Fourier Diffusion Models: A Method to Control MTF and NPS in Score-Based Stochastic Image Generation

Score-based diffusion models are new and powerful tools for image generation. They are based on a forward stochastic process where an image is degraded with additive white noise and optional input scaling. A neural network can be trained to estimate the time-dependent score function, and used to run the reverse-time stochastic process to generate new samples from the training image distribution. However, one issue is that sampling the reverse process requires many passes of the neural network. In this work we present Fourier Diffusion Models which replace the scalar operations of the forward process with linear shift invariant systems and additive spatially-stationary noise. This allows for a model of continuous probability flow from true images to measurements with a specific modulation transfer function (MTF) and noise power spectrum (NPS). We also derive the reverse process for posterior sampling of high-quality images given blurry noisy measurements. We conducted a computational experiment using the Lung Image Database Consortium dataset of chest CT images and simulated CT measurements with correlated noise and system blur. Our results show that Fourier diffusion models can improve image quality for supervised diffusion posterior sampling relative to existing conditional diffusion models.