Cascadic multilevel methods for fast nonsymmetric blur- and noise-removal

Image deblurring is a discrete ill-posed problem. This paper discusses cascadic multilevel methods designed for the restoration of images that have been contaminated by nonsymmetric blur and noise. Prolongation is carried out by nonlinear edge-preserving and noise-reducing operators, while restrictions are determined by weighted local least-squares approximation. The restoration problem is on each level solved by an iterative method, with the number of iterations determined by the discrepancy principle. The performance of several iterative methods is compared. Computed examples demonstrate the effectiveness of the image restoration methods proposed. The discrepancy principle requires that an estimate of the norm of the noise in the contaminated image be available. We illustrate how such an estimate can be computed with the aid of the nonlinear Perona-Malik diffusion equation.

[1]  Fiorella Sgallari,et al.  Semi-implicit complementary volume scheme for solving level set like equations in image processing and curve evolution , 2003, Numerische Mathematik.

[2]  Raymond H. Chan,et al.  A Fast Algorithm for Deblurring Models with Neumann Boundary Conditions , 1999, SIAM J. Sci. Comput..

[3]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[4]  A. Nemirovskii.  The regularizing properties of the adjoint gradient method in ill-posed problems , 1987 .

[5]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[6]  Dianne P. O'Leary,et al.  Deblurring Images: Matrices, Spectra, and Filtering (Fundamentals of Algorithms 3) (Fundamentals of Algorithms) , 2006 .

[7]  P. Hansen,et al.  Subspace Preconditioned LSQR for Discrete Ill-Posed Problems , 2003 .

[8]  Thomas Huckle,et al.  Multigrid Preconditioning and Toeplitz Matrices , 2002 .

[9]  Thomas Brox,et al.  PDE-Based Deconvolution with Forward-Backward Diffusivities and Diffusion Tensors , 2005, Scale-Space.

[10]  Michael A. Saunders,et al.  LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares , 1982, TOMS.

[11]  Serena Morigi,et al.  Cascadic Multiresolution Methods for Image Deblurring , 2008, SIAM J. Imaging Sci..

[12]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[13]  Lothar Reichel,et al.  Cascadic multilevel methods for ill-posed problems , 2010, J. Comput. Appl. Math..

[14]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[15]  Dianne P. O'Leary,et al.  Deblurring Images: Matrices, Spectra and Filtering , 2006, J. Electronic Imaging.

[16]  Tony F. Chan,et al.  Image processing and analysis , 2005 .

[17]  Lothar Reichel,et al.  On the regularizing properties of the GMRES method , 2002, Numerische Mathematik.

[18]  Jean-Michel Morel,et al.  The staircasing effect in neighborhood filters and its solution , 2006, IEEE Transactions on Image Processing.

[19]  Jitendra Malik,et al.  Scale-Space and Edge Detection Using Anisotropic Diffusion , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[20]  Lothar Reichel,et al.  Arnoldi-Tikhonov regularization methods , 2009 .

[21]  Jianping Zhu,et al.  Multilevel grid method for history matching multi-dimensional multi-phase reservoir models , 1992 .

[22]  Lothar Reichel,et al.  Restoration of images with spatially variant blur by the GMRES method , 2000, SPIE Optics + Photonics.

[23]  P. Favati,et al.  A framework for studying the regularizing properties of Krylov subspace methods , 2006 .

[24]  Mårten Gulliksson,et al.  Modifying the QR-Decomposition to Constrained and Weighted Linear Least Squares , 1992, SIAM J. Matrix Anal. Appl..

[25]  P. Hansen,et al.  Iterative regularization with minimum-residual methods , 2007 .

[26]  Stefano Serra Capizzano,et al.  On the Regularizing Power of Multigrid-type Algorithms , 2005, SIAM J. Sci. Comput..

[27]  Marco Donatelli,et al.  FILTER FACTOR ANALYSIS OF AN ITERATIVE MULTILEVEL REGULARIZING METHOD , 2007 .

[28]  M. Hanke.  Conjugate gradient type methods for ill-posed problems , 1995 .

[29]  Lothar Reichel,et al.  GMRES, L-Curves, and Discrete Ill-Posed Problems , 2002 .

[30]  David G. Vass,et al.  Proceedings of the Society of Photo-optical Instrumentation Engineers (SPIE) , 1997 .

[31]  Tony F. Chan,et al.  Image processing and analysis - variational, PDE, wavelet, and stochastic methods , 2005 .

[32]  Richard S. Varga,et al.  Matrix Iterative Analysis , 2000, The Mathematical Gazette.

[33]  J. Navarro-Pedreño.  Numerical Methods for Least Squares Problems , 1996 .

[34]  Max A. Viergever,et al.  Efficient and reliable schemes for nonlinear diffusion filtering , 1998, IEEE Trans. Image Process..

[35]  Lothar Reichel,et al.  On the Choice of Subspace for Iterative Methods for Linear Discrete Ill-Posed Problems , 2001 .

[36]  Qiang Ye,et al.  Breakdown-free GMRES for Singular Systems , 2005, SIAM J. Matrix Anal. Appl..