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A lightweight approach to repairing digitized polygon meshes
Marco Attene IMATI-GE / CNR
2010
Presented by Naitsat Alexander
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Desirable meshes
• Closed watertight mesh• Single oriented manifold• No degenerated elements
Type of defects :– Defects in topology– Defects in geometry
2
link: [2]
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Topological defects
3
[3]
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Geometrical defects
4
[3]
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Importance of mesh repairing
5
• Slicing for 3D printing • Partition into voxels
– Inside test for voxels
• Generation of tetrahedral meshes– Tetgen library
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Volumetric algorithms + More robust
- Produce a new mesh
Surface algorithms+ Less aggressive
- Can’t handle too much defects
6
Repairing methods
[3]
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Volumetric algorithms + More robust
- Produce a new mesh
Surface algorithms+ Less aggressive
- Can’t handle too much defects
7
Repairing methods
[6]
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Volumetric algorithms + More robust
- Produce a new mesh
Surface algorithms+ Less aggressive
- Can’t handle too much defects
8
Repairing methods
[3]
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General steps in repairing
9
Polygonal mesh
Triangulation &
Preprocessing Fixing topology Fixing
Geometry
Triangle mesh of closed
manifold
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Simplicial complexes
= set of subsets of {1,…n}
Realization of f:V R3
10
1
2
3
4
5
6
7
8
9
10
11
12
13 14
(1,5,6) (6,5,7) (9,6,8)l(6)=
l(6*)= {(1,5,6),(6,5,7),(9,6,8)}
V = {1,2,…,14}E = {{1,2},{1,3}, …,{1,6},
{2,3}…}F = {(1,2,3),(1,3,4),(1,4,2),…}
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Simplicial complexes
• Simplicial neighborhood of order k (k-ring)
11
0 ( )t N t 1( )N t 2 ( )N t
t t t
0
1
( )
( ) | share vertex of ( )k k
N S S
N S t F t N S
[1]
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Repairing topology
12
Polygonal mesh
Triangulation & Data init
Remove topological singularities
Orient meshno cuts
Triangle mesh of closed
manifold
Patch mesh holes
Keep the largest connected component
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Repairing topology
13
Polygonal mesh
Triangulation & Data init
Remove topological singularities
Orient meshno cuts
Triangle mesh of closed
manifold
Patch mesh holes
Keep the largest connected component
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Repairing topology
14
“Cutting and Stitching”
A. Gueziec, G. Taubin, F. Lazarus 2001
Polygonal mesh
Triangulation & Data init
Remove topological singularities
Orient meshno cuts
Triangle mesh of closed
manifold
Patch mesh holes
Keep the largest connected component
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v
Cutting and Stitching
1. Locating singularities• Singular edges• Singular isolated vertices
15
l(v)
e
rank(e)>2
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Cutting and Stitching
1. Locating singularities• Singular edges• Singular isolated vertices
2. Cutting• Global and local
16
e3
l(v2)
l(v2)
v1
v2
e2e1
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Local and Global cutting
17
local global
[4]
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Cutting and Stitching
1. Locating singularities• Singular edges• Singular isolated vertices
2. Cutting• Global and local
3. Stitching
18
v1
l(v2)
e1
e2
v2
l(v1)
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Stitching
19
• Pinching two adjacent edges that
were cut
[4]
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Repairing topology
20
Polygonal mesh
Triangulation & Data init
Remove topological singularities
Orient meshno cuts
Triangle mesh of closed
manifold
Patch mesh holes
Keep the largest connected component
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Repairing topology: long gaps
21
Polygonal mesh
Triangulation & Data init
Remove topological singularities
Orient meshno cuts Patch the
holes
Keep the largest connected component
Triangle mesh of closed
manifold
?
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Repairing topology
22
Polygonal mesh
Triangulation & Data init
Remove topological singularities
Orient meshno cuts
Triangle mesh of closed
manifold
Patch mesh holes
Keep the largest connected component
Stitch components
[5]
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Repairing topology
23
1. Orient the seed triangle
2. Propagate orientation to entire mesh
3. Cut through inconsistently orientated triangles
4. If cuts were necessary then terminate
Polygonal mesh
Triangulation & Data init
Remove topological singularities
Orient meshno cuts
Triangle mesh of closed
manifold
Patch mesh holes
Keep the largest connected component
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Repairing topology
24
Polygonal mesh
Triangulation & Data init
Remove topological singularities
Orient meshno cuts
Triangle mesh of closed
manifold
Patch mesh holes
Keep the largest connected component
P. Liepa. “Filling holes in meshes” 2003
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Filling holes in meshes
• Main stages of the algorithm1. Compute a coarse triangulation to fill a hole
• Based on Barequet et al. 1995
2. Refine and smooth triangulation to match the surrounding density and geometry
25
[3]
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Filling holes in meshes
Assumptions: • Holes are connected components of the boundary• Holes are simply connected polygons• No islands in holes
26
n verticesn-2 triangles
Self-intersections cannot be
excluded
Quality depends on boundary
distortion
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Filling holes in meshes
Weight w(T) of triangulation T
27
( ) ( )
( ) area( )
( ) maximal dihedral angle
mixture of dihedral angle and area
t Tw T w t
w t t
w t
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Filling holes in meshes
Weight w(T) of triangulation T
28
( ) ( )
( ) area( )
( ) maximal dihedral angle
mixture of dihedral angle and area
t Tw T w t
w t t
w t
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Filling holes in meshes
• w(i,j) = minimal weight in triangulating sub-polygon i,i+1,…,j
w(2,9) = ?
29
[3]
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Filling holes in meshes
30
• w(i,j) = minimal weight in triangulating sub-polygon i,i+1,…,j
w(2,9) = min ( w((2,3,9)) + w(3,9),
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Filling holes in meshes
31
• w(i,j) = minimal weight in triangulating sub-polygon i,i+1,…,j
w(2,9) = min ( w((2,3,9)) + w(3,9),
w(2,4)+w((2,4,9)) + w(4,9),
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Filling holes in meshes
32
• w(i,j) = minimal weight in triangulating sub-polygon i,i+1,…,j
w(2,9) = min ( w((2,3,9)) + w(3,9),
w(2,4)+w((2,4,9)) + w(4,9),w(2,5)+w((2,5,9)) + w(5,9),
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Filling holes in meshes
33
• w(i,j) = minimal weight in triangulating sub-polygon i,i+1,…,j
w(2,9) = min ( w((2,3,9)) + w(3,9),
w(2,4)+w((2,4,9)) + w(4,9),w(2,5)+w((2,5,9)) + w(5,9),w(2,6)+w((2,6,9)) + w(6,9),
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Filling holes in meshes
34
• w(i,j) = minimal weight in triangulating sub-polygon i,i+1,…,j
w(2,9) = min ( w((2,3,9)) + w(3,9),
w(2,4)+w((2,4,9)) + w(4,9),w(2,5)+w((2,5,9)) + w(5,9),w(2,6)+w((2,6,9)) + w(6,9),w(2,7)+w((2,7,9)) + w(7,9),
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Filling holes in meshes
• w(i,j) = minimal weight in triangulating sub-polygon i,i+1,…,j
w(2,9) = min ( w((2,3,9)) + w(3,9),
w(2,4)+w((2,4,9)) + w(4,9),w(2,5)+w((2,5,9)) + w(5,9),w(2,6)+w((2,6,9)) + w(6,9),w(2,7)+w((2,7,9)) + w(7,9),w(2,8)+w((2,8,9)) )
35
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Filling holes in meshes
• w(0,n) = minimal weight in triangulating the boundary
• Recursive formula
• O(n3) complexity
(Dynamic programming)
36
0
(0, ) min (0, ) (0, , ) ( , )
( , 1) 0m n
w n w m w m n w m n
w i i
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Filling holes : results
37
[3]
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Degeneracy removal
38
Mesh Mk←0
T← narrow trianglesk← k+1
Fixed mesh
Remove Nk(T) & Path holes
T Fix triangles &update T
k> kmax ?
T
Failure YesNo
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Repairing geometry
39
Mesh Mk←0
T← narrow trianglesk← k+1
Fixed mesh
Remove Nk(T) & Path holes
• Epsilon geometry to locate narrow triangles• Repairing by swap & collapse method
T Fix triangles &update T
k> kmax ?
T
Failure YesNo
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Swap and Collapse
• Nearly flat triangle with >-– Swap edge against
• Other narrow triangles with <– Collapse edge against
• Check if swap/collapse is legal
40
=arcsin(10-
5) [1]
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Degeneracy removal
41
Mesh Mk←0
T← narrow trianglesk← k+1
Fixed mesh
Remove Nk(T) & Path holes
• Maximal number of iterations = 3
T Fix triangles &update T
k> kmax ?
T
Failure YesNo
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Degeneracy removal
42
Mesh Mk←0
T← narrow trianglesk← k+1
Fixed mesh
Remove Nk(T) & Path holes
• Remove possible disconnected components• Next loop of repairing is made only for effected
triangles
T Fix triangles &update T
k> kmax ?
T
Failure YesNo
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Removal of self intersections
43
Mesh Mk←0 Init voxels Fixed
mesh
Remove Nk(T) & Path holes
T T← self intersecting trianglesk← k+1
k> kmax ?
T
Failure YesNo
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Removal of self intersections
44
Mesh Mk←0 Init/update voxels Fixed
mesh
Remove Nk(T) & Path holes
• Building 100x100x100 voxels• After 1st loop storing only voxels of effected
triangles
T T← self intersecting trianglesk← k+1
k> kmax ?
T
Failure YesNo
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Removal of self intersections
45
Mesh Mk←0 Init/update voxels Fixed
mesh
Remove Nk(T) & Path holes
• Maximal number of iterations = 3• Remove possible disconnected components
T T← self intersecting trianglesk← k+1
k> kmax ?
T
Failure YesNo
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Repairing geometry : flow
46
Mesh M
Degeneracy removal with
kmax= 3
Fixed mesh
M’
Self intersection removal with
kmax= 3
T← narrow triangles
T
T success
failure
maximum of 10 iterations
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Results
47
topology
geometry
[1]
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Results
48
geometrytopology
[1]
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Comparison
Other methods for mesh repairing
1. S. Bischoff and L. Kobbelt. “Structure preserving cad model repair” 2005
2. F. Nooruddin and G. Turk. “Simplification and repair of polygonal models using volumetric techniques” 2003
3. T. Ju. “Robust repair of polygonal models” 2004
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Comparison
50
11534 faces 12634 faces 17174 faces 110908 faces 363776 faces
Original Current method Method 1 Method 2 Method 3
[1]
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Failure cases
51[1]
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Improvements
• Parallel computations• More efficient search for self-intersection
– Adapting voxel size / octree resolution to the mesh
• Repairing topology and geometry in a single loop– There is overlap between the topology and the geometry
stages
52
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References
1) A lightweight approach to repairing digitized polygon meshes. Marco Attene.
2) Polygon Mesh Repairing: An Application Perspective. Marco Attene, Marcel Campen and Leif Kobbelt. 2012
3) Model Repair. Leif Kobbelt RWTH Aachen University .
4) Cutting and stitching : Converting sets of polygons to manifold surfaces. A. Gueziec, G. Taubin, F. Lazarus, and B. Horn.
5) Automatic mesh-healing technique for model repair and finite element model generation. C.S. Chong, A. Senthil Kumar, H.P. Lee.
6) A Practical Guide to Polygon Mesh Repairing. Marcel Campen, Marco Attene and Leif Kobbelt.
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Thank you
Questions ?
54