Simplification des cartes géographiques par minimisation de la déformation locale

In cartography, the geographic regions are usually represented using regular dense maps corresponding to heights values associated with the nodes of a regular grid of R2. The simplification of such maps is an absolute requirement in order to make storage, simulation and display possible. In this Note, we propose a new simplification method based on a measure of the local deformation of the surface. The latter allows, in particular, minimization of the approximation error during the simplification. A numerical example is provided to emphasize the efficiency of this approach. To cite this article: P.J. Frey, H. Borouchaki, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 227–232.  2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS Abridged English version Maps are involved in numerous numerical or graphical simulations. In general, in this kind of application, the geometric model (a terrain, for instance) is represented by a triangulation or a grid that may contain several millions triangles or quadrilaterals. The simplification of such maps is thus required, in order to facilitate storage and transmission, as well as to make a finite element simulation or the display more efficient. Adresses e-mail : [email protected] (P.J. Frey); [email protected] (H. Borouchaki).  2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés S1631-073X(02)02250-1/FLA 227 P.J. Frey, H. Borouchaki / C. R. Acad. Sci. Paris, Ser. I 334 (2002) 227–232 Several simplification methods for surface triangulations have been developed (see [5] for a survey), that differ mainly by the type of data and the criteria used to measure the deviation of the simplified triangulation to the initial one. Here, we consider the following problem: given T the canonical triangulation of a map (its vertices are the images of the nodes of a regular grid G of size n × m from N2 to R3 along given heights), find a simplified triangulation Ts based on a ‘minimal’ subset V of the vertices of T in which the ‘distance’ to the map T is less than or equal to a given threshold value. This gap can be quantified in two ways: continuous or discrete. The continuous measure consists in formulating this gap using the Hausdorff distance from Ts to T [1]. The computation of the Hausdorff measure seems here too expensive. However, the discrete measure consists in considering the gap between the vertices of the map T and their orthogonal projections on the triangulation Ts (this being a simplification of the continuous Hausdorff measure and seems more appropriate for terrain simplification). Formally speaking, let z(p) be the height of a node p of the grid and let uT the vector of dimension n×m of components z(p), p covering the nodes of the grid and let vT ,Ts be the vector of dimension n×m of components zs(p), the height of the orthogonal projection of P = (p, z(p)) of T on Ts . The problem is then to find a subset of minimal cardinality W of the nodes of the grid G and a triangulation Ts based on the vertices images of the nodes in W, such that δTs = ‖uT − vT ,Ts‖ is less than or equal to the threshold δ, where ‖ · ‖ denotes a norm of R3. As shown by [6], the triangulation Ts can be considered as the image of the Delaunay triangulation of the nodes of W. Thus, the simplification problem is reduced to finding the nodes of W. Our approach, different from methods based on incremental insertion (greedy insertion, cf. [3], for instance), aims at constructing the simplified triangulation Ts (and thus the nodes of W) by removing iteratively the vertices of the initial triangulation T . At each stage, the vertex to be removed is that for which the ‘current’ deformation measure given by ε(P ) = maxi d(Pi, (P )) is minimal, d(·, ·) denoting the distance from a point to a plane, (P) being the tangent plane at P and Pi , the vertices adjacent to P . The point removal operation consists in removing the triangles incident to this vertex (i.e., the ball of the vertex) and to retriangulate the resulting cavity using the Delaunay criterion. The simplification procedure stops as soon as the deformation measure becomes greater than a given threshold, at each vertex of the current triangulation.