Efficient nonlocal linear image denoising: Bilevel optimization with Nonequispaced Fast Fourier Transform and matrix-free preconditioning

TL;DR

This paper introduces an efficient nonlocal image denoising method using bilevel optimization, Nonequispaced FFT, and matrix-free preconditioning, enabling rapid processing of large images with low memory usage.

Contribution

It proposes a novel bilevel optimization framework with advanced numerical techniques for large-scale nonlocal image denoising, improving speed and memory efficiency.

Findings

Effective denoising of large images demonstrated

The method converges rapidly with theoretical guarantees

Low storage requirements achieved in experiments

Abstract

We present a new approach for nonlocal image denoising, based around the application of an unnormalized extended Gaussian ANOVA kernel within a bilevel optimization algorithm. A critical bottleneck when solving such problems for finely-resolved images is the solution of huge-scale, dense linear systems arising from the minimization of an energy term. We tackle this using a Krylov subspace approach, with a Nonequispaced Fast Fourier Transform utilized to approximate matrix-vector products in a matrix-free manner. We accelerate the algorithm using a novel change of basis approach to account for the (known) smallest eigenvalue-eigenvector pair of the matrices involved, coupled with a simple but frequently very effective diagonal preconditioning approach. We present a number of theoretical results concerning the eigenvalues and predicted convergence behavior, and a range of numerical experiments which validate our solvers and use them to tackle parameter learning problems. These demonstrate that very large problems may be effectively and rapidly denoised with very low storage requirements on a computer.