colorization. In all these tasks, our method (namely DMPS) demonstrates highly competitive or even better reconstruction performances while being significantly faster than all the baseline methods.

Motivation & Objective

• Motivate and address linear inverse problems with additive Gaussian noise in a Bayesian framework.
• Leverage diffusion models as implicit priors to reconstruct signals from noisy measurements.
• Provide a closed-form approximation for the intractable noise-perturbed likelihood score that enables efficient posterior sampling.
• Demonstrate DMPS across multiple image restoration tasks and compare against state-of-the-art diffusion-based methods.

Proposed method

• Model noisy linear inverse problems as p(x|y) using an implicit diffusion-model prior p(x).
• Derive a noise-perturbed pseudo-likelihood score ∇x_t log p̃(y|x_t) under an uninformative prior assumption for p(x0).
• Provide a closed-form solution for the pseudo-likelihood score that reduces to a tractable expression involving A, y, x_t, σ², and ᾱ_t (Theorem 1).
• Compute the score efficiently via a single SVD of A (Corollary 1.2).
• Integrate the prior score from a pre-trained diffusion model with the pseudo-likelihood score in a reverse-diffusion sampling scheme (Algorithm 1).
• Introduce a scaling parameter λ for guidance analogous to classifier guidance in diffusion models.

Experimental results

Research questions

• RQ1 Can an uninformative-prior-based approximation of the noise-perturbed likelihood score yield accurate posterior sampling for noisy linear inverse problems?
• RQ2 Does DMPS provide faster inference and competitive restoration quality compared to existing diffusion-based posteriors (e.g., DPS, DDNM+) across tasks like super-resolution, denoising, deblurring, and colorization?
• RQ3 How does the DMPS framework perform with real-world datasets and out-of-distribution samples when using pre-trained diffusion models?
• RQ4 What are the robustness properties of DMPS to varying noise levels and measurement operators A?

Key findings

• DMPS achieves competitive or superior PSNR, FID, and LPIPS metrics across noise-affected tasks such as super-resolution, denoising, deblurring, and colorization.
• DMPS runs about 3 times faster per function evaluation than the diffusion posterior sampling (DPS) method.
• The method demonstrates robustness to noise and yields more natural colorization than DPS in qualitative comparisons.
• DMPS maintains strong performance on both in-distribution and out-of-distribution datasets when evaluated with pre-trained ADM diffusion models.