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Score-based stochastic denoising models have recently been demonstrated aspowerful machine learning tools for conditional and unconditional imagegeneration. The existing methods are based on a forward stochastic processwherein the training images are scaled to zero over time and white noise isgradually added such that the final time step is approximately zero-meanidentity-covariance Gaussian noise. A neural network is then trained toapproximate the time-dependent score function, or the gradient of the logarithmof the probability density, for that time step. Using this score estimator, itis possible to run an approximation of the time-reversed stochastic process tosample new images from the training data distribution. These score-basedgenerative models have been shown to out-perform generative adversarial neuralnetworks using standard benchmarks and metrics. However, one issue with thisapproach is that it requires a large number of forward passes of the neuralnetwork. Additionally, the images at intermediate time steps are not useful,since the signal-to-noise ratio is low. In this work we present a new methodcalled Fourier Diffusion Models which replaces the scalar operations of theforward process with shift-invariant convolutions and the additive white noisewith additive stationary noise. This allows for control of MTF and NPS atintermediate time steps. Additionally, the forward process can be crafted toconverge to the same MTF and NPS as the measured images. This way, we can modelcontinuous probability flow from true images to measurements. In this way, thesample time can be used to control the tradeoffs between measurementuncertainty and generative uncertainty of posterior estimates. We compareFourier diffusion models to existing scalar diffusion models and show that theyachieve a higher level of performance and allow for a smaller number of timesteps.

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