Radiosity

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Introduction

•Ray tracing best with many highly specular surfaces

Not real scenes

•Rendering equation describes general shading problem

•Radiosity solves rendering equation for perfectly diffuse surfaces

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Terminology

•Energy ~ light (incident, transmitted) Must be conserved

•Energy flux = luminous flux = power = energy/unit time

Measured in lumens

Depends on wavelength so can integrate over spectrum using luminous efficiency curve of sensor

•Energy density (Φ) = energy flux/unit area

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Terminology

Intensity ~ brightness/lightness Brightness/lightness perceptual

= flux/area-solid angle = power/unit projected area per solid angle

Measured in candela

Φ = ∫ ∫ I dA dω

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Rendering Eqn (Kajiya)

•Consider point on surface

N

Iout(Φout)Iin(Φin)

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Rendering Equation

•Outgoing light from two sources Emission

Reflection of incoming light

•Must integrate over all incoming light Integrate over hemisphere

•Must account for foreshortening of incoming light

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Rendering Equation

Iout(Φout) = E(Φout) + ∫ 2πRbd(Φout, Φin )Iin(Φin) cos θ dω

bidirectional reflection coefficient

angle between normal and Φinemission

Note that angle is really two angles in 3D and wavelength is fixed

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Rendering Equation

• Rendering equation = energy balance Energy in = energy out

• Integrate over hemisphere

• Fredholm integral equation Cannot solve analytically in general

• Various approximations of Rbd give standard rendering models

• Should also add occlusion term in front of right side to account for other objects blocking light from reaching surface

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Another version

Consider light at point p arriving from p’

i(p, p’) = υ(p, p’)(ε(p,p’) + ∫ ρ(p, p’, p’’)i(p’, p’’)dp’’

occlusion = 0 or 1/d2emission from p’ to p

light reflected at p’ from all points p’’ towards p

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Radiosity

•Consider objects to be broken up into flat patches

may correspond to polygons in model

•Assume patches = perfectly diffuse reflectors

•Radiosity = flux = energy/unit area/ unit time leaving patch

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Notation

n patches numbered 1 to n

bi = radiosity of patch i

ai = area of patch i

bi ai = total intensity leaving patch i

ei ai = emitted intensity from patch i

ρi = reflectivity of patch i

fij = form factor = fraction of energy leaving patch j that reaches patch i

form factor

In physics, (usually written e) of material

sometimes called emissivity

= proportion of energy transmitted by object

that can be transferred to another objecten.wikipedia.org/wiki/Form_factor_(radiative_transfer)

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Radiosity Equation

energy balance

biai = eiai + ρi ∑ fjibjaj

reciprocity

fijai = fjiaj

radiosity equation

bi = ei + ρi ∑ fijbj

emitted reflected

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Matrix Form

b = [bi]

e = [ei]

R = [rij] rij = ρi if i ≠ j rii = 0

F = [fij]

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Matrix Form

b = e - RFb

formal solution

b = [I-RF]-1e

Not useful since n usually very largeAlternative: use observation that F is sparse

Will consider determination of form factors later

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Solving the Radiosity Equation

For sparse matrices, iterative methods usuallyrequire only O(n) operations per iteration

Jacobi’s method

bk+1 = e - RFbk

Gauss-Seidel: use immediate updates

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Series Approximation

1/(1-x) = 1 + x + x2+ ……

b = [I-RF]-1e = e + RFe + (RF)2e +…

[I-RF]-1 = I + RF +(RF)2+…

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Rendered Image

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Patches

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Computing Form Factors

•Consider two flat patches

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Using Differential Patches

foreshortening

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Form Factor Integral

fij = (1/ai) ∫ai ∫ai (oij cos θi cos θj / πr2 )dai daj

occlusion

foreshortening of patch i

foreshortening of patch j

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Solving the Intergral

•Very few cases where integral has (simple) closed form solution

Occlusion further complicates solution

•Alternative: use numerical methods•Two step process similar to texture mapping

Hemisphere

Hemicube

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Form Factor Examples 1

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Form Factor Examples 2

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Form Factor Examples 3

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Hemisphere

•Use illuminating hemisphere•Center hemisphere on patch with normal pointing up

•Must shift hemisphere for each point on patch

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Hemisphere

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Hemicube

•Easier to use hemicube instead of hemisphere

•Rule each side into “pixels”•Easier to project on pixels give delta form factors can be added up to give desired form factor

•To get delta form factor need only cast ray through each pixel

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Hemicube

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Instant Radiosity

•Want to use graphics system if possible•Suppose make one patch emissive•Light from this patch distributed among other patches

•Shade of other patches ~ form factors•Must use multiple OpenGL point sources to approximate uniformly emissive patch