Abstract
Regularization may be regarded as diffusion filtering with an implicit time discretization where one single step is used. Thus, iterated regularization with small regularization parameters approximates a diffusion process. The goal of this paper is to analyse relations between noniterated and iterated regularization and diffusion filtering in image processing. In the linear regularization framework, we show that with iterated Tikhonov regularization noise can be better handled than with noniterated. In the nonlinear framework, two filtering strategies are considered: the total variation regularization technique and the diffusion filter technique of Perona and Malik. It is shown that the Perona-Malik equation decreases the total variation during its evolution. While noniterated and iterated total variation regularization is well-posed, one cannot expect to find a minimizing sequence which converges to a minimizer of the corresponding energy functional for the Perona–Malik filter. To overcome this shortcoming, a novel regularization technique of the Perona–Malik process is presented that allows to construct a weakly lower semi-continuous energy functional. In analogy to recently derived results for a well-posed class of regularized Perona–Malik filters, we introduce Lyapunov functionals and convergence results for regularization methods. Experiments on real-world images illustrate that iterated linear regularization performs better than noniterated, while no significant differences between noniterated and iterated total variation regularization have been observed.
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References
R. Acar and C.R. Vogel, “Analysis of bounded variation penalty methods for ill-posed problems,” Inverse Problems, Vol. 10, pp. 1217–1229, 1994.
R.A. Adams, Sobolev Spaces, Academic Press: New York, San Francisco, London, 1975.
G.I. Barenblatt, M. Bertsch, R. Dal Passo, and M. Ughi, “A degenerate pseudoparabolic regularization of a nonlinear forward-backward heat equation arising in the theory of heat and mass exchange in stably stratified turbulent shear flow,” SIAM J. Math. Anal., Vol. 24, pp. 1414–1439, 1993.
M. Bertero, T.A. Poggio, and V. Torre, “Ill-posed problems in early vision,” Proc. IEEE, Vol. 76, pp. 869–889, 1988.
F. Catté, P.L. Lions, J.M. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal., Vol. 32, pp. 1895–1909, 1992.
A. Chambolle and P.L. Lions, “Image recovery via total variation minimization and related problems,” Numer. Math., Vol. 76, pp. 167–188, 1995.
T.F. Chan, G.H. Golub, and P. Mulet, “A nonlinear primal-dual method for total-variation based image restoration,” in ICAOS '96: Images, Wavelets and PDEs, M.O. Berger, R. Deriche, I. Herlin, J. Jaffré, and J.M. Morel (Eds.), Lecture Notes in Control and Information Sciences, Vol. 219, Springer, London, 1996, pp. 241–252.
T.F. Chan and C.K. Wong, “Total variation blind deconvolution,” IEEE Trans. Image Proc., Vol. 7, pp. 370–375, 1998.
P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud, “Two deterministic half-quadratic regularization algorithms for computed imaging,” in Proc. IEEE Int. Conf. Image Processing (ICIP-94, Austin, Nov. 13–16, 1994), IEEE Computer Society Press, Los Alamitos, 1994, Vol. 2, pp. 168–172.
B. Dacorogna, Weak Continuity and Weak Lower Semicontinuity of Non-Linear Functionals, Lecture Notes in Mathematics, Springer, Berlin, 1982.
B. Dacorogna, Direct Methods in the Calculus of Variations, Springer: Berlin, 1989.
D. Dobson and O. Scherzer, “Analysis of regularized total variation penalty methods for denoising,” Inverse Problems, Vol. 12, pp. 601–617, 1996.
D.C. Dobson and C.R. Vogel, “Convergence of an iterative method for total variation denoising,” SIAM J. Numer. Anal., Vol. 34, pp. 1779–1791, 1997.
H.W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer, Dordrecht, 1996.
L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press: Boca Raton, 1992.
L. Florack, Image Structure, Kluwer: Dordrecht, 1997.
D. Geman and C. Yang, “Nonlinear image recovery with half-quadratic regularization,” IEEE Transactions on Image Processing, Vol. 4, pp. 932–945, 1995.
P.J. Green, “Bayesian reconstructions from emission tomography data using a modified EM algorithm,” IEEE Trans. Medical Imaging, Vol. 9, pp. 84–93, 1990.
C.W. Groetsch, The Theory of Tikhonov regularization for Fredholm Equations of the First Kind, Pitman: Boston, 1984.
C.W. Groetsch, “Spectral methods for linear inverse problems with unbounded operators,” J. Approx. Th., Vol. 70, pp. 16–28, 1992.
C.W. Groetsch and O. Scherzer, “Optimal order of convergence for stable evaluation of differential operators,” Electronic Journal of Differential Equations (http://ejde.math. unt.edu), No. 4, pp. 1–10, 1993.
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order. Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer, Berlin, 2nd ed., 1994.
E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser: Basel, 1984.
B.M. ter Haar Romeny (Ed.), Geometry-Driven Diffusion in Computer Vision, Kluwer: Dordrecht, 1994.
M. Hanke and C.W. Groetsch, “Nonstationary iterated Tikhonov regularization,” J. Optim. Theory and Applications, Vol. 98, pp. 37–53, 1998.
G. Helmberg, Introduction to Spectral Theory in Hilbert Space, Amsterdam, London, 1969.
K. Ito, and K. Kunisch, “An active set strategy based on the augmented Lagrangian formulation for image restoration,” RAIRO Math. Mod. and Num. Analysis, Vol. 33, pp. 1–21, 1999.
Y. Li and F. Santosa, “A computational algorithm for minimizing total variation in image enhancement,” IEEE Trans. Image Proc., Vol. 5, pp. 987–995, 1996.
T. Lindeberg, Scale-Space Theory in Computer Vision, Kluwer: Boston, 1994.
J.M. Morel and S. Solimini, Variational Methods in Image Segmentation, Birkhäuser: Boston, 1995.
V.A. Morozov, Methods for Solving Incorrectly Posed Problems, Springer: New York, 1984.
M.Z. Nashed and O. Scherzer, “Least squares and bounded variation regularization with nondifferentiable functionals,” Numer. Funct. Anal. and Optimiz., Vol. 19, pp. 873–901, 1998.
M. Nielsen, L. Florack, and R. Deriche, “Regularization, scale-space and edge detection filters,” J. Math. Imag. Vision, Vol. 7, pp. 291–307, 1997.
M. Nitzberg and T. Shiota, “Nonlinear image filtering with edge and corner enhancement,” IEEE Trans. Pattern Anal. Mach. Intell., Vol. 14, pp. 826–833, 1992.
N. Nordström, “Biased anisotropic diffusion-a unified regularization and diffusion approach to edge detection,” Image and Vision Computing, Vol. 8, pp. 318–327, 1990.
S. Osher and L.I. Rudin, “Feature-oriented image enhencement using shock filters,” SIAM J. Numer. Anal., Vol. 27, pp. 919–940, 1990.
P. Perona and J. Malik, “Scale space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Mach. Intell., Vol. 12, pp. 629–639, 1990.
E. Radmoser, O. Scherzer, and J. Weickert, “Scale-space properties of regularization methods,” in M. Nielsen, P. Johansen, O.F. Olsen, and J. Weickert (Eds.), Scale-Space Theories in Computer Vision, Lecture Notes in Computer Science, Springer, Berlin, Vol. 1682, pp. 211–222, 1999.
F. Rieszm and B. Sz-Nagy, Functional Analysis, Ungar, New York, 1955.
L.I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D, Vol. 60, pp. 259–268, 1992.
O. Scherzer, “Stable evaluation of differential operators and linear and nonlinear multi-scale filtering,” Electronic Journal of Differential Equations (http://ejde.math.unt.edu), No. 15, pp. 1–12, 1997.
O. Scherzer, “Denoising with higher order derivatives of bounded variation and an application to parameter estimation,” Computing, Vol. 60, pp. 1–27, 1998.
C. Schnörr, “Unique reconstruction of piecewise smooth images by minimizing strictly convex non-quadratic functionals,” J. Math. Imag. Vision, Vol. 4, pp. 189–198, 1994.
J. Sporring, M. Nielsen, L. Florack, and P. Johansen (Eds.), Gaussian Scale-Space Theory, Kluwer: Dordrecht, 1997.
D.M. Strong and T.F. Chan, “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.
A.N. Tikhonov and V.Y. Arsenin, Solutions of Ill-Posed Problems, John Wiley & Sons, Washington, D.C., 1977, Translation editor Fritz John (English).
V. Torre and T.A. Poggio, “On edge detection,” IEEE Trans. Pattern Anal. Mach. Intell., Vol. 8, pp. 148–163, 1986.
C.R. Vogel and M.E. Oman, “Fast, robust total variation-based reconstruction of noisy, blurred images,” IEEE Trans. Image Proc., Vol. 7, pp. 813–824, 1998.
J. Weickert, “Anisotropic diffusion filters for image processing based quality control,” in Proc. Seventh European Conf. on Mathematics in Industry, A. Fasano and M. Primicerio (Eds.), Teubner, Stuttgart, 1994, pp. 355–362.
J. Weickert, “A review of nonlinear diffusion filtering,” in Scale-Space Theory in Computer Vision, B. ter Haar Romeny, L. Florack, J. Koenderink and M. Viergever (Eds.), Lecture Notes in Computer Science, Vol. 1252, Springer, Berlin, 1997, pp. 3–28.
J. Weickert, Anisotropic Diffusion in Image Processing, Teubner: Stuttgart, 1998.
J. Weickert and B. Benhamouda, “A semidiscrete nonlinear scale-space theory and its relation to the Perona-Malik paradox,” in Advances in Computer Vision, F. Solina, W.G. Kropatsch, R. Klette, and R. Bajcsy (Eds.), Springer, Wien, 1997, pp. 1–10.
J. Weidmann, Lineare Operatoren in Hilberträumen, Teubner: Stuttgart, 1976.
J. Wloka, Partielle Differentialgleichungen, Teubner: Stuttgart, 1982.
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Scherzer, O., Weickert, J. Relations Between Regularization and Diffusion Filtering. Journal of Mathematical Imaging and Vision 12, 43–63 (2000). https://doi.org/10.1023/A:1008344608808
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DOI: https://doi.org/10.1023/A:1008344608808