Abstract
A noise removal technique using partial differential equations (PDEs) is proposed here. It combines the Total Variational (TV) filter with a fourth-order PDE filter. The combined technique is able to preserve edges and at the same time avoid the staircase effect in smooth regions. A weighting function is used in an iterative way to combine the solutions of the TV-filter and the fourth-order filter. Numerical experiments confirm that the new method is able to use less restrictive time step than the fourth-order filter. Numerical examples using images with objects consisting of edge, flat and intermediate regions illustrate advantages of the proposed model.
Similar content being viewed by others
References
Bertozzi, A.L. and Greer, J.B. 2003. Low curvature image simplifiers: Global regularity of smooth solutions and laplacian limiting schemes. Tech. rep. (03–26), UCLA, Applied mathematics.
Chambolle, A. and Lions, P-L. 1997. Image recovery via total variation minimization and related problems. Numerische Matematik, 76:167–188.
Chan, T., Marquina, A., and Mulet, P. 2000. High-order total variation-based image restoration. SIAM Journal on Scientific Computation, 22:503–516.
Giusti, E. 1998. Minimal Surface and Functions of Bounded Variations. Boston, Birkhäuser.
Greer, J.B. and Bertozzi, A.L. 2004. H1 solutions of a class of fourth order nonlinear equations for image processing. Discrete and continuous dynamical systems 2004, special issue in honor of Mark Vishik, Editors: V. Chepyzhov, M. Efendiev, Alain Miranville and Roger Temam, 1–2(10): 349–366.
Greer, J.B. and Bertozzi, A.L. 2003. Traveling wave solutions of fourth order pdes for image processing. Tech. rep. (03–25), UCLA, Applied mathematics.
Hinterberger, W. and Scherzer, O. 2004. Variational methods on the space of functions of bounded Hessian for convexification and denoising Preprint.
Ito, K. and Kunisch, K. 2000. BV-type regularization methods for convoluted objects with edge, flat and grey scales. Inverse Problems, 16:909–928.
Karkkainen, T. and Majava, K. 2000. SAC-methods for image restoration. In World Scientific and Engineering Society, Greece, pp. 162–167.
Lysaker, M., Lundervold, A., and Tai, X.-C. 2003. Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time. IEEE Trans. Image Processing, 12(12):1579–1590.
Lysaker, M., Osher, S., and Tai, X.-C. 2004. Noise removal using smoothed normals and surface fitting. IEEE Trans. Image Processing, 13:(10)1345–1357.
Nagao, M. and Matsuyama, T. 1979. Edge preserving smoothing, computer graphics and image processing. Computer Graphics and Image Processing, 9(4):394–407.
Osher, S., Sole, A., and Vese, L. 2003. Image decomposition and restoration using total variation minimization and the H-1 norm. Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, 1:(3)349–370.
Ring, W. 2000. Structural properties of solutions to total variation regularization problems. M2AN Mathematical Modeling and Numerical Analyses. 34(4):799–810.
Rudin, L.I., Osher, S., and Fatemi, E. 1992. Nonlinear total variation based noise removal algorithms Physica D, 60:259–268.
Scherzer, O. 1998. Denoising with higher order derivatives of bounded variation and an application to parameter estimation. Computing, 60:1–27.
Terzopoulos, D. 1986. Regularization of inverse visual problems involving discontinuities. IEEE Transactions on Pattern Analysis and Machine Intelligence, 8(4):413–424.
Terzopoulos, D. 1988. The computation of visible-surface representations. IEEE Trans Pattern Anal Mach Intell, 10(4):417–438.
Weickert, J. 1998. Anisotropic Diffusion in Image Processing, Stutgart, B.G. Teubner.
You, Y-L. and Kaveh, M. 2000. Fourth-order partial differential equation for noise removal. IEEE Transactions on Image Processing, 9(10):1723–1730.
Ziemer, W. P. 1989. Weakly Differentiable Functions, vol. 120 of Graduate Texts in Mathematics. Springer-Verlag, New York, Sobolev spaces and functions of bounded variation.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work has been supported by the Research Council of Norway.
Rights and permissions
About this article
Cite this article
Lysaker, M., Tai, XC. Iterative Image Restoration Combining Total Variation Minimization and a Second-Order Functional. Int J Comput Vision 66, 5–18 (2006). https://doi.org/10.1007/s11263-005-3219-7
Received:
Revised:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11263-005-3219-7