The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Interactive Visualization and Collision Detection using Dynamic Simplification and Cache-Coherent Layouts

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL

Interactive Visualization and Collision Detection using Dynamic Simplification and Cache-Coherent Layouts

Sung-Eui YoonDissertation defense talk

Advisor: Prof. Dinesh Manocha

2 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL

Goal

• Design algorithms for interactive visualization and collision detection between massive models


Interactive Visualization

• Walkthrough♦ large man-made structures

• Investigate scientific simulation data


Collision Detection

• Main component of:♦ Dynamic simulation♦ Navigation and path planning♦ Haptic rendering♦ Virtual prototyping


Main Requirements

• Generality♦ Handle any kind of polygonal models♦ (e.g., CAD, scanned, isosurface

models)

• Interactivity♦ Provide at least 10 frames per

second


Challenges

• Complex and massive models♦ Ever-increasing model complexity

Bunny model,70K St. Matthew,

372M (10GB)

Puget sound,400M+


Challenges


Power plant, 12M

Double eagle tanker, 82M

Boeing 777,350M


Challenges


472M

Isosurface from a turbulence simulation

(LLNL)


Different Approaches

• Approximate algorithms• Cache-coherent algorithms• Output-sensitive algorithms• Parallel algorithms




Use simplification while minimizing error




CPU or GPU

DiskCaches Memory

Minimize cache misses




ViewerVisible

trianglesInvisible triangles






• Approximate algorithms♦ Dynamic simplifications

• Cache-coherent algorithms♦ Cache-oblivious layouts

• Output-sensitive algorithms• Parallel algorithms, etc


View-Dependent Rendering• [Clark 76, Funkhouser and Sequin

93]

• Static level-of-details (LODs)• Dynamic (or view-dependent)

simplification

ViewerObject

Lower resolution


Static LODs

50,00050,000 facesfaces10,000 faces10,000 faces2,000 faces2,000 faces

poppop poppop

Courtesy of [Hoppe 97]


Courtesy of [Hoppe 97]

Dynamic Simplification

• Provides smooth and varying LODs over the mesh

1st person’s view 3rd person’s view


Dynamic Simplification

• Progressive mesh [Hop96]• View-dependent progressive

meshes [Hoppe 97]• Merge tree [Xia and Varshney 97]• Octree-based vertex clustering

[Luebke and Erikson 97]• View-dependent tree [El-Sana

and Varshney 99]• Out-of-core approaches [Decoro

and Pajarola 02; Lindstrom 03]


Vertex Hierarchy

Edge CollapseVa

Vb

VaVc

Vertex Split

Vertex Hierarchy

Vc

Va Vb


View-Dependent Refinement

Vertex Hierarchy

Front representing a LOD mesh


Dynamic Simplification: Issues

• Representation• Construction• Runtime computation• Integration with other

acceleration techniques





22+GB for 100M triangles [Hoppe 97]









Rendering throughput of 3M triangles per sec

[Lindstrom 03]





GPUSmall vertexcache

Many cache misses






Parallel View-Dependent Rendering [Baxter et al. 02]

Static LODs

SGI RealityMonster

32 processors8 GPUs16GB RAM

(3)(2)


Live Demo – View-Dependent Rendering

Pentium4

GeForce Go 6800 Ultra

1GB RAM

20 Pixels of error


Thesis Statement

We can design interactive visualization and collision

detection algorithms between massive models by using dynamic simplification and cache-coherent

layouts


Thesis Statement



layouts


Thesis Statement



layouts


Outline

• Dynamic simplification• Approximate collision

detection• Cache-oblivious layouts• Conclusion & future work


Outline




New Results

• New dynamic simplification algorithm

Clustered hierarchy of progressive meshes (CHPM)

Out-of-core construction


Outline

• Dynamic simplification♦ CHPM representation♦ Construction♦ Results

• Approximate collision detection

• Cache-oblivious layouts• Conclusion & future work


Outline





Clustered Hierarchy of Progressive Meshes (CHPM)• Novel dynamic simplification

representation ♦ Cluster hierarchy♦ Progressive meshes

PM1

PM3

PM2


Clustered Hierarchy of Progressive Meshes (CHPM)• Cluster hierarchy

♦ Clusters are spatially localized regions of the mesh

♦ Used for visibility computations and out-of-core rendering


Clustered Hierarchy of Progressive Meshes (CHPM)• Progressive mesh (PM)

[Hoppe 96]♦ Each cluster contains a PM as an LOD

representation

PM:

Base mesh

Vertex split 0

Vertex split 1

Vertex split n

…..


Two-Levels of Refinement at Runtime

• Coarse-grained view-dependent refinement ♦ Provided by selecting a front in the

cluster hierarchy♦ Inter-cluster level refinements

Front



• Coarse-grained view-dependent refinement ♦ Provided by selecting a front in the


Cluster-split



• Coarse-grained view-dependent refinement♦ Provided by selecting a front in the


Cluster-split Cluster-collapse



• Fine-grained local refinement♦ Supported by performing vertex

splits in PMs♦ Intra-cluster refinements

Vertex split 0

Vertex split 1

Vertex split n

…..PM


Main Properties of CHPM

• Low refinement cost♦ 1 or 2 order of magnitude lower than

a vertex hierarchy

• Alleviates visual popping artifacts♦ Provides smooth transition between

different LODs


Video – Comparison of CHPM with Vertex Hierarchy

4X overall frame rate speedup

38X refinement speedup


Outline





Overview of Building a CHPM

Cluster decomposition

Input model

Cluster hierarchy generation

Hierarchical simplification

CHPM

Performedout-of-core




Input model



CHPM




Input model



CHPM




Input model



CHPM


Out-of-Core Hierarchical Simplification

• Simplifies clusters bottom-up


Out-of-Core Hierarchical Simplification

• Simplifies clusters bottom-up

PM PM

PM


Boundary Simplification

CB

Cluster hierarchy

A D

E F

dependency

B C DA

Boundary constraints


Boundary Simplification

CB

Cluster hierarchy

A D

E F

dependency

B C DA

FE


Boundary Constraints

• Common problem in many hierarchical simplification algorithms♦ [Hoppe 98; Prince 00; Govindaraju et

al. 03]






Cluster Dependencies

• Replaces preprocessing constraints with runtime dependencies



CB

Cluster hierarchy

A D

E F

dependency

B C DA

E F



CB

Cluster hierarchy

A D

E F

dependency

B C DA

E F


Cluster Dependencies at Runtime

CB

Cluster hierarchy

A D

E F

dependencyCluster-splitForce

cluster-split


Chained Dependency

• Inappropriate for refinements





227K triangles 19K triangles92% triangles reduced

After creatingcluster

dependencies


Outline





Processing Time and Storage Requirements

• Process 30M triangles per hour

• CHPM requires 88MB per million vertices♦ 224MB (Hoppe’s VDPM) [Hoppe 97]♦ 108MB (XFastMesh) [DeCoro and

Pajarola 02]


Runtime Performance

ModelPixels

of errorFrame rate

Mem. footpri

nt

Refinement time

Power plant

1 28 400MB 1%

Isosurface (100M)

20 27 600MB 2%

St. Matthew

1 29 600MB 2%

512x512 image resolution, GeForce 5950FX


Limitations

• Long preprocessing time♦ 12 hours for St. Matthew model

consisting of 372M triangles

• Performance depends on computed clusters


Summary

• Low refinement time• Combines out-of-core, VDR,

and visibility culling• No assumption on input mesh

♦ Applied to various polygonal meshes (e.g., CAD, scanned, isosurface models)


Outline




Goal

•Fast algorithm for collision detection between complex and massive models♦ Handle any kind of polygonal

models (e.g., polygon soup)


Problems

• High computation cost of exact query♦ Depends on input model complexity

and the number of colliding triangles

• High memory requirements♦ 40GB for OBB-tree of 100M triangles

[Gottschalk et al. 1996]


Our Solution

• Perform approximate query based on a simplified mesh


LOD-based Query

Object A

Object B

Exact colliding regions


LOD-based Query

Object A Simplified object A

Object B Simplified object B

Simplified colliding regions



LOD-based Query

Object A Simplified object A

Object B Simplified object B


Simplified colliding regions


New Results

• Conservative LOD-based collision detection algorithm between massive models

Clustered hierarchy of progressive meshes (CHPM)

Conservative error metric


Outline


detection♦ Conservative error metric♦ Results



Outline





CHPM Representation

• Serve as a dual hierarchy for collision detection♦ LOD hierarchy♦ Bounding volume hierarchy

• Unified representation for rendering and collision detection


Main Properties of CHPM• Allows a dramatic

acceleration of collision detection♦ Reduces the number of overlap tests

• Alleviates simulation discontinuities♦ Provides smooth transition between

colliding triangles of LODs


• Given two original models, A0 and B0, and a separation distance,

Exact Collision Detection Query

• Computes triangle pairs (tA0, tB0) such that tA A0, tB B0 and dist (tA0, tB0) <


LOD-based Query

• Given two CHPM representations, A and B, and a separation distance, with distance error bound,


such that dist (tA0, tB0) < where tA0 simplifies into tA and tB0 simplifies into tB

• Computes triangle pairs (tA, tB) such that tA A, tB B and dist (tA, tB) <

LOD-based Query

Original meshes Simplified meshes

Simplification


Conservative Error Metric

• Dynamic simplification♦ Determined by the user specified

error,

Original meshes Simplified meshes

Simplification

• Conservative algorithm


Outline





Benchmark Models – Dynamic Simulation

Lucy model: 28M triangles

Turbine model: 1.7M triangles

Impulse based rigid body simulation

[Mirtich and Canny 1995]


Live Demo – Rigid Body Simulation

Error bound:0.1% of width of Lucy model

Average query time:18ms


Limitations

• Can be very conservative and compute many “false positives”♦ Two concentric spheres in parallel

proximity with a separation with d


Summary

• Generality• Conservative collision

queries with reduced discontinuities


Outline




Goal

• Compute cache-coherent layouts of polygonal meshes ♦ For visualization and collision

detection♦ Handle any kind of polygonal models

(e.g., irregular geometry)


Motivation

• High growth rate of computational power of CPUs and GPUs

Improvementfactor

during 1993 - 2004

Courtesy: Anselmo Lastra and http://www.hcibook.com/e3/online/moores-law/

during 99 - 04


Memory Hierarchies and Caches: I/O Model [Aggarwal and Vitter 88]

CPU or GPU

Fast memory or cache

Slow memory

Blocktransfer

Disk

106nsAccess time: 102ns100ns


Cache-Coherent Layouts

• Cache-Aware♦ Optimized for particular cache

parameters (e.g., block size)

• Cache-Oblivious♦ Minimizes data access time without

any knowledge of cache parameters♦ Directly applicable to various

hardware and memory hierarchies










Low Computation Speed

• Rendering throughput♦ GPU capable of 100M+ triangles per

sec♦ Only achieved 20M triangles per sec

• Low cache utilization♦ Cannot efficiently use triangle strips

for dynamically generated geometry


New Results

• Algorithm to compute cache-oblivious layouts

Cache-oblivious metric

Multilevel optimization framework

Applicable to multiresolution mesh


Realtime Captured Video – St. Matthew Model


Related Work

• Cache-coherent algorithms• Mesh layouts


Cache-Coherent Algorithms

• Cache-aware [Coleman and McKinley 95, Vitter 01, Sen et al. 02]

• Cache-oblivious [Frigo et al. 99, Arge et al. 04]

Focus on specific problems such as sorting and linear algebra computations!


Mesh Layouts

• Rendering sequences♦ Triangle strips♦ [Deering 95, Hoppe 99, Bogomjakov

and Gotsman 02]

• Processing sequences♦ [Isenburg and Gumhold 03, Isenburg

and Lindstrom 04]

Assume that globally the access pattern follows the layout order!


Mesh Layouts

• Space-filling curves♦ [Sagan 94, Velho and Gomes 91,

Pascucci and Frank 01, Lindstrom and Pascucci 01, Gopi and Eppstein 04]

Assume geometric regularity!

Z-curve:


Outline


detection• Cache-oblivious layouts

♦ Overview♦ Results

• Conclusion & future work


Outline






Overview

Multilevel optimizationCache-oblivious metric

Local permutations

va

vb vd

vc

Input graph

va vb vd vc

Result 1D layout


Graph-based Representation

• Undirected graph, G = (V, E)♦ Represents access patterns of

applications

• Vertex♦ Data element ♦ (e.g., mesh vertex or mesh triangle)

• Edge♦ Connects two vertices if they are

likely to be accessed sequentially

va

vb vd

vc


Problem Statement

• Vertex layout of G = (V, E)♦ One-to-one mapping of vertices to

indices in the 1D layout

• Compute a that minimizes the expected number of cache misses

: |}|, ... ,1{ VVva

vb vd

vc

va vb vd vc

1 2 3 4


Local Permutation

va

vb vd

vc

va vb vd vc

va vb vc vd

Layout computation


Edge Span

va

vb vd

vc

va vb vd vc

va vb vc vd

21

Absolute value of index gap of two vertices in the layout

1 2 3 4Indices of vertices


Expected Number of Cache Misses

Cache miss probability function

Edge span

va

vb vd

vc

va vb vd vc

va vb vc vd

Differences of histogramsof edge spans

Edge span

Number of edges

Local permutation

(dot product)




Edge span

?

• Assume monotonicity


Edge span

Number of edges


Probability Computation


Edge span

?

• Compute a probability that cache misses decrease


Edge span

Number of edges


Geometric Formulation

Cache miss probability function (pl)

Edge span = l

?

p2

p10

Closed hyperspace

Half-hyperspace mapping to probability

functions reducing cache misses


Edge span

Number of edges

2D example


Geometric Volume Computations

• Time complexity♦ Exact: [Lasserre and Zeron

01]♦ Approximate: [Kannan et al. 97]

where n is the number of hyperplanes

)( 1nnO)( 5nO


Fast and Approximate Volume Comparison

• Propose an approximate volume comparison♦ Has linear time complexity♦ Has very small error (0.2% compared

to exact method)


Layout Optimization

• Find an optimal layout that minimizes our metric♦ Combinatorial optimization problem

[Diaz et al. 2002]


Multilevel Minimization

Step 1: Coarsening



Step 2: Ordering of coarsest graph



Step 3: Refinement and

local optimization


Outline






Layout Computation Time

• Process 140 million triangles per hour♦ Takes 2.6 hours to lay out St.

Matthew model (372 million triangles)

♦ 2.4GHz of Pentium 4 PC with 1 GB main memory


Edge Span Histograms of Different Layouts

Cache-oblivious layout

Spectral layout

Original layout

Edge span

Nu

mb

er o

f ed

ges

>

[Isenburg and Lindstrom 05]


Applications

• View-dependent rendering• Collision detection• Isocontour extraction


View-Dependent Rendering

• Layout vertices and triangles of CHPM♦ Reduce misses in GPU vertex cache


View-Dependent Rendering

Models # of Tri.Our

layoutCHPM layout

St. Matthew

372M 106 M/s 23 M/s

Isosurface 100M 90 M/s 20 M/s

Double eagle tanker

82M 47 M/s 22 M/s

4.5X

2.1X

Peak performance: 145 M tri / s on GeForce 6800 Ultra


Comparison with Optimal Cache Miss Ratio

Our layout

CHPM layout

Optimal cache miss ratio

[Bar-Yehuda and Gotsman 96]

Test model: Bunny model


Comparison with Hoppe’s Rendering Sequence

Our layout

[Hoppe 99]

Optimized for 16 vertex cache sizewith FIFO replacement

Optimized for no particular cache size

Test model: Bunny model


Comparison with Space Filling Curve on Power Plant Model

Our layout

Space filling curve (Z-curve) [Sagan 94]


Collision Detection

• Bounding volume hierarchies♦ Widely used to accelerate the

performance of collision detection♦ Traversed to find contacting area♦ Uses pre-computed layouts of OBB

trees [Gottschalk et al. 96]


Rigid Body Simulation


Collision Detection Time

2X on average

Depth-first layout

Cache-oblivious layout


Isocontour Extraction

• Contour tree [van Kreveld et al. 97]

• Use mesh as the input graph

• Extract an isocontour that is orthogonal to z-axis

Puget sound, 134 M triangles

Isocontourz(x,y) = 500m


Comparison – FirstExtraction of Z(x,y) = 500m

Relative Performance

overZ-axis sorted

layout

Nearly optimized for particular isocontour

2

21

13

1

Disk access time is bottleneck


Comparison – Second Extraction of Z(x,y) = 500m

Relative Performance

overZ-axis sorted

layout

2

21

13

379

212

10.8

Memory and L1/L2 cache access times are bottleneck


Limitations

• Monotonicity assumption♦ May not work well for all applications

• Does not compute global optimum♦ Greedy solution


Summary

• General ♦ Applicable to all kinds of polygonal models♦ Works well for various applications

• Cache-oblivious♦ Can have benefit for all levels of the memory

hierarchy (e.g. CPU/GPU caches, memory, and disk)

• No modification of runtime applications♦ Only layout computation

Source codes are available as a library called

OpenCCL


Outline

• Dynamic simplification & visibility culling

• Cache-oblivious layouts• Approximate collision

detection• Conclusion & future work


Conclusion

• Dynamic simplification and cache-oblivious layouts♦ Applied them to visualization and

collision detection♦ Demonstrated with a wide variety of

polygonal models♦ Achieved interactive performance on

commodity hardware


New Results

• Dynamic simplification method♦ CHPM representation♦ Out-of-core construction method♦ Application to collision detection

• Cache-oblivious layout algorithm♦ Cache-oblivious metric♦ Multilevel minimization


Future Work on Visualization

• Achieve end-to-end interactivity♦ Requires no or minimal

preprocessing

• Handle time-varying geometry

Just one instance among 27K time steps

during simulation


Future Work on Collision Detection

• Handle dynamically deformable models (e.g. cloth simulation)♦ Requires no or minimal

preprocessing

• Support penetration depth computations Obj 1 Obj 2


Future Work on Cache-Coherent Layouts

• Develop cache-aware layouts• Investigate optimality• Apply to other applications

and other representations♦ Shortest path computation, etc.

• Provide multiresolution functionality from layouts♦ [Pascucci and Frank 01]


Publications

S.-E. Yoon, B. Salomon, R. Gayle, and D. Manocha,

Quick-VDR: Interactive View-Dependent Rendering of Massive Models

IEEE Vis. 04 &

Invited at IEEE Tran. on Vis. and Comp. Graphics 05

S.-E. Yoon, B. Salomon, and D. Manocha,Interactive View-Dependent Rendering with

Conservative Occlusion Culling in Complex EnvironmentIEEE Vis. 03


Publications

S.-E. Yoon, P. Lindstrom, V. Pascucci, and D. Manocha,Cache-Oblivious Mesh Layouts

ACM SIGGRAPH (ACM Tran. on Graphics) 05

S.-E. Yoon, B. Salomon, M. Lin and D. Manocha,Fast Collision Detection between Massive Models

using Dynamic SimplificationEurographics Sym. of Geometric Processing 04


Acknowledgements

• Advisor, Dinesh Manocha• Other committee members

♦ Anselmo Lastra, Ming C. Lin, Peter Lindstrom, Valerio Pascucci

• Other faculty and staff members


Acknowledgements

• Coauthors and colleagues♦ Bill Baxter, Russ Gayle, Naga

Govindaraju, Martin Isenburg, Jayeon Jung, Ted Kim, Young Kim, Brandon Lloyd, Miguel Otaduy, Stephane Redon, Brian Salomon, Avneesh Sud, Gokul Varadhan, and Kelly Ward

• English support♦ Elise London, Charlotte Powell, and

Mary Wakeford


Acknowledgements

• Model contributors♦ Boeing company♦ Digital Michelangelo Project♦ Lawrence Livermore National

Laboratory♦ Newport News Shipbuilding♦ Power plant donor


Acknowledgements

• Funding agencies♦ Army Research Office♦ Defense Advance Research Projects

Agency ♦ Ilju foundation♦ Intel company♦ Lawrence Livermore National

Laboratory♦ National Science Foundation♦ Office of Naval Research


Acknowledgements

• My parents, brother, sister, and parents-in-law


Acknowledgements

• My wife, Dawoon Jung

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL

Questions?


Backup Slides - Collision


LOD-based Query

• Given two bounding volumes, BVi and BVj

• ConservBVTest (BVi, BVj, , )♦ Similar to LodCollide in a BV level


Conservative Collision Formulation

• Transforms the distance query with into an intersection query

2

2

• Considers the dilated BV, dilated (BV, )♦ Defined as the Minkowski sum of BV

with a sphere of radius



• Defines♦ Hausdorff distance between a

bounding volume, BV, and the original mesh

)(ˆ BVh

BV

Contained original mesh

Hausdorff distance


Lemma 1

• If dilated (BVi , ) and dilated (BVj , ) do not intersect, the distance is greater than

2

2


Lemma 2

• If there is an intersection between dilated (BVi , ) and dilated (BVj , ), the distance has an upper bound of

2

)(ˆ)(ˆ ji BVhBVh

2

)(ˆ jBVh

)(ˆ iBVh


Lemma 2

• If there is an intersection between dilated (BVi , ) and dilated (BVj , ), the distance has an upper bound of

2

)(ˆ)(ˆ ji BVhBVh

2



ConservBVTest (BVi, BVj, , ) =

NoCollision, if

Collision, if

Potential collision, if


Cull and Refine Operations

• Culling operation♦ BV pairs whose distance is greater

than are culled

• Refining operation♦ If there is collision and sum of

Hausdorff distance is bigger than , further refinement is performed

LodCollide (A, B, , ) =


Collision Detection between CHPMs

Cluster Level Culling using Bounding Volume Test Tree

PM Refinements

GPU basedTriangle Level Culling

Exact Intersection Tests

Two CHPMs

Potentially colliding clusters

Colliding triangles

Potentially colliding triangles (PCTs)

Reduced number ofPCTs


BVTT using CHPM

• Recursively perform culling and refining operations♦ Computes potentially colliding

clusters♦ performs coarse-grained LOD

refinement over regions of the mesh in collision


BVTT using CHPM

• Recursively perform culling and refining operations♦ Computes potentially colliding

clusters♦ performs coarse-grained LOD

refinement over regions of the mesh in collision Culling

operationRefining operation


Benchmark Models – Navigation

Power plant model: 12M triangles

Dragon model: 0.8M triangles


Contact-Dependent LODs [Otaduy and Lin 2003]• Uses hierarchies of static

LODs♦ Has relatively small runtime

overhead♦ Can create “popping” or

discontinuity ♦ Assumes a closed and manifold

model because they use a hierarchy of convex hull


Limitations

• Relies on high temporal coherence

• Can be very conservative and return many “false positives”♦ Two objects in parallel proximity

with a separation with d


Backup Slides – Cache-Oblivious Mesh Layouts


Terminology

• Edge span distribution ♦ where i is in [1, n]|| iE

1|| 3 E1|| 2 E

1|| 4 E

4|| 1 E

Edge span1

Number of edges

2 3 4

1

1

1

1

4

2

3

4

1


Cache Miss Ratio Function (CMRF),

• Probability of a cache miss for a given edge span i

ip

0

1Cache miss ratio =Probability to have

a cache miss

Edge span

ip

1 n-1i


Number of Cache Misses at Runtime

• Estimated by multiplying two factors♦ Runtime edge span distribution♦ CMRF

1D Layout:

Edge span 2 Edge span 4 Edge span 2

2p 2p4p+ + ( 2 1, () 2p 4p, )( )


Number of Cache Misses at Runtime

1D Layout:

Edge span 2 Edge span 4 Edge span 2

2p 2p4p+ + ( 2 1, () 2p 4p, )

Runtime edge span distribution CMRF

( )



♦ Approximate runtime edge span distribution with one of the layout

1

1

||n

iii pE

Edge span distribution of the layout

The number of vertices


Does a Local Permutation Decrease Cache Misses?

1

1

||n

iii pE

1

1

|)||(|n

iiii pEE

|||| ii EE || iE

?


Does a Local Permutation Decrease Cache Misses?

1

1

||n

iii pE

1

1

|)||(|n

iiii pEE

0||1

1

n

iii pE


Exact Cache-Oblivious Metric

0||1

1

n

iii pE

where

All the possible cache configurations

1...0 1221 nn pppp

Monotonicity of CMRF


Geometric Formulation

where

0||1

1

n

iii pE

1...0 1221 nn pppp

Half hyperspacep2

p10

Closed hyperspace1 n

p2

p10


Geometric Volume Computation

• Assume each CMRF to be equally likely

• Half hyperspace (blue area)♦ Space of CMRFs that reduce cache

misses

p2

p10where

0||1

1

n

iii pE

1...0 1221 nn pppp


p2

p10


• Define a top polytope in closed hyperspace

• Compute the centroid, C, of the top polytope

Top polytope Centroid, C


p2

p10


• Use the centroid for approximate volume comparison♦ The volume containing the centroid is

likely to be larger

Centroid, C


Final Approximate Metric

0||1

)(

m

jjl jE

Centroid

Pack non-zero to 1,…, m || iE


Performance during View-Dependent Rendering

Our layout

[Hoppe 99]

Optimized for various resolutions

Optimized for full resolution

Documents

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Interactive Visualization and Collision Detection using Dynamic Simplification and Cache-Coherent Layouts