Abstract
The Beltrami flow is an efficient non-linear filter, that was shown to be effective for color image processing. The corresponding anisotropic diffusion operator strongly couples the spectral components. Usually, this flow is implemented by explicit schemes, that are stable only for small time steps and therefore require many iterations. In this paper we introduce a semi-implicit scheme based on the locally one-dimensional (LOD) and additive operator splitting (AOS) schemes for implementing the anisotropic Beltrami operator. The mixed spatial derivatives are treated explicitly, while the non-mixed derivatives are approximated in a semi-implicit manner. Numerical experiments demonstrate the stability of the proposed scheme. Accuracy and efficiency of the splitting schemes are tested in applications such as the scale-space analysis and denoising. In order to further accelerate the convergence of the numerical scheme, the reduced rank extrapolation (RRE) vector extrapolation technique is employed.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Rosman, G., Dascal, L., Kimmel, R., Sidi, A.: Efficient beltrami image filtering via vector extrapolation methods. SIAM J. Imag. Sci. (2008) (submitted)
Mešina, M.: Convergence acceleration for the iterative solution of the equations X = AX + f. Comp. Meth. Appl. Mech. Eng. 10, 165–173 (1977)
Eddy, R.: Extrapolating to the limit of a vector sequence. In: Wang, P. (ed.) Information Linkage Between Applied Mathematics and Industry, New York, pp. 387–396. Academic Press, London (1979)
Spira, A., Kimmel, R., Sochen, N.A.: A short-time Beltrami kernel for smoothing images and manifolds. IEEE Trans. Image Process. 16(6), 1628–1636 (2007)
Smith, S.M., Brady, J.: Susan - a new approach to low level image processing. Intl. J. of Comp. Vision 23, 45–78 (1997)
Aurich, V., Weule, J.: Non-linear gaussian filters performing edge preserving diffusion. In: Mustererkennung 1995, 17. DAGM-Symposium, London, UK, pp. 538–545. Springer, Heidelberg (1995)
Tomasi, C., Manduchi, R.: Bilateral filtering for gray and color images. In: Proceedings of IEEE International Conference on Computer Vision, pp. 836–846 (1998)
Sochen, N., Kimmel, R., Bruckstein, A.M.: Diffusions and confusions in signal and image processing. J. of Math. Imag. and Vision 14(3), 195–209 (2001)
Elad, M.: On the bilateral filter and ways to improve it. IEEE Trans. Image Process. 11(10), 1141–1151 (2002)
Barash, D.: A fundamental relationship between bilateral filtering, adaptive smoothing and the nonlinear diffusion equation. IEEE Trans. Image Process. 24(6), 844–847 (2002)
Buades, A., Coll, B., Morel, J.M.: A review of image denoising algorithms, with a new one. SIAM Interdisciplinary Journal 4, 490–530 (2005)
Lu, T., Neittaanmaki, P., Tai, X.C.: A parallel splitting up method and its application to Navier-Stokes equations. Applied Mathematics Letters 4(2), 25–29 (1991)
Lu, T., Neittaanmaki, P., Tai, X.C.: A parallel splitting up method for partial differential equations and its application to Navier-Stokes equations. RAIRO Mathematical Modelling and Numerical Analysis 26(6), 673–708 (1992)
Weickert, J., Romeny, B.M.T.H., Viergever, M.A.: Efficient and reliable schemes for nonlinear diffusion filtering. IEEE Trans. Image Process. 7(3), 398–410 (1998)
Peaceman, D.W., Rachford, H.H.: The numerical solution of parabolic and elliptic differential equations. Journal Soc. Ind. Appl. Math. 3, 28–41 (1955)
Yanenko, N.N.: The method of fractional steps. The solution of problems of mathematical physics in several variables. Springer-Verlag, New York (1971)
Barash, D., Schlick, T., Israeli, M., Kimmel, R.: Multiplicative operator splittings in nonlinear diffusion: from spatial splitting to multiple timesteps. J. of Math. Imag. and Vision 19(16), 33–48 (2003)
Kimmel, R., Malladi, R., Sochen, N.: Images as embedding maps and minimal surfaces: Movies, color, texture, and volumetric medical images. Intl. J. of Comp. Vision 39(2), 111–129 (2000)
Sochen, N., Kimmel, R., Maladi, R.: From high energy physics to low level vision. In: ter Haar Romeny, B.M., Florack, L.M.J., Viergever, M.A. (eds.) Scale-Space 1997. LNCS, vol. 1252, pp. 236–247. Springer, Heidelberg (1997)
Sochen, N., Kimmel, R., Maladi, R.: A general framework for low level vision. IEEE Trans. Image Process. 7, 310–318 (1998)
Yezzi, A.J.: Modified curvature motion for image smoothing and enhancement. IEEE Trans. Image Process. 7(3), 345–352 (1998)
Polyakov, A.M.: Quantum geometry of bosonic strings. Physics Letters 103 B, 207–210 (1981)
Rudin, L., Osher, S., Fatemi, E.: Non-linear total variation based noise removal algorithms. Physica D Letters 60, 259–268 (1992)
Yanenko, N.N.: About implicit difference methods of the calculation of the multidimensional equation of thermal conductivity. In: Proceedings of VUZ. Series of Mathematics, vol. 23(4), pp. 148–157 (1961)
Andreev, V.B.: Alternating direction methods for parabolic equations in two space dimensions with mixed derivatives. Zhurnal Vychislitelnoi Matematiki i Matematicheskoi Fiziki 7(2), 312–321 (1967)
Mckee, S., Mitchell, A.R.: Alternating direction methods for parabolic equations in three space dimensions with mixed derivatives. The Computer Journal 14(3), 25–30 (1971)
Weickert, J.: Coherence-enhancing diffusion filtering. Intl. J. of Comp. Vision 31(2/3), 111–127 (1999)
Dascal, L., Rosman, G., Kimmel, R.: Efficient Beltrami filtering of color images via vector extrapolation. In: Sgallari, F., Murli, A., Paragios, N. (eds.) SSVM 2007. LNCS, vol. 4485, pp. 92–103. Springer, Heidelberg (2007)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Dascal, L., Rosman, G., Tai, XC., Kimmel, R. (2009). On Semi-implicit Splitting Schemes for the Beltrami Color Flow. In: Tai, XC., Mørken, K., Lysaker, M., Lie, KA. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2009. Lecture Notes in Computer Science, vol 5567. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02256-2_22
Download citation
DOI: https://doi.org/10.1007/978-3-642-02256-2_22
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02255-5
Online ISBN: 978-3-642-02256-2
eBook Packages: Computer ScienceComputer Science (R0)