Shape from LIDAR Data. Univ. of Florida

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1 Shape from LIDAR Data Yusuf Sahillioğlu Alper Üngör Univ. of Florida 1. Introduction LIght Detection And Ranging systems, LIDAR, are capable of acquiring data to produce accurate digital elevation models by allowing the positioning of the footprint of a laser beam as it hits an object. The laser produces an optical pulse that is transmitted, reflected off an object, and returned to the receiver. The receiver accurately measures the travel time of the pulse from its start to its return. With the pulse traveling at the speed of light, the receiver senses the return pulse before the next pulse is sent out. Since the speed of light is known, the travel time can be converted to a range measurement. Combining the laser range, laser scans angle, laser position (from GPS), and laser orientation (from INS), accurate x, y, z object coordinates can be calculated for each laser pulse [1]. While we focus on forest modeling in this project, there are plenty of other applications that are fed by LIDAR data, such as flood risk mapping, oil/gas exploration surveys, real estate development, coastal zone mapping, and urban modeling. Our problem is, given a set of unorganized and noisy 3D points acquired by a LIDAR system, approximate the underlying surface by a 2-manifold triangular mesh. The approach we present employs a surface deformation framework and advertises ending up with a lower resolution mesh compared to the high resolution of the LIDAR points. Although this property could also be obtained by a triangulation of the LIDAR points followed by edge collapses that rely on edge length and/or quadric [2] error metrics, our results introduce a different perspective. Besides, thanks to our silhouette based approach, one can eliminate the necessity of initial triangulation of LIDAR points, though it is not trivial. 2. Related Work LIDAR becomes an important and convenient data source. Many researchers are developing algorithms to extract a bare-earth model and building boundaries from LIDAR data. The traditional method for building reconstruction with LIDAR data uses filtered LIDAR data and combines cadastral building boundaries data. Another method, [6], uses the Voronoi Diagram to trace building outlines, for which they extrude buildings via Computer Aided Design (CAD)-type Euler Operators (to create a TIN model) and then used the operators to modify the TIN, e.g. extrude buildings, interactive editing or further spatial analysis. Most researchers, on the other hand, perform 2D triangulation of the projection of LIDAR data, followed by edge collapses with convenient error metrics to build fair reconstructions in sufficiently low resolutions

2 3. Algorithm We have constructed a surface deformation framework that is guided by synthetic silhouettes, and tangent planes attached to some of the LIDAR points. Local mesh transform operations [3], and mesh fairing [4] keep the evolving mesh in good shape. The generic algorithm follows: While active triangles Move each vertex P with velocity in its normal direction Regularize/Fair the mesh Update guide LIDAR point for each P //not in silhouette phase Deactivate triangle T if it has a small velocity (due to its vertices) Apply local mesh transform operations Deformation framework is initially guided by multi-view projections of triangulated LIDAR model. Intersecting cones of back-projections of each silhouette image generates the underlying visual hull (Figure 1). This technique does not use any heuristics except moving the evolving mesh vertices in their normal directions, and hence is advertised as a robust one. Besides, it is quite efficient since we simply eliminate guide LIDAR update step of the generic algorithm. Figure 1: Visual hull by back-projecting images via camera calibration parameters. We represent binary silhouettes as bitmaps (like this figure); 1 for IN, 0 for OUT pixels Tangent Plane Computation The more accurate the tangent planes, the more robust the mesh evolution gets. We have used two strategies while recommending the latter one Tangent Plane by k-nearest Neighbors Having subdivided the bounding box of LIDAR points into cubes, we select up to m points from each cube that are closest to the cube center. Finding the k-nearest neighbors of a selected point p lets us fit a plane K to that set of k+1 points in the least squares sense, where K gets assigned to p. For efficiency reasons, we further deny the tangent plane computation for the points which already appear in the k-nearest neighbor list of

3 any other point by assuming that there already exists a close tangent plane that can do the equivalent job. Note also that, the tangent planes in use may be inconsistent oriented [5]. Figure 2: Subdivided LIDAR space (left), and assigned tangent planes (right) Tangent Plane by Auxiliary Mesh So-called auxiliary mesh is obtained by triangulating [7] the orthogonal projections of LIDAR points (to xy-plane), followed by an elevation (using the back-projected/original z-coordinates). Auxiliary mesh has accurate and consistent plane information for each LIDAR point used during elevation process. This mesh is also ready to serve as a synthetic silhouette provider for section 3.4. Figure 3: Assigned tangent planes Velocity Computation w/o Silhouettes Velocity of a vertex P should be high enough to yield fast convergence, and low enough to prevent over-carving. Therefore, we prefer to use a mixture of two velocities, which both rely on the fact that each triangle has a guide LIDAR point/tangent plane.

4 Figure 4: Two alternatives for an estimate distance between mesh and LIDAR boundary. Omitted in Figure 4 for visual convenience, we also consider neighborhood of P in velocity computation to emphasize the local effect. To do so, signed distances are computed for each triangle neighbor T of P, and then multiplied by normal of T, and then the vectors at hand are averaged. Finally, the length of the projection of the averaged vector onto normal of P is obtained via dot product, and this value becomes a signed velocity. We call this velo1 or velo2 if it is obtained by using dist1 or dist2, respectively, of Figure 4. Now, velocity of P can be defined solidly. velo1 if abs( velo1) < abs( velo2) velocity( P) = velo2 otherwise Note that, this signed velocity is capable of making both shrinking and growing effect. If tangent planes, however, are computed with k-nearest neighbor tactic, of section 3.1.1, then the inconsistent plane orientation will block us making benefit of signed distances Geometric Interpretation Both alternatives involve computing an estimate of the distance of the deformable mesh to the LIDAR boundary. The second alternative relies on the distance between a vertex of the deformable mesh and the LIDAR surface. Hence, it locally approximates the LIDAR surface with the tangent plane of the LIDAR point. However, this is not always a reliable approximation and may yield over-carving, especially if the guide LIDAR point is distant to the vertex and the vertex does not evolve by deformation towards that LIDAR point, as it usually happens with oblique LIDAR points. The first alternative, on the other hand, relies on the distance between a LIDAR point and the deformable surface. This is not as accurate as the second one since it does not make use of the available LIDAR surface orientation. However, it can be used to constrain the magnitude of the velocity in order to avoid over-carving in cases where the second alternative fails. Thus, we end up with the third alternative which is actually a

5 combination of the first two; we use the one that gives a velocity with smaller magnitude. This formulation is the one that we have used in our experiments since it yields more reliable reconstructions as compared to the others Velocity Constraints There is an upper bound of E min / 4 in velocity of P, where Emin is the minimum edge length guaranteed by edge collapse operation. Not exceeding this value prevents birth of non-manifold triangles during evolution. Besides, velocity computed in section 3.2 is subject to a weighting factor of planefitquality * (1 - triangletotangentplanedistance), where we use center of gravity of triangle and LIDAR point of tangent plane in the computation of the normalized distance, and we simply neglect fit quality if tangent planes are computed using auxiliary mesh of section Velocity Computation w/ Silhouettes For silhouette based algorithm, we establish the connection between vertices of the evolving mesh and silhouettes (taken from different views) of the LIDAR points by the following formulation: velocity P) = E f ( P) = E min{ G[Pr oj ( P)] 0.5}, ( min min where Pr oji n ( P) is the projection of the vertex P(x, y, z) to I n, the n th binary image (0 for outside, 1 for inside) in the sequence, and G( x', y') = (1 α)((1 β ) I ( x', y' ) + βi ( x', y' + 1) + α((1 β ) I ( n In x' + 1, y' ) + βi ( x' + 1, y' + 1), where ( x ', y' ) denotes the integer part, and ( α, β ) is the fractional part of the coordinate (x, y ) in the binary discrete image I. The function G, taking values between 0 and 1, is the bilinear interpolation of the projection (x, y ) of the vertex P. Thus, the isovalue function f(p) takes on values between -0.5 and 0.5, and the zero crossing of this function reveals the isosurface. The isovalue of the vertex P is provided by the image of the silhouette that is farthest away from the point, or in other words, where the interpolation function G assumes its minimum value. This discussion follows that, we classify P as OUT if f(p) is negative, ON if it is 0, and IN if it is positive. Not surprisingly, velocity (P) becomes 0 (freezing) if P is ON the surface described by LIDAR silhouettes. 4. Experimental Results Figure 5 shows the auxiliary mesh of three different sets of input LIDAR points we used. Following figures are pairs of resulting meshes; one after deformation framework is run in the guidance of synthetic silhouettes (left), and one in the guidance of tangent planes (right). Not surprisingly, silhouette based models lack details due to the hidden concavity problem coming with the all shape from silhouette based approaches. Thanks to the tangent plane guidance, we recover those details as much as possible, and without facing over-carving. Starting tangent plane phase with the mesh silhouette phase left leads efficiency (less guide LIDAR updates), and accuracy (close start).

6 Figure 5: Auxiliary meshes by elevated triangulation of (left), (right), and (bottom) LIDAR input points. Figure 6: (# of vertices, # of triangles) is (3989, 7810) at left, and (4756, 9340) at right. Figure 7: (# of vertices, # of triangles) is (2710, 5233) at left, and (3382, 6558) at right.

7 Figure 8: (# of vertices, # of triangles) is (2755, 5368) at left, and (3309, 6462) at right. 5. Future Work The silhouette based approach is promising. Although we have used the triangulation of LIDAR points, which we call auxiliary mesh, to obtain synthetic silhouettes, we hope to do this without ever considering that triangulation. We plan to project 3D LIDAR points directly to the image planes, and then estimate the underlying 0-1 bitmap silhouettes as accurately as possible; i.e. by finding the closed contour via active contours/snakes approach, and then fill inside. This will, eventually, clear the need of messing up with noisy data triangulation to reconstruct the LIDAR surface. 6. Conclusion We have described a generic surface deformation framework that is capable of being fed with both LIDAR data synthetic silhouettes, and tangent planes. This framework evolves the initial mesh iteratively until underlying surface of LIDAR points is recovered. While silhouette based phase leaves a good starting mesh for the more accurate tangent plane phase, it also winks for the future by its potential of omitting initial noisy LIDAR triangulation. 7. References [1] Mosaic Mapping System Inc., A White Paper on LIDAR Mapping, (2001). [2] M. Garland, P. Heckbert, Surface Simplification Using Quadric Error Metrics, SIGGRAPH 97, (1997). [3] H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, W. Stuetzle, Mesh Optimization, SIGGRAPH 93, (1993), [4] G. Taubin, A Signal Processing Approach To Fair Surface Design, Computer Graphics Proceedings, (1995), [5] H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, W. Stuetzle, Surface Reconstruction from Unorganized Points, Proceedings of the 19th annual conference on Computer graphics and interactive techniques, (1992), [6] R. Tse, M. Dakowicz, C. Gold, D. Kidner, Building Reconstruction Using LIDAR Data, Dynamic and Multi-dimensional GIS 2005, (2005),

8 [7] P. Bourke, Triangulate: Efficient Triangulation Algorithm Suitable for Terrain Modelling, Pan Pacific Computer Conference, Beijing, China, (1989). [8] Puget Sound Lidar Consortium (PSLC), data source [9] Y. Sahillioglu, Y. Yemez, A Surface Deformation Framework for 3D Shape Recovery, Multimedia Content Representation, Classification and Security, International Workshop (MRCS), (2006),