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