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Best Kronecker Product Approximation of The Blurring Operator in Three Dimensional Image Restoration Problems

Mansoor Rezghi, S. Mohammad Hosseini, Lars Eldén
2014 SIAM Journal on Matrix Analysis and Applications  
In this paper, we propose a method to find the best Kronecker product approximation of the blurring operator which arises in three dimensional image restoration problems.  ...  This approximation can be used as a preconditioner in solving image restoration problems with iterative methods.  ...  The authors wish to express their gratitude to the referees for their helpful remarks. The first author also thanks Dr. M. Amirmazlaghani for worthy remarks.  ... 
doi:10.1137/130917260 fatcat:bymfktq7kvdahmjmzgm6topwn4

Kronecker product approximations for image restoration with whole-sample symmetric boundary conditions

Xiao-Guang Lv, Ting-Zhu Huang, Zong-Ben Xu, Xi-Le Zhao
2012 Information Sciences  
It assumes that the true infinite scene can be represented as a mosaic of a single finite dimensional image, repeated periodically in all directions.  ...  It extends the pixel values across the boundary in such a way that continuity of the image and of the normal derivative are preserved at the boundary.  ...  Acknowledgments The authors would like to express their great thankfulness to the referees and Prof. Pedrycz for much constructive, detailed and helpful advice regarding revising this manuscript.  ... 
doi:10.1016/j.ins.2011.09.026 fatcat:d6sj6a7gmjh2pnb5vw67kuz7ve

Image restoration with shifting reflective boundary conditions

Jie Huang, TingZhu Huang, XiLe Zhao, ZongBen Xu
2011 Science China Information Sciences  
A Kronecker product approximation of the corresponding blurring matrix is then provided, regardless of symmetry requirement of the PSF.  ...  In this paper, we first propose shifting reflective BCs which preserve the continuity at the boundaries and, therefore, reduce ringing effects in the restored image.  ...  In the following, we illustrate the efficiency of our Kronecker product approximation in an SVD-based regularization algorithm with the proposed shifting reflective BCs for image restoration problems.  ... 
doi:10.1007/s11432-011-4425-2 fatcat:wias4euqlzgkhjttnqelutkxqm

"Plug-and-Play" Edge-Preserving Regularization [article]

Donghui Chen, Misha E. Kilmer, Per Christian Hansen
2014 arXiv   pre-print
In many inverse problems it is essential to use regularization methods that preserve edges in the reconstructions, and many reconstruction models have been developed for this task, such as the Total Variation  ...  We present a simpler approach that relies only on standard computational building blocks in matrix computations, such as orthogonal transformations, preconditioned iterative solvers, Kronecker products  ...  Also, in some of our experiments we assume that the singular vectors are approximated by a Kronecker product, which might be not accurate.  ... 
arXiv:1406.1001v1 fatcat:nitpovxlzff3zg5dc7bp6hnaau

A Framework for Regularization via Operator Approximation

Julianne M. Chung, Misha E. Kilmer, Dianne P. O'Leary
2015 SIAM Journal on Scientific Computing  
We demonstrate the effectiveness of our method in computations using operator approximations such as sums of Kronecker products, block circulant with circulant blocks matrices, and Krylov subspace approximations  ...  In this paper, we present a framework that uses operator approximations to efficiently obtain good regularization parameters without an SVD of the original operator.  ...  Kronecker product approximation. In image deblurring or image deconvolution, the blurring process can be described using a point spread function (PSF) .  ... 
doi:10.1137/130945363 fatcat:mopwfvki5vfyhlqjzagpudnspq

Singular Value Decomposition Approximation via Kronecker Summations for Imaging Applications [article]

Clarissa Garvey, Chang Meng, James G. Nagy
2018 arXiv   pre-print
By first decomposing the matrix into a sum of Kronecker products, our approach can be used to approximate a large number of singular values and vectors more efficiently than other well known schemes, such  ...  We provide theoretical results and numerical experiments to demonstrate the accuracy of our approximation and show how the approximation can be used to solve large scale ill-posed inverse problems, either  ...  For example, in image restoration d is a vector representation of an observed blurred image, x is a vector representation of the corresponding clean image, and K models the blurring operation.  ... 
arXiv:1803.11525v2 fatcat:lvvv4j47sfeq3chmxcx66p2uiq

Mathematics for demosaicking

H.J. Trussell, R.E. Hartwig
2002 IEEE Transactions on Image Processing  
Digital color cameras sample the continuous color spectrum using three or more filters; however, each pixel represents a sample of only one of the color bands. This arrangement is called a mosaic.  ...  To produce a full-resolution color image, the recorded image must be processed to estimate the values of the pixels for all the other color bands.  ...  The problem of restoration of subsampled color images is called demosaicking. 1 This problem is fundamental to the operation of current color digital cameras.  ... 
doi:10.1109/tip.2002.999681 pmid:18244649 fatcat:ww4fddl5brhc5pa4tmehmkvn7a

Where Is Mathematics? [Point of View]

Robert Sinclair
2014 Proceedings of the IEEE  
Digital color cameras sample the continuous color spectrum using three or more filters; however, each pixel represents a sample of only one of the color bands. This arrangement is called a mosaic.  ...  To produce a full-resolution color image, the recorded image must be processed to estimate the values of the pixels for all the other color bands.  ...  The problem of restoration of subsampled color images is called demosaicking. 1 This problem is fundamental to the operation of current color digital cameras.  ... 
doi:10.1109/jproc.2013.2291291 fatcat:hlz4k6goe5avfhcj2t4xa46za4

Multiple-image deblurring with spatially-variant point spread functions

R. Vio, J. Nagy, W. Wamsteker
2005 Astronomy and Astrophysics  
The algorithm was developed in the context of a least-square (LS) approach, to estimate the image corresponding to a given object when a set of observed images are available with different and spatially-invariant  ...  In an appendix we also present a novel, computationally efficient deblurring algorithm that is based on a Singular Value Decomposition (SVD) approximation of the variant PSF, and which is usable with any  ...  However, here we show that the SVD approximation based on a Kronecker product factorization of K can be extended to the case of space variant blurs.  ... 
doi:10.1051/0004-6361:20035754 fatcat:beotgefssrdjlijqqak3zcdpcu

Tikhonov regularization via flexible Arnoldi reduction

Lothar Reichel, Xuebo Yu
2015 BIT Numerical Mathematics  
Moreover, computed examples show that our method can give approximate solutions of higher accuracy than available direct methods for small-scale problems.  ...  Flexible GMRES, introduced by Saad, is a generalization of the standard GMRES method for the solution of large linear systems of equations.  ...  We would like to thank the referees for comments that lead to improvements of the presentation. This research is supported in part by NSF grant DMS-1115385.  ... 
doi:10.1007/s10543-014-0542-9 fatcat:blernibkcncmlloppr3dva7enu

Sparse Representation of a Blur Kernel for Blind Image Restoration [article]

Chia-Chen Lee, Wen-Liang Hwang
2015 arXiv   pre-print
Blind image restoration is a non-convex problem which involves restoration of images from an unknown blur kernel.  ...  As a demonstration, we construct a dictionary formed by basic patterns derived from the Kronecker product of Gaussian sequences.  ...  ACKNOWLEDGMENT The authors would like to thank...  ... 
arXiv:1512.04418v1 fatcat:kagvwlioozbw3czoll7zsoweom

Extensions of the Justen–Ramlau blind deconvolution method

Tristan A. Hearn, Lothar Reichel
2013 Advances in Computational Mathematics  
Blind deconvolution problems arise in many image restoration applications. Most available blind deconvolution methods are iterative.  ...  The method was derived under the assumption of periodic boundary conditions. These boundary conditions may introduce oscillatory artifacts into the computed restoration.  ...  In Section 2, the convolution operation and the effects of the boundary conditions on the solution of the associated inverse problem are discussed in terms of finite-dimensional linear algebra.  ... 
doi:10.1007/s10444-012-9290-z fatcat:eqe4olaborhzvedsqmizsvbm7a

Improved image deblurring with anti-reflective boundary conditions and re-blurring

M Donatelli, C Estatico, A Martinelli, S Serra-Capizzano
2006 Inverse Problems  
In [24] the approach firstly proposed in [5] for the Richardson-Lucy method was applied to least-squares image deconvolution problems and compared with the BCs approach in terms of the quality of the restored  ...  In real applications the quality of the restored images is slightly better than that with AR-BCs.  ...  Acknowledgements Warm thanks to Jim Nagy and to the referees for very pertinent and useful remarks. The work of all the authors was partially supported by MIUR, grant number 2004015437.  ... 
doi:10.1088/0266-5611/22/6/008 fatcat:u4sgs6pm5ffazdejjmbvzvlqdm

A generalized global Arnoldi method for ill-posed matrix equations

A. Bouhamidi, K. Jbilou, L. Reichel, H. Sadok
2012 Journal of Computational and Applied Mathematics  
We consider problems in which the coefficient matrix is the sum of Kronecker products of matrices and present a generalized global Arnoldi method, that respects the structure of the equation, for the solution  ...  of the regularized problem.  ...  Acknowledgments We would like to thank the referees for comments that improved the presentation.  ... 
doi:10.1016/j.cam.2011.09.031 fatcat:kjgqs4phc5cptmjnkvnsjspal4

Convex constrained optimization for large-scale generalized Sylvester equations

A. Bouhamidi, K. Jbilou, M. Raydan
2009 Computational optimization and applications  
Second, we apply the new approach, combined with a Tikhonov regularization term, to restore some blurred and highly noisy images.  ...  First, we apply it to solve the GSE that appear after applying left an right preconditioning schemes to the linear problems associated with the discretization of classical PDE problems.  ...  In the context of image restoration, when the point spread function (PSF) is separable, the blurring matrix H is given as a Kronecker product H = H 2 ⊗ H 1 of two blurring matrices where H 1 and H 2 are  ... 
doi:10.1007/s10589-009-9253-6 fatcat:hamjyh4ryfcrvozorucnasbehy
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