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## PowerPoint Slideshow about ' Mesh Coarsening' - kay-mccormick

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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

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

- Define Quadric of each vertex

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

- 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

- Efficiency, local, extremely fast
- Quality, maintain high fidelity to the original mesh
- Generality, can join unconnected regions of original mesh together

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

- let N be the average of all Ci

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

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 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

- (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

Area-based Remeshing

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

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

- 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

- 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 Diagram

- Centroidal Voronoi Diagrams minimize the Energy given as:
- On mesh, Energy above becomes to

Construct CVD

- Here
- Construct CVD based on global minimization of the Energy term E2

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
- High quality of result
- Optimization of original mesh
- Cons
- Slow
- Global

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