Measurement Score-Based Diffusion Model

Diffusion models have achieved remarkable success in tasks ranging from image generation to inverse problems. However, training diffusion models typically requires clean ground-truth images, which are unavailable in many applications. We introduce the Measurement Score-based diffusion Model (MSM), a novel framework that learns partial measurement scores directly from noisy and subsampled measurements. By aggregating these scores in expectation, MSM synthesizes fully sampled measurements without requiring access to clean images. To make this practical, we develop a stochastic sampling variant of MSM that approximates the expectation efficiently and analyze its asymptotic equivalence to the exact formulation. We further extend MSM to posterior sampling for linear inverse problems, enabling accurate image reconstruction directly from partial scores. Experiments on natural images and multi-coil MRI demonstrate that MSM achieves state-of-the-art performance in unconditional generation and inverse problem solving---all while being trained exclusively on degraded measurements.