Blind hierarchical deconvolution

TL;DR

This paper introduces a blind hierarchical deconvolution method that jointly estimates the convolution kernel and signal, allowing for accurate reconstruction without prior knowledge of the kernel or regularity, using an efficient empirical Bayes approach.

Contribution

It proposes a novel framework that parametrizes and jointly estimates the kernel and regularity, overcoming limitations of traditional deconvolution methods.

Findings

Accurate reconstructions of signals with unknown kernels.

Efficient two-step empirical Bayes estimation process.

Handles signals with varying regularity.

Abstract

Deconvolution is a fundamental inverse problem in signal processing and the prototypical model for recovering a signal from its noisy measurement. Nevertheless, the majority of model-based inversion techniques require knowledge on the convolution kernel to recover an accurate reconstruction and additionally prior assumptions on the regularity of the signal are needed. To overcome these limitations, we parametrise the convolution kernel and prior length-scales, which are then jointly estimated in the inversion procedure. The proposed framework of blind hierarchical deconvolution enables accurate reconstructions of functions with varying regularity and unknown kernel size and can be solved efficiently with an empirical Bayes two-step procedure, where hyperparameters are first estimated by optimisation and other unknowns then by an analytical formula.