Volume radiosity

Michal Roušal

University of West Bohemia, Plzeň

Czech republic

Index

Introduction to the radiosity method– Lightning model– Form factors – Radiosity equation

Radiosity for the volume and its specifics– Lightning model for volume absorbing and

scatering

Introduction to radiosity method

Global visualisation method, presented about “1985”

Based on finite element method Using physical based lightning model Subdividing scene into small patches (border

representation) Scene should be”closed” for energy

General rendering equation

Radiance of patch P is:

Radiance at point x for non transparent patches

surfacesall

q)dqp,)scatered(radiance(q emited(p) )radiance(p

')',(),'()',,(),(),( ' dxxxGxLxfxLxLS

e

Lightning model

Represented by BRDF function: f(,x, ’)

Reciprocity:

Energy conservation:

),,(),,( '' xfxf

1),,( ' xf

x

’nx

Most used lightning models

Diffuse model:

Modified Phong model:

h halfway vector between and ‘.

.),( ' konstf

)(cos),( 'h

nsd kkf

Form factors

Geometric characteristic of the patches visibility in the scene.

Analytical/geometrical approach Generally we can define:

i jA Ai

ij dxdxxxGA

F ''),(1

Analytical approach

Probability definition: – We define Fij as probability that random particle

shot from patch i will hit path j. (particle tracing)

Monte Carlo integration methods Global Lines

– Fast but less accurate

Geometrical approach

Nusselt‘s analog (projection on disk)– high computation complexity

Projection on hemicube– can be used hardware

Projection on tetrahedra– can be used hardware– less faces

Radiosity equation (1)

Works not with the point x but with the patch i Only for diffuse reflectance and uniform

emittance we can write.

– Ei is self-emitted constant radiance of patch i

ie

i

ExExL

xxinconstxf

)(),(

)(.),,( '

Radiosity equation (2)

We define the radiance of the path B as:

For all patches we can write:

j

ijjiii FBEB

NNNNNNNNN

N

N

jiijjii

E

E

E

B

B

B

FFF

FFF

FFF

EFBB

2

1

2

1

21

22222212

11121111

1

1

1

Solving radiosity equation

Easy to solve by numerical algorithms (gathering radiosity)– Gauss O(N3)– Gauss-Seidel iteration O(N2) per step– Time and memory consuming

Progressive refinement (shooting radiosity)– Solving equation by columns with highest Bj

– O(N2) complexity but not so much memory needed

Some radiosity improvements

Hierarchical radiosity– Adaptive subdivision of the patches due to error

Clustering– Using clusters of patches to reduce complexity of

scene. Combination of radiosity with other global

visualisation algorithms etc.

Volume radiosity

More complex scenes Usually combination of volume and surface

elements Lightning model for participating media

– Not only reflectance but also volume absorption and scattering

Visualisation of volume objects/scenes

Lightning model (1)

We define: a(x) – coefficient of absorption

s(x) – scattering coefficient

e(x) – extinction coefficient

- scattering albedo

)()()( xxx sae

e

s

Lightning model (2)

Integral equation

– transmittance

'

''''

',

x

x

e dxx

exx

''''00

'''

,,,,,

:,

,,,,,

0

2

dxxLxxxxLxxxL

radianceSourcexL

dxLxfxxLxL

S

x

x

e

S

S

pseS

Solving volume radiosity equation Zonal method

– More-less similar to classic approach using Gauss-Seidel iteration

Progressive refinement + Hierarchical radiosity

Finite element method approach Random (Monte Carlo) based methods

My interest

Combination of volume and surface elements Multiple and not only diffuse scattering Clustering of patches based on initial scene

geometry (probably no adaptive clustering during computation or hierarchical subdivision – orientation on volume)

Parallel and maybe distributed implementation for applications with dynamic environment

Possible application

Simulation of visibility from a car with headlights in fog/dense rain or other volume/participating media based environment

Our approach (theory)

Main aim is to create fast algorithm with decent accuracy

Progressive refinement with randomized approach (Monte Carlo method) for shooting strategy

Clustering:– Some clusters we can get from the scene model– Analyze the geometry of scene and create clusters

of voxels

Clustering

Some clusters we can get from the scene model

Analyze the geometry of scene and create clusters of voxels

Parallelization

Divide the scene by planes

Each part of the scene can be computed by different processor/thread