Triangular Mesh Decimation

Martin Franc, Václav Skalamarty|skala@kiv.zcu.czhttp://herakles.zcu.cz

University of West Bohemia in PlzenCzech Republic

Contents

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

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

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

Decimation

• General scheme– mesh element importance evaluation– element removal– arising “hole” triangulation

• Non-trivial triangulation in 3D

Decimation

• Vertex decimation– vertex topology assessment

– importance evaluation– the least vertex removal– triangulation

Decimation

• Edge decimation– edge importance evaluation– the least important edge removal– triangulation

• Patch decimation– patch importance evaluation– the least important removal– triangulation

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)

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

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

Data structure

• Winged edge modification

1 ,2 ,5,4 ,3 ,7,6n

ne igh bo urin g t rian g le s

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

First results

• Speedup Time ratio of various parts of the algorithm

0

1

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5

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8

9

0 1 2 3 4 5 6 7 8 9

number of threads

spee

du

p Imp. Eval.

Decimation

IDEAL CASE

Vertex Importance Evaluation

35%

Sort (Quick Sort)41%

Independent Set19%

Decimation5%

Vertex Importance Evaluation

49%

Sort (Quick Sort)25%

Independent Set11%

Decimation15%

Algorithm modification 1

• Vertices sorting removal– remove vertices under some importance threshold

only

kx

xy

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1.5 -1 -0.5 0 0.5 1 1.5

k=0.1k=0.5k=1

importance of a vertex (x)

maximum vertex importance in whole set

Algorithm modification 1

• Vertices evaluation• topology assessment• importance evaluation

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

Algorithm modification 1

• + Speedup• – Higher approximation error• – Independent set of vertices

Algorithm modification 2

• Hash function– basic function

– modification

0%

5%

10%

15%

20%

25%

30%

35%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

relative importance [0.1%]

per

cen

t o

f o

ccu

rren

ce

kx

xy

kx

xCy

Algorithm modification 2

• Idea

0

1

kx

xCy

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

a b

C1

1% 10% 100%

kCa

ak

1,)1(

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

Results

• Mentioned approaches comparison

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180

Teeth Bunny Horse Bone Hand Dragon Happy Buddha Turbine blade

model name

tim

e [s

ec]

Hash function Thresholding Sort

Results

• Time comparison (rough)

Test / machine SGI R10000 DELL 410 Precision k (DELL/SGI)SPECfp95 8.77 13.1 1.49SPECint95 10.1 17.6 1.74

Algorithm Reduction time from 69.451 to 1.000 triangles [sec]Presented algorithm 3.9 * k = 3.9* 1.49 = 5.81Garland & Heckbert 10.4Lindstrom & Turk 2585Hoppe 500JADE 325

Results

• Example of non-trivial data

Results

• Example of reduced data

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

Results

• Example of reduced data

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

Results

• Example of reduced data

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

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