A CRYSTALLOGRAPHIC APPROACH TO SIMPLIFY 3D BUILDING
B. Poupeau, A. Ruas
The geometric simplification of complex objects is well-studied in computer graphics in order to improve rendering performance, to shorten geometric computations and to reduce storage space (Heckbert and Garland, 1997; Puppo and Scopigno, 1997). These algorithms are dedicated to polygonal and triangular meshes and give very good results on smooth and complex objects made from hundreds of thousands to millions of primitives, but they are not optimal for 3D objects with low geometric complexity (hundreds of primitives). Others studies, based on half space modelling (Kada, 2005), scale-space theory (Forber, 2004) and CSG tree (Thiermann, 2004), have been performed to simplify 3D buildings according to their symmetrical properties. These studies try to preserve the special geometrical characteristics of buildings (orthogonality, parallelism, colinearity,...) during the simplification.
This paper presents a new approach for building simplification based on the crystallographic description of objects. Each 3D building is analysed in a bounding sphere with a centre O. The poles (i.e. the intersection between S and the normals to faces, passing by O, of the 3D building) are used to enable the symmetric analysis (Mirror, Axes, Centre) of the 3D building by means of a stereographic projection. This description enables to extract parallel, colinear and orthogonal faces and is useful to define at least three levels of details :
1. The first one is the geographical object itself.
2. The second one is the regularised 3D object, called crystallographic form, rebuilt from the normals of the symmetric description. Between these two representations, it is possible to get step-by-step others simplifications by merging colinear and parallel faces according to the distance between the bounding sphere and the centre, and according to area criteria. Furthermore, if the 3D object is too complex, a subdivision into convex objects is computed with the help of the stereographic projection and a crystallographic analysis is performed on each 3D convex object.
3. Last, when all simplifications, based on previously defined criteria, are finished, a lattice unit is deduced from the crystallographic form. This last simplification is an ideal crystallographic form with the maximum number of all symmetric elements, limited to seven lattice units: cubic, hexagonal, trigonal, monoclinic, triclinic tetragonal and orthogonal.
In this paper we present this algorithm of simplification which has been implemented in our 3D GIS prototype Cristage (Poupeau, 2006) based upon GeOxygene and we illustrate the results on different buildings shape. Eventually, we discuss the advantages and drawbacks of this crystallographic approach and give some directions for future improvements.
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