A Morphological Interpolation Approach — Geodesic Set Definition in Case of Empty Intersection

Abstract

This article presents a morphological approach for the generation of intermediate 2D objects, using others 2D objects, as initial ones. The approach is useful for a smooth realistic 3D object’s visualisation, in case of shortage input information. The study is based on mathematical morphology concepts, such as geodesic distance and geodesic set definition, dealing with the case of empty intersection between the objects, in a orthogonal projection over a plane. A classical approach is used to define spatial polynomial curves, interpolating the extreme left and right sets of visible essential border points of the initial objects. Further, the arc of each curve bordered between a couple of initial 2D objects is orthogonal projected over the lower plane together with the upper 2D object. The projections and the object, all over the same plane, are used to create the geodesic set. Then, a set of intermediate sections, between each couple of initial objects, is defined by the application of a morphological linear interpolation. Experiments were performed in order to validate the theory. Real data obtained by well logs performed in Vale de Milhaços (Setúbal - Portugal) was used for this propose, and the results are given in the article.