Image restoration subject to a total variation constraint

Total variation has proven to be a valuable concept in connection with the recovery of images featuring piecewise smooth components. So far, however, it has been used exclusively as an objective to be minimized under constraints. In this paper, we propose an alternative formulation in which total variation is used as a constraint in a general convex programming framework. This approach places no limitation on the incorporation of additional constraints in the restoration process and the resulting optimization problem can be solved efficiently via block-iterative methods. Image denoising and deconvolution applications are demonstrated.

[1]  Ilya Pollak.  Nonlinear multiscale filtering , 2002 .

[2]  M. Nikolova.  An Algorithm for Total Variation Minimization and Applications , 2004 .

[3]  Alexander M. Bronstein,et al.  Reconstruction in diffraction ultrasound tomography using nonuniform FFT , 2002, IEEE Transactions on Medical Imaging.

[4]  P. L. Combettes,et al.  The Convex Feasibility Problem in Image Recovery , 1996 .

[5]  L. Rudin,et al.  Feature-oriented image enhancement using shock filters , 1990 .

[6]  Patrick L. Combettes,et al.  An adaptive level set method for nondifferentiable constrained image recovery , 2002, IEEE Trans. Image Process..

[7]  D. Youla,et al.  Image Restoration by the Method of Convex Projections: Part 1ߞTheory , 1982, IEEE Transactions on Medical Imaging.

[8]  Stanley Osher,et al.  Total variation based image restoration with free local constraints , 1994, Proceedings of 1st International Conference on Image Processing.

[9]  Curtis R. Vogel,et al.  Ieee Transactions on Image Processing Fast, Robust Total Variation{based Reconstruction of Noisy, Blurred Images , 2022 .

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

[11]  Yves Meyer,et al.  Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures , 2001 .

[12]  Heinz H. Bauschke,et al.  On Projection Algorithms for Solving Convex Feasibility Problems , 1996, SIAM Rev..

[13]  W. Ring.  Structural Properties of Solutions to Total Variation Regularization Problems , 2000 .

[14]  Tony F. Chan,et al.  High-Order Total Variation-Based Image Restoration , 2000, SIAM J. Sci. Comput..

[15]  D. B. Graham,et al.  Characterising Virtual Eigensignatures for General Purpose Face Recognition , 1998 .

[16]  B. R. Hunt,et al.  The Application of Constrained Least Squares Estimation to Image Restoration by Digital Computer , 1973, IEEE Transactions on Computers.

[17]  Stanley Osher,et al.  Multiplicative Denoising and Deblurring: Theory and Algorithms , 2003 .

[18]  Guy Demoment,et al.  Image reconstruction and restoration: overview of common estimation structures and problems , 1989, IEEE Trans. Acoust. Speech Signal Process..

[19]  E. Giusti.  Minimal surfaces and functions of bounded variation , 1977 .

[20]  Mila Nikolova,et al.  Local Strong Homogeneity of a Regularized Estimator , 2000, SIAM J. Appl. Math..

[21]  François Malgouyres,et al.  Minimizing the total variation under a general convex constraint for image restoration , 2002, IEEE Trans. Image Process..

[22]  Patrick L. Combettes,et al.  Strong Convergence of Block-Iterative Outer Approximation Methods for Convex Optimization , 2000, SIAM J. Control. Optim..

[23]  D. B. Gerham.  Characterizing virtual eigensignatures for general purpose face recognition , 1998 .

[24]  Aria Abubakar,et al.  Total variation as a multiplicative constraint for solving inverse problems , 2001, IEEE Trans. Image Process..

[25]  R. Temam,et al.  Analyse convexe et problèmes variationnels , 1974 .

[26]  Stephen L. Keeling,et al.  Total variation based convex filters for medical imaging , 2003, Appl. Math. Comput..

[27]  Yann Gousseau,et al.  Are Natural Images of Bounded Variation? , 2001, SIAM J. Math. Anal..

[28]  P. Lions,et al.  Image recovery via total variation minimization and related problems , 1997 .

[29]  Patrick L. Combettes,et al.  The use of noise properties in set theoretic estimation , 1991, IEEE Trans. Signal Process..

[30]  Patrick L. Combettes,et al.  A block-iterative surrogate constraint splitting method for quadratic signal recovery , 2003, IEEE Trans. Signal Process..

[31]  Patrick L. Combettes,et al.  Convex set theoretic image recovery by extrapolated iterations of parallel subgradient projections , 1997, IEEE Trans. Image Process..

[32]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[33]  Dudley,et al.  Real Analysis and Probability: Measurability: Borel Isomorphism and Analytic Sets , 2002 .

[34]  H. Trussell,et al.  The feasible solution in signal restoration , 1984 .

[35]  Joachim Weickert,et al.  Anisotropic diffusion in image processing , 1996 .

[36]  Dominikus Noll,et al.  Adaptive Image Reconstruction Using Information Measures , 2000, SIAM J. Control. Optim..

[37]  B. Frieden,et al.  Image recovery: Theory and application , 1987, IEEE Journal of Quantum Electronics.

[38]  Michel Barlaud,et al.  Variational approach for edge-preserving regularization using coupled PDEs , 1998, IEEE Trans. Image Process..