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Abstract. The Beltrami image flow is an effective nonlinear filter, often used in color image processing. It was shown to be closely related to the median, total variation, and bilateral filters. It treats the image as a two-dimensional manifold embedded in a hybrid spatial-feature space. Minimization of the image surface area yields the Beltrami flow. The corresponding diffusion operator is anisotropic and strongly couples the spectral components. Thus, there is so far no implicit or operator-splittingbased numerical scheme for the partial differential equation that describes the Beltrami flow in color. Usually, this flow is implemented by explicit schemes, which are stable only for very small time steps and therefore require many iterations. At the other end, vector extrapolation techniques accelerate the convergence of vector sequences, without explicit knowledge of the sequence generator. In this paper, we propose using vector extrapolation techniques for accelerating the convergence of the explicit schemes for the Beltrami flow. Experiments demonstrate fast convergence and efficiency compared to explicit schemes.Key words. diffusion, partial differential equation, Beltrami, extrapolation, filtering, denoising AMS subject classifications. 60J60, 58J35, 65B05DOI. 10.1137/0807283911. Introduction. The Beltrami framework, introduced in [38,39,44], is based on a nonlinear flow that was applied as an edge preserving denoising and deblurring algorithm for signals and especially multichannel images; see, for example, [3]. Unlike related nonlinear filters such as the total variation (TV) filter [23,1,8], which can be computed efficiently using semiimplicit schemes [43], the Beltrami flow is usually implemented by an explicit finite-difference approximation of the characterizing partial differential equation (PDE). Standard explicit finite-difference schemes require small time steps for stability that lead to a large number of iterations required for convergence to the desired solution. So far, there is no implicit scheme for the Beltrami flow, due to the strong coupling of the color components and its anisotropic nature. Our goal is to accelerate the slow convergence of the explicit schemes, for which we propose employing vector extrapolation techniques.As an alternative to the explicit scheme, an approximation using the short time kernel for the Beltrami operator was suggested in [40]. This method is still computationally demanding, since computing the kernel operation involves geodesic distance computation around each pixel. A semi-implicit scheme has been devised in [11] for an approximation of the Beltrami flow. This approximation is not, however, consistent with the PDE characterizing the Beltrami flow. Rather, it discretizes a slightly different PDE. The Beltrami flow is also strongly linked

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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Abstract. The Beltrami image flow is an effective non-linear filter, often used in color image processing. It was shown to be closely related to the median, total variation, and bilateral filters. It treats the image as a 2D manifold embedded in a hybrid spatial-feature space. Minimization of the image area surface yields the Beltrami flow. The corresponding diffusion operator is anisotropic and strongly couples the spectral components. Thus, there is so far no implicit nor operator splitting based numerical scheme for the PDE that describes Beltrami flow in color. Usually, this flow is implemented by explicit schemes, which are stable only for very small time steps and therefore require many iterations. At the other end, vector extrapolation techniques accelerate the convergence of vector sequences, without explicit knowledge of the sequence generator. In this paper, we propose to use the minimum polynomial extrapolation (MPE) and reduced rank extrapolation (RRE) vector extrapolation methods for accelerating the convergence of the explicit schemes for the Beltrami flow. Experiments demonstrate their stability and efficiency compared to explicit schemes.