Fast transforms for high order boundary conditions
For image deblurring, this work provides a faster and more accurate boundary treatment than existing methods, though it is an incremental extension of prior antireflective models.
This paper extends antireflective boundary conditions for blurring models to preserve higher-degree polynomials while maintaining FFT-level computational complexity, and demonstrates effectiveness in Tikhonov regularization with generalized cross validation.
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.