Image Synthesis

computer graphics & visualization

Image Synthesis

Basics of global illumination


Image Synthesis – WS 07/08Dr. Jens Krüger – Computer Graphics and Visualization Group

Global illumination• Local lighting

• Light mapping (environment maps, illumination maps) approximates global illumination on static scene geometry • Does not address dynamic objects that move through the scene• No motion parallax

• Non-local lighting

• Area light sources• Shadows• Inter-object reflections• Subsurface scattering

• Simulated via ray-tracing or radiosity

• Not yet possible in real-time



Global illumination

• Solution:

• Precomputed Irradiance Volumes for static scenes and Precomputed Radiance Transfer for objects within those scenes



Photorealistic image synthesis• Rendering in interactive computer graphics

• Hardware oriented: pipeline approach (OpenGL/DirectX)• Plausible, but no allways photorealistic

• Physics based rendering - photorealism

• Geometry, animation, illumination, rendering

• Current trends

• Photorealism in real-time via graphics hardware• Shading Languages



Photorealistic image synthesis

• Light transport

• Photons from light sources into the scene• Global problem: every point illuminates every other point• Interaction between matter and light• Visibility • Result: incoming light at every point

• Scene description

• Geometry: surface and volumes• Light sources: position, orientation, power• Surface properties: reflection properties




• Simulation of the interaction between light and matter

• In volumes: Volume Rendering (Participating Media)• At surfaces: Reflection and refraction

• Traditional computer graphics:

• Surface graphics with vacuum in between, no interaction• Scattering only at surfaces




• Light as physical quantity: Radiometry

• Radiometry: physical measurement of electromagnetic energy• Photometry: psychophysical measurement of the visual sensation produced by the electromagnetic spectrum • Radiometric quantities have to be converted to photometric quantities

• Photons have energy: E=hn

• Planck constant: h• Frequency of the light wave: n



Radiometric quantitiesStrahlungsenergie: radiant energy

Q in Joule [J] Strahlungsleistung oder -fluss: radiant flux

in Watt [W=J/s]

Einfallende Flussdichte: irradiance (incident)

power per area in [W/m2]

ausgehende Flussdichte: radiosity (radiant exitance)

power per area in [W/m2]

dtdQ

dAdE

dAdB



Radiometric quantitiesStrahlungsintensität

power per solid angle in [W/sr]ddI



Strahldichte: RadianceCombination of flux and intensityStrahldichte = Radiance

Central quantity in physics based images synthesisUnits: [W/(m2sr)]Power per unit solid angle per projected unit area

ddA

dddAd

ddAdL N cos

222

N

d

dNdA

dA



RadianceRelation between irradiance and radiance

dxLxE i cos),()(



BRDFBRDF

Bidirectional reflection distribution function

Proportionality constant fr(i ,x, r ) [1/sr]

iiii

rrrir dxL

xdLxf

cos),(),(),,(



Reflection equationDifferential reflected radiance from incoming radiance using BRDF

Integration over all directions

Integral equation for one unknown based on relation between Li(i) and Lr(r)

iiiirirrr dxLxfxdLi

cos),(),,(),( from,

iiiirirrr dxLxfxL

cos),(),,(),(



RadiosityRadiosity equationForm factorsSolution methods



Solving the rendering equation

• Monte-Carlo techniques– See course Computer Graphics

• Finite-Elemente techniques– Radiosity technique– Projection of equations with infinite dimension onto

functions space with finite dimension– Results in a linear system of equations– Efficient for smooth illumination and reflection

Sy

yirirrer dAyxGyxVyLxfxLxL

),(),(),(),,(),(),(



Example



Example



Example



Considerations• Energy conservation in closed scene

• Energy equilibrium

• Origin: heat simulation

• Application in CG by Goral 1984

• Subdidision of the scene into planar patches

• Diffuse Reflection (Lambertian reflector)



ConsiderationsSubdivision of the scene into planar patchesDiffuse Reflection (Lambertian reflector)



BRDF: Diffuse ReflectionRadiosity and reflectance

from radiance

we get

EfLddL

dL

dLB

diffusrdiffusrrrrrdiffusr

rrdiffusr

rrdiffusr

,,

2

0 0,

,

,

2

sincos

cos

cos

diffus

diffusrdiffusrdiffus ffEB

,,

EB

diffus N

LiLr



Continuous radiosity equationFrom rendering equation

Assumption: diffuse Reflection

Integration over outgoing directions

from

10 ,1),,(

rir xf

Sy

ye dAyxGyxVyBxxBxB

),(),()()()()(

Sy

yirirrer dAyxGyxVyLxfxLxL

),(),(),(),,(),(),(

x

y

x

y

)(),( xBxL r



Continuous radiosity equationRadiosity equation

with new geometry factor

2||||coscos

),(),('yx

yxVyxG yx

Sy

ye dAyxGyBxxBxB

),()()()()(

x

y

x

y



Classical radiosity equationRadiosity equation

Assumption: surface-patches i with constant Bi(x)

Averaging: integration of all Bi and division through Ai

ij

i j

F

Axx

Ayy

ij ji

eii dAdAyxG

ABBB

),(1

Sy

ye dAyxGyBxxBxB

),()()()()(

iAx

xi

i dAxBA

B

)(1



Classical radiosity equationClassical discrete Radiosity equation

Form factors

Fraction of energy that leaves element i and directly arrives at element j

ijj jieii FBBB

i jAx

xAy

yyx

iij dAdA

yxyxV

AF

2||||coscos

),(1



“Direction” of Form factorsFraction of energy that leaves element i and directly arrives at element j

jjij jiieiii

iijj jiieiii

ijj jieii

Axx

Ayy

yx

iij

AFBABAB

AFBABAB

FBBB

dAdAyx

yxVA

Fi j

2||||coscos

),(1

iijjji AFAF



Form factor computationForm factor computation methods



Form factorsNusselt-Analog

– Geometric interpretation of form factorsfrom differential area dAi to element Aj

– Proportional to the area of doubleprojection onto base of hemisphere

– First projection:– Second projection:

Pi

Pj

Projectiononto hemisphere

Cylinder projectiononto circle area

2/cos rj /cos i



Form factors

• The Nusselt analog

• First projection gives the solid angle subtended by the element• Second projection accounts for area foreshortening on the receiver • Fraction with respect to the area of the unit circle implicitely accounts for

the division through



Form factor computation

• The hemicube algorithm

• Nusselt for hemicube• The dAiAj form factor remains constant for arbitrary elements as long as the projected area is unchanged• Idea:

• Pre-compute ´delta form factors´ for a number of ´well-defined´ elements • Determine which of these elements are covered by the projection • Sum up the contributions to obtain the form factor



Form factor computation• The hemicube algorithm

– Fij = q Fq




• Implementing the hemicube algorithm

• Subdivide the hemicube into regular elements• The finer the refinement the more accurate the form factor can be approximated

• Compute the delta form factors (analytically) and store them in a lookup table • Determine all facets covered by the projection of an element• Count the covered facets and sum up the delta form factors



Form factor computation• Hemicube method

222

SIDE

222

TOP

222

1

;1

1coscos

1coscos

yzAzF

yxAF

r

yxrr

F

q

qqi

qiq

x

z

A

A




• Hardware accelerated hemicube algorithm

• Process each face of the hemicube separately• Select the center of projection as the camera point• Define the current face as the view plane

• The viewport determines the size of facets• Render each element using this view

• Color elements with a unique id• Grab the color buffer• Count the number of colored pixels and sum up the corresponding form factors

• Visibility test is performed by the depth test




Hemicube-Verfahren: Simulated Steel Mill (Feldman, Wallace)55 000 Patches, gerechnet auf VAX 8700



Computing the radiosity• Discretize the scene

• Issue emission values and reflectivities for each element

• Compute the Form Factors based on the geometric relationship between elements

• Solve the radiosity system

• Derive shading values from the radiosity and render the elements



Computing the radiosity

Be>0ijj ji

eii FBBB



Smooth solutionSolution yields constant radiosities per patchSolution is independent of view pointInterpolate per-vertex valuesGouraud-Shading for interactive walk-throughsAlternatively: Radiosity-Texture



Radiosity solution techniques• Classical radiosity (often Ei instead of Bi

e)

n

T

nnnn

n

n

nnn

ijj jiii

B

BB

FFF

FFFFFF

E

EE

B

BB

FBEB

2

1

21

22221

11211

2

1

2

1

2

1

00

0000

GeometryMaterial



Radiosity solution techniques• Direct solution

• Gauß elimination: matrix inversion– complexity = O(n3) for n patches

ETB

BTETBEB

1)1(

)1(



Linear system of equationsFor n patches: a system for n unknowns Bi n2 matrix elements from form factorsMatrix elements (1-T)ij,i!=j = 0, if V(i,j) = 0

dominant diagonalMatrix 1

1

11

2

1

2

1

21

22222212

11121111

EBT

E

EE

B

BB

FFF

FFFFFF

nnnnnnnnn

n

n



Iterative solution methodsB = E + TB

= E + T(E+TB) = E + TE + T2B

= ... = T0E + T1E + T2E + T3E + ... = B(0) + B(1) + B(2) + B(3) + ...

2-times reflected light 1-times reflected light 0-time reflected light



Jacobi iterationIteration 0












GatheringOne step gathers energy from all other patches and generates one new valueSelect patches consecutively independent of their contributionPhysical interpretationof Gauß-Seidel

B;output }

;:

)(each for { converged)(not while

guess; starting ) (allfor

,1

n

ijj ii

jijii

i

TBT

EB

i

Bi



Shooting Select patches with regard to importance, distribute energy to all others, shot „unshot“ radiosity

B;output }

;: ) (allfor

;

;

largest; is such that pick { converged)(not while

}; ;0

{ ) (allfor

temprrj

rtemp

BB

ri

ErBi

ii

ji

ii

i

TT

jj

i

Tr

ii

i

iii



Solution methods summary

• So far, solution methods rely on a static subdivision of the domain

• Scene is subdivided into patches• Constant radiosity is assumed for each patch

• Hierarchical methods rely on a dynamic subdivision of the scene

• Refinement is adapted with respect to the local radiosity distribution



Radiosity vs. ray-tracing• Ray-tracing only accounts for specular reflection and refraction

• Too many rays would have to be traced

• Diffuse interactions can´t be simulated easily

• Sharp specular reflections can´t be simulated

• Radiosity only accounts for diffuse interactions

• Both methods are global, but differ significantly in the visual result



Ray-tracing vs. RadiosityExamples

Documents

Image Synthesis