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Mesh Simplification

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Mesh Simplification

Simplifier12,000 2,000 300

3D

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Motivation

Reduce information contentAccelerate renderingMulti-resolution models

size

error

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Refined mesh for close objectsSimplified mesh for far

Level of Detail (LOD)

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Performance RequirementsOffline

Generate model at given level(s) of detailEmphasis on quality

Real-timeGenerate model at given level(s) of detailEmphasis on speedRequires preprocessingTime/space/quality tradeoff

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Quality (e.g. Crease Preservation)

original

(12,946 faces)

192 faces 1,070 faces PM (200 faces) PM (1,000 faces)

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MethodologySequence of local operations

Involve near neighbors - only small patch affected in each operationEach operation introduces error Find and apply operation which introduces the least error

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Simplification Operations (1)Decimation

Vertex removal: v ← v-1f ← f-2

Remaining vertices are subset of original vertex set

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Simplification Operations (2)Decimation

Edge collapsev ← v-1f ← f-2

Triangle collapsev ← v-2f ← f-4

Vertices may move

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Simplification Operations (3)Contraction

Pair contraction

Cluster contraction(set of vertices)

Vertices may move

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Error ControlLocal error: Compare new patch with previous iteration

FastAccumulates error Memory-less

Global error: Compare new patch with original meshSlowBetter quality control Can be used as termination conditionMust remember the original mesh throughout the algorithm

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Local vs. Global Error

488 faces2000 faces 488 faces

Original GlobalLocal

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Σα / 2π

Simplification Error MetricsMeasures

Distance to planeCurvature

Usually approximatedAverage planeDiscrete curvature

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The Basic AlgorithmRepeat

Select the element with minimal errorPerform simplification operation (remove/collapse)Update error (local/global)

Until mesh size / quality is achieved

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Implementation Details

Vertices/Edges/Faces data structure

Easy access from each element to neighboring elements

Use priority queue (e.g. heap)

Fast access to element with minimal errorFast update

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Vertex Removal Algorithm

Simplification operation: Vertex removal

Error metric: Distance to average plane

May preserve mesh features (creases)

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Algorithm Outline

Characterize local topology/geometryClassify vertices as removable or notRepeat

Remove vertexTriangulate resulting holeUpdate error of affected vertices

Until reduction goal is met

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Characterizing Local Topology/Geometry

SimpleSimple

BoundaryBoundary

ComplexComplex

InteriorInterior

CornerCorner

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Decimation CriterionEmax – user defined parameterSimple vertex:

Distance of vertex to the face loop average plane < Emax

Boundary vertices:Distance of the vertex to the new boundary edge < Emax

DistanceDistance

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Triangulating the HoleVertex removal produces non-planar loop

Split loop recursivelySplit plane orthogonal to the average plane

Control aspect ratio of triangles3D triangulation may be invalid

Vertex is not removed

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Example

Simplifier

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Pros and ConsPros:

EfficientSimple to implement and use

Few input parameters to control qualityReasonable approximationWorks on very large meshesPreserves topologyVertices are a subset of the original mesh

Cons:Error is not bounded

Local error evaluation causes error to accumulate

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Edge Collapse Algorithm - QSlim

Simplification operation:Pair contraction

Error metric:Distance, Pseudo-global

Simplifies also topology

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Distance Metric: Quadrics

Choose point closest to set of planes (triangles)

Sum of squared distances to set of planes is quadratic ⇒has a global minimum

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QuadricsPlane

Ax + By + Cz + D = 0, where A2 + B2 + C2 = 1p = [A, B, C, D], v = [x, y, z, 1], v pT = 0

Squared distance between v and p:∆p(v) = (v pT)2

= (v pT) (p vT) = v (pTp) vT

= v KP vT

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⎤

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⎣

⎡

2

2

2

2

DCDBDADCDCBCACBDBCBABADACABA

KP =

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Distance to Set of Planes

Tv

T

planes(v)pp

planes(v)p

Tp

planes(v)pp

vQv

v)K( v

)vK (v

)v()v(

=

=

=

∆=∆

∑

∑

∑

∈

∈

∈

After v1, v2 are contracted to v,Qv ← Qv1+Qv2

Pseudo-global

All original planes persist during the entire simplification process

V1 V2 V

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Contracting Two VerticesGoal: Given edge e = (v1,v2), find contracted

v = (x,y,z) that minimizes ∆(v):

∂∆/∂x = ∂∆/∂y = ∂∆/∂z = 0

Solve system of linear normal equations:

If no solution - select the edge midpoint

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

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1000

v

1000qqqqqqqqqqqq

34333231

24232221

14131211

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Selecting Valid Pairs for Contraction

Edges:{(v1,v2) : (v1v2) is in the mesh}

Close vertices:{(v1,v2) : ||v1 - v2|| < T}

Threshold T is input parameter

T

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AlgorithmCompute QV for all the mesh vertices Identify all valid pairsCompute for each valid pair (v1,v2) the contracted vertex v and its error ∆(v)Store all valid pairs in a priority queue (according to ∆(v))While reduction goal not met

Contract edge (v1,v2) with the smallest ∆(v)Update the priority queue with new valid pairs

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ExamplesDolphin (Flipper)

Original - 12,337 faces 2,000 faces 300 faces (142 vertices)

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ExamplesBuddha

Simplifier

Original - 12,000 2,000 faces 298 faces (140 vertices)

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Pros and ConsPros

Error is boundedAllows topology simplificationHigh quality resultQuite efficient

ConsDifficulties along boundariesDifficulties with coplanar planesIntroduces new vertices not present in the original mesh

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View-Dependent Simplification

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View-Dependent RefinementProblem:

While rendering for each viewpoint there are always hidden faces Using traditional simplification techniques those faces rendered in the same LOD as complete mesh

Outside View-Frustum

Backface

Distant

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View-Dependent RefinementGoal:

Generate progressive representation of mesh s.t. only some faces are simplified & others are fully detailed

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Refinement CriteriaView-Frustum

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τ = 0% τ = 0.33%

Refinement CriteriaScreen-space geometric error

Refine mesh only if distance between approximated & original surfaces when projected on the screen is larger than screen-space tolerance τ

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Simplification Algorithms

Mn 13,133

Mc1

5038,903

Mc2 Mc3

10,103

489

Continuous

Mn

13,133

M0

5038,90310,103

Mc1 Mc2 Mc3

Selective

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Examples

Original Simplified Top-View

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Examples (cont’)

Original Simplified Top-View