Semi-implicit complementary volume scheme for solving level set like equations in image processing and curve evolution

Abstract

We introduce linear semi-implicit complementary volume numerical scheme for solving level set like nonlinear degenerate diffusion equations arising in image processing and curve evolution problems. We study discretization of image selective smoothing equation of mean curvature flow type given by Alvarez, Lions and Morel ([3]). Solution of the level set equation of Osher and Sethian ([26], [30]) is also included in the study. We prove L ∞ and W 1,1 estimates for the proposed scheme and give existence of its (generalized) solution in every discrete time-scale step. Efficiency of the scheme is given by its linearity and stability. Preconditioned iterative solvers are used for computing arising linear systems. We present computational results related to image processing and plane curve evolution.

Discrete duality finite volume schemes for Leray−Lions−type elliptic problems on general 2D meshes

“…Other possibilities exist; among them, let us mention the schemes studied in [56] (see also [3,12,8]), in [10], in [42] (see also [47]), and in [52] (see also [50,51,53]). All these schemes possess some variant of the "integration-by-parts" property of Proposition 4.1.…”

“…This is also the case of the "complementary volumes" schemes as described in [56]. In 2D, ideas quite similar to that of [56] were used to construct the schemes of [3,12,8].…”

“…In 2D, ideas quite similar to that of [56] were used to construct the schemes of [3,12,8]. All these schemes work on meshes dual to conformal triangular 2D meshes, and the discrete gradient is reconstituted by affine per triangle interpolation.…”

“…The key feature of the 2D schemes of [3,56,12,10,8] (see also [40]) lies in the fact that the fluxes across interfaces are reconstructed "manually". The approaches of Droniou and Eymard [47,42] and those of the HVF, SUCCES and SUSHI schemes of Eymard, Gallouët and Herbin [50,51,53,52] are different; they rely on introducing additional unknowns (either for the fluxes, or for the values on some of the edges) and on careful penalization of the finite differences.…”

Stability and consistency of the semi-implicit co-volume scheme for regularized mean curvature flow equation in level set formulation

“…Our semi-implicit scheme leads to solution of linear systems in every discrete time step (for other semi-implicit approaches to solving nonlinear diffusion see e.g. [18,22,16,17]), so it is much more efficient than a fully implicit nonlinear scheme [33], and it is unconditionally stable without any restriction to time step in spite of many other explicit schemes [30,32,29,31]. Consistency and stability are two properties, in the theory of Barles and Souganidis [3], which are used to show convergence of a numerical scheme to solution of fully nonlinear second order partial differential equations and we discuss them in this paper.…”