Semi-implicit finite volume scheme for solving nonlinear diffusion equations in image processing. (English) Zbl 1013.65094

Mikula, Karol; Ramarosy, Narisoa

Numer. Math. 89, No. 3, 561-590 (2001).

Summary: We propose and prove a convergence of the semi-implicit finite volume approximation scheme for the numerical solution of the modified [in the sense of F. Catté, P.-L. Lions, J.-M. Morel and T. Coll, SIAM J. Numer. Anal. 29, No. 1, 182-193 (1992; Zbl 0746.65091)] nonlinear image selective smoothing equation (called anisotropic diffusion in the image processing) studied by P. Perona and J. Malik [Scale space and edge detection using anisotropic diffusion. In Proc. IEEE Computer Socity Workshop on Computer Vision (1987)]. The proof is based on \(L_2\) a priori estimates and Kolmogorov’s compactness theorem. The implementation aspects and computational results are discussed.

MSC:

65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K55	Nonlinear parabolic equations
94A08	Image processing (compression, reconstruction, etc.) in information and communication theory

Keywords:

numerical examples; semi-implicit finite volume scheme; nonlinear diffusion equations; image processing; convergence