Fast transforms for high order boundary conditions

TL;DR

This paper introduces fast transform methods for high-order boundary conditions that enhance the precision of blurring models while maintaining computational complexity similar to the Fast Fourier Transform, applicable to regularization techniques.

Contribution

It extends spectral decomposition techniques for blurring matrices with antireflective boundary conditions to higher polynomial degrees and nonsymmetric cases, enabling faster computations.

Findings

Effective high-order boundary condition transforms demonstrated

Improved accuracy in blurring models with maintained computational efficiency

Numerical experiments confirm the proposed methods' effectiveness

Abstract

We study strategies for increasing the precision in the blurring models by maintaining a complexity in the related numerical linear algebra procedures (matrix-vector product, linear system solution, computation of eigenvalues etc.) of the same order of the celebrated Fast Fourier Transform. The key idea is the choice of a suitable functional basis for representing signals and images. Starting from an analysis of the spectral decomposition of blurring matrices associated to the antireflective boundary conditions introduced in [S. Serra Capizzano, SIAM J. Sci. Comput. 25-3 pp. 1307--1325], we extend the model for preserving polynomials of higher degree and fast computations also in the nonsymmetric case. We apply the proposed model to Tikhonov regularization with smoothing norms and the generalized cross validation for choosing the regularization parameter. A selection of numerical experiments shows the effectiveness of the proposed techniques.