Radon needlet thresholding

TL;DR

This paper introduces a novel thresholding algorithm based on Radon needlets for noisy image reconstruction, achieving minimax optimality across various loss functions and adapting to object regularity.

Contribution

It presents a new localized basis and thresholding method for Radon transform inversion, establishing minimax bounds and demonstrating adaptation to object sparsity and smoothness.

Findings

Achieves minimax bounds for -loss and other metrics.

Demonstrates adaptation to object regularity and inhomogeneous smoothness.

Provides a numerical study with an averaging procedure akin to cycle spinning.

Abstract

We provide a new algorithm for the treatment of the noisy inversion of the Radon transform using an appropriate thresholding technique adapted to a well-chosen new localized basis. We establish minimax results and prove their optimality. In particular, we prove that the procedures provided here are able to attain minimax bounds for any $\mathbb {L}_p$ loss. It s important to notice that most of the minimax bounds obtained here are new to our knowledge. It is also important to emphasize the adaptation properties of our procedures with respect to the regularity (sparsity) of the object to recover and to inhomogeneous smoothness. We perform a numerical study that is of importance since we especially have to discuss the cubature problems and propose an averaging procedure that is mostly in the spirit of the cycle spinning performed for periodic signals.