3D reconstruction of complex objects with the help of simplified Voronoi diagrams: an application to 3d interpolation of geological sections
Our aim is the problem of the three-dimensional reconstruction of complex objects from cross-sectional contours. In the frrst part, we show that the 3D geological modelling requires a various set of tools for modelling and interpolating geological bodies from various types of data (chapter 1). Then, we detail the set of problems of the 3D reconstruction and we propose a full review of the various approaches ( chapter 2). In the second part of the manuscript, we propose an original method of 3D reconstruction. Our approach allows to deal with all the topological variations as multibranching, holes, and islands, in an automatic and easy way (chapter 3). It builds intermediate sections in a flexible way by interpolating initial contours. This process uses a simplified Voronoi diagram, the bisector network, as a 2D interpolating tool. We propose a full geometrie description of the bisector network, and show its low algebraic complexity (linear segments in 2D, portions of plane in 3D). Then, we propose two algorithms for building the internai and extemal bisector networks of a polygonal shape with holes. They run respectively in O(n2 log n) and O(n2) and the memory storage is in O(n2) and O(n) respectively ( chapter 4 ). In the chapter 5 we discuss the correspondance problem and we suggest sorne solutions for critical problems. The building of this structure allows to obtain a valid surface by meshing in a simple way all points of the various sections. Furthermore, by adding automatically portions of intermediate contours in the areas of topological and morphological variations, the method allows a best definition of the reconstructed surface ( chapter 6). ln the last part we discuss the results of the 3D reconstruction and we compare it with results of others methods (chapter 7).