1. reconstruction process, RANSAC, primitive shapes, alpha-shapes
LiDAR Data Processing
reconstruction process, RANSAC, primitive shapes, alpha-shapes

2. LiDAR data 3D points Irregular Dense (50-100/m2) Accurate (5 cm)
Explain the zooming process they will see before clicking it.

3. Reconstruction Building a 3D model from the point cloud
A polyhedral model uses edges and facets (like triangles), it should “fit well” with the points
Types of models/approaches:
Meshes: many (small) triangles that use the points of the point cloud as vertices (like a TIN but not just for a bivariate function)
Parametric models: pre-defined shapes where certain parameters can be set for the best fit (e.g. the 6 planes of a house with a roof)
Hybrid methods

4. Reconstruction methods
Iso-surfaces and marching cubes
Medical imaging
DEMs, and parametric building models
Geosciences
Region growing and surface extraction
Urban reconstruction

5. A reconstruction pipeline

6. Phase 1: Clustering

7. Clustering using RANSAC
RANdom SAmple Consensus
Select random sample set (size of sample is 3 for finding a plane)
Build candidate plane
Compute plane quality (support)
Keep best plane
Iterate

8. Terminating RANSAC Stop criterion: probability of success
: outlier ratio; fraction not in plane to be found
: samples needed to build plane = 3
: number of RANSAC iterations
In practice: unknown; set
: support of best candidate so far
: complete point set

9. Specializing RANSAC Problems Solutions Very large point sets
RANSAC finds only one plane
Solutions
Sampling strategy
Efficient quality estimation
Approximate probability of success
Repeat after finding best plane
[Schnabel et al.: Efficient RANSAC…, 2007]

10. Sampling strategy Early candidate elimination Octree localization
Approximate point normals
Over-sampling
Octree localization
Local vs. Global
Local: more likely in same plane
Global: less impact of noise
Sample at optimal octree depth

11. Efficient quality estimation
Quality = |support|
Maximum distance to plane
Maximum normal deviation
Connected component
Lazy support estimation
Subset of data
Confidence interval of support

12. Approximate probability of success
RANSAC iterations
Taylor approximation

13. Identify multiple models
After iterations
Refit best candidate on support using lazy support estimation
Remove points near candidate
Repeat until probability of finding plane with support too small ( is minimal support; pre-specified)

14. Efficient RANSAC results

15. RANSAC results Planes with ~100 points or more are found
Points in vegetation and in small planes remain unclustered; some of these points may still help in the reconstruction

16. Phase 2: Modeling surfaces

17. Bound and connect surfaces
Known shapes
Identify surfaces bounded by shape
Unknown shapes
Base boundary on local point density
Base boundary on nearby surfaces

18. Identifying rectangular regions
Find rectangles containing support
Edges of different surfaces parallel
Related to Rotating Calipers
Time complexity O(n log n)

19. Rotating calipers Rotate parallel lines around convex hull
Find optimal rotation
Width
Diameter
Area

20. Requiring coverage Well covered by data (δ-coverage):
The rectangle is δ-covered by the point set if it is covered by the union of all δ-disks centered on a point in the set

21. Identifying well covered rectangles
Rectangle as it is rotated
Trajectory on circles
Thales’ theorem
[Thales’ theorem image: Wikipedia]

22. Identifying well covered rectangles
Two ways to leave the allowed region
Edge passes over vertex
O(n) time
Corner passes over arc
O(n) time

23. Extensions to algorithm
Handling outliers
Handling noise

24. Bounding unknown shape
Problems
More difficult shapes
A priori knowledge required
Solution: base shape on data
Use point density
Use other surfaces

25. Using point density Shape unknown Estimated local density
Non-empty circles

26. α-Shape Non-convex hull α determines detail
Corners are points from set
Edges between points on boundary of empty α-disk
α determines detail

27. Using other surfaces Intersect other, nearby surfaces in 3D
Construct 2D boundary candidates from intersections
Identify segments near points
Adjust segments

28. Using other surfaces Many boundaries not closed Close using α-shape
Add boundary end-points

29. Phase 3: Assembling objects

30. Geometric objects Objects Topology graph Bounded by surfaces Connected
Has topology
Topology graph
Intersection lines

31. Gap filling
Determine facets that might be used to fill holes in the model, for example the triangles in a Delaunay tetrahedrilization (constrained to include known surfaces)
Define a graph dual to the terahedrilization (tetrahedra are nodes, adjacent tetrahedra give arcs)
Define sources and sinks, make known surfaces “free” in the graph and let the cost of other triangles be their area
Then use graph-cut algorithm on the graph dual to the tetrahedrilization to select facets to add to the model

32. Summary LiDAR specification Reconstruction pipeline Efficient RANSAC
Identify known shapes
Bounding using α-Shapes
Bounding using surfaces
Assembling objects
Closing scene
Texturing surfaces
Verifying model