Fast Difference Schemes for Edge Enhancing Beltrami Flow

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 from 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 problemto 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.

[1]  A. Polyakov.  Quantum Geometry of Bosonic Strings , 1981 .

[2]  R. Malladi,et al.  Fast difference scheme for anisotropic Beltrami smoothing and edge contrast enhancement of gray level and color images , 2001 .

[3]  Joachim Weickert,et al.  Anisotropic diffusion in image processing , 1996 .

[4]  Ron Kimmel,et al.  A Natural Norm for Color Processing , 1998, ACCV.

[5]  William H. Press,et al.  Numerical recipes in C , 2002 .

[6]  Joachim Weickert,et al.  Scale-Space Theories in Computer Vision , 1999, Lecture Notes in Computer Science.

[7]  Roman Goldenberg,et al.  Fast Geodesic Active Contours , 1999, Scale-Space.

[8]  James A. Sethian,et al.  Image Processing: Flows under Min/Max Curvature and Mean Curvature , 1996, CVGIP Graph. Model. Image Process..

[9]  G I Barenblatt.  Self-similar intermediate asymptotics for nonlinear degenerate parabolic free-boundary problems that occur in image processing , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[10]  Ron Kimmel,et al.  Images as Embedded Maps and Minimal Surfaces: Movies, Color, Texture, and Volumetric Medical Images , 2000, International Journal of Computer Vision.

[11]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[12]  Guillermo Sapiro,et al.  Vector-valued active contours , 1996, Proceedings CVPR IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[13]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[14]  Ron Kimmel,et al.  A general framework for low level vision , 1998, IEEE Trans. Image Process..

[15]  Curtis F. Gerald.  Applied numerical analysis , 1970 .

[16]  Bart M. ter Haar Romeny,et al.  Geometry-Driven Diffusion in Computer Vision , 1994, Computational Imaging and Vision.

[17]  P. Lions,et al.  Image selective smoothing and edge detection by nonlinear diffusion. II , 1992 .

[18]  R Malladi,et al.  Image processing via level set curvature flow. , 1995, Proceedings of the National Academy of Sciences of the United States of America.

[19]  Anthony J. Yezzi,et al.  Modified curvature motion for image smoothing and enhancement , 1998, IEEE Trans. Image Process..

[20]  Jitendra Malik,et al.  Scale-Space and Edge Detection Using Anisotropic Diffusion , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[21]  Max A. Viergever,et al.  Efficient and reliable schemes for nonlinear diffusion filtering , 1998, IEEE Trans. Image Process..

[22]  Ron Kimmel,et al.  From High Energy Physics to Low Level Vision , 1997, Scale-Space.