Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie a Matematicas, 2005
The Beltrami framework for image processing and analysis introduces a non-linear parabolic proble... more The Beltrami framework for image processing and analysis introduces a non-linear parabolic problem, called in this context the Beltrami flow. We study in the framework for functions of bounded variation, the well-posedness of the Beltrami flow in the one-dimensional case. We prove existence and uniqueness of the weak solution using lower semi-continuity results for convex functions of measures. The solution is defined via a variational inequality, following Temam's technique for the evolution problem associated with the minimal surface equation.
We study, in this work, the maximum principle for the Beltrami color flow and the stability of th... more We study, in this work, the maximum principle for the Beltrami color flow and the stability of the flow's numerical approximation by finite difference schemes. We discuss, in the continuous case, the theoretical properties of this system and prove the maximum principle in the strong and the weak formulations. In the discrete case, all the second order explicit schemes, that are currently used, violate, in general, the maximum principle. For these schemes we give a theoretical stability proof, accompanied by several numerical examples.
We propose to solve the signal/image denoising problem by minimizing the total variation of the s... more We propose to solve the signal/image denoising problem by minimizing the total variation of the signal and forcing the residual between the estimated and the measured signal to be statistically noise-similar; we thus project the residual on a set of directions in signal space and require these projections to be 4σ limited and to have appropriate empirical nonlinear moments. Experimental results on denoising 1-D signals demonstrate the efficiency of the method.
The Beltrami image flow is an effective non-linear filter, often used in color image processing. ... more 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 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, nor 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 to use 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.
From the introduction: We are interested in large time behaviour of the solution to the Cauchy Pr... more From the introduction: We are interested in large time behaviour of the solution to the Cauchy Problem: ∂ t u=u∂ xx u-(∂ x u) 2 ,x∈ℝ,t≥0u(x,0)=φ(x),x∈ℝ In B. Meerson, P. V. Sasorov and K. Sekimoto [Logarithmically slow expansion of hot bubbles in gases, Phys. Rev. E, 61, 1403–1406 (2000)] it was shown by formal asymptotic expansion that if the initial temperature profile decays at large |x| as e -k|x| , k>0, then the hot bubble, while cooling down should expand logarithmically slowly. This fact represents a new feature of the cool conductive flow. It was also shown that for smooth initial data with compact support [-l,l] a long time asymptotics is given by a separable solution w(x,t)=θ l (x) t where θ l (x)=a 2 2cos 2 x a,for|x|≤l0,for|x|>l and a=2 πl. The purpose of this paper is to give a rigorous mathematical justification of these facts and to consider a more general problem, namely the following Cauchy problem: ∂ t u=uΔu-γ|∇u| 2 ,x∈ℝ N ,t≥0u(x,0)=φ(x),x∈ℝ N where γ≥0 is a...
The Beltrami framework for image processing and analysis introduces a non-linear parabolic proble... more The Beltrami framework for image processing and analysis introduces a non-linear parabolic problem, called in this context the Beltrami flow. We study in the framework for functions of bounded variation, the well-posedness of the Beltrami flow in the one-dimensional case. We prove existence and uniqueness of the weak solution using lower semi-continuity results for convex functions of measures. The solution
The Laplace-Beltrami operator is an extension of the Laplacian from flat domains to curved manifo... more The Laplace-Beltrami operator is an extension of the Laplacian from flat domains to curved manifolds. It was proven to be useful for color image processing as it models a meaningful coupling between the color channels. This coupling is naturally expressed in the Beltrami framework in which a color image is regarded as a two dimensional manifold embedded in a hybrid, five-dimensional, spatialchromatic (x, y, R, G, B) space.
The Beltrami image flow is an effective non-linear filter, often used in color image processing. ... more 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.
The Beltrami image flow is an effective non-linear filter, often used in color image processing. ... more 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 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, nor 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 to use 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.
We study, in this work, the maximum principle for the Beltrami color flow and the stability of th... more We study, in this work, the maximum principle for the Beltrami color flow and the stability of the flow's numerical approximation by finite difference schemes. We discuss, in the continuous case, the theoretical properties of this system and prove the maximum principle in the strong and the weak formulations. In the discrete case, all the second order explicit schemes, that are currently used, violate, in general, the maximum principle. For these schemes we give a theoretical stability proof, accompanied by several numerical examples.
The Beltrami flow is an efficient nonlinear filter, that was shown to be effective for color imag... more 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.
We propose to solve the signal/image denoising problem by minimizing the total variation of the s... more We propose to solve the signal/image denoising problem by minimizing the total variation of the signal and forcing the residual between the estimated and the measured signal to be statistically noise-similar; we thus project the residual on a set of directions in signal space and require these projections to be 4σ limited and to have appropriate empirical nonlinear moments. Experimental results on denoising 1-D signals demonstrate the efficiency of the method.
We analyze the discrete maximum principle for the Beltrami color flow. The Beltrami flow can disp... more We analyze the discrete maximum principle for the Beltrami color flow. The Beltrami flow can display linear as well as nonlinear behavior according to the values of the parameter which represents the ratio between spatial and color distances. The standard schemes fail, in general, to satisfy the discrete maximum principle. In this work we show that a nonnegative second order difference scheme can be built for this flow only for small values of this parameter named β, i.e. linear diffusion. This limitation on the parameter β makes the nonnegative scheme unpractical. We construct a novel finite difference scheme, which is not nonnegative yet satisfies the discrete maximum principle for all values of β. Numerical results support the analysis.
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