Mesh Coarsening

zhenyu shu

2007.5.12

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

Mesh Coarsening

Size, quality and speed

Mesh optimization

Many simplification methods now

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

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

Quadric

Define Quadric of each vertex

2Ti i i i

i i i

n v d Q v Q v

Pair Contraction

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

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

Advantage

Efficiency, local, extremely fast

Quality, maintain high fidelity to the original mesh

Generality, can join unconnected regions of original mesh together

Result

Original model An approximation

with 69451 triangles with 1000 triangles

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.

Edge collapsing

Edge swapping

Edge smoothing

let N be the average of all Ci

Data Structure of mesh model

A type of

data structure to

present mesh

model for

reference

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

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

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

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

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)

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

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

Area-based Remeshing

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

Result

Result

CVD

Valette S, Chassery J M. Approximated Centroidal Voronoi Diagrams for Uniform Polygonal Mesh Coarsening[J]. Computer Graphics Forum. 2004, 23(3): 381-389

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),(),(

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

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

Here

Construct CVD based on global minimization of the Energy term E2

Construct CVD

j

j

C

j

C

xdx

dx

j jarea C

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

Clustering

Triangulate

Sample

Sample

Result Quality and Speed

Pros and Cons

Pros High quality of result Optimization of original mesh

Cons Slow Global

Thanks