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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 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 Q A, b, c nn , dn, d T Q v v Av 2b v c T T 2 Quadric Define Quadric of each vertex n v d T i i i 2 Qi v Qi v i i Pair Contraction Pair Selection Condition v1 , v2 is an edge or v1 v2 t , where t is a threshold When performing v1 , v2 v , Q Q1 Q2 Choose position of v minimizingQ v Q v 0 v A1b 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 targetv for each valid pair v1 , v2 Place all the pairs in a heap keyed on cost with the minimum cost pair at the top. Iteratively remove the pair v1 , v2 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 Result Original model with 69451 triangles triangles An approximation with 1000 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 v with neighbors v1 , v2 , , vk Find new location of v to satisfy some constraints, e.g. improving the angles of the triangles incident on v Vertex Relocation Map these vertices into a plane, v is mapped to the origin, v1 , v2 , , vk satisfy v vi 0 vinew ,1 i k The angles of all triangles at v are proportional to the corresponding angles and sum to 2 Vertex Relocation Let new position of v be the average of v1 , v2 , , vk to improve the angles of the adjacent faces Bring new position of v 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 1 , 2 , , i Area-based Remeshing k x, y arg min Ai x, y i A 2 i 1 Here Ai is the area of triangle p, pi , pi 1 , A is the area of polygon p1 , , pk Area-based 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 as1/ K v H 2 v 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 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: Vi x d ( x, z i ) d ( x, z j ) j 0,..., n 1, j i 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: zi x ( x)dx ( x)dx V V here ( x ) is a density function of Vi Centroidal Voronoi Diagram Centroidal Voronoi Diagrams minimize the Energy given as: E n 1 i 0 Vi ( x) x z i dx 2 On mesh, Energy above becomes to n 1 E 2 C V j j j i i 0 2 C j Vi j j C j Vi j 2 Construct CVD Here j Cj Cj xdx j area C j dx 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 Clustering Triangulate Sample Sample Result Quality and Speed Pros and Cons Pros High quality of result Optimization of original mesh Cons Slow Global Thanks