  Diffusion models have recently been shown to excel in many image
reconstruction tasks that involve inverse problems based on a forward
measurement operator. A common framework uses task-agnostic unconditional
models that are later post-conditioned for reconstruction, an approach that
typically suffers from suboptimal task performance. While task-specific
conditional models have also been proposed, current methods heuristically
inject measured data as a naive input channel that elicits sampling
inaccuracies. Here, we address the optimal conditioning of diffusion models for
solving challenging inverse problems that arise during image reconstruction.
Specifically, we propose a novel Bayesian conditioning technique for diffusion
models, BCDM, based on score-functions associated with the conditional
distribution of desired images given measured data. We rigorously derive the
theory to express and train the conditional score-function. Finally, we show
state-of-the-art performance in image dealiasing, deblurring, super-resolution,
and inpainting with the proposed technique.
