Fast Difference Schemes for Edge Enhancing Beltrami Flow

摘要

The Beltrami flow [13,14] is one of the most effective denoising algorithms in image processing. For gray-level images, we show that the Beltrami flow equation can be arranged in a reaction-diffusion form. This reveals the edge-enhancing properties of the equation and suggests the application of additive operator split (AOS) methods [4,5] for faster convergence. As we show with numerical simulations, the AOS method results in an unconditionally stable semi-implicit linearized difference scheme in 2D and 3D. The values of the edge indicator function are used form the previous step in scale, while the pixel values of the next step are used to approximate the flow. The optimum ratio between the reaction and diffusion counterparts of the governing PDE is studied, in order to achieve a better quality of segmentation. The computational time decreases by a factor of ten, as compared to the explicit scheme. For 2D color images, the Beltrami flow equations are coupled, and do not yield readily to the AOS technique. However, in the proximity of an edge, the cross-products of color gradients nearly vanish, and the coupling becomes weak. The principal directions of the edge indicator matrix are normal and tangent to the edge. Replacing the action of the matrix on the gradient vector by an action of its eigenvalue, we reduce the color problem to the gray level case with a reasonable accuracy. The scalar edge indicator function for the color case becomes essentially the same as that for the gray level image, and the fast implicit technique is implemented.