Building Reconstruction - Outside and In

The modelling or reconstruction of buildings has two aspects – on the one hand we need a data structure and the associated geometric information, and on the other hand we need a set of tools to construct the building incrementally. This paper discusses both of these aspects, but starts from the simpler exterior model and geometry determination, and then looks at representations of the building interiors. Our starting point is a set of raw LIDAR data, as this is becoming readily available for many areas. This is then triangulated in the x-y plane using standard Delaunay techniques to produce a TIN. The LIDAR values will then show buildings as regions of high elevation compared with the ground. Our initial objective is to extrude these buildings from the landscape in such a manner that they have well defined wall and roof planes. We may have already been provided with the building footprint from national mapping information, or we may need to extract it from the triangulation. We do this by superimposing a coarse Voronoi cell structure on the data, and identifying wall segments within each. We then examine the triangulated interior (roof) data, identify planar segments and connect them to form the final surface model of the building embedded in the terrain. This is done using Euler Operators and Quad-Edges. Building interiors are added by using an extension of these. 1. Previous work We have previously demonstrated (Tse and Gold, 2002) that a TIN may be represented with advantage using the Quad-Edge data structure of Guibas and Stolfi (1985), and that this structure is closely related to the basic Euler Operators used in CAD systems for boundary representation (b-rep) modelling of exterior surfaces (e.g. Mantyla, 1988; Lee, 1999). Other structures may also be used (e.g. Baumgart, 1972; Weiler, 1986). Based on this equivalence the TIN model may be modified using these Euler Operators, to permit the modelling of seamless exterior surfaces of buildings or other structures, for example by the splitting or merging of faces, the creation or deletion of bridges or tunnels, etc. Fig. 1 shows the basic Quad-Edge element, and Fig. 2 shows the elementary Splice Operator. Fig. 3 shows a simple triangulation network, the Quad-Edges and the topological loops around faces and nodes. Note that this represents both the primal and the dual structure: loops around Delaunay nodes are equivalent to Voronoi cell boundaries, and loops around Voronoi nodes are equivalent to Delaunay triangle boundaries. Quad-Edge data structure Fig. 1: Quad-Edge structure Fig. 2: Splice operation Fig. 3: Quad-Edge navigation Euler Operators to create a tunnel or bridge In the CAD industry the most common elementary operations on surface models (b-reps) are called Euler Operators. These have been shown (Tse and Gold, 2002) to be simply constructed from the Quad-Edge operations Make-Edge (for a new edge) and Splice (to join or split two Quad-Edges). Each Euler Operator has an inverse. Shown here are MEV (Make Edge and Vertex, Fig. 4), MEF (Make Edge and Face, Fig. 5) and how to construct a tunnel or a bridge (Fig. 6). Fig. 7 shows two simple examples. Fig. 4: Connecting two Quad-Edges with MEV Fig. 5: Splitting a polygon with MEF Fig. 6: Tunnel Construction Fig. 7: Bridges and tunnels Building Extraction with Provided Mapping Information If we are given the building footprint we may insert points into the terrain model along these boundary lines, and use the Euler Operators to extrude the building vertically (Tse and Gold, 2001). This will give a flat roof at the average height of the LIDAR data within the boundary, with the building modelled by Quad-Edges. Fig. 8 shows the terrain with the building footprints, and Fig. 9 shows the extruded buildings. Fig. 8: Building footprints on terrain Fig. 9: Extruded buildings 2. Building Construction from LIDAR data alone In principle it should be possible to extract good approximations of buildings from a sufficiently dense set of elevation data. In practice this is difficult. There are two steps: firstly to extract the vertical walls, and then to model the roof. There are two basic approaches: to attempt to fit a pre-defined template to the data (e.g. Vosselman, 2003; Rottensteiner and Briese, 2003); or to attempt to construct a building-like shape by extracting features from elevation data (e.g. vertical walls or roof planes). The first approach is limited by the models included in the system, while the second will only approximate the building form, and will often need subsequent rectification. We take the second approach, and only assume properties of “buildings” when absolutely necessary. Automatic Wall Extraction While it is not difficult to identify the near-vertical triangles in the TIN it is not a simple task to form a complete building from these segments. The remote sensing literature has many examples of attempts to first detect line segments and then glue them together. Our approach is always to preserve a tessellation model, with connectivity, rather than attempting to connect line segments. We apply a coarse Voronoi diagram over the original data, with perhaps 50-100 LIDAR points in each cell. We then attempt to modify these cells so that the building boundaries (defined as a partition between “high” points and “low” points”) are a subset of the Voronoi cell edges. We can split cells along the high/low edge to achieve this. Proposition 1: Buildings are collections of contiguous elevations that are higher than the surrounding terrain. Their boundaries are “walls”. The approach is based on calculating the eigenvalues and eigenvectors of the 3 x 3 variance-covariance matrix of the coordinates of the points within a cell. The first eigenvector (with the largest eigenvalue) “explains” as much of the overall variance as possible, the second (perpendicular) eigenvector explains as much as possible of what is left, and the third (perpendicular to the other two) contains the residue. (For example, a wrinkled piece of paper might have the first eigenvector oriented along the length of the paper, the second along its width, and the third “looking” along the wrinkles.) Thus the eigenvector of the smallest eigenvalue indicates the orientation of a wall segment, if present, and looks along it. The next step is to locate the line parallel to the smallest eigenvector that best separates “high” elevations from “low” ones within each Voronoi cell. This is achieved iteratively, by testing various positions of this line in order to find the greatest difference between the means of the elevation values in the Voronoi cell that are on each side of the line. (In order to minimize the effect of sloping roofs or terrain, only those elevations close to the line are used.) If this maximum difference is not sufficiently large then no wall segment was detected. Proposition 2: Walls have a specified minimum height, and this height difference is achieved within a very few “pixels”. The Voronoi cells are then split along these lines, by adding a generator on each side of this line, at the mid-point. This gives a set of “high” Voronoi cells surrounded by “low” ones. Building boundaries are then determined by walking around the cells forming the high region, using the topological consistency of the Voronoi tessellation. This must form a closed region, or else the high region is not considered to be a building. The building outline is then estimated from the Voronoi boundary segments – but only those that were created with the eigenvector technique, not those Voronoi cell boundaries that only connect them. Fig. 10 shows the Voronoi structure and the “high” LIDAR points, as well as the wall segments detected. Fig. 11 shows the “high” Voronoi cells before they are split, and Fig. 12 shows the approximation of the walls based on the split cells. Proposition 3: A building consists of a high region entirely surrounded by walls. Fig. 10: Voronoi cells and wall segments Fig. 11: Initial “high” Voronoi cells Fig. 12: “High” Voronoi cells after splitting