Abstract
The Beltrami flow is an efficient nonlinear 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 very small time steps and therefore require many iterations. In this paper we introduce a semi-implicit Crank-Nicolson scheme based on locally one-dimensional (LOD)/additive operator splitting (AOS) for implementing the anisotropic Beltrami operator. The mixed spatial derivatives are treated explicitly, while the non-mixed derivatives are approximated in an implicit manner. In case of constant coefficients, the LOD splitting scheme is proven to be unconditionally stable. Numerical experiments indicate that the proposed scheme is also stable in more general settings. Stability, accuracy, and efficiency of the splitting schemes are tested in applications such as the Beltrami-based scale-space, Beltrami denoising and Beltrami deblurring. In order to further accelerate the convergence of the numerical scheme, the reduced rank extrapolation (RRE) vector extrapolation technique is employed.
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This research was partly supported by United States–Israel Binational Science Foundation grant No. 2004274, ISF grant No. 1551/09, by ONR grant No. N00014-06-1-0978, the Ministry of Science grant No. 3-3414 and by Rubin Scientific and Medical Fund and the Elias Fund for Medical Research. The research is also supported by MOE (Ministry of Education) Tier II project T207N2202 and IDM project NRF2007IDMIDM002-010.
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Rosman, G., Dascal, L., Tai, XC. et al. On Semi-implicit Splitting Schemes for the Beltrami Color Image Filtering. J Math Imaging Vis 40, 199–213 (2011). https://doi.org/10.1007/s10851-010-0254-y
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DOI: https://doi.org/10.1007/s10851-010-0254-y