Product of Gaussian Mixture Diffusion Models

Martin Zach, Antonin Chambolle, Thomas Pock, Erich Kobler

Abstract:

In this work we tackle the problem of estimating the density f_X of a random variable X by successive smoothing, such
that the smoothed random variable Y fulfills the diffusion partial differential equation (∂_t− ∆_1)fY (· , t) = 0 with
initial condition f_Y (·, 0) = f_X . We propose a product-of-experts-type model utilizing Gaussian mixture experts and
study configurations that admit an analytic expression for f_Y (· , t). In particular, with a focus on image processing, we
derive conditions for models acting on filter-, wavelet-, and shearlet-responses. Our construction naturally allows the
model to be trained simultaneously over the entire diffusion horizon using empirical Bayes. We show numerical results
for image denoising where our models are competitive while being tractable, interpretable, and having only a small
number of learnable parameters. As a byproduct, our models can be used for reliable noise level estimation, allowing
blind denoising of images corrupted by heteroscedastic noise.