Lorina Dascal

Efficient Beltrami filtering of color images via vector extrapolation

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.

Polyakov action minimization for efficient color image processing

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, spatial-chromatic (x,y,R,G,B) space. The Beltrami filter defined by this framework minimizes the Polyakov action, adopted from high-energy physics, which measures the area of the image manifold. Minimization is usually obtained through a geometric heat equation defined by the Laplace-Beltrami operator. Though efficient simplifications such as the bilateral filter have been proposed for the single channel case, so far, the coupling between the color channel posed a non-trivial obstacle when designing fast Beltrami filters. Here, we propose to use an augmented Lagrangian approach to design an efficient and accurate regularization framework for color image processing by minimizing the Polyakov action. We extend the augmented Lagrangian framework for total variation (TV) image denoising to the more general Polyakov action case for color images, and apply the proposed framework to denoise and deblur color images.

Efficient Beltrami Image Filtering via Vector Extrapolation Methods

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-splitting-based 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.

On Semi-implicit Splitting Schemes for the Beltrami Color Image Filtering

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.

On Semi-implicit Splitting Schemes for the Beltrami Color Flow

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.

A Maximum Principle for Beltrami Color Flow

We study, in this work, the maximum principle for the Beltrami color flow and the stability of the flows 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.