Partial differential equations and mathematical morphology

AbstractIn the past few years, nonlinear parabolic PDEs have been introduced in image analysis. A complete classification of these equations is now established with the geometrical invariance properties that may be required. 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 Matheron yields a complete classification of morphological operators that is monotone, translation invariant and contrast invariant functions operators. In this paper, we prove that any adequately scaled and iterated affine invariant, morphological operator converges to the semi-group associated with the unique affine invariant PDE of the classification. In a second part, by using again Matheron's characterization, we give a new proof of the convergence of other morphological operators, the weighted median filters, towards the Mean Curvature Motion