Mesh Coarsening
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Mesh Coarsening
zhenyu shu
2007.5.12
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Mesh Coarsening
Large meshes are commonly used in numerous application area
Modern range scanning devices are used High resolution mesh model need more time
and more space to handle Large meshes need simplification to improve
speed and reduce memory storage
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Mesh Coarsening
Size, quality and speed
Mesh optimization
Many simplification methods now
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QEM
Garland M, Heckbert P. Surface simplification using quadric error metrics. In: Proceedings of the Computer Graphics, Annual Conference Series. Los Angeles: ACM Press, 1997. 209~216
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QEM Quadric Error Metric method Using Pair Contraction to simplify the mesh Minimize Quadric function when contracting Define Quadric
2, , , ,
2
T
T T
Q A b c nn dn d
Q v v Av b v c
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Quadric
Define Quadric of each vertex
2Ti i i i
i i i
n v d Q v Q v
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Pair Contraction
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Pair Selection
Condition is an edge or , where t is a threshold
When performing , Choose position of minimizing
If A is not invertible, choose among two endpoints and midpoint of two endpoints
1 2,v v
1 2v v t
1 2,v v v 1 2Q Q Q
v Q v
10Q v v A b
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Algorithm Summary
Compute the Q matrices for all the initial vertices. Select all valid pairs. Compute the optimal contraction target for each v
alid pair Place all the pairs in a heap keyed on cost with the
minimum cost pair at the top. Iteratively remove the pair of least cost from
the heap, contract this pair, and update the costs of all valid pairs involving v1.
v 1 2,v v
1 2,v v
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Advantage
Efficiency, local, extremely fast
Quality, maintain high fidelity to the original mesh
Generality, can join unconnected regions of original mesh together
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Result
Original model An approximation
with 69451 triangles with 1000 triangles
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Topology manipulation
Hattangady N V. A fast, topology manipulation algorithm for compaction of mesh/faceted models[J]. Computer-Aided Design. 1998, 30(10): 835-843.
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Edge collapsing
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Edge swapping
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Edge smoothing
let N be the average of all Ci
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Data Structure of mesh model
A type of
data structure to
present mesh
model for
reference
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Remeshing
Surazhsky V, Gotsman C. Explicit surface remeshing[C]. Aachen, Germany: Eurographics Association, 2003
Improve mesh quality by a series of local modification of the mesh geometry and connectivity
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Vertex Relocation
with neighbors
Find new location of to satisfy some constraints, e.g. improving the angles of the triangles incident on
v 1 2, , , kv v v
v
v
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Vertex Relocation
Map these vertices into a plane, is mapped to the origin, satisfy
The angles of all triangles at are proportional to the corresponding angles and sum to
v1 2, , , kv v v
0 ,1i inewv v v i k
v
2
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Vertex Relocation
Let new position of be the average of
to improve the angles of the adjacent faces
Bring new position of back to the original surface by maintain same barycentric coordinate
v1 2, , , kv v v
v
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Detail
(c) is original mesh, (b) is new mesh, (d) is 2D mesh which defines a parameterization of (c)
Use the same barycentric coordinates in (a) and (d)
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Area-based Remeshing
Area equalization is done iteratively by relocating every vertex such that the areas of the triangles incident on the vertex are as equal as possible
Extending method above to relocating vertices such that the ratios between the areas are as close as possible to some specified values 1 2, , , i
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Area-based Remeshing
Here is the area of triangle , is the area of polygon
2
1
, arg min ,k
i ii
x y A x y A
iA A1, ,i ip p p
1, , kp p
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Area-based Remeshing
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Curvature sensitive remeshing More curved region contain small triangles an
d a dense vertex sampling, while almost flat regions have large triangles
Define density function as
here K and H are approximated discrete Gaussian and mean curvatures
Meyer M, Desbrun M, Schroder P, et al. Discrete differential geometry operator for triangulated 2-manifolds [A]. In: Proceedings of Visual Mathe
matics'02, Berlin, 2002. 35~ 57
21/ K v H v
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Result
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Result
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CVD
Valette S, Chassery J M. Approximated Centroidal Voronoi Diagrams for Uniform Polygonal Mesh Coarsening[J]. Computer Graphics Forum. 2004, 23(3): 381-389
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Voronoi Diagram
Given an open set of Rm, and n different points zi; i=0,...,n-1, the Voronoi Diagram can be
defined as n different regions Vi such that:
where d is a function of distance.
ijnjzxdzxdxV jii ,1,...,0),(),(
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Centroidal Voronoi Diagram
A Centroidal Voronoi Diagram is a Voronoi Diagram where each Voronoi site zi is also the
mass centroid of its Voronoi Region:
here is a density function of
( )
( )V
i
V
x x dxz
x dx
( )x iV
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Centroidal Voronoi Diagram
Centroidal Voronoi Diagrams minimize the Energy given as:
On mesh, Energy above becomes to
1
0
2)(
n
iV
ii
dxzxxE
2
1 22
0
j i
j i
j i
n j jC V
j jC Vi jC V
E
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Here
Construct CVD based on global minimization of the Energy term E2
Construct CVD
j
j
C
j
C
xdx
dx
j jarea C
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Algorithm Summary
Randomly choose n different cells in mesh and these cells form n regions
Cluster all cells in mesh by extending these regions and choosing correct cells’ owner to minimize the energy term E2
Now calculate each center of these regions and replace each region with it’s center
Triangulate and get new mesh
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Clustering
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Triangulate
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Sample
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Sample
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Result Quality and Speed
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Pros and Cons
Pros High quality of result Optimization of original mesh
Cons Slow Global
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Thanks