Reconstruction of Solid Models from Oriented Point Sets

1
Reconstruction of Solid Models from Oriented
Point Sets

• Misha Kazhdan
• Johns Hopkins University

2
Shape Spectrum

• There are many ways to represent a shape
• Point Set
• Polygon Soup
• Polygonal Mesh
• Solid Model

3
Equivalence of Representations

• There are many ways to represent a shape
• Point Set
• Polygon Soup
• Polygonal Mesh
• Solid Model

?
In one direction, the transition between
representations is straight-forward
4
Equivalence of Representations

• There are many ways to represent a shape
• Point Set
• Polygon Soup
• Polygonal Mesh
• Solid Model

?
?
In one direction, the transition between
representations is straight-forward
The challenge is to transition in the other
direction
5
Equivalence of Representations

• There are many ways to represent a shape
• Point Set
• Polygon Soup
• Polygonal Mesh
• Solid Model

?
The goal of this work is to define a method for
computing solid models from oriented point sets.
6
Applications

• Surface Blending

Disjoint Model
Zippered Model
7
Applications

• Surface Blending
• Hole-Filling

Model with Hole
Water-Tight Model
8
Applications

• Surface Blending
• Hole-Filling
• Compression

Geometry Topology Representation
Geometry Representation
9
Applications

• Surface Blending
• Hole-Filling
• Compression
• Simplification

Original Model871,000 Triangles
Simplified Model95,000 Triangles
10
Related Work

• Three general approaches
• Computational Geometry
• Boissonnat, 1984 Edelsbrunner, 1984
• Amenta et al., 1998 Dey et al., 2003
• Surface Fitting
• Terzopoulos et al., 1991 Chen et al., 1995
• Implicit Function Fitting
• Hoppe et al., 1992 Curless et al., 1996
• Whitaker, 1998 Carr et al., 2001
• Davis et al., 2002 Ohtake et al., 2004
• Turk et al., 2004 Shen et al., 2004

11
Related Work

• Implicit Function Fitting
• Use the point samples to define an function whose
  values at the sample positions are zero.

lt0
0
gt0
Sample Points
F(x,y)
12
Related Work

• Implicit Function Fitting
• Use the point samples to define an function whose
  values at the sample positions are zero.
• Extract the iso-surface with iso-value equal to
  zero.

lt0
F(x,y) 0
0
F(x,y)gt0
F(x,y)lt0
gt0
Sample Points
F(x,y)0
13
Related Work

• Implicit Function Fitting
• Use the point samples to define an function whose
  values at the sample positions are zero.
• Extract the iso-surface with iso-value equal to
  zero.

gt0
F(x,y) 0
0
F(x,y)lt0
F(x,y)gt0
How should we define the implicit function so
that the reconstruction fits the samples?
lt0
Sample Points
F(x,y)0
14
Outline

• Introduction
• Related Work
• Approach
• The Divergence Theorem
• Reduction to Volume Integration
• Implementation
• Results
• Conclusion

15
Divergence Theorem

• Given a vector field F and a region V
• The volume integral of ??F over V and the surface
  integral of F over ?V are equal

?V
V
16
Reduction to Volume Integration (Step 1)

• Characteristic Function
• The characteristic function ?V of a solid V is
  the function

V
17
Reduction to Volume Integration (Step 2)

• Fourier Coefficients
• The Fourier coefficients of the characteristic
  function give an expression of ?V as a sum of
  complex exponentials

18
Reduction to Volume Integration (Step 3)

• Volume Integration
• The Fourier coefficients of the characteristic
  function ?V can be obtained by integrating

19
Reduction to Volume Integration (Step 3)

• Volume Integration
• The Fourier coefficients of the characteristic
  function ?V can be obtained by integrating
• since the characteristic function is one inside
  of V and zero everywhere else.

20
Applying the Divergence Theorem

• Surface Integration
• If Flmn(x,y,z) is any function whose divergence
  is equal to the (l,m,n)-th complex
  exponentialapplying the Divergence Theorem,
  the volume integral can be expressed as a surface
  integral

21
Reconstruction Algorithm

• Given an oriented point sample (pi,ni)
• Compute a Monte-Carlo approximation of the
  Fourier coefficients of the characteristic
  function
• Apply the inverse Fourier Transform to obtain the
  characteristic function.
• Extract the reconstruction an iso-surface of the
  characteristic function

22
Algorithm Implementation

• Efficiency
• We show that the computation of the Fourier
  coefficients is actually a convolution, reducing
  the reconstruction complexity O(R5) ?
  O(R3logR).
• Non-Uniformity
• We provide a simple heuristic for assigning
  weights to samples that may be non-uniformly
  distributed.

23
Outline

• Introduction
• Related Work
• Approach
• Results
• Resolution
• Sample Count
• Non-Uniformity
• Related Work
• Conclusion

24
Results (Resolution)
100,000 Points
100,000 Points
100,000 Points
res1283 tris49,008 time001
res2563 tris199,796 time007
res643 tris11,672 timelt001
25
Results (Sample Count)
100,000 Points
1000 Points
10,000 Points
res2563 tris200,704 time007
res2563 tris206,216 time007
res2563 tris199,796 time007
26
Results (Non-Uniform Sampling)
100,000 Points
100,000 Points
100,000 Points
res2563 tris111,680 time009
res2563 tris220,324 time009
res2563 tris199,712 time009
27
Results(Non-Uniform Sampling / Related Work)
100,000 Points
100,000 Points
100,000 Points
RBF Reconstruction
MPU Reconstruction
Our Reconstruction
res2563 tris302,000 time523
res2563 tris288,000 time039
res2563 tris286,916 time009
Ohtake et al. ACM TOG 03
Carr et al. SIGGRAPH 01
28
Results (Noise / Related Work)
Original
RBF Reconstruction
Our Reconstruction
MPU Reconstruction
res2563 tris200,000 time2410 points100,000
res2563 tris205,000 time214 points100,000
res2563 tris174,824 time007 points100,000
Ohtake et al. ACM TOG 03
Carr et al. SIGGRAPH 01
29
Outline

• Introduction
• Related Work
• Approach
• Results
• Conclusion

30
Conclusion

• Properties
• Fast and simple to compute
• Independent of topology
• Robust to non-uniform sampling
• Robust to noise
• O(R3) memory footprint for O(R2) reconstruction

31
Conclusion

• Theoretical Contribution
• Transformed the surface reconstruction problem
  into a Volume integral
• Used the Divergence Theorem to express the
  integral as a surface integral
• Used Monte-Carlo integration to approximate the
  surface integral as a summation over an oriented
  point sample

32
Conclusion

• We presented an algorithm for reconstruction that
  proceeds in three simple steps

33
Conclusion

• We presented an algorithm for reconstruction that
  proceeds in three simple steps
• Splat the oriented points into a voxel grid

34
Conclusion

• We presented an algorithm for reconstruction that
  proceeds in three simple steps
• Splat the oriented points into a voxel grid
• Convolve with a fixed filter

35
Conclusion

• We presented an algorithm for reconstruction that
  proceeds in three simple steps
• Splat the oriented points into a voxel grid
• Convolve with a fixed filter
• Extract the iso-surface

36
Thank You
Source and Executables http//www.cs.jhu.edu
/misha/Reconstruct3D
37
Choosing the Functions Flmn
38
Choosing the Functions Flmn

• There are many solutions to the equation

39
Choosing the Functions Fl,m,n

• There are many solutions to the equation
• Examples

40
Choosing the Functions Fl,m,n

• There are many solutions to the equation
• Examples

41
Choosing the Functions Fl,m,n

• There are many solutions to the equation
• In our implementation, we choose the function
• This is the unique definition of Flmn with the
  property that the reconstruction commutes with
  translation and rotation.

42
Choosing the Functions Fl,m,n
0o
30o
45o
Does not Commute
Commutes
43
Choosing the Functions Fl,m,n
Does not Commute
Commutes
44
Non-Uniform Sampling
45
Non-Uniform Samples

• Challenge
• In a direct implementation of Monte-Carlo
  integration, it is assumed that the samples are
  uniformly distributed

46
Non-Uniform Samples

• Challenge
• In a direct implementation of Monte-Carlo
  integration, it is assumed that the samples are
  uniformly distributed
• However, often the oriented point samples may not
  be uniformly distributed over the surface
• Parts of scans may overlap
• Faces parallel to the view plane may be more
  densely sampled
• Compressed representations may store fewer points
  in regions of low curvature

47
Non-Uniform Samples

• Challenge
• If we have a sampling density ?i associated to
  each sample xi, we can modify the summation

48
Non-Uniform Samples

• Challenge
• If we have a sampling density ?i associated to
  each sample xi, we can modify the summation
• However, when we get an oriented point sample, we
  usually arent given the sampling density at each
  sample.

49
Non-Uniform Samples

• Challenge
• If we have a sampling density ?i associated to
  each sample xi, we can modify the summation
• However, when we get an oriented point sample, we
  usually arent given the sampling density at each
  sample.

Non-Uniform Samples
Unweighted Reconstruction
50
Non-Uniform Samples

• Approach
• Compute the sampling density at each sample by
  counting the number of samples around it

51
Non-Uniform Samples

• Approach
• Compute the sampling density at each sample by
  counting the number of samples around it

?i1/2
52
Non-Uniform Samples

• Approach
• Compute the sampling density at each sample by
  counting the number of samples around it

?i1/1
53
Non-Uniform Samples

• Approach
• Compute the sampling density at each sample by
  counting the number of samples around it

?i1/3
54
Non-Uniform Samples

• Approach
• Compute the sampling density at each sample by
  counting the number of samples around it

?i1/4
55
Non-Uniform Samples

• Approach
• Compute the sampling density at each sample by
  counting the number of samples around it.
• To do this efficiently, we splat the points
  into a voxel grid and convolve with a low-pass
  filter.

56
Non-Uniform Samples

• Approach
• Compute the sampling density at each sample by
  counting the number of samples around it.
• To do this efficiently, we splat the points
  into a voxel grid and convolve with a low-pass
  filter.
• We set ?i equal to the reciprocal of the value of
  the convolution at point pi.

57
Non-Uniform Samples

• Approach
• Compute the sampling density at each sample by
  counting the number of samples around it.
• To do this efficiently, we splat the points
  into a voxel grid and convolve with a low-pass
  filter.
• We set ?i equal to the reciprocal of the value
  convolution at point pi.

Non-Uniform Samples
Unweighted Reconstruction
Weighted Reconstruction
58
Completing the Characteristic Function
59
Completing the Characteristic Function

• In the generation of the characteristic function
  we use the functions Flmn with
• And we approximated an integral using the
  Monte-Carlo approximation

60
Completing the Characteristic Function

• In the generation of the characteristic function
  we use the functions Flmn with
• And we approximated an integral using the
  Monte-Carlo approximation

The function Flmn is not defined when lmn0.
So, we can only reconstruct the characteristic
function up to an additive constant additive.
61
Completing the Characteristic Function

• In the generation of the characteristic function
  we use the functions Flmn with
• And we approximated an integral using the
  Monte-Carlo approximation

To approximate the surface integral, we need to
know the area of the surface ?V. Since this is
not given, we can only reconstruct the
characteristic function up to a multiplicative
constant.
62
Completing the Characteristic Function

• To address the missing constants, we sample the
  reconstructed characteristic function at the
  sample points and compute the average value

63
Completing the Characteristic Function

• To address the missing constants, we sample the
  reconstructed characteristic function at the
  sample points and compute the average value
• Then we can set the values of the characteristic
  function to be

64
More Results
65
Results (Positional Noise)
disp0 res2563 tris141,808 time007 points100,
000
dispradius/64 res2563 tris134,848 time007 poi
nts100,000
dispradius/128 res2563 tris139,468 time007 po
ints100,000
dispradius/32 res2563 tris124,300 time007 poi
nts100,000
66
Results (Normal Noise)
angle0o res2563 tris141,808 time007 points10
0,000
angle30o res2563 tris141,876 time007 points1
00,000
angle15o res2563 tris141,776 time007 points1
00,000
angle45o res2563 tris142,048 time007 points1
00,000
67
Results (Holes)
Original
Reconstruction
res2563 tris267,736 time007 points25,000