1. Point Set Processing and Surface Reconstruction (www.cgal.org)
Nader
Nader Salman, INRIA
In collaboration with
Pierre Alliez, INRIA
Laurent Saboret, INRIA
Gaël Guennebaud, INRIA Bordeaux
9th ACCV Xi’an China, Sep

2. Outline
Common pipeline
Structure
Documentation
Demo
Roadmap

3. Common Pipeline Physical Acquisition Registration Pre-processing
Reconstruction
Digital
Geometry

4. Example Pipeline (1) laser range data
Acquisition
Registration
Pre-processing
(simplification, denoising, smoothing, …)
Reconstruction

5. Example Pipeline (2) multi-view passive stereo
Acquisition
& calibration
Point cloud generation
Pre-processing
(simplification, denoising, smoothing, …)
Reconstruction

6. Structure (pipeline-oriented)
Point set
Analysis
Processing
Normals
Reconstruction
Contouring
Bounding box
Bounding sphere
Centroid
Average spacing
Simplification
Outlier removal
Smoothing
Estimation
Orientation
Poisson
Algebraic point set surfaces
Surface mesh generator
(grayed out = in other CGAL packages)

7. Entering the Pipeline…
Point set
Clean point set
Clean points with oriented normals
Analysis
Processing
Normals
Reconstruction
Contouring
We can enter the pipeline later if we have a clean point set, or points with oriented normals, or points with unoriented normals, etc.
Bounding box
Bounding sphere
Centroid
Average spacing
Simplification
Outlier removal
Smoothing
Estimation
Orientation
Poisson
Algebraic point set surfaces
Surface mesh generator
Points with unoriented normals
(grayed out = in other CGAL packages)

8. Clean points with oriented normals
Output…
Point set
Analysis
Processing
Normals
Reconstruction
Contouring
Bounding box
Bounding sphere
Centroid
Average spacing
Simplification
Outlier removal
Smoothing
Estimation
Orientation
Poisson
Algebraic point set surfaces
Surface mesh generator
Clean points with oriented normals
Implicit function
Surface triangle mesh
(grayed out = in other CGAL packages)

9. Point_set_processing_3 Introduction
Left: 275k points sampled on an elephant (Minolta laser scanner)
Right: point set cleaned and simplified to 17K points

10. Analysis Average spacing Bounding box Bounding sphere Centroid
reuse orthogonal search for K nearest neighbors
used by reconstruction / contouring algorithms
takes an iterator range of 3D points and parameter K

11. Processing Simplification Outlier removal Smoothing
by random selection
by clustering (regular grid)
Outlier removal
sort w.r.t. sum of squared distances to KNN and cut at specified percentile.
Smoothing
jet fitting + projection

12. Example Simplification
from 100K to 50k points by random simplification

13. Example Simplification
from 100K to 50k points by clustering

14. Example Outlier Removal
(a lion and a bunch of outliers)

15. Example Smoothing For each point find KNN
fit jet (smooth parametric surface)
project onto jet
(noisy point set)
(smoothed point set)

16. Normals Estimation (no orientation) Orientation
KNN + PCA (fit a plane)
KNN + jet fitting (better for noisy data)
Orientation
KNN + BGL MST (Minimum Spanning Tree) [Hoppe 92]

17. Example Normal Estimation
Normals estimation through PCA (7 KNN)
Orientation through MST (7 KNN)

18. Example Normal Orientation
Normal Orientation through MST

19. Surface_reconstruction_points_3 Introduction
reconstructed surface using Poisson
17K points sampled on an elephant (Minolta laser scanner)
reconstructed surface using APSS (in progress)

20. Reconstruction (1)
Poisson surface reconstruction [Kazhdan-Bolitho-Hoppe, SGP 2006]
Solves for an implicit function (~indicator function)
Isosurface extracted by CGAL surface mesh generator

21. Poisson Surface Reconstruction
Reconstruct the surface of the model by solving for the indicator function of the shape. M
Indicator function 1 1 1

22. Poisson Surface Reconstruction
There is a relationship between the normal field and gradient of indicator function
M
Indicator gradient
points + oriented normals

23. Poisson Surface Reconstruction
Represent the points by a vector field
Find the function whose gradient best approximates :
Applying the divergence operator, we can transform this into a Poisson problem: 

24. Poisson Surface Reconstruction
We solve for the Poisson equation onto the vertices of a (refined) 3D Delaunay triangulation [TAUCS linear solver]

25. Example Poisson

26. Example Poisson
Left: 120K points sampled on child statue (Minolta laser scanner)
Right: reconstructed surface

27. Example Poisson
Left: 120K points sampled on a statue (Minolta laser scanner)
Right: reconstructed surface

28. Example Poisson

29. Example Poisson Left: 70K points with (emphasized) outliers
Right: reconstructed surface

30. Example Poisson
Left: Bimba 120K reconstructed with distance = 0.25*average spacing
Right: Bimba 120K reconstructed with distance = 0.15*average spacing

31. Right: reconstructed surface
Example Poisson
Left: 65K points sampled on a hand with no data at the wrist base (Kreon laser scanner)
Right: reconstructed surface

32. Example Poisson

33. Example Poisson Left: 50K points sampled on Neptune trident
Right: point set simplified to 1K then reconstructed

34. Example Poisson Left: points sampled on a sphere with flipped normals
Right: reconstructed surface

35. Example Poisson
Left: 4K points sampled on a mechanical piece with sharp edges
Right: reconstructed surface

36. Poisson duration wrt #input points

37. Surface meshing duration and error wrt approximation distance

38. Memory wrt #input points

39. Reconstruction error wrt #input points

40. Reconstruction (2)
Algebraic point set surfaces (APSS) [Guennebaud, SIGGRAPH 2007]
Evaluates an implicit function on the fly (~signed distance function)
Isosurface extracted by CGAL surface mesh generator

41. APSS
Based on Moving Least Squares fitting on algebraic spheres

42. APSS
Point projection

43. APSS

44. APSS
can evaluate an implicit function at any point

45. APSS

46. Poisson vs APSS?
INPUT = 275K points sampled on an elephant (minolta laser scanner)

47. Poisson
APSS
Approximation error in surface mesh generator: 0.004

48. Poisson
APSS
Approximation error in surface mesh generator: 0.002

49. Poisson APSS Approximation error in surface mesh generator: 0.001
always a bit smoother
one extra component

50. Documentation Demo Current: 98 pages Reference manual available online
User manual available online
Demo
Current: 3D point set demo (QT4 + QGLViewer)

51. The End