@article {12440,
	title = {Example-Driven Manifold Priors for Image Deconvolution},
	journal = {Image Processing, IEEE Transactions on},
	volume = {20},
	year = {2011},
	month = {2011/11//},
	pages = {3086 - 3096},
	abstract = {Image restoration methods that exploit prior information about images to be estimated have been extensively studied, typically using the Bayesian framework. In this paper, we consider the role of prior knowledge of the object class in the form of a patch manifold to address the deconvolution problem. Specifically, we incorporate unlabeled image data of the object class, say natural images, in the form of a patch-manifold prior for the object class. The manifold prior is implicitly estimated from the given unlabeled data. We show how the patch-manifold prior effectively exploits the available sample class data for regularizing the deblurring problem. Furthermore, we derive a generalized cross-validation (GCV) function to automatically determine the regularization parameter at each iteration without explicitly knowing the noise variance. Extensive experiments show that this method performs better than many competitive image deconvolution methods.},
	keywords = {Bayesian, cross-validation, data;Bayes, deconvolution;image, determination;deblurring, estimation;, framework;GCV, function;automatic, function;image, image, image;patch-manifold, manifold, method;iteration, method;natural, methods;deconvolution;image, methods;natural, parameter, prior;generalized, prior;unlabeled, problem, regularization, regularization;example-driven, restoration, restoration;iterative, scenes;parameter},
	isbn = {1057-7149},
	doi = {10.1109/TIP.2011.2145386},
	author = {Ni,Jie and Turaga,P. and Patel, Vishal M. and Chellapa, Rama}
}
@article {15823,
	title = {Tikhonov Regularization and Total Least Squares},
	journal = {SIAM Journal on Matrix Analysis and Applications},
	volume = {21},
	year = {1999},
	month = {1999///},
	pages = {185 - 194},
	abstract = {Discretizations of inverse problems lead to systems of linear equations with a highly ill-conditioned coefficient matrix, and in order to computestable solutions to these systems it is necessary to apply regularization methods. We show how Tikhonov{\textquoteright}s regularization method, which in its original formulation involves a least squares problem, can be recast in a total least squares formulation suited for problems in which both the coefficient matrix and the right-hand side are known only approximately. We analyze the regularizing properties of this method and demonstrate by a numerical example that, in certain cases with large perturbations, the new method is superior to standard regularization methods.},
	keywords = {bidiagonalization, discrete ill-posed problems, regularization, total least squares},
	doi = {10.1137/S0895479897326432},
	url = {http://link.aip.org/link/?SML/21/185/1},
	author = {Golub, Gene H. and Hansen,Per Christian and O{\textquoteright}Leary, Dianne P.}
}
@article {15800,
	title = {Restoring Images Degraded by Spatially Variant Blur},
	journal = {SIAM Journal on Scientific Computing},
	volume = {19},
	year = {1998},
	month = {1998///},
	pages = {1063 - 1082},
	abstract = {Restoration of images that have been blurred by the effects of a Gaussian blurring function is an ill-posed but well-studied problem. Any blur that is spatially invariant can be expressed as a convolution kernel in an integral equation. Fast and effective algorithms then exist for determining the original image by preconditioned iterative methods. If the blurring function is spatially variant, however, then the problem is more difficult. In this work we develop fast algorithms for forming the convolution and for recovering the original image when the convolution functions are spatially variant but have a small domain of support. This assumption leads to a discrete problem involving a banded matrix. We devise an effective preconditioner and prove that the preconditioned matrix differs from the identity by a matrix of small rank plus a matrix of small norm. Numerical examples are given, related to the Hubble Space Telescope (HST) Wide-Field/Planetary Camera. The algorithms that we develop are applicable to other ill-posed integral equations as well.},
	keywords = {convolution, discrete ill-posed problems, first-kind integral equations, image restoration, regularization, spatially variant point spread function},
	doi = {10.1137/S106482759528507X},
	url = {http://link.aip.org/link/?SCE/19/1063/1},
	author = {Nagy,James G. and O{\textquoteright}Leary, Dianne P.}
}
@article {15796,
	title = {Regularization by Truncated Total Least Squares},
	journal = {SIAM Journal on Scientific Computing},
	volume = {18},
	year = {1997},
	month = {1997///},
	pages = {1223 - 1241},
	abstract = {The total least squares (TLS) method is a successful method for noise reduction in linear least squares problems in a number of applications. The TLS method is suited to problems in which both the coefficient matrix and the right-hand side are not precisely known. This paper focuses on the use of TLS for solving problems with very ill-conditioned coefficient matrices whose singular values decay gradually (so-called discrete ill-posed problems), where some regularization is necessary to stabilize the computed solution. We filter the solution by truncating the small singular values of the TLS matrix. We express our results in terms of the singular value decomposition (SVD) of the coefficient matrix rather than the augmented matrix. This leads to insight into the filtering properties of the truncated TLS method as compared to regularized least squares solutions. In addition, we propose and test an iterative algorithm based on Lanczos bidiagonalization for computing truncated TLS solutions.},
	keywords = {bidiagonalization, discrete ill-posed problems, regularization, total least squares},
	doi = {10.1137/S1064827594263837},
	url = {http://link.aip.org/link/?SCE/18/1223/1},
	author = {Fierro,R. D. and Golub, G. H and Hansen,P. C. and O{\textquoteright}Leary, Dianne P.}
}
@article {15822,
	title = {The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems},
	journal = {SIAM Journal on Scientific Computing},
	volume = {14},
	year = {1993},
	month = {1993///},
	pages = {1487 - 1503},
	abstract = {Regularization algorithms are often used to produce reasonable solutions to ill-posed problems. The L-curve is a plot{\textemdash}for all valid regularization parameters{\textemdash}of the size of the regularized solution versus the size of the corresponding residual. Two main results are established. First a unifying characterization of various regularization methods is given and it is shown that the measurement of {\textquotedblleft}size{\textquotedblright} is dependent on the particular regularization method chosen. For example, the 2-norm is appropriate for Tikhonov regularization, but a 1-norm in the coordinate system of the singular value decomposition (SVD) is relevant to truncated SVD regularization. Second, a new method is proposed for choosing the regularization parameter based on the L-curve, and it is shown how this method can be implemented efficiently. The method is compared to generalized cross validation and this new method is shown to be more robust in the presence of correlated errors.},
	keywords = {discrepancy principle, generalized cross validation, ill-posed problems, L-curve, parameter choice, regularization},
	doi = {10.1137/0914086},
	url = {http://link.aip.org/link/?SCE/14/1487/1},
	author = {Hansen,Per Christian and O{\textquoteright}Leary, Dianne P.}
}
@article {15696,
	title = {A Bidiagonalization-Regularization Procedure for Large Scale Discretizations of Ill-Posed Problems},
	journal = {SIAM Journal on Scientific and Statistical Computing},
	volume = {2},
	year = {1981},
	month = {1981///},
	pages = {474 - 489},
	abstract = {In this paper, we consider ill-posed problems which discretize to linear least squares problems with matrices $K$ of high dimensions. The algorithm proposed uses $K$ only as an operator and does not need to explicitly store or modify it. A method related to one of Lanczos is used to project the problem onto a subspace for which $K$ is bidiagonal. It is then an easy matter to solve the projected problem by standard regularization techniques. These ideas are illustrated with some integral equations of the first kind with convolution kernels, and sample numerical results are given.},
	keywords = {first kind integral equation, ill-posed problems, Lanczos algorithm, regularization},
	doi = {10.1137/0902037},
	url = {http://link.aip.org/link/?SCE/2/474/1},
	author = {O{\textquoteright}Leary, Dianne P. and Simmons,John A.}
}
