Wavelets and subband coding
Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion
Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion
A Fast Algorithm for Deblurring Models with Neumann Boundary Conditions
SIAM Journal on Scientific Computing
A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way
A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way
ForWaRD: Fourier-wavelet regularized deconvolution for ill-conditioned systems
IEEE Transactions on Signal Processing
De-noising by soft-thresholding
IEEE Transactions on Information Theory
Directional Multiscale Processing of Images Using Wavelets with Composite Dilations
Journal of Mathematical Imaging and Vision
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Regularization is used in order to obtain a reasonable estimate of the solution to an ill-posed inverse problem. One common form of regularization is to use a filter to reduce the influence of components corresponding to small singular values, perhaps using a Tikhonov least squares formulation. In this work, we break the problem into subproblems with narrower bands of singular values using spectrally defined windows, and we regularize each subproblem individually. We show how to use standard parameter-choice methods, such as the discrepancy principle and generalized cross-validation, in a windowed regularization framework. A perturbation analysis gives sensitivity estimates. We demonstrate the effectiveness of our algorithms on deblurring images and on the backward heat equation.