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A Multigrid Method for Total Variation-Based Image Denoising

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Computation and Control IV

Part of the book series: Progress in Systems and Control Theory ((PSCT,volume 20))

Abstract

Byimage reconstruction we mean obtaining the solution u of an operator equation of the form

$$Au = z,$$
((1.1))

where A is typically (but not necessarily) linear, and the dataz is assumed to be contaminated by error. Two important special cases are: (i) Denoising, or noise removal, whereAu =u; and (ii)Deblurring, whereA is a Fredholm first kind integral operator

$$(Au)(x) = \int_\Omega k (x,y)u(y)dy,x \in \Omega $$
((1.2))

which models “blurring” processes involved in image formation. See [9] for details.

Supported in part by NSF under Grant DMS-9303222

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References

  1. R. Acar and C. R. Vogel,Analysis of Total Variation penalty methods for ill-posed problems, Inverse Problems, to appear.

    Google Scholar 

  2. O. Axelsson and V.A. Barker, Finite Element Solution of Boundary Value Problems, Academic Press, 1984.

    Google Scholar 

  3. J. H. Bramble, J. E. Pasciak, and J. Xu,The analysis of multigrid algorithms with non-nested spaces or non-inherited quadratic forms, Math. Comp., vol. 56 (1991), pp. 1 – 34.

    Article  MathSciNet  MATH  Google Scholar 

  4. J. H. Bramble, R. E. Ewing, J. E. Pasciak, and J. Shen,The analysis of multigrid algorithms for cell centered finite difference methods, preprint, Institute for Scientific Computation, Texas A & M University (1994).

    Google Scholar 

  5. D. Dobson and S. Santosa,Recovery of blocky images from noisy and blurred data, Tech. Report No. 94–7, Center for the Mathematics of Waves, University of Delaware (1994).

    Google Scholar 

  6. R. E. Ewing and J. Shen,A discretization scheme and error estimate for second-order elliptic problems with discontinuous coefficients, preprint, Institute for Scientific Computation, Texas A & M University.

    Google Scholar 

  7. R. E. Ewing and J. Shen,A multigrid algorithm for the cell-centered finite difference scheme, in the Proceeding of the 6th Copper Mountain Conference on Multigrid Methods, April 1993.

    Google Scholar 

  8. K. Ito and K. Kunisch, An active set strategy for image restoration based on the augmented Lagrangian formulation, preprint, Center for Research in Scientific Computing, North Carolina State University (1994).

    Google Scholar 

  9. A. K. Jain,Fundamentals of Digital Image Processing, Prentice-Hall, 1988.

    Google Scholar 

  10. Y. Li and F. Santosa,An affine scaling algorithm for minimizing Total Variation in image enhancement, preprint (1994).

    Google Scholar 

  11. S. F. McCormick, ed.,Multigrid Methods, SIAM, 1987.

    MATH  Google Scholar 

  12. M. E. Oman,Fast multigrid techniques in Total Variation-based image reconstruction, preprint, Montana State University (1994).

    Google Scholar 

  13. M. E. Oman and C. R. Vogel,Iterative methods for Total Variation denoisingPreprint, Montana State University (1994).

    Google Scholar 

  14. L. I. Rudin, S. Osher, and E. Fatemi,Nonlinear Total Variation Based Noise Removal Algorithms, Physica D, vol 60 (1992), pp. 259 – 268.

    Article  MATH  Google Scholar 

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© 1995 Birkhäuser Boston

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Vogel, C.R. (1995). A Multigrid Method for Total Variation-Based Image Denoising. In: Computation and Control IV. Progress in Systems and Control Theory, vol 20. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2574-4_21

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  • DOI: https://doi.org/10.1007/978-1-4612-2574-4_21

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-1-4612-7586-2

  • Online ISBN: 978-1-4612-2574-4

  • eBook Packages: Springer Book Archive

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