Cascadic Multiresolution Methods for Image Deblurring

This paper investigates the use of cascadic multiresolution methods for image deblurring. Iterations with a conjugate gradient-type method are carried out on each level, and terminated by a stopping rule based on the discrepancy principle. Prolongation is carried out by nonlinear edge-preserving operators, which are defined via PDEs associated with Perona-Malik or total variation-type models. Computed examples demonstrate the effectiveness of the methods proposed.

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

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

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

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

[5]  P. Hansen.  Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion , 1987 .

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

[7]  Fabiana Zama,et al.  Iterative methods for ill-posed problems and semiconvergent sequences , 2006 .

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

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

[10]  Seongjai Kim,et al.  Edge-Forming Methods for Image Zooming , 2006, Journal of Mathematical Imaging and Vision.

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

[12]  Stanley Osher,et al.  Explicit Algorithms for a New Time Dependent Model Based on Level Set Motion for Nonlinear Deblurring and Noise Removal , 2000, SIAM J. Sci. Comput..

[13]  Curtis R. Vogel,et al.  Iterative Methods for Total Variation Denoising , 1996, SIAM J. Sci. Comput..

[14]  Uno Hämarik,et al.  On rules for stopping the conjugate gradient type methods in ill‐posed problems , 2007 .

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

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

[17]  Per Christian Hansen,et al.  REGULARIZATION TOOLS: A Matlab package for analysis and solution of discrete ill-posed problems , 1994, Numerical Algorithms.

[18]  Hassane Sadok,et al.  A new L-curve for ill-posed problems , 2008 .

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

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

[21]  P. Deuflhard,et al.  The cascadic multigrid method for elliptic problems , 1996 .

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

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