Truncated decompositions and filtering methods with Reflective/Anti-Reflective boundary conditions: a comparison

TL;DR

This paper compares spectral filtering methods for image deblurring under Reflective and Anti-Reflective boundary conditions, showing spectral decompositions, especially Anti-Reflective SD, offer high-quality restoration with lower computational costs.

Contribution

It provides a comparative analysis of spectral filtering methods with new insights into their effectiveness under specific boundary conditions, including extensions to color images.

Findings

Spectral decompositions yield good image restoration quality.

Anti-Reflective SD performs well despite loss of orthogonality.

Computational costs are comparable to existing methods and lower than SVD.

Abstract

The paper analyzes and compares some spectral filtering methods as truncated singular/eigen-value decompositions and Tikhonov/Re-blurring regularizations in the case of the recently proposed Reflective [M.K. Ng, R.H. Chan, and W.C. Tang, A fast algorithm for deblurring models with Neumann boundary conditions, SIAM J. Sci. Comput., 21 (1999), no. 3, pp.851-866] and Anti-Reflective [S. Serra Capizzano, A note on anti-reflective boundary conditions and fast deblurring models, SIAM J. Sci. Comput., 25-3 (2003), pp. 1307-1325] boundary conditions. We give numerical evidence to the fact that spectral decompositions (SDs) provide a good image restoration quality and this is true in particular for the Anti-Reflective SD, despite the loss of orthogonality in the associated transform. The related computational cost is comparable with previously known spectral decompositions, and results substantially lower than the singular value decomposition. The model extension to the cross-channel blurring phenomenon of color images is also considered and the related spectral filtering methods are suitably adapted.