Triangular Mesh Decimation

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Triangular Mesh Decimation

• Martin Franc, Václav Skala
• martyskala_at_kiv.zcu.cz
• http//herakles.zcu.cz
• University of West Bohemia in Plzen
• Czech Republic

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Contents

• Motivation
• Decimation
• Previous work
• Algorithm modification
• Results
• Conclusion

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Motivation

• fast and interactive visualization of large and
  complex data (CAD, 3D scanner, CT, MRI)
• reduction of the number of triangles preserving
  important details of the model

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Decimation

• W. Schroeder 1992
• Simplification methods based on a specific mesh
  element removal
• vertex decimation
• edge decimation edge contraction
• patch decimation vertex clustering
• Fast and simple method
• Can be generalized to 3D

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Decimation

• General scheme
• mesh element importance evaluation
• element removal
• arising hole triangulation
• Non-trivial triangulation in 3D

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Decimation

• Vertex decimation
• vertex topology assessment
• importance evaluation
• the least vertex removal
• triangulation

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Decimation

• Edge decimation
• edge importance evaluation
• the least important edge removal
• triangulation
• Patch decimation
• patch importance evaluation
• the least important removal
• triangulation

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Previous work

• Vertex and edge decimation combination
• vertices evaluation
• the least important vertex selection
• adjacent edges importance evaluation
• triangulation
• Parallelization
• independent set of vertices
• multithread programming (no critical sections)

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Previous work

• Edge contraction criterion
• minimal length
• minimal surface area after triangulation
• Independent set of vertices
• independent set of vertices
• super independent set of vertices

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Algorithm

• Vertices evaluation
• topology assessment
• importance evaluation
• The least important one search
• vertices sorting (according to their importance)
• super independent set creation
• Vertex removal
• edges evaluation (optimal edge selection)
• consistency check
• triangulation

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Data structure

• Winged edge modification

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Implementation

• Symmetrical multiprocessor (shared memory)
• Windows NT
• Threads
• no critical sections
• as many threads as free processors
• procedure
• get number of free processors (P)
• divide a set of vertices onto P parts
• run P threads, each with its own subset of
  vertices
• Parallel section
• vertices importance evaluation
• vertices removal
• No load balancing

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First results

• Speedup Time ratio of various parts
  of the algorithm

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Algorithm modification 1

• Vertices sorting removal
• remove vertices under some importance threshold
  only

importance of a vertex (x)
maximum vertex importance in whole set
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Algorithm modification 1

• Vertices evaluation
• topology assessment
• importance evaluation
• Vertices removal
• threshold function (bucketing instead of sorting)
• independent set of vertices
• edges evaluation
• triangulation

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Algorithm modification 1

• Speedup
• Higher approximation error
• Independent set of vertices

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Algorithm modification 2

• Hash function
• basic function
• modification

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Algorithm modification 2

• Idea

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Algorithm modification 2

• Function coefficients
• Application of the principle of indep. sets
• vertices impacted by previous decimation step are
  moved to the end of the cluster

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Algorithm modification 2

• Preprocessing
• vertices evaluation
• clusters creation (according the importance)
• Processing of the least important cluster
• vertex removal
• triangulation
• new evaluation of surrounded vertices which are
  moved to the end of proper cluster

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Results

• Mentioned approaches comparison

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Results

• Time comparison (rough)

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Results

• Example of non-trivial data

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Results

• Example of reduced data

871,414 triangles 430,000 triangles 87,000
triangles
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Results

• Example of reduced data

137,072 triangles 13,706 triangles
6,854 triangles 1,248 triangles
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Results

• Example of reduced data

58,328 triangles 29,000 triangles 6,000
triangles
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Conclusion

• Fast parallel algorithm for simplification of
  large triangular meshes
• Efficient sequential algorithm
• non-manifold meshes reduction
• simple triangulation function
• Future work
• triangulation method improvement
• decimation controlled by the approximation error
• volume decimation

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