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Iterative Methods for Total Variation Denoising

Published: 01 January 1996 Publication History
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  • Abstract

    Total variation (TV) methods are very effective for recovering “blocky,” possibly discontinuous, images from noisy data. A fixed point algorithm for minimizing a TV penalized least squares functional is presented and compared with existing minimization schemes. A variant of the cell-centered finite difference multigrid method of Ewing and Shen is implemented for solving the (large, sparse) linear subproblems. Numerical results are presented for one- and two-dimensional examples; in particular, the algorithm is applied to actual data obtained from confocal microscopy.

    References

    [1]
    R. Acar, C. R. Vogel, Analysis of bounded variation penalty methods for ill-posed problems, Inverse Problems, 10 (1994), 1217–1229
    [2]
    O. Axelsson, V. A. Barker, Finite element solution of boundary value problems, Computer Science and Applied Mathematics, Academic Press Inc., Orlando, FL, 1984xviii+432
    [3]
    Ingrid Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math., 41 (1988), 909–996
    [4]
    John E. Dennis, Jr., Robert B. Schnabel, Numerical methods for unconstrained optimization and nonlinear equations, Prentice Hall Series in Computational Mathematics, Prentice Hall Inc., Englewood Cliffs, NJ, 1983xiii+378
    [5]
    D. Dobson, F. Santosa, Recovery of Blocky Images from Noisy and Blurred Data, Tech. Report, 94-7, Center for the Mathematics of Waves, University of Delaware, Newark, DE, 1994
    [6]
    David L. Donoho, Iain M. Johnstone, Ideal spatial adaptation by wavelet shrinkage, Biometrika, 81 (1994), 425–455
    [7]
    R. E. Ewing, J. Shen, A Discretization Scheme and Error Estimate for Second-Order Elliptic Problems with Discontinuous Coefficients, Institute for Scientific Computation, Texas A & M University, College Station, preprint
    [8]
    R. E. Ewing, J. Shen, A multigrid algorithm for the cell-centered finite difference scheme, Proc. 6th Copper Mountain Conference on Multigrid Methods, NASA Conference Publication 3224, 1993, April
    [9]
    S. Fucik, A. Kufner, Nonlinear differential equations, Studies in Applied Mechanics, Vol. 2, Elsevier Scientific Publishing Co., Amsterdam, 1980, 359–
    [10]
    Gene H. Golub, Charles F. Van Loan, Matrix computations, Johns Hopkins Series in the Mathematical Sciences, Vol. 3, Johns Hopkins University Press, Baltimore, MD, 1989xxii+642, 2nd ed.
    [11]
    D. Luenberger, Introduction to Linear and Nonlinear Programming, Addison–Wesley, Reading, MA, 1965
    [12]
    S. F. McCormick, Multigrid methods, Frontiers in Applied Mathematics, Vol. 3, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1987xviii+282
    [13]
    Y. Meyer, Wavelets: Algorithms and Applications, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1993xii+133
    [14]
    G. P. Nason, B. W. Silverman, The discrete wavelet transform in S, J. Comput. Graphical Statistics, 3 (1994), 163–191
    [15]
    L. I. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), 259–268
    [16]
    Gilbert Strang, Wavelet transforms versus Fourier transforms, Bull. Amer. Math. Soc. (N.S.), 28 (1993), 288–305
    [17]
    C. R. Vogel, K. Bowers, J. Lund, A multigrid method for total variation-based image denoisingComputation and control, IV (Bozeman, MT, 1994), Progr. Systems Control Theory, Vol. 20, Birkhäuser Boston, Boston, MA, 1995, 323–331
    [18]
    Grace Wahba, Spline models for observational data, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 59, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990xii+169
    [19]
    T. Wilson, Confocal Microscopy, Academic Press, New York, 1990

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    Published In

    cover image SIAM Journal on Scientific Computing
    SIAM Journal on Scientific Computing  Volume 17, Issue 1
    * Special Issue on Iterative Methods in Numerical Linear Algebra
    Jan 1996
    305 pages

    Publisher

    Society for Industrial and Applied Mathematics

    United States

    Publication History

    Published: 01 January 1996

    Author Tags

    1. 65F10
    2. 65N55

    Author Tags

    1. total variation
    2. denoising
    3. image reconstruction
    4. multigrid methods
    5. confocal microscopy
    6. fixed point iteration

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