R ECONSTRUCTING P OWER C ABLES F ROM LIDAR D ATA U SING E IGENVECTOR S TREAMLINES OF THE P OINT D ISTRIBUTION T ENSOR F IELD M ARCEL R ITTER ( SPEAKER ), W ERNER B ENGER WSCG2012 Plzen, Czech Rep., Center for Computation and Technology IBK

Overview Motivation Methodology – The Point Distribution Tensor – Weighting Functions – Eigenvector Streamlines Implementation and Verification – Comparison Meshfree/Uniform Grid – Test Cases Application Conclusion and Future Work

Arose from an airborne light detection and ranging (LIDAR) application Earth surface scanned by laser pulses  point cloud Motivation

LIDAR point cloud: Motivation

Reconstruct linearly distributed points of the LIDAR laser scan as lines Power cable detection applicable for companies to maintain power lines Alternative method to current approaches Motivation

Based on previous work – Direct visualization of the point distribution tensor – Streamline integration – Inspired by diffusion tensor fiber tracking StreamlinesPoint Distribution Tensor Field [RBBPML12]

Methodology Computing the point distribution tensor

Tensor analysis: – Shape factors by [Westin97] – S(P i ) is a 3x3 symmetric tensor and positive definite – 3 Eigen-Values: – Shape factors: Methodology [BBHKS06]

Tensor visualization: – Ellipsoids representing the shape factors – Tensor Splats [BengerHege04] Methodology -> barycentric [BBHKS06]

Points Tensor Splats Tensor Splats of a rectangular point distribution Methodology

Distribution tensor of airborne LIDAR data Methodology

Weighting functions: – 7 different weighting functions were implemented Methodology

Weighting functions: Methodology

Influence of weighting on the resulting tensor Distribution tensor and its linearity of a rectangular point distribution

Methodology Influence of weighting on the resulting tensor Distribution tensor and its linearity of a rectangular point distribution

Methodology Influence of weighting on the resulting tensor Distribution tensor and its linearity of a rectangular point distribution

Methodology Influence of weighting on the resulting tensor Distribution tensor and its linearity of a rectangular point distribution

Methodology Streamlines – Common tool for flow visualization – Curve q on Manifold M with s the curve parameter – Vector field v with Tp(M) an element of the tangential space at point P on M – Streamline as curve tangential to the vector field

Methodology Eigen-Streamlines – Must be able to follow against the vector field Tensor Valid major eigenvectors Streamline Eigen-Streamline

Implementation and Verification Implemented in the VISH visualization shell Allows to implement visualization modules C++/OpenGL/OpenCL  Network of modules 2 Eigen-streamline modules Uniform/Curviliear grid Meshfree grid Distribution tensor module extended by the weighting functions

Eigen-streamline module KDTree for neighborhood search All vectors are aligned to the direction of the first vector in the neighborhood Interpolation of the vector is done using one of the weighing functions The Eigen-vector is reversed when the dot product to the last tangent vector is negative C++ templates used to switch weighting functions Implementation and Verification

Verification of Meshfree Approach – Eigenvector field of MRI brain scan, [BBHKS06] – Converted uniform grid data to meshfree grid – Compare streamlines computed on both grids Implementation and Verification

Verification of Meshfree Approach – Trilinear interpolation on uniform grid – ω 2 slinear interpolation on meshless grid – 81% of 144 short streamlines coincide well Meshfree Uniform Grid Implementation and Verification

Circle Integration – Tested numerical integration schemes Explicit Euler DOP853 (Runge-Rutta order 8) Implementation and Verification

Rectangle Integration – Tested different weighting functions for vector interpolation – Horizontal distance of integration start to endpoint as error measure Implementation and Verification

Error of rectangle reconstruction integration Implementation and Verification

Application LIDAR cable reconstruction

Application LIDAR cable reconstruction

Application

LIDAR cable reconstruction

Conclusion Computed Eigen- streamlines in a mesh free point distribution tensor field Verification on simple test geometries Reconstructed a power cable LIDAR dataset 280m cable

Future Work Investigate other weighting functions and weighting combinations Automatically find an optimal combination of weightings Investigate other data setsImprove seeding  automatic seedingDo not follow Eigen-vectors in non linear regionsBetter interpolate tensors directly and not Eigen-vectors during integrationIntegrate in 2 directions simultaneously from given seeding point

References [RBBPML12] Ritter M., Benger W., Biagio C., Pullman K., Moritsch H., Leimer W., Visual Data Mining Using the Point Distribution Tensor, IARIS The First International Workshop on Computer Vision and Computer Graphics - VisGra 2012, February 29 - March 5, Saint Gilles, Reunion Island, France [Taubin95] G. Taubin, “Estimating the tensor of curvature of a surface from a polyhedral approximation,” in Proceedings of the Fifth International Conference on Computer Vision, ser. ICCV ’95. Washington, DC, USA: IEEE Computer Society, 1995, pp. 902–. [Westin97] C. Westin, S. Peled, H. Gudbjartsson, R. Kikinis, and F. Jolesz, “Geometrical diffusion measures for MRI from tensor basis analysis,” in Proceedings of ISMRM, Fifth Meeting, Vancouver, Canada, Apr. 1997, p [BengerHege04] W. Benger and H.-C. Hege, “Tensor splats,” in Conference on Visualization and Data Analysis 2004, vol Proceedings of SPIE Vol. #5295, 2004, pp. 151–162. [BBHKS06] Benger, W., Bartsch, H., Hege, H.-C., Kitzler, H., Shumilina, A. & Werner, A. (2006). Visualizing Neuronal Structures in the Human Brain via Diffusion Tensor MRI, International Journal of Neuroscience 116(4): pp. 461–514.

Marcel Ritter 1) Werner Benger 2,3) 1) Institute for Basic Sciences in Civil Engineering, University of Innsbruck, Austria 2) Center for Computation & Technology, Louisiana State University, Baton Rouge, USA 3) Institute for Astro- and Particle Physics, University of Innsbruck, Austria