THE MORPHOLOGICAL STRUCTURE OF IMAGES - THE DIFFERENTIAL-EQUATIONS OF MORPHOLOGICAL SCALE-SPACE

In this paper we introduce a class of nonlinear differential equations that are solved using morphological operations. The erosion and dilation act as morphological propagators propagating the initial condition (original image in computer vision terminology) into the ''scale-space,'' much like the Gaussian convolution is the Propagator for the linear diffusion equation. Analysis starts in the set domain, resulting in the description of erosions and dilations in terms of contour propagation. We show that the structuring elements to be used must have the property that at each point of the contour there is a well-defined and unique normal vector. Then given the normal at a point of the dilated contour we can find the corresponding point (point-of-contact) on the original contour. In some situations we can even link the normal of the dilated contour with the normal in the point-of-contact of the original contour. The results of the set domain are then generalized to grey value images. The role of the normal is replaced with the function gradient. The same analysis also holds for the erosion. Using a family of increasingly larger structuring functions we are then able to link infinitesimal changes in grey value (resulting from the use of an infinitesimally larger structuring function) with the gradient in the image. The obtained differential equations bear great resemblance to the nonlinear differential equation, Burgers' equation, describing the propagation of a shock-wave. In the discussion we indicate that the results of this paper provide the theoretical basis to analyze morphological scale-space in much greater depth.