Radiosity for Point-Sampled Geometry

Hokkaido University

Radiosity forRadiosity forPoint-Sampled GeometryPoint-Sampled Geometry

Tsuyoshi YamamotoTsuyoshi Yamamoto

Tomoyuki NishitaTomoyuki Nishita((The University of TokyoThe University of Tokyo))

Yoshinori DobashiYoshinori Dobashi

((Hokkaido UniversityHokkaido University))

Hokkaido University

OverviewOverview

• Introduction• Background and Motivation• Previous Work

• Radiosity Method• Proposed Method

• Basic Idea• Computation of Effective Area• Construction of Hierarchy• Radiosity Calculation• Adaptive Addition of Surfels

• Examples• Conclusions and Future Work





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OverviewOverview









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Background and MotivationBackground and Motivation• Point-sampled Geometry• Point-sampled Geometry

- Points sampled on surfaces of an object- Points sampled on surfaces of an object

PointsPointsObjectObject PolygonsPolygons

- No information on connectivity- No information on connectivity

- Used for objects with complex shapes- Used for objects with complex shapes

- Many researches on point-sampled geometry- Many researches on point-sampled geometry

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- Points as display primitive [Levoy85]- Points as display primitive [Levoy85]

- Ray tracing [Shaufler00]- Ray tracing [Shaufler00]

- Surfel [Pfister00], Q-splat [Rusinkiewica00]- Surfel [Pfister00], Q-splat [Rusinkiewica00]

- Point sample rendering [Grossman98]- Point sample rendering [Grossman98]

- Surface splatting [Zwicker01]- Surface splatting [Zwicker01]

- Hardware-acceleration [Ren02]- Hardware-acceleration [Ren02]

• Previous methods for displaying point- sampled geometry• Previous methods for displaying point- sampled geometry

• No method for radiosity (Finite element approach)• No method for radiosity (Finite element approach)

Background and MotivationBackground and Motivation

- Points as display primitive [Levoy85]- Points as display primitive [Levoy85]

- Ray tracing [Shaufler00]- Ray tracing [Shaufler00]

- Surfel [Pfister00], Q-splat [Rusinkiewica00]- Surfel [Pfister00], Q-splat [Rusinkiewica00]

- Point sample rendering [Grossman98]- Point sample rendering [Grossman98]

- Surface splatting [Zwicker01]- Surface splatting [Zwicker01]

- Hardware-acceleration [Ren02]- Hardware-acceleration [Ren02]

(Monte-Carlo approach)(Monte-Carlo approach)

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- Radiosity method for point-sampled

geometry

• Goal of this research• Goal of this research

- Diffuse surfaces

• Condition• Condition

Background and MotivationBackground and Motivation

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OverviewOverview









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Radiosity MethodRadiosity Method

light

directindirect

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• Energy transfer between patches

Bi: radiosity

Fij: form factor

j: diffuse reflectance

Vij: visibility

light

patch i

patch j

• A linear system

n

ijjjijjii BFEB

,1

j iA A

jiijij

ji

iij dAdAV

rAF 2

coscos1


j

i

rij

Aj

Ai

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Bi: radiosity

Fij: form factor


Vij: visibility

light

patch i

patch j

• A linear system

n

ijjjijjii BFEB

,1


j iA A

jiijij

ji

iij dAdAV

rAF 2

coscos1

j

i

rij

Aj

Ai

Hokkaido University

n

ijjjijjii BFEB

,1


Bi: radiosity

Fij: form factor


Vij: visibility

light

patch i

patch j

• A linear system


j

i

rij

Aj

Ai

jij

jiij A

rF

2

coscos

ijV

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OverviewOverview



• Computation of Effective Area• Construction of Hierarchy• Radiosity Calculation• Adaptive Addition of Surfels




• Computation of Effective Area• Construction of Hierarchy• Radiosity Calculation• Adaptive Addition of Surfels


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Basic IdeaBasic Idea

original surface

point xi

Ri

normal ni

max. distance

between points in

small neighborhood

• Use of surfels to represent object• Use of surfels to represent object [Pfister00][Pfister00]

tangent disk

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1. Computation of effective area1. Computation of effective area

2. Construction of hierarchy2. Construction of hierarchy

3. Computation of interreflection3. Computation of interreflection

4. Creation of final images4. Creation of final images

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- Form factor:- Form factor: jij

jiij A

rF

2

coscos

(area)(area)

Area of tangent diskArea of tangent disk Effective areaEffective area

Overestimationdue to overlapOverestimationdue to overlap

Considering overlapConsidering overlap

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- Large number of surfels- Large number of surfels

- Increase in computation time- Increase in computation time

- Efficient computation using hierarchy- Efficient computation using hierarchy

Grouping neighborsGrouping neighbors

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- Hierarchical radiosity with global refinement- Hierarchical radiosity with global refinement[Stamminger00]

- Adaptive addition of surfels- Adaptive addition of surfels

4. Creation of final images4. Creation of final images- Hardware accelerated surface splatting- Hardware accelerated surface splatting [Ren02]

- Original points may not be sufficient- Original points may not be sufficient

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Hokkaido University






Hokkaido University






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Effective Area ComputationEffective Area Computation• Definition: effective area of surfel i • Definition: effective area of surfel i

y

A(y)

i: Influence functioni: Influence function

A: Differential areaA: Differential area

ri• Condition of i :• Condition of i :

- Decreasing with distance- Decreasing with distance

yy dArS iii )()( yy dArS iii )()(

Use of exponential func.Use of exponential func.

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A(y)

Effective Area ComputationEffective Area Computation• Definition: effective area of surfel i • Definition: effective area of surfel i

yy dArS iii )()( yy dArS iii )()(

i: Influence functioni: Influence function

A: Differential areaA: Differential area

- Decreasing with distance- Decreasing with distance

- Taking into account overlap- Taking into account overlap

ri• Condition of i :• Condition of i :y

Use of exponential func.Use of exponential func.

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Effective Area ComputationEffective Area Computation

A

• Simple case:• Simple case: Same normal vectors (coplanar disks)

i

iii

i dArS yy)()(Sum of effective areas:Sum of effective areas:

yy dAStotal )(

Total area:Total area:

== ++

==

0.1)()()( 21 yyy m

y

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Effective Area ComputationEffective Area Computation• Simple case:• Simple case: Same normal vectors (coplanar disks)

(2) Taking into account overlap(2) Taking into account overlap

(1) Decreasing with distance (exp func.)(1) Decreasing with distance (exp func.)

0.1)()()( 21 yyy m

)exp()(,)(/)( iii

iii rrwrwrw )exp()(,)(/)( iii

iii rrwrwrw • We use following influence function• We use following influence function

for cond. (1)for cond. (1)for cond. (1)for cond. (1) for cond. (2)for cond. (2)for cond. (2)for cond. (2)

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• General case• General case


yy dArS iii )()( yy dArS iii )()( - Effective area:

Ai

- Problem: how to compute A- Problem: how to compute A

Aj

- Use of weighted average of Ai and Aj- Use of weighted average of Ai and Aj

(Aj = Ai/cos )(Aj = Ai/cos )

ri

rjdisk j disk i

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• General case• General case


• Effective area for general case• Effective area for general case

m

j j

m

j jj

iirw

ArwAdArS

1

1

)(

)(,)( y

m

j j

m

j jj

iirw

ArwAdArS

1

1

)(

)(,)( y for details,

see paper

• Use of graphics hardware• Use of graphics hardware

yy dArS iii )()( yy dArS iii )()( - Effective area:

- Problem: how to compute A- Problem: how to compute A

- Use of weighted average of Ai and Aj- Use of weighted average of Ai and Aj

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Effective Area ComputationEffective Area Computation• Use of hardware-accelerated surface splatting• Use of hardware-accelerated surface splatting

virtual camera

m

j j

m

j jj

iirw

ArwAdArS

1

1

)(

)(,)( y

[Ren02][Ren02]

virtual screen

m: number of neighboring surfels

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virtual screen


m

j j

m

j jj

iirw

ArwAdArS

1

1

)(

)(,)( y

[Ren02][Ren02]

R: w(r)A

: w(r)additive blending

virtual camera

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m

j j

m

j jj

iirw

ArwAdArS

1

1

)(

)(,)( y

[Ren02][Ren02]

R

m

j jj Arw1

)(

m

j jrw1 )(


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[Ren02][Ren02]

R

m

j jj Arw1

)(

m

j jrw1 )(÷

A

=

m

j j

m

j jj

iirw

ArwAdArS

1

1

)(

)(,)( y

. . . . .


Hokkaido University






Hokkaido University

Computation of InterreflectionComputation of Interreflection• Hierarchical radiosity with global refinement• Hierarchical radiosity with global refinement

- Algorithm- Algorithm[Stamminger00]

1. Compute intereflection1. Compute intereflection

2. Evaluate errors2. Evaluate errors

3. Replace surfels with their children3. Replace surfels with their children

4. If error > , go to step 14. If error > , go to step 1(: user-specified threshold)(: user-specified threshold)

• Original points may not be sufficientOriginal points may not be sufficient• Original points may not be sufficientOriginal points may not be sufficient

• Addition of new surfelsAddition of new surfels• Addition of new surfelsAddition of new surfels

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Adaptive Addition of SurfelsAdaptive Addition of Surfels

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• Two conditions• Two conditions

shadow boundary

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(1) Place surfels according to intensity gradient


shadow boundary

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(2) Place surfels as uniformly as possible

shadow boundary

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• Local coordinate based on intensity gradient• Local coordinate based on intensity gradient


- Compute error vector- Compute error vector

j ij

ijijij d

xx

xxe

)(

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• Local coordinate based on intensity gradient• Local coordinate based on intensity gradient


- Compute error vector- Compute error vector

j ij

ijijij d

xx

xxe

)(

- Define local coordinate- Define local coordinate

- Add four surfels at:- Add four surfels at:(-cRi , cRi , 0.0), (-cRi , cRi , 0.0),

(cRi , -cRi , 0.0), (-cRi , -cRi , 0.0)

(-cRi , cRi , 0.0), (-cRi , cRi , 0.0),

(cRi , -cRi , 0.0), (-cRi , -cRi , 0.0)

Ri ： radius of surfel iRi ： radius of surfel i c ： user-specified cnst.c ： user-specified cnst.

Normal vetor, reflectance: interpolationNormal vetor, reflectance: interpolationEffective area: x0.25Effective area: x0.25Disk size: x0.5Disk size: x0.5

uu

vv ww

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(2) Place surfels as uniformly as possible(2) Place surfels as uniformly as possible

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• Use of point repulsion method• Use of point repulsion method


- Compute repulsive forces- Compute repulsive forces

[Turk92][Pauly03]

(Newly added surfels only)(Newly added surfels only)

- Move surfels- Move surfels

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OverviewOverview

• Introduction• Background and Motivation• Previous Methods


• Basic Idea• Computing Representative Area• Construction of Hierarchy• Computing Interreflection using Progressive Radiosity• Adaptive addition of surfels


• Introduction• Background and Motivation• Previous Methods


• Basic Idea• Computing Representative Area• Construction of Hierarchy• Computing Interreflection using Progressive Radiosity• Adaptive addition of surfels


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Experimental ResultsExperimental Results

• Three spheres in a simple room• Three spheres in a simple room

3.5m x 3.5m, 400 surfels

0.2m0.4m

0.8m

642 surfels

- Sum of effective areas:- Sum of effective areas:

- True value:- True value:

8.23 m2 8.23 m2

8.04 m2 8.04 m2

- Relative error:- Relative error:

2.63 % 2.63 %

Very accurateVery accurate

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Experimental ResultsExperimental Results

• Three spheres in a simple room• Three spheres in a simple room

ResultResult(15,748 surfels)(15,748 surfels)

Distribution of surfelsDistribution of surfels

Computer: Pentium 4 (2.8GHz), GPU: nVidia Geforce4Computation time (radiosity): 38 sec.

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Complex ResultsComplex Results

ResultResult

• Two bunnies in a simple room• Two bunnies in a simple room

Distribution of surfelsDistribution of surfels(72,068 surfels)(72,068 surfels)

Computer: Pentium 4 (2.8GHz), GPU: nVidia Geforce4Computation time (radiosity): 47 sec.

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Complex ResultsComplex Results

• Gallery of statues• Gallery of statues

ResultResult Distribution of surfelsDistribution of surfels(315,686 surfels)(315,686 surfels)

Computer: Pentium 4 (2.8GHz), GPU: nVidia Geforce4Computation time (radiosity): 2,541 sec.

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ConclusionsConclusions

• Interreflection of light for point- sampled geometry• Interreflection of light for point- sampled geometry

- Efficient computation of effective areas- Efficient computation of effective areas

- Adaptive addition of surfels- Adaptive addition of surfels

Future WorkFuture Work

• Specular reflection• Specular reflection

Documents

Radiosity for Point-Sampled Geometry