3D Building Model Fitting Using A New Kinetic Framework

arXiv:0805.0648v1 [cs.CG] 6 May 2008 3D Building Model Fitting Using A New Kinetic Framework Mathieu Br´edif a,b, Didier Boldo a, Marc Pierrot-Deseilligny a, Henri Maˆıtre b a IGN (French Mapping Agency), MATIS, 2/4 avenue Pasteur, 94165 Saint-Mand´e, Cedex, France b Institut TELECOM - TELECOM ParisTech, LTCI UMR 5141 - CNRS, 46 rue Barrault, 75013 Paris, France Abstract We describe a new approach to fit the polyhedron describing a 3D building model to the point cloud of a Digital Elevation Model (DEM). We introduce a new kinetic framework that hides to its user the combinatorial complexity of determining or maintaining the polyhedron topology, allowing the design of a simple variational optimization. This new kinetic framework allows the manipulation of a bounded polyhedron with simple faces by specifying the target plane equations of each of its faces. It proceeds by evolving continuously from the polyhedron defined by its initial topology and its initial plane equations to a polyhedron that is as topologically close as possible to the initial polyhedron but with the new plane equations. This kinetic framework handles internally the necessary topological changes that may be required to keep the faces simple and the polyhedron bounded. For each intermediate configurations where the polyhedron looses the simplicity of its faces or its boundedness, the simplest topological modification that is able to reestablish the simplicity and the boundedness is performed. Key words: Kinetic Data Structure, Polyhedron, Computational Geometry, Fitting, 3D Modeling, Building, Digital Elevation Model. 1. Introduction To satisfy the growing demand for 3D city models of increasingly better accuracy and higher level of detail, there is a solid research effort to model the real world using lidar data or satellite, aerial or ground imagery. To drive costs down, the algorithm must be automatic and make use of the coarser or inaccurate 3D city models that may already exist. This fitting seems compulsory for state of the art build- ing reconstruction methods[1,2,3] that proceed by first de- tecting a static set of geometric features like planes, points or lines and then determine the building topology using a combinatorial exploration. As this combinatorial explo- ration time explodes when the number of initial geometric features increases, the number of such detected features is kept low. Together with the fact that the geometry of the detected features is static, this small number of possibly in- accurate geometric features yield approximate reconstruc- tions. Fitting these approximate geometries to the avail- able data as a post process will improve the geometric ac- curacy of the reconstructed models. This paper presents a fully automatic method to fit an existing inaccurate build- ing model to a real world data set. Previous works[4] mainly focus on optimizing the point coordinates of the model and do not question the topology of the 3D model. They keep the topology fixed and either fail or stop before conver- gence when it occurs that the initial topology was wrong as illustrated in Figure 2. To achieve 3D building model fitting without having to keep the topology fixed, this paper introduces a new generic framework to fit a polyhedron model to data. It makes use of an initial polyhedron and a mapping from the initial supporting planes of its faces to a set of target support- ing planes. It continuously interpolates the representation from the initial polyhedron to a novel polyhedron built on the target plane equations, with minimal topological mod- ifications. During this continuous evolution, the algorithm is able to handle implicitly the topological changes that are required to keep a 3D model well formed when the geom- etry of the 3D model is altered, as illustrated in figure 1. This makes the design of a variational shape optimization possible. To our knowledge, only the variational shape ap- proximation approach[5] achieves the same goal but it only maintains a partition of the fitted data. The resulting poly- hedron is only exported as a post process. Furthermore, it cannot guarantee that a partition cell is exported as a sin- gle face or even that the exported faces will be supported by a common plane. By contrast, our approach maintains a polyhedron throughout the optimization, thus avoiding the final export. Besides, the guarantee of our framework that data that has been fitted to a plane will be represented by Preprint submitted to Elsevier 27 October 2018 a) b) c) Fig. 1. a) A polyhedron with two faces that are evolving with an increasing slope. b) The resulting polyhedron if no topological mod- ification is performed: faces are no longer simple and an inside-out tetrahedron is present. c) At the time in the evolution when the problematic edge length was null, a topological flip of this edge has been performed to keep the faces simple just after the singularity. Fig. 2. A simple applicative instance of the problem described in Fi

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