Fast Wavelet Decomposition of Linear Operators through Product-Convolution Expansions

TL;DR

This paper introduces fast algorithms for decomposing linear operators using product-convolution expansions, enabling efficient computation for large-scale problems in imaging and wave simulations.

Contribution

It presents novel quasi-linear time algorithms for operator decomposition via product-convolution expansions, improving efficiency over traditional wavelet methods.

Findings

Algorithms run in quasi-linear time.

Numerical experiments demonstrate effectiveness in imaging with space-varying blurs.

Decomposition is efficient with limited impulse response data.

Abstract

Wavelet decompositions of integral operators have proven their efficiency in reducing computing times for many problems, ranging from the simulation of waves or fluids to the resolution of inverse problems in imaging. Unfortunately, computing the decomposition is itself a hard problem which is oftentimes out of reach for large scale problems. The objective of this work is to design fast decomposition algorithms based on another representation called product-convolution expansion. This decomposition can be evaluated efficiently assuming that a few impulse responses of the operator are available, but it is usually less efficient than the wavelet decomposition when incorporated in iterative methods. The proposed decomposition algorithms, run in quasi-linear time and we provide some numerical experiments to assess its performance for an imaging problem involving space varying blurs.