Multi-resolution deconvolution

TL;DR

This paper extends classical deconvolution to handle spatially varying resolution using a pseudodifferential-like operator, providing theoretical guarantees and numerical demonstrations of its effectiveness.

Contribution

It introduces a generalized deconvolution framework with a symbol depending on both base and cotangent variables, enabling spatially adaptive deconvolution with stability.

Findings

Proves consistency, convergence, and stability of the new method.

Provides convergence rates for the proposed deconvolution.

Numerical examples demonstrate the advantages of the generalized approach.

Abstract

We extend the classical deconvolution framework in Rn to the case with a pseudodifferential-like solution operator with a symbol depending on both the base and cotangent variable. Our framework enables deconvolution with spatially varying resolution while maintaining a set global stability, and it additionally allows rather general distributional convolution kernels. We provide consistency, convergence and stability results, as well as convergence rates. Finally, we include numerical examples supporting our results and demonstrating advantages of the generalized framework.