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GR2Advanced Computer

GraphicsAGR

GR2Advanced Computer

GraphicsAGR

Lecture 15Radiosity

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ReviewReview

First a review of the two rendering approaches we have studied:– Phong reflection model– ray tracing

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Phong Reflection ModelPhong Reflection Model

This is the most common approach to rendering– objects represented as polygonal faces– intensity of faces calculated by Phong

locallocal illumination model

– polygons projected to viewplane– Gouraud shading applied with Z buffer

to determine visibility

I() = Ka()Ia() + ( Kd()( L . N ) + Ks( R . V )n ) I*() / dist

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Phong Reflection ModelPhong Reflection Model

Strengths– simple and efficient– models ambient, diffuse and specular

reflection Limitations

– only considers light incident from a light source, and not inter-object reflections - ie it is a local illumination method (ambient term is approximation to global illumination

– empirical rather than theoretical base– objects typically have plastic appearance

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Ray TracingRay Tracing

Ray traced from viewpoint through pixel until first object intersected

Colour calculated as summation of:– local Phong reflection at that point– specularly reflected light from

direction of reflection– transmitted light from refraction

direction if transparent

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Ray TracingRay Tracing

This is done recursively Colour of light incoming along

reflection direction found by:– tracing ray back until it hits an

object– colour of light emitted by object is

itself summation of local component, reflected component and transmitted component

– and so on

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Ray Tracing - Strengths and Weaknesses

Ray Tracing - Strengths and Weaknesses

Advantages– increased realism through ability to

handle inter-object reflection Disadvantages

– much more expensive than local reflection

– still empirical– only handles specular inter-object

reflection– entire calculation is view-dependent

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RadiosityRadiosity

Based on the physics of heat transfer between surfaces

Developed in 1980s at Cornell University in US (Cohen, Greenberg)

Determine energy balance of light transfer between all surfaces in an enclosed space– equilibrium reached between emission of

light and partial absorption of light Assume surfaces are opaque, are perfect

diffusediffuse reflectors and are represented as sets of rectangular patches Ai

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Radiosity ExamplesRadiosity Examples

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Radiosity - DefinitionRadiosity - Definition

Radiosity defined as:– energy (Bi ) per unit area leaving a

surface patch (Ai ) per unit time

Bi Ai = Ei Ai + Ri ( Fj-i Bj Aj) i=1,2,..Nj

Ei is the light energy emitted by Ai per unit areaRi is the fraction of incident light reflected in all directionsFj-i is the fraction of energy leaving Aj that reaches Ai

lightemitted

lightreflected

lightleaving

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Radiosity - Pictorial Definition

Radiosity - Pictorial Definition

Ai

Aj

r

Ni

Nj

i

j

Bi Ai = Ei Ai + Ri (Fj-i Bj Aj)

Form factor Fj-i is fraction ofenergy leaving Aj that reachesAi. It is determined by therelative orientation of thepatches and distance rbetween them

Form factors arehard to calculate!Let’s assume for nowwe can do it.

j

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Simplifying the EquationSimplifying the Equation

Bi Ai = Ei Ai + Ri (Fj-i Bj Aj)ieBi Ai = Ei Ai + Ri (Bj Fj-i Aj)

There is a reciprocity relationship:Fj-i Aj = Fi-jAi

Bi Ai = Ei Ai + Ri (Bj Fi-j Ai)

Hence:

Bi = Ei + Ri ( Bj Fi-j )

So the radiosity of patch Ai is given by:

j

j

j

j

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Creating a System of Equations

Creating a System of Equations

We get one equation for each patch– assume we can calculate form factors

– then N equations for N unknowns B1,B2,..BN

First equation is:

(1-R1F1-1)B1 - (R1F1-2)B2 - (R1F1-3)B3 - .. - (R1F1-N)BN = E1

..and we get in all N equations like this. Generally Fi-i will be zero - why? Most of the Ei will be zero - why?

B1 = E1 + R1 ( Bj F1-j )j

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Solving the EquationsSolving the Equations

Equations are solved iteratively - ie a first guess chosen, then repeatedly refined until deemed to converge

Suppose we guess solution as:B1

(0), B2(0), .. BN

(0)

Rewrite first equation as:B1 = E1 + (R1F1-2)B2 + .. + (R1F1-N)BN

Then we can ‘improve’ estimate of B1 by:B1

(1) = E1 + (R1F1-2)B2(0) + .. + (R1F1-N)BN

(0)

.. and so on for other Bi

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Solving the EquationsSolving the Equations

This gives an improved estimate:B1

(1), B2(1), .. BN

(1)

and we can continue until the iteration converges.

This is known as the JacobiJacobi iterative method

An improved method is Gauss-SeidelGauss-Seidel iteration– this always uses best available values

– eg B1(1) (rather than B1

(0) ) used to calculate B2

(1), etc

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RenderingRendering

What have we calculated? The Bi are the intensities of light

emanating from each patch– the form factors do not depend on

wavelength , but the Ri do

– thus Bi depend on , so we need to calculate Bi

RED, BiGREEN, Bi

BLUE

We get vertex intensities by averaging the intensities of surrounding faces

Then we can pass to Gouraud renderer code for interpolated shading

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Pause at this stagePause at this stage

Number of equations can be very large

Calculation is not view dependent

Only diffuse reflection We still have not seen how to

calculate the form factors! Their calculation dominates

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Form FactorsForm Factors

The calculation of the form factors is unfortunately quite hard

We begin by looking at the form factor between two infinitesimal areas on the patches

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Form Factors - NotationForm Factors - Notation

Ai

Ni

dAi

Nj

Aj

dAj

i

j

Form factor Fdi-dj gives the fractionof energy reaching dAj from dAi.

r

r is distance between elementsNi, Nj are the normalsi, j are angles made with normalsby line joining elements

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Form Factor Calculation : 2D Cross Section View

Form Factor Calculation : 2D Cross Section View

Draw in 2D but imagine this in 3D:* create unit hemisphere with dAi at centre* light emits/reflects equally in alldirections from dAi * form factor Fdi-dj is fraction of energyreaching dAj from dAi

dAi

Aj

dAj

Ai

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Form Factor Calculation : 2D Cross Section View

Form Factor Calculation : 2D Cross Section View

We begin calculation by* projecting dAj onto the surfaceof the hemisphere (blue)* this resulting area is (cos j / r2 ) dAj

This gives us the relative area of light energy reaching dAj from dAi

dAi

Aj

dAj

j

r

Nj

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Form Factor Calculation : 2D Cross Section View

Form Factor Calculation : 2D Cross Section View

But we need a measure perunit area of Ai so need to adjustfor orientation of Ai

This gives a ‘corrected’ area as:( cos i cos j / r2 ) dAj

dAi

Aj

dAj

i

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Form Factor Calculation : 2D Cross Section View

Form Factor Calculation : 2D Cross Section View

dAi

Total energy comes from integratingover whole hemisphere - this comes to this comes to

Hence form factor is given by:Fdi-dj = ( cos i cos j ) / ( r2 )dAj

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Form Factor Calculation : 2D Cross Section View

Form Factor Calculation : 2D Cross Section View

Finally we need to sum up for ALL dAj. This means integrating over the whole of the patch:Fdi-j = (cos i cos j ) /( r2 ) dAj

dAi

Aj

Ai

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Form Factor CalculationForm Factor Calculation

dAi

Aj

Ai

Strictly speaking, we should nowintegrate over all dAi. In practice,we take Fdi-j as representative ofFi-j and this assumption:

Fi-j = Fdi-j

works OK in practice.

However the calculation of the integral (cos i cos j ) /( r2 ) dAj

is extremely difficult. Next lecture will discuss an approximation.