Quantisation Scale-Spaces

Abstract

Probabilistic diffusion models excel at sampling new images from learned distributions. Originally motivated by drift-diffusion concepts from physics, they apply image perturbations such as noise and blur in a forward process that results in a tractable probability distribution. A corresponding learned reverse process generates images and can be conditioned on side information, which leads to a wide variety of practical applications. Most of the research focus currently lies on practice-oriented extensions. In contrast, the theoretical background remains largely unexplored, in particular the relations to drift-diffusion. In order to shed light on these connections to classical image filtering, we propose a generalised scale-space theory for probabilistic diffusion models. Moreover, we show conceptual and empirical connections to diffusion and osmosis filters.

Generalised Diffusion Probabilistic Scale-Spaces

“…With our previous conference publication [21], we made first steps to bridge this gap between the scale-space and deep learning communities. To this end, we introduced first generalised scale-space concepts for diffusion probabilistic models.…”

“…Scale-spaces have a long tradition in visual computing. Most of them rely on partial differential [17][18][19][20]32] or pseudo-differential equations [33,34], but they have also been considered for wavelets [35], sparse inpainting-based image representations [36], or hierarchical quantisation operators [37]. Such classical scalespaces describe the evolution of an input image over multiple scales, which gradually simplifies the image.…”

“…Moreover, a comprehensive theory for shape analysis exists in the form of morphological scale-spaces [17,[50][51][52][53][54]. Wavelet shrinkage as a form of blurring [35] and sparse image representations [36,37] have been considered from a scale-space perspective as well. Among this wide variety of different options, for us, the original Gaussian scale-space is still the most relevant.…”

“…In parallel to our conference publication [21], Zach et al [59] have used homogeneous diffusion scale-spaces on probability densities. However, they learn image priors via denoising score matching with practical applications to image denoising.…”

“…5. In contrast to our earlier conference publication [21], we consider a more general transition probability than the original model of Sohl-Dickstein et al [1]. Relying on the model of Kingma et al [6], we use Gaussian distributions of the type…”