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Scale-Space Properties of Regularization Methods

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Scale-Space Theories in Computer Vision (Scale-Space 1999)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1682))

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Abstract

We show that regularization methods can be regarded as scale-spaces where the regularization parameter serves as scale. In analogy to nonlinear diffusion filtering we establish continuity with respect to scale, causality in terms of a maximum/minimum principle, simplifica- tion properties by means of Lyapunov functionals and convergence to a constant steady-state. We identify nonlinear regularization with a single implicit time step of a diffusion process. This implies that iterated regu- larization with small regularization parameters is a numerical realization of a diffusion filter. Numerical experiments in two and three space dimen- sions illustrate the scale-space behaviour of regularization methods.

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References

  1. Acar, R.; Vogel, C.R.: Analysis of bounded variation penalty methods for ill-posed problems, Inverse Problems, 10, 1217–1229, 1994

    MATH  MathSciNet  Google Scholar 

  2. Alvarez, L.; Guichard, F.; Lions, P.L.; Morel, J.M., Axioms and fundamental equations of image processing, Arch. Rat. Mech. Anal., 16, 200–257, 1993

    MathSciNet  Google Scholar 

  3. Chambolle, A.; Lions P.-L.: Image recovery via total variation minimization and related problems, Numer. Math., 76, 167–188, 1995

    MathSciNet  Google Scholar 

  4. Chan, T.F.; Golub, G.H.; Mulet, P.: A nonlinear primal-dual method for total-variation based image restoration, in Berger, M.O.; Deriche, R.; Herlin, I.; Jaffré, J.; Morel, J.M. (Eds.): ICAOS’ 96: Images, wavelets and PDEs, Lecture Notes in Control and Information Sciences, 219, Springer, London, 241–252, 1996

    Google Scholar 

  5. Charbonnier, P.; Blanc-féraud, L.; Aubert, G.; Barlaud, M.: Two deterministic half-quadratic regularization algorithms for computed imaging, Proc. IEEE Int. Conf. Image Processing (Austin, Nov. 13–16, 1994), Vol. 2, 168–172, 1994

    Google Scholar 

  6. D_hlen, M.; Tveito, A.: Numerical Methods and Software Tools in Industrial Mathematics, Birkhäuser, Boston, 1997

    Google Scholar 

  7. Engl, H.W.; Hanke, M.; Neubauer, A.: Regularization of Inverse Problems, Kluwer, Dordrecht, 1996

    MATH  Google Scholar 

  8. Florack, L.: Image Structure, Kluwer, Dordrecht, 1997

    Google Scholar 

  9. Geman, D.; Yang, C.: Nonlinear image recovery with half-quadratic regularization, IEEE Transactions on Image Processing, 4, 932–945, 1995

    Google Scholar 

  10. Ter Haar Romeny, B.M. (Ed.): Geometry-Driven Diffusion in Computer Vision, Kluwer, Dordrecht, 1994

    MATH  Google Scholar 

  11. Hummel, R.A.: Representations based on zero crossings in scale space, Proc. IEEE Comp. Soc. Conf. Computer Vision and Pattern Recognition (Miami Beach, June 22–26, 1986), 204–209, 1986.

    Google Scholar 

  12. Morel, J.F.; Solimini S.: Variational Methods in Image Segmentation; Birkhäuser, Boston, 1995

    Google Scholar 

  13. Ito, K.; Kunisch, K.: An active set strategy for image restoration based on the augmented Lagrangian formulation, Preprint No. 410/1994, Fachbereich Mathematik (3), Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany, 1994, submitted to SIAM J. Optimization

    Google Scholar 

  14. Lindeberg, T.: Scale-Space Theory in Computer Vision, Kluwer, Boston, 1994

    Google Scholar 

  15. Nielsen, M.; Florack, L.; Deriche, R.: Regularization, scale-space and edge detection filters, J. Math. Imag. Vision, 7, 291–307, 1997

    MathSciNet  Google Scholar 

  16. Niessen, W.J., Vincken, K.L., Weickert, J., Viergever, M.A.: Nonlinear multi-scale representations for image segmentation, Computer Vision and Image Understanding, 66, 233–245, 1997

    Google Scholar 

  17. Nashed, M.Z.; Scherzer O.: Least squares and bounded variation regularization with nondifferentiable functionals, Numer. Funct. Anal. and Optimiz., 19, 873–901, 1998

    MATH  MathSciNet  Google Scholar 

  18. Nordström, N.: Biased anisotropic diffusion a unified regularization and diffusion approach to edge detection, Image and Vision Computing, 8, 318–327, 1990

    Google Scholar 

  19. Perona, P.; Malik, J.: Scale space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12, 629–639, 1990

    Google Scholar 

  20. Radmoser, E.; Scherzer, O.; Weickert, J.: Lyapunov functionals for nonstationary regularization, in preparation, 1999.

    Google Scholar 

  21. Rudin, L.I.; Osher, S.; Fatemi, E.: Nonlinear total variation based noise removal algorithms, Physica D, 60, 259–268, 1992

    MATH  Google Scholar 

  22. Scherzer, O.: Stable evaluation of differential operators and linear and nonlinear multi-scale filtering, Electronic Journal of Differential Equations (http://ejde.math.unt.edu), No. 15, 1–12, 1997

  23. Scherzer, O.; Weickert, J.: Relations between regularization and diffusion filtering, J. Math. Imag. Vision, to appear.

    Google Scholar 

  24. Schnörr, C.: Unique reconstruction of piecewise smooth images by minimizing strictly convex non-quadratic functionals, J. Math. Imag. Vision, 4, 189–198, 1994

    Google Scholar 

  25. Sporring, J.; Nielsen, M.; Florack, L.; Johansen, P. (Eds.): Gaussian Scale-Space Theory, Kluwer, Dordrecht, 1997

    MATH  Google Scholar 

  26. Sporring, J.; Weickert, J.: Information measures in scale-spaces, IEEE Trans. Information Theory, 45, 1051–1058, 1999

    MATH  MathSciNet  Google Scholar 

  27. Strong, D.M.; Chan, T.F.: Relation of regularization parameter and scale in total variation based image denoising, CAM Report 96-7, Dept. of Mathematics, Univ. of California, Los Angeles, CA 90024, U.S.A., 1996

    Google Scholar 

  28. Torre, V.; Poggio, T.A.: On edge detection, IEEE Trans. Pattern Anal. Mach. Intell., 8, 148–163, 1986

    Google Scholar 

  29. Weickert, J.: Anisotropic Diffusion in Image Processing, Teubner, Stuttgart, 1998

    MATH  Google Scholar 

  30. Weickert, J.; Heers, J.; Schnörr, C.; Zuiderveld, K.J.; Scherzer, O.; Stiehl, H.S.: Fast parallel algorithms for a broad class of nonlinear variational diffusion approaches, Report 5/99, Computer Science Series, Dept. of Mathematics and Computer Science, University of Mannheim, 68131 Mannheim, Germany, 1999, submitted.

    Google Scholar 

  31. Weickert, J.; Zuiderveld, K.J.; Ter haar romeny, B.M.; Niessen, W.J.: Parallel implementations of AOS schemes: A fast way of nonlinear diffusion filtering, Proc. 1997 IEEE International Conference on Image Processing (Santa Barbara, Oct. 26–29, 1997), Vol. 3, 396–399, 1997.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Industrial Mathematics Institute, University of Linz, Altenberger Str. 69, A-4040, Linz, Austria

    Esther Radmoser & Otmar Scherzer

  2. Computer Vision, Graphics and Pattern Recognition Group, Department of Mathematics and Computer Science, University of Mannheim, D-68131, Mannheim, Germany

    Joachim Weickert

Authors
  1. Esther Radmoser
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  2. Otmar Scherzer
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  3. Joachim Weickert
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Editor information

Editors and Affiliations

  1. The IT University in Copenhagen, Glentevej 67, DK-2400, Copenhagen NV, Denmark

    Mads Nielsen

  2. Department of Computer Science, University of Copenhagen, Universitetsparken 1, DK-2100, Copenhagen OE, Denmark

    Peter Johansen  & Ole Fogh Olsen  & 

  3. Department of Mathematics and Computer Science, University of Mannheim, D-68131, Mannheim, Germany

    Joachim Weickert

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© 1999 Springer-Verlag Berlin Heidelberg

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Radmoser, E., Scherzer, O., Weickert, J. (1999). Scale-Space Properties of Regularization Methods. In: Nielsen, M., Johansen, P., Olsen, O.F., Weickert, J. (eds) Scale-Space Theories in Computer Vision. Scale-Space 1999. Lecture Notes in Computer Science, vol 1682. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48236-9_19

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  • DOI: https://doi.org/10.1007/3-540-48236-9_19

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-66498-7

  • Online ISBN: 978-3-540-48236-9

  • eBook Packages: Springer Book Archive

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