Random tree Besov priors -- Towards fractal imaging

TL;DR

This paper introduces a novel wavelet-based fractal prior for inverse problems that preserves edges and has well-defined infinite-dimensional properties, with applications in denoising.

Contribution

It presents a new Besov space-based prior with fractal characteristics and an efficient algorithm for MAP estimation in denoising tasks.

Findings

Realisations of the prior lie in Besov spaces with controlled singularities.

The prior exhibits fractal behavior with small Hausdorff dimension.

An efficient MAP estimation algorithm was developed for denoising.

Abstract

We propose alternatives to Bayesian a priori distributions that are frequently used in the study of inverse problems. Our aim is to construct priors that have similar good edge-preserving properties as total variation or Mumford-Shah priors but correspond to well defined infinite-dimensional random variables, and can be approximated by finite-dimensional random variables. We introduce a new wavelet-based model, where the non zero coefficient are chosen in a systematic way so that prior draws have certain fractal behaviour. We show that realisations of this new prior take values in some Besov spaces and have singularities only on a small set $\tau$ that has a certain Hausdorff dimension. We also introduce an efficient algorithm for calculating the MAP estimator, arising from the the new prior, in denoising problem.