800 research outputs found
Truncated decompositions and filtering methods with Reflective/Anti-Reflective boundary conditions: a comparison
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.Comment: 22 pages, 10 figure
V-cycle optimal convergence for DCT-III matrices
The paper analyzes a two-grid and a multigrid method for matrices belonging
to the DCT-III algebra and generated by a polynomial symbol. The aim is to
prove that the convergence rate of the considered multigrid method (V-cycle) is
constant independent of the size of the given matrix. Numerical examples from
differential and integral equations are considered to illustrate the claimed
convergence properties.Comment: 19 page
Rapid deconvolution of low-resolution time-of-flight data using Bayesian inference
The deconvolution of low-resolution time-of-flight data has numerous advantages, including the ability to extract additional information from the experimental data. We augment the well-known Lucy-Richardson deconvolution algorithm using various Bayesian prior distributions and show that a prior of second-differences of the signal outperforms the standard Lucy-Richardson algorithm, accelerating the rate of convergence by more than a factor of four, while preserving the peak amplitude ratios of a similar fraction of the total peaks. A novel stopping criterion and boosting mechanism are implemented to ensure that these methods converge to a similar final entropy and local minima are avoided. Improvement by a factor of two in mass resolution allows more accurate quantification of the spectra. The general method is demonstrated in this paper through the deconvolution of fragmentation peaks of the 2,5-dihydroxybenzoic acid matrix and the benzyltriphenylphosphonium thermometer ion, following femtosecond ultraviolet laser desorption
Regularization matrices for discrete ill-posed problems in several space-dimensions
Many applications in science and engineering require the solution of large linear discrete ill-posed problems that are obtained by the discretization of a Fredholm integral equation of the first kind in several space dimensions. The matrix that defines these problems is very ill conditioned and generally numerically singular, and the right-hand side, which represents measured data, is typically contaminated by measurement error. Straightforward solution of these problems is generally not meaningful due to severe error propagation. Tikhonov regularization seeks to alleviate this difficulty by replacing the given linear discrete ill-posed problem by a penalized least-squares problem, whose solution is less sensitive to the error in the right-hand side and to roundoff errors introduced during the computations. This paper discusses the construction of penalty terms that are determined by solving a matrix nearness problem. These penalty terms allow partial transformation to standard form of Tikhonov regularization problems that stem from the discretization of integral equations on a cube in several space dimensions
Algorithms for Toeplitz Matrices with Applications to Image Deblurring
In this thesis, we present the O(n(log n)^2) superfast linear least squares Schur algorithm (ssschur). The algorithm we will describe illustrates a fast way of solving linear equations or linear least squares problems with low displacement rank. This program is based on the O(n^2) Schur algorithm speeded up via FFT. The algorithm solves a ill-conditioned Toeplitz-like system using Tikhonov regularization. The regularized system is Toeplitz-like of displacement rank 4. We also show the effect of choice of the regularization parameter on the quality of the image reconstructed
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