Solving Inverse Problems with Latent Diffusion Models via Hard Data Consistency

Diffusion models have recently emerged as powerful generative priors for solving inverse problems. However, training diffusion models in the pixel space are both data intensive and computationally demanding, which restricts their applicability as priors in domains such as medical imaging. Latent diffusion models, which operate in a much lower-dimensional space, offer a solution to these challenges. Though, their direct application to solving inverse problems remains an unsolved technical challenge due to the nonlinearity of the encoder and decoder. To address this issue,we propose ReSample, an algorithm that solves general inverse problems with pre-trained latent diffusion models. Our algorithm incorporates data consistency by solving an optimization problem during the reverse sampling process, a concept that we term as hard data consistency. Upon solving this optimization problem, we propose a novel resampling scheme to map the measurement-consistent sample back onto the correct data manifold. Our approach offers both memory efficiency and considerable flexibility in the sense that (1) it can be readily adapted to various inverse problems using the same pre-trained model as it does not assume any fixed forward measurement operator during training, and (2) it can be generalized to different domains by simply fine-tuning the latent diffusion model with a minimal amount of data samples. Our empirical results on both linear and non-linear inverse problems demonstrate that our approach can reconstruct high-quality images even compared to state-of-the-art works that operate in the pixel space.