RADIOSITY

RADIOSITYRADIOSITY

Submitted by

CASULA, BABUPRIYANK. N

Computer GraphicsComputer Graphics

Computer Graphics

Application

Image Synthesis

Animation

Hardware &Architecture

Image Synthesis Image Synthesis

Image Synthesis

Modeling•2d/3d Rendering

•Radiosity•Illumination models•Visibility•Ray Tracing•Texture Mapping

Viewing2d/3d

RadiosityRadiosity

Surfaces in a scene reflect & emit light.– Some of this light reaches the viewer; this

makes the surface visible.– But much of this reflected/emitted light will

illuminate other surfaces.– This light will then reflect of these other

surfaces; in fact, every surface in a scene will illuminate other surfaces in the scene.

SamplesSamples

Background needed… Background needed…

LightLight TransportRadiometryReflection Functions

LightLight

The visible light

can be polarizedOptics is the area

that studies about

these radiations

OpticsOptics

Optics

Geometric Physical Quantum

Shadows Optical Interference Photons

laws

To study radiosity Geometric Optics is needed

Light TransportationLight Transportation

Light travels in the form of particles(photons)

Total number of particles in a small differential volume dV is

P(x) = p(x) dV particle density

P(x) = p(x) (v dt cos()) dA

Light Transportation contd..Light Transportation contd..

Not all particles flow with the same speed and same direction.

The particle density is now a function of two independent variables x, .

Then we have

P(x, ) = p(x, ) cos d dA

Here d is called the differential solid angle.

AnglesAngles

2D angle 3D/Solid Angle

Solid AngleSolid Angle

Definition :The SA subtended by an object from a point P is the area of projection of the object onto the unit sphere centered at P.

Area (dA) = (r d) (r sin d)= r2 sin d d

The differential solid angle :

d = dA cos / r2 = cos sin d d

Radiant Energy QRadiant Energy Q

Rendering systems consider the stuff that flows as radiant energy or radiant power()

The radiant energy per unit volume is the photon volume density times the energy of a single photon(hc/).

L(x,) = p(x, , ) (hc/) dL is called radiance

RadiometryRadiometry

Science of Measuring light

Analogous science called Photometry is based on human perception

Radiometry contd..Radiometry contd..The radiometric quantities that characterize

the distribution of light in the environment are:

Radiant EnergyRadianceRadiant PowerIrradianceRadiosityRadiant Intensity

RadianceRadiance

Radiance (L) is the flux that leaves a surface, per unit projected area of the surface, per unit solid angle of direction.

n

dA

L

RadianceRadiance

For computer graphics the basic particle is not the photon and the energy it carries but the ray and its associated radiance.

n

dA

L d

Radiance is constant along a ray.

Properties of Radiance

1)Fundamental quantity -all other quantities derived from it2) Invariant along a ray - quantity used by ray tracers3) Sensor response is proportional to

radiance -eye/camera response depends on

radiance

Radiant Power(Radiant Power())

Flow of energy.Power is the energy per unit time.Also called as radiant flux. = dQ/dt.The differential flux is the radiance in small

beam with cross sectional area dA and solid angle d

d = L(x, ) cos d dA

Invariance of Radiance

dL1 ddA1 = L2 ddA2

ddA2 /r2 and ddA1 /r2

Throughput T = ddA1 = ddA2

= dA1 dA2/ r2

Irradiance

Irradiance: Radiant power per unit area

incident on a surface

E = Li(x,cos d

RadiosityOfficial term : Radiant Exitance

Radiosity: Radiant power per unit area

exiting a surface

B = Lo(x,cos d

Radiant Intensity

Radiant Intensity: Radiant power per solid angle of a point source

I() = d()/d()

= I() d()

For an isotropic point source: I() =

Irradiance due to a Point Light

Irradiance on a differential surface due to

an isotropic point light source is

E = ddA

= I() d()

dA

= cos(

x – xs|2

Reflection FunctionsReflection Functions

Reflection is defined as the the process by which the light incident on a surface leaves the surface from the same side.

The nomenclature and the general properties of reflection functions are discussed.

BRDFBRDF

Bidirectional Reflection Distribution Function

f(x, i , r) =Lr(x,r)/ dEi(x,r)

In short this is the ratio of radiance in a reflected direction to the differential irradiance that created

ir

Incident ray

Reflected ray

Illumination hemisphere

f(p, i , r )

Properties of the BRDF

1)Reciprocity

f(x, i , r) = f(x, r , i)

2)Anisotropy

If the incident and the reflected light are fixed and the underlying surface is rotated about the surface normal, the percentage of light reflected may change.

Reflectance Equation

The BRDF allows us to calculate outgoing light, given incoming light:

Lr(x,r)= f(x, i , r) * dEi(x,r)

= f(x, i , r) * Li(xi,cos di

Integrating over the hemisphere gives thereflectance equation:

Lr(x,r)= f(x, i , r) * Li(xi,cos di

ReflectanceReflectanceReflectance: ratio of reflected flux to incident

flux

= dr/ do= Lr(r) cos r dr

r

Li(i) cos i di

i

Reflectance is always between 0 and 1

but depends on incident radiance distribution

Lambertian Diffuse Reflection

Reflection is equal in all directions

f r ,diffuse (x, i , r) is constant.

Lr(x,r)= f r ,diffuse(x, i , r) * Li(xi,cos di

= f r ,diffuse(x, i , r) Li(xi,*cos di

= f r ,diffuse(x, i , r) E


Reflected radiance is independent of direction

Therefore the radiosity is simply:

B = Lr,diffuse(x,cos d

Lr,diffuse

f r ,diffuse(x, i , r) E


= dr/ do= Lr(r) cos r dr

r

Li(i) cos i di

i

= Lr,diffuse cos r dr

r

E

= f r ,diffuse

Global Illumination

Radiance is invariant along a ray

Li(x`, i) = Lr(x, r) V(x,x`)

V(x,x`) is the visibility from point x to x’Converting the directional integralinto a surface integral

di = cos o dA |x-x`| 2

The Projected solid angle is

cos i di = cosi coso dA |x-x`| 2

Global IlluminationGeometry term:G(x,x1) = cosi coso

|x-x`| 2

cosi di = G(x,x1) dA Rewriting the reflectance equation:

Lr(x`,`)= f(x, -, `)L(xi,G(x,x’)V(x,x`)dAs

Global IlluminationReparameterizing gives:Lr(x`,`)= f(x, -, `)L(xi,G(x,x’)V(x,x`)dAs

Lr(x`,x``)= f(x x` x``)L(x x`G(x,x’)V(x,x`)dA

Lr(x`,x``)=Lr(x`,x``)+ x L (x x`G(x,x’)V(x,x`)dAs

s

The radiance sent from x’ to x’’ is simply the amount of radiance sent from all other visible points xin the scene and then reflected to x’’

Rendering Equation

Adding in the radiance directly emitted from x’ to x’’ yields the rendering equation:

Lr(x`,x``)=Lr(x`,x``)+ f(x x` x``)L(x x`G(x,x’)V(x,x`)dAs

The radiance sent from x’ to x’’is simply the amount of radiancedirectly emitted from x’ to x’’ plusthe radiance sent from all othervisible points x in the sceneand then reflected to x’’

Radiosity EquationRadiosity Equation

More importantly the outgoing radiance is same in all directions and in fact equals B/

B(x) = E(x) +x B(x`) G(x,x’)V(x,x`)dA

s

AdvantagesAdvantages

1)Highly realistic quality of the resulting images by calculating the diffuse interreflection of light energy in an environment.

2)Accurate simulation of energy transfer. 3)The viewpoint independence of the basic

radiosity algorithm provides the opportunity for interactive "walkthroughs" of environments.

4)Soft shadows and diffuse interreflection.

DisadvantagesDisadvantages

1)Large computational and storage costs for form factors.

2)Must preprocess polygonal environments.

3)Non-diffuse components of light not represented.

4)Will be very expensive if object(s) is moving in the scene.

ReferencesReferences

Radiosity Papers are available here 1)http://www.scs.leeds.ac.uk/cuddles/rover/radpap.htm

2)SIGGRAPH 1993 Education Slide Set, by Stephen Spencer http://www.education.siggraph.org/materials/HyperGraph/radioity/overview_1.htm

Books: 1)Radiosity and Global Illumination Sillion and Puech ISBN 1-55860-277-1 2)Radiosity and Realistic Image Synthesis. Cohen and Wallace .ISBN 0-12-178270-0 Software: 1)www.acurender.com

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RADIOSITY