A morphological scheme for mean curvature motion and applications to anisotropic diffusion and motion of level sets

Abstract

We introduce in this paper a discrete scheme using a morphological image processing approach. After briejly presenting the axiomatic approach of image processing and the mean curvature motion PDE, the properties of the proposed scheme are studied. We show that this morphological scheme performs mean curvature evolution on a gray level image. ,Then we notice that the same scheme can be applied to forms. To conclude we present experimental results.

A semi-Lagrangian scheme for the curve shortening flow in codimension-2

“…This scheme (which has been proposed in a very similar form by Catté et al in [9]) has the advantage of detecting the gradient direction without requiring an approximation of the gradient itself, although it is more expensive due to the presence of the min-max operator. Construction and stochastic interpretation of the codimension-2 version of schemes (6) and (8) will be presented in Section 3.…”

An Automatic Left Ventricle Segmentation on Echocardiogram Exams via Morphological Geodesic Active Contour with Adaptive External Energy

“…His work was pioneer in the use successives mutiscale dilatations and erosions morphological [15], in order to offer a stable and efficient numerical scheme to solving PDEs. Posteriorly, Catté et al [6] proved that successives applications of the mean operator, for a structuring element sufficiently small is equivalent to the curvature flow of the PDE at Equation 6.…”

Efficient and reliable schemes for nonlinear diffusion filtering

“…Schemes that inherit a large number of the properties of their continuous counterparts have also been proposed in the context of curvature-based nonlinear scale-spaces [8], [10], [11], [13]. Sophisticated algorithms for such processes comprise fast level set methods [40], high-order ENO schemes [41], and implicit algorithms for mean curvature motion [2], [31].…”

“…In the dimensional case each inner pixel has neighbors with which it is connected via nonvanishing entries in the th row of . From (11) we see that we can estimate (49) where denote the dimensions of an dimensional pixel. With and , restriction (48) may be replaced by (50) Thus, the allowed step size of the explicit scheme becomes even smaller for higher dimensions.…”