Multi-resolution parameter choice method for total variation regularized tomography

TL;DR

This paper introduces a multi-resolution computational method for selecting the regularization parameter in total variation regularized tomography, which adapts to resolution and noise without requiring prior sparsity information.

Contribution

The paper presents a novel multi-resolution approach for automatic parameter selection in TV tomography that does not need prior sparsity assumptions and is mathematically analyzed for convergence.

Findings

Method effectively adapts to resolution and noise levels

Results are comparable to the S-curve method

Mathematical proof of TV norm convergence

Abstract

A computational method is introduced for choosing the regularization parameter for total variation (TV) regularization. The approach is based on computing reconstructions at a few different resolutions and various values of regularization parameter. The chosen parameter is the smallest one resulting in approximately discretization-invariant TV norms of the reconstructions. The method is tested with X-ray tomography data measured from a walnut and compared to the S-curve method. The proposed method seems to automatically adapt to the desired resolution and noise level, and it yields useful results in the tests. The results are comparable to those of the S-curve method; however, the S-curve method needs a priori information about the sparsity of the unknown, while the proposed method does not need any a priori information (apart from the choice of a desired resolution). Mathematical analysis is presented for (partial) understanding of the properties of the proposed parameter choice method. It is rigorously proven that the TV norms of the reconstructions converge with any choice of regularization parameter.