High-Quality Simplification with Generalized Pair Contractions Pavel Borodin,* Stefan Gumhold, # Michael Guthe,* Reinhard Klein* *University of Bonn, Germany

High-Quality High-Quality Simplification with Simplification with Generalized Pair Generalized Pair

ContractionsContractions

Pavel Borodin,*Pavel Borodin,* Stefan Stefan GumholdGumhold,,##

Michael Guthe,Michael Guthe,** Reinhard Reinhard Klein*Klein*

*University of Bonn, Germany*University of Bonn, Germany##University of TUniversity of Tuebingenuebingen, ,

GermanyGermany

P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality SimplificatioP. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractionsn with Generalized Pair Contractions

University of Bonn University of Bonn Computer Graphics Group Computer Graphics Group22

OutlineOutline IntroductionIntroduction Related WorkRelated Work Generalized Pair Generalized Pair

ContractionsContractions Spatial Search Spatial Search Data Data

StructureStructure Simplification AlgorithmSimplification Algorithm Applications and ResultsApplications and Results ConclusionConclusion



IntroductionIntroduction Polygon models are widely Polygon models are widely

usedused Permanent growth of model Permanent growth of model

complexitycomplexity Reduction is often Reduction is often

necessarynecessary Many mesh simplification Many mesh simplification

algorithmsalgorithms

Original Original modelmodel(34834 (34834

vertices)vertices)

Simplified Simplified modelmodel(3483 (3483

vertices)vertices)

More More simplificatiosimplificatio

nn(1000 (1000

vertices)vertices)



IntroductionIntroduction Many models Many models

don't have don't have consistent consistent connectivityconnectivity T-verticesT-vertices self-self-

intersectionsintersections gaps and gaps and

small holessmall holes close, but not close, but not

connected connected surface partssurface parts



IntroductionIntroduction Problem: vertex pair Problem: vertex pair

contractions not always sew contractions not always sew together geometrically close but together geometrically close but not incident surface partsnot incident surface parts

Unnecessarily large complexity Unnecessarily large complexity of simplified models / of simplified models / unnecessarily large errorsunnecessarily large errors

Need of more general operationsNeed of more general operations

??



IntroductionIntroduction Problem: quadric error Problem: quadric error

metric does not preserve metric does not preserve sharp featuressharp features

SimplifiedSimplified200 vertices200 vertices

OriginalOriginal1972 vertices1972 vertices

QQff11

QQff22

ee QQee = Q= Qff11 + Q + Qff

22



Edge collapse simplificationEdge collapse simplification[[Hoppe 1996]Hoppe 1996]

contracts twocontracts two adjacent adjacent verticesvertices performs no topology alterationperforms no topology alteration

Related WorkRelated Work

vvnewnewvv22

vv11



Vertex pair contraction Vertex pair contraction simplificationsimplification

[Popovic and Hoppe[Popovic and Hoppe / / Garland and Garland and HeckbertHeckbert 1997] 1997]

the contracted vertices do not the contracted vertices do not necessarily lie on a common edgenecessarily lie on a common edge

allows topology modificationsallows topology modifications sews together unconnected parts and sews together unconnected parts and

closes small gapscloses small gaps


vvnewnew

vv11vv22



Vertex-edge contractions on Vertex-edge contractions on boundariesboundaries

[[Borodin et al. 2002 - Progressive Gap Borodin et al. 2002 - Progressive Gap Closing]Closing]

generalizes generalizes vertex pair vertex pair contractioncontraction contracts boundary vertex with contracts boundary vertex with

boundary edgeboundary edge improves the sewing potential of improves the sewing potential of

vertex pair contractionvertex pair contraction


vvnewnew

vvvvintint

vv

ee



Generalized Pair Generalized Pair ContractionsContractions

Three new operations:Three new operations: vertex-edge contractionvertex-edge contraction vertex-triangle contractionvertex-triangle contraction edge-edge contractionedge-edge contraction

Sufficient to connect the Sufficient to connect the closest closest points of two objects points of two objects or parts of an objector parts of an object

PPerform no reduction, buterform no reduction, butincrease the connectedness increase the connectedness of the modelof the model




Vertex-edge contractionVertex-edge contraction

project the vertex onto the project the vertex onto the corresponding edgecorresponding edge

insert an insert an intermediateintermediate vertex at the vertex at the projection pointprojection point ( (split split the edge and the edge and all all triangletriangles incident to its incident to it))

contract intermediate and contract intermediate and contraction verticescontraction vertices



Vertex-triangle contractionVertex-triangle contraction

project the vertex onto the project the vertex onto the corresponding corresponding triangletriangle

insert an insert an intermediateintermediate vertex at vertex at the projection pointthe projection point ( (split split the the triangle)triangle)

contract intermediate and contract intermediate and contraction verticescontraction vertices





Edge-edge contractionEdge-edge contraction

find the closest pointsfind the closest points on the on the corresponding edgecorresponding edgess

insert two insert two intermediateintermediate verticesvertices at at thesethese point pointss((split split both edges and all both edges and all triangletriangles s incident to themincident to them))

contract two intermediate verticescontract two intermediate vertices




Ordering of operationsOrdering of operations Quadric error metric Quadric error metric [[Garland Garland

& Heckbert& Heckbert 1997] 1997] Problem: no control of the Problem: no control of the real real geometric errorgeometric error

Problem: Problem: error accumulation error accumulation at at flat regions with noiseflat regions with noise

vv00 vv11 vv44 vv55 vv88 vv99

vv22 vv33 vv66 vv77

vv00 vv11 vv44 vv55 vv88 vv99

vv1010vv1111

QQ10 10 = Q= Q22 + Q + Q33

QQ1111 = Q = Q66 + Q + Q77

QQ1212 = Q= Q11 + Q + Q1010

QQ1313 = Q = Q55 + Q + Q1111

QQ1414 = Q = Q1212 + Q + Q4 4 = =

= = QQ11 + Q + Q22 + Q+ Q33 + Q+ Q44

QQ1515 = Q = Q1313 + Q + Q88 = =

= = QQ55 + Q + Q66 + Q+ Q77 + Q+ Q88

vv00 vv99

vv1414 vv1515

vv00 vv44 vv88 vv99

vv1212 vv1313




Ordering of operationsOrdering of operations Non-accumulating Non-accumulating error error

quadricsquadrics Preprocessing phase:Preprocessing phase:calculate initial error quadrics for calculate initial error quadrics for all verticesall vertices

Intermediate vertex created:Intermediate vertex created:calculate its quadric from the calculate its quadric from the incident trianglesincident triangles

Contraction operation performed:Contraction operation performed:calculate quadric of a new vertex,calculate quadric of a new vertex,recalculate quadrics of all recalculate quadrics of all adjacent verticesadjacent vertices




Ordering of operationsOrdering of operations Error testError test before the operation: before the operation:

calculate the local one-sided calculate the local one-sided Hausdorff distanceHausdorff distance after the after the operation from the original to the operation from the original to the simplified meshsimplified mesh

if this error exceeds the predefined if this error exceeds the predefined threshold threshold ddmaxmax, the operation is , the operation is rejectedrejected

Guarantees maximum geometric Guarantees maximum geometric distance between original and distance between original and simplified modelsimplified model




Ordering of operationsOrdering of operations Approximation of the Hausdorff distance is Approximation of the Hausdorff distance is

used to order the possible operations:used to order the possible operations:

- accumulated errors of simplices - accumulated errors of simplices ss11 and and ss22

- local one-sided Hausdorff - local one-sided Hausdorff distancedistance before the operation before the operation

- quadric error of the operation- quadric error of the operation - approximation of the distance - approximation of the distance

betweenbetween meshes before and after the meshes before and after the operationoperation

( ) 1/ 21 2

s s s qd = max d ,d + e

( )1 2s smax d ,d

1 2s sd ,d

qe1/ 2 qe




Handling of sharp featuresHandling of sharp features Feature edgesFeature edges and and feature feature

verticesvertices: all incident faces are : all incident faces are inside chosen small angle inside chosen small angle maxmax

For each feature:For each feature: generategenerate average average plane plane PPaa of all of all

incident facesincident faces generategenerate constraint constraint plane plane PPcc running running

through the feature and through the feature and perpendicular to perpendicular to PPaa

add the quadric of add the quadric of PPcc to the quadrics to the quadrics of of appropriateappropriate vertices vertices

<< maxmax

PPaa PPaa

PPcc




Handling of sharp featuresHandling of sharp features

200 vertices, features 200 vertices, features not preservednot preserved

200 vertices,200 vertices, features features preservedpreserved

Original with detected Original with detected featuresfeatures



Spatial Search Spatial Search Data Data StructureStructure

Queries:Queries: for given vertex find closest non-for given vertex find closest non-incident simplexincident simplex

for given edge find closest non-for given edge find closest non-incident edgeincident edge

need of spatial search need of spatial search data data structurestructure

During simplification simplices are During simplification simplices are eliminated and created eliminated and created support of support of fast dynamic updatesfast dynamic updates

Regular gridRegular grid average edge length as cell side average edge length as cell side lengthlength



Simplification AlgorithmSimplification Algorithm Preprocessing: Preprocessing:

Initialization of the gridInitialization of the grid Identification of corresponding Identification of corresponding

pairspairs For For eacheach vertex vertex the corresponding the corresponding simplexsimplex(vertex, edge or triangle) is found(vertex, edge or triangle) is found

For For somesome edges edges the corresponding the corresponding edge is foundedge is found

The references to the found The references to the found simplices are storedsimplices are stored

Pairs are inserted into the Pairs are inserted into the priority queuepriority queue according to according to their errorstheir errors



Simplification AlgorithmSimplification Algorithm Decimation loopDecimation loop

Pop next operation from the Pop next operation from the priority queuepriority queue

Before the operation:Before the operation: normal testnormal test minimal angle testminimal angle test error testerror test collision testcollision test

After the operation:After the operation: update the gridupdate the grid update all affected simplex pairsupdate all affected simplex pairs update the priority queueupdate the priority queue



Applications and ResultsApplications and Results Controlled topology Controlled topology

modifying simplificationmodifying simplificationand mesh repairand mesh repair

Original model with Original model with 4288 vertices4288 verticesModel courtesy of Model courtesy of

DaimlerChrysler AGDaimlerChrysler AG





500 vertices500 verticesvertex pair vertex pair

contractionscontractions

500 vertices500 verticesgeneralized pair generalized pair








125 vertices125 verticesgeneralized pair generalized pair




Applications and ResultsApplications and Results

4467 vertices4467 verticesoriginaloriginal

250 vertices250 verticesgener. pair gener. pair






Applications and ResultsApplications and Results Out-of-core simplificationOut-of-core simplification

Cutting the model into subparts Cutting the model into subparts and simplifying each subpart and simplifying each subpart independentlyindependently

Gaps are automatically closed Gaps are automatically closed when subparts are simplified when subparts are simplified togethertogether

Vertex pair Vertex pair contractionscontractions

Generalized pair Generalized pair contractionscontractions



Applications and ResultsApplications and Results

OriginalOriginal1 087 716 1 087 716

facesfaces

OEMM-OEMM-QEMQEM

18 338 18 338 facesfaces

Stream Stream decimationdecimation

18 486 18 486 facesfaces

Our Our methodmethod18 338 18 338 facesfaces




OriginalOriginal1 087 716 1 087 716

facesfaces

OEMM-QEMOEMM-QEM18 338 faces18 338 faces

error = error = 0.82%0.82%


18 486 faces18 486 faceserror = error = 0.82%0.82%

Our methodOur method18 338 faces18 338 faces

error = error = 0.26%0.26%




Error quadrics are not Error quadrics are not accumulatedaccumulated

better simplification of flat better simplification of flat regions with noiseregions with noise

OEMM-QEMOEMM-QEM18 338 faces18 338 faces

error = 0.82%error = 0.82%


18 486 faces18 486 faceserror = 0.82%error = 0.82%

Our methodOur method18 338 faces18 338 faces

error = 0.26%error = 0.26%



ConclusionConclusion A strategy to generate high A strategy to generate high

quality approximations for models quality approximations for models with arbitrary topologywith arbitrary topology

Generalized pair contraction Generalized pair contraction operations introducedoperations introduced

Allow to remove gaps and holes Allow to remove gaps and holes and integrate the automatic and integrate the automatic connection of close surface partsconnection of close surface parts

Guaranteed maximum geometric Guaranteed maximum geometric distance between original and distance between original and simplified modelsimplified model

Simplification controlled by a Simplification controlled by a geometric errorgeometric error



Future Work (already Future Work (already done)done)

Intersection free Intersection free simplificationsimplification

Out-of-core simplificationOut-of-core simplification View-dependent out-of-core View-dependent out-of-core

visualizationvisualization



Thank youThank you

Documents

High-Quality Simplification with Generalized Pair Contractions Pavel Borodin,* Stefan Gumhold, # Michael Guthe,* Reinhard Klein* *University of Bonn, Germany