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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

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Mesh coarsening

Mesh Coarsening

zhenyu shu

2007.5.12


Mesh coarsening1
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 coarsening2
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


Quadric
Quadric

  • Define Quadric of each vertex



Pair selection
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


Algorithm summary
Algorithm Summary

  • Compute the Q matrices for all the initial vertices.

  • Select all valid pairs.

  • Compute the optimal contraction target for each valid pair

  • Place all the pairs in a heap keyed on cost with the minimumcost 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.


Advantage
Advantage

  • Efficiency, local, extremely fast

  • Quality, maintain high fidelity to the original mesh

  • Generality, can join unconnected regions of original mesh together


Result
Result

Original model An approximation

with 69451 triangles with 1000 triangles


Topology manipulation
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 smoothing
Edge smoothing

  • let N be the average of all Ci


Data structure of mesh model
Data Structure of mesh model

  • A type of

    data structure to

    present mesh

    model for

    reference


Remeshing
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
Vertex Relocation

  • with neighbors

  • Find new location of to satisfy some constraints, e.g. improving the angles of the triangles incident on


Vertex relocation1
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


Vertex relocation2
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


Detail
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-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


Area based remeshing1
Area-based Remeshing

  • Here is the area of triangle , is the area of polygon



Curvature sensitive remeshing
Curvature sensitive remeshing

  • More curved region contain small triangles and 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 Mathematics'02, Berlin, 2002. 35~57




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
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 Visuch that:

    where d is a function of distance.


Centroidal voronoi diagram
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


Centroidal voronoi diagram1
Centroidal Voronoi Diagram

  • Centroidal Voronoi Diagrams minimize the Energy given as:

  • On mesh, Energy above becomes to


Construct cvd
Construct CVD

  • Here

  • Construct CVD based on global minimization of the Energy term E2


Algorithm summary1
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







Pros and cons
Pros and Cons

  • Pros

    • High quality of result

    • Optimization of original mesh

  • Cons

    • Slow

    • Global



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