Abstract
Error minimization of global functionals provides a natural setting for analyzing image processing and regularization. This approach leads to scale spaces, which in the continuous formulation are the solution of nonlinear partial differential equations. In this work we derive properties for a class of inverse scale space methods. The main contribution of this paper is the development of a proof that the methods considered are convergent for convex regularization operators. The proof is based on energy methods and the Bregman distance. Further, estimates for convergence toward a clean image with noisy forcing data is provided in terms of both the L 2 norm and Bregman distances. This leads to natural estimates of optimal stopping scale for the inverse scale space method. These analytical results are discussed in the context of a numerical example.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. Attouch, G. Buttazzo, and G. Michaille, Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization, vol. 6 of MPS-SIAM Series on Optimization. SIAM, 2006.
G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, vol. 147 of Applied Mathematical Sciences. Springer-Verlag, 2002.
L.M. Bregman, “Relaxation method for finding common points of convex sets and its application in optimization,” Reports of the USSR Akademy of Sciences, Vol. 171, No. 5, pp. 1019–1022, 1966 (in Russian).
M. Burger, G. Gilboa, S. Osher, and J. Xu, “Nonlinear inverse scale space methods,” Communications in Mathematical Sciences, Vol. 4, No. 1, pp. 179–212, 2006.
M. Burger, S. Osher, J. Xu, and G. Gilboa, “Nonlinear inverse scale space methods for image restoration.” Lecture Notes in Computer Science, Vol. 3752, pp. 25–36, 2005.
T.F. Chan and S. Esedoglu, “Aspects of total variation regularized L 1 function approximation,” SIAM Journal of Applied Mathematics, Vol. 65, No. 5, pp. 1817–1837, 2005.
T.F. Chan and J. Shen, Image Processing And Analysis: Variational, PDE, Wavelet, And Stochastic Methods, SIAM, 2005.
G. Chen and M. Teboulle, “Convergence analysis of a proximal-like minimization algorithm using Bregman functions,” SIAM J. Optimization, Vol. 3, No. 3, pp. 538–543, 1993.
L.C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics. AMS, 1998.
G. Gilboa, N. Sochen, and Y.Y. Zeevi, “Estimation of optimal PDE-based denoising in the SNR sense,” UCLA Compuational and Applied Mathematics Report, 05-48, 2005.
M. Hanke and P.C. Hansen, “Regularization methods for large-scale problems,” Surveys Math. Indust., Vol. 3, No. 4, pp. 253–315, 1993.
P.C. Hansen, “Deconvolution and regularization with toepliz matrices,” Numerical Algorithms, Vol. 29, pp. 323–378, 2002.
R. Kimmel, R. Malladi, and N. Sochen, “Image processing via the Beltrami operator,” in Proc. of Third Asian Conf. on Computer Vision, Hong Kong, January 1998.
M. Lysaker, Variational PDE Techniques for Image Processing Problems, PhD thesis, 2004.
M. Lysaker, S. Osher, and X.-C. Tai, “Noise removal using smoothed normals and surface fitting,” IEEE Trans. Image Processing, Vol. 13, No. 10, pp. 1345–1357, 2004.
J. Nocedal and S.J. Wright, Numerical Optimization, Springer Series in Operations Research. Springer-Verlag: New York, 1999.
S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “Using geometry and iterated refinement for inverse problems (1): Total variation based image restoration,” UCLA Computational and Applied Mathematics Report, 04-13, 2004.
S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Model. Simul., Vol. 4, No. 2, pp. 460–489, 2005.
L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithm,” Physica D., Vol. 60, pp. 259–268, 1992.
G. Sapiro, Geometric Partial Differential Equations and Image Analysis, Cambridge University Press, 2001.
O. Scherzer and C. Groetsch, “Inverse scale space theory for inverse problems,” in Scale-Space and Morphology in Computer Vision : Third International Conference, Scale-Space 2001, M. Kerckhove (Ed.), volume 2106 of Lecture Notes in Computer Science. Springer-Verlag GmbH, 2001.
D. Strong and T.F. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Problems, Vol. 19, pp. 165–187, 2003.
Author information
Authors and Affiliations
Corresponding author
Additional information
Johan Lie received his Cand. Scient. (M.S.) degree from the University of Bergen in 2003. He is currently pursuing his PhD degree in applied mathematics within the topic of Diffusion Tensor Imaging of the human brain. His main research interest is image processing using partial differential equations and transform based methods.
Jan M. Nordbotten received his Cand. Scient. (M.S.) and Dr. Scient. (Ph.D.) degrees from the University of Bergen in 2002 and 2004, respectively. For his work in his doctorate degree, he was awarded the Lauritz Meltzer award for young scientists. In 2004 and 2005 Nordbotten spent time as a Post. Doc. at Princeton University and the University of Bergen before taking an Associate Professor position in the Department of Mathematics at the University of Bergen in 2006. His main research focus lies in the analysis of partial differential equations arising in the environmental sciences, in particular multi-phase flow in porous media and eco-hydrology.
Rights and permissions
About this article
Cite this article
Lie, J., Nordbotten, J.M. Inverse Scale Spaces for Nonlinear Regularization. J Math Imaging Vis 27, 41–50 (2007). https://doi.org/10.1007/s10851-006-9694-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10851-006-9694-9