Conditional gradient Tikhonov method for a convex optimization problem in image restoration

In this paper, we consider the problem of image restoration with Tikhonov regularization as a convex constrained minimization problem. Using a Kronecker decomposition of the blurring matrix and the Tikhonov regularization matrix, we reduce the size of the image restoration problem. Therefore, we apply the conditional gradient method combined with the Tikhonov regularization technique and derive a new method. We demonstrate the convergence of this method and perform some numerical examples to illustrate the effectiveness of the proposed method as compared to other existing methods.

[1]  K. Siddaraju,et al.  DIGITAL IMAGE RESTORATION , 2011 .

[2]  Michael K. Ng,et al.  Kronecker Product Approximations forImage Restoration with Reflexive Boundary Conditions , 2003, SIAM J. Matrix Anal. Appl..

[3]  G. Golub,et al.  Estimation of the L-Curve via Lanczos Bidiagonalization , 1999 .

[4]  D. Kumar.  OPTIMIZATION METHODS , 2007 .

[5]  Leiba Rodman,et al.  Algebraic Riccati equations , 1995 .

[6]  Lothar Reichel,et al.  TIKHONOV REGULARIZATION WITH NONNEGATIVITY CONSTRAINT , 2004 .

[7]  Philip Wolfe,et al.  An algorithm for quadratic programming , 1956 .

[8]  Florin Popentiu,et al.  Iterative identification and restoration of images , 1993, Comput. Graph..

[9]  K. Schittkowski,et al.  NONLINEAR PROGRAMMING , 2022 .

[10]  Per Christian Hansen,et al.  Regularization methods for large-scale problems , 1993 .

[11]  J. Nagy,et al.  KRONECKER PRODUCT AND SVD APPROXIMATIONS IN IMAGE RESTORATION , 1998 .

[12]  K. Jbilou,et al.  Sylvester Tikhonov-regularization methods in image restoration , 2007 .

[13]  Gene H. Golub,et al.  Generalized cross-validation as a method for choosing a good ridge parameter , 1979, Milestones in Matrix Computation.

[14]  Abderrahman Bouhamidi,et al.  Convex constrained optimization for large-scale generalized Sylvester equations , 2011, Comput. Optim. Appl..

[15]  Gene H. Golub,et al.  Linear algebra for large scale and real-time applications , 1993 .

[16]  Dianne P. O'Leary,et al.  The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems , 1993, SIAM J. Sci. Comput..

[17]  Serena Morigi,et al.  An interior-point method for large constrained discrete ill-posed problems , 2010, J. Comput. Appl. Math..

[18]  Per Christian Hansen,et al.  Analysis of Discrete Ill-Posed Problems by Means of the L-Curve , 1992, SIAM Rev..

[19]  J. Nagy,et al.  Enforcing nonnegativity in image reconstruction algorithms , 2000, SPIE Optics + Photonics.

[20]  Reginald L. Lagendijk,et al.  Iterative Identification and Restoration of Images (The International Series in Engineering and Computer Science) , 2001 .

[21]  C. Loan,et al.  Approximation with Kronecker Products , 1992 .

[22]  D. Calvetti,et al.  Tikhonov regularization and the L-curve for large discrete ill-posed problems , 2000 .

[23]  G. Wahba.  Practical Approximate Solutions to Linear Operator Equations When the Data are Noisy , 1977 .

[24]  Anil K. Jain.  Fundamentals of Digital Image Processing , 2018, Control of Color Imaging Systems.

[25]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[26]  H. Tolle,et al.  Optimization Methods , 1975 .

[27]  Raymond H. Chan,et al.  Journal of Computational and Applied Mathematics a Reduced Newton Method for Constrained Linear Least-squares Problems , 2022 .

[28]  Trond Steihaug,et al.  An interior-point trust-region-based method for large-scale non-negative regularization , 2002 .

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