• CAD/CAM
• Visualization
• Unoriented polygons
• Intersecting polygons
• Collision detection
• Lighting simulation
• Missing polygons
Courtesey of Steve Fortune
Challenges
Consistent
Solid and Boundary
Representations
• T−junctions
• Unconnected polygons
• Overlapping polygons
Model may be non−manifold:
Arbitrary
Set of Polygons
?
Aim to reconstruct consistent solid and boundary
representations for the objects modeled by a
set of polygons.
• Physical simulation
• Finite element methods
Applications
Thomas A. Funkhouser
Princeton University
Reconstructing
Consistent 3D Models
From Polygon Soup
Goal
Baum et al. ‘91
• Solid region labeling
• Solid region labeling
• Baum et al. ‘91
• Bohn & Wozny ‘93
• Makela & Dolenc ‘93
• Sheng & Meier ‘95
• Baraquet & Sharir ‘95
• Butlin & Stops ‘96
• Baraquet & Kumar ‘97
• Gueziec et al. ‘97
• Solid region labeling
• Boundary resampling
• Boundary stitching
• Szeliski ‘93
Three Approaches
• Boundary resampling
• Boundary resampling
Boundary Stitching
• Boundary stitching
Three Approaches
• Boundary stitching
Three Approaches
Input
Cell Regions
1) Spatial
Subdivision
Three Steps:
Cell Solidities
2) Solid
Determination
Murali et al. ‘97
Szeliski et al. ‘93
Output
3) Model
Output
Solid Region Labeling
Boundary Resampling
A
7
C
D
3
1
E
F
2
5
G
4
Binary Space Partition
B
6
Input Polygons
H
8
J
Spatial
Subdivision
G
G
B
F
B
7
1
H
2
A
D
J
H
A
D
J
Cell Regions
4
3
6
E
E
C
Cell Adjacency Graph
F
5
8
C
• Thibault & Naylor ‘87
• Teller ‘92
• Murali et al. ‘97
Spatial Subdivision
• Solid region labeling
• Boundary resampling
• Boundary stitching
Three Approaches
D
A
B
H
F
E
D
A
B
J
F
H
E
C
M
x
SA
SB
SC
SD
SE
SF
SG
SH
SJ
=
b
Mi,j = oi,j − ti,j
bi = Σk (ti,j − oi,j) for all k
where Ck is unbounded
positive
• negative
•
Mi,i = Ai
Linear system of equations:
G
C
Solid Determination
G
J
• Unbounded cells are not solid
• Adjacent cells sharing a transparent boundary
should have the same solidity
• Adjacent cells sharing an opaque boundary
should have opposite solidities
Intuition:
Solid Determination
Σj
Ai
M
x
SA
SB
SC
SD
SE
SF
SG
SH
SJ
=
b
These properties imply
that M has an inverse,
and the system of
equations is solvable!
• Mi,i > 0
• Mi,j > 0.0 indicates Li,j is mostly opaque
• Mi,j < 0.0 indicates Li,j is mostly transparent
• M is weakly diagonal dominant, Mi,i ≥ Σj |Mi,j|
• M is symmetric, Mi,j = Mj,i
Linear system of equations:
Solid Determination
Si = −1.0
Si = Solidity of cell Ci [−1.0 : 1.0]
Li,j = Boundary between Ci and Cj
ti,j = transparent area of Li,j
oi,j = opaque area of Li,j
Ai = Σj (ti,j + oi,j)
(ti,j − oi,j) Sj
• For unbounded cells:
Si =
• Cell ‘‘solidity’’ relationship for bounded cells:
Formalism:
Solid Determination
D
A
F
B
H
Cell Solidities
Input Model
Honda Clutch
G
J
E
C
Cell Solidities
Input Model
Telephone
Results
Cell Solidities
Input Model
Building
Output a polygon for each
boundary separating a solid
cell from a non−solid cell
(oriented away from solid)
Model Output
Solid Cells
Summary
Cell Solidities
Output Model
+ Global solution
+ Finds topology automatically
+ Constructs consistent solid representation
− Does not work for non−physical objects
− Depends on spatial subdivision constructed
Solid region labeling:
+ Global solution
+ Finds topology automatically
+ Works for non−physical objects
− Approximate reconstruction
Boundary resampling:
+ Relatively simple, local operations
+ Works for non−physical objects
− Uses heuristics based on distance tolerances
− Does not find global solution
Boundary stitching:
Input Model
Results
• Group vertices, edges, polygons
• Split intersecting polygons
• Merge coplanar polygons
• Subdivide large polygons
• Example: Baum et al. ‘91
• Apply sequence of filters
General Idea