RECONSTRUCTING POWER CABLES FROM
LIDAR DATA
USING EIGENVECTOR STREAMLINES
OF THE
POINT DISTRIBUTION TENSOR FIELD
marcel.ritter@uibk.ac.at
MARCEL RITTER (SPEAKER),
WERNER BENGER
WSCG2012
Plzen, Czech Rep., 26.6. 2012
Center for Computation
and Technology
ASTRO@UIBK
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
Motivation
• 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
Point Distribution Tensor Field
[RBBPML12]
Streamlines
Methodology
• Computing the point distribution tensor
Methodology
• Tensor analysis:
– Shape factors by [Westin97]
– S(Pi) is a 3x3 symmetric tensor and positive definite
– 3 Eigen-Values:
– Shape factors:
[BBHKS06]
Methodology
• Tensor visualization:
– Ellipsoids representing the shape factors
– Tensor Splats [BengerHege04]
-> barycentric
[BBHKS06]
Methodology
• Tensor Splats of a rectangular point distribution
Points
Tensor Splats
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
𝜔1
𝜔2
Distribution tensor and its linearity of a rectangular point distribution
Methodology
• Influence of weighting
on the resulting tensor
𝜔3
𝜔4
Distribution tensor and its linearity of a rectangular point distribution
Methodology
• Influence of weighting
on the resulting tensor
𝜔5
𝜔6
Distribution tensor and its linearity of a rectangular point distribution
Methodology
• Influence of weighting
on the resulting tensor
𝜔7
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
Streamline
Valid major eigenvectors
Eigen-Streamline
Implementation and Verification
Implemented
in the VISH
visualization
shell
Allows to implement
visualization modules
C++/OpenGL/OpenCL
2 Eigen-streamline
modules
Uniform/Curviliear grid
Distribution tensor
module extended by
the weighting functions
 Network of modules
Meshfree grid
Implementation and Verification
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
– ω2 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
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
Application
• LIDAR cable reconstruction
Application
• LIDAR cable reconstruction
Application
• LIDAR cable reconstruction
– Manual seeding position and direction
– Tested 41 different combinations of different
weighting functions and neighborhood radii
• Tensor computation (r = 0.5, 1.0, 2.0 [m])
• Vector interpolation (r = 0.25, 0.5, 1.0, 2.0, 3.0 [m])
Tensor computation with 𝜔3 ,r=2.0, and interpolation
with 𝜔7 , r=3.0, worked best in this scenario
Reconstructed 280m of cable with an error of 80cm
Application
• LIDAR cable reconstruction
Tensor 𝜔1,r=2, DOP853 Eigen-streamlines 𝜔3,r=1
Tensor 𝜔3,r=2, DOP853 Eigen-streamlines 𝜔7,r=1
Conclusion
Computed Eigenstreamlines 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 sets
Improve seeding  automatic seeding
Do not follow Eigen-vectors in non linear regions
Better interpolate tensors directly and not Eigen-vectors during integration
Integrate 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, 2012 - 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. 1742.
[BengerHege04]
W. Benger and H.-C. Hege, “Tensor splats,” in Conference on
Visualization and Data Analysis 2004, vol. 5295. 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