PARTIAL-DIFFERENTIAL EQUATIONS AND MATHEMATICAL MORPHOLOGY

In the past few years, nonlinear parabolic PDEs have been introduced i n image analysis. A complete classification of these equations is now established with the geometrical invariance properties that may be req uired. An important result is that there exists a unique second order parabolic equation which is invariant with respect to contrast changes and affine distorsions. On the other hand, a classical result by Math eron yields a complete classification of morphological operators that is monotone, translation invariant and contrast invariant functions op erators. In this paper, we prove that any adequately scaled and iterat ed affine invariant, morphological operator converges to the semi-grou p associated with the unique affine invariant PDE of the classificatio n. In a second part, by using again Matheron's characterization, we gi ve a new proof of the convergence of other morphological operators, th e weighted median filters, towards the Mean Curvature Motion. (C) Else vier, Paris.