  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 and develop
empirically effective algorithms for solving noiseless and noisy image
inpainting tasks. We validate our proposed algorithms with extensive
experiments across diverse mask settings.
