Inverse Problems with Diffusion Models: A MAP Estimation Perspective

Abstract

Inverse problems have many applications in science and engineering. In Computer vision several image restoration tasks such as inpainting deblurring and super-resolution can be formally modeled as inverse problems. Recently methods have been developed for solving inverse problems that only leverage a pre-trained unconditional diffusion model and do not require additional task-specific training. In such methods however the inherent intractability of determining the conditional score function during the reverse diffusion process poses a real challenge leaving the methods to settle with an approximation instead which affects their performance in practice. Here we propose a MAP estimation framework to model the reverse conditional generation process of a continuous time diffusion model as an optimization process of the underlying MAP objective whose gradient term is tractable. In theory the proposed framework can be applied to solve general inverse problems using gradient-based optimization methods. However given the highly non-convex nature of the loss objective finding a perfect gradient-based optimization algorithm can be quite challenging nevertheless our framework offers several potential research directions. We use our proposed formulation to develop empirically effective algorithms for image restoration. We validate our proposed algorithms with extensive experiments over multiple datasets across several restoration tasks.