Download - Rendering III : Radiosity
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Rendering III :Radiosity
July 20, 1999
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July 20, 1999
Motivational Film
✔More Bells and Whistles– Computer Graphics’ first music video
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July 20, 1999
Logistics
✔Paper summaries on Radiosity– Any takers?
✔Projects– Updates due today
– Sign up for presentations now open
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July 20, 1999
Logistics
✔Many thanks to Max and Rob for takingover for me last week.
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July 20, 1999
Photography and Light
pho•tog•ra•phy, n., the process or art ofproducing images of objects by the action oflight on a sensitized surface, esp, a film in acamera.
Sure, we know, it’s all aboutlight…
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July 20, 1999
Computer Graphics as Virtual Photography
camera(captureslight)
syntheticimage
cameramodel
(focusessimulatedlighting)
processing
photoprocessing
tonereproduction
realscene
3Dmodels
Photography:
ComputerGraphics:
Photographicprint
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July 20, 1999
Today’s Class
✔Radiosity– Basics
– The Radiosity Equation– Form Factors
• What they are• The hemicube solution
– Solving the Radiosity Equation– Rendering
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July 20, 1999
Radiosity - Basics
✔The problem with ray tracing– Great for specular type reflections
– Awful for diffuse reflections.
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July 20, 1999
Radiosity - Basics
✔Based on the theory of heat transfer✔Calculate lighting in a steady state✔Assumes that all surfaces are perfectly
diffuse✔Formulates a large systems of linear
equations✔Solution of these equations give you
radiant exitance at each point
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July 20, 1999
Radiosity - Basics
✔Radiant exitance - radiant flux out
dAdA
dM
Φ=
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July 20, 1999
Radiosity - Basics
✔Image is created by using calcuatedradiant existance values in standardrendering process.– In essence, radiosity defines a “made to fit”
texture mapping
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July 20, 1999
Radiosity - Basics
✔Program Flow
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July 20, 1999
Radiosity - Basics
✔View dependence vs viewindependence– Radiosity provides a view independent
solution
– Scene needs to be further rendered from agiven view point.
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July 20, 1999
Radiosity - Basics
✔Not points -- But patches– Scene is subdivided into patches
– Radiant exitance will be calculated for eachpatch
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July 20, 1999
Radiosity - Basics
✔Patches
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July 20, 1999
Radiosity - Basics
✔Basic idea– Each patch will receive a certain amount of
light from the environment– It will reflect fraction back into the
environment
– Keep track of amount of light reflectedback
– Continue till all light has been exhasted.
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July 20, 1999
Radiosity - Basics
✔Key idea– Since all objects are perfectly diffuse, we
can determine where light is coming from(and going) by simply considering thegeometry of the scene.
– Calculation of radiant exitance per patch isgiven by the radiosity equation.
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July 20, 1999
The Radiosity Equation
✔For each patch i, the RE gives you theradiant exitance of that patch.
✔All surfaces are perfectly diffuse (I.e.reflectivity is equal in all directions)– Reflectance for each patch is a constant ρI
✔Geometry term Fi,j indicates the fractionof flux leaving patch i and arriving atpatch j.
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July 20, 1999
Radiosity Equation
{ { {43421
surfaceother fromlight
1ereflectancoutflux init exitance
∑=
+=n
jijjioii FMMM ρ
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July 20, 1999
Radiosity Equation
✔Radiosity vs Rendering Equation
[ ]∫ ′′′′′′′′+′′=′S
xdxxIxxxxxxxgxxI ),(),,(),(),(),( ρε
∑=
+=n
jijjioii FMMM
1
ρ
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July 20, 1999
Radiosity System of Equations
✔Rearrange terms a bit
∑=
+=n
jijjioii FMMM
1
ρ
∑=
−=n
jijjiioi FMMM
1
ρ
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July 20, 1999
Radiosity System of Equations
✔Expand this out
)(
)(
)(
22111
22222212222
11221111111
nnnnnnnnnon
nnno
nnno
FMFMFMMM
FMFMFMMM
FMFMFMMM
ρρρ
ρρρρρρ
+++−=
+++−=+++−=
L
K
L
L
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July 20, 1999
Radiosity System of Equations
✔Put into matrix form
−−−
−−−−−−
=
nnnnnnnn
n
n
on
o
o
M
M
M
FFF
FFF
FFF
M
M
M
L
L
LLLL
L
L
L2
1
21
22222212
11121111
2
1
1
1
1
ρρρ
ρρρρρρ
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July 20, 1999
Radiosity System of Equations
✔Put in matrix notation
MTIM o )( −=Mo = initial n x 1 exitance vector
I = n x n identity matrix
M = final n x 1 exitance vector
T = n x n matrix whose i,j element is ρiFij
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July 20, 1999
Radiosity System of Equations
✔We have Mo
✔We have I
✔We have ρI
✔We’re solving for M✔We need this Fij -- Form Factor
✔Any questions?
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July 20, 1999
Radiosity - Form Factors
✔Form factors are a geometric term:– Given the radiant exitance of a patch Ei,
what fraction of its flux is received byanother patch Ej.
– Remember: All patches are Lambertiansurfaces and distribute flux equally in alldirections.
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July 20, 1999
Radiosity - Form Factors
✔ Flux emitted by Ei:
✔ Flux arriving at Ej
iii AME =
iijij F Φ=Φ
i
ijijF
ΦΦ
=
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July 20, 1999
Radiosity - Form Factors
✔Physically based geometry– Points are actually areas
– Rays are actually solid angles.
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July 20, 1999
Radiosity - Form Factors
✔Solid angles
2r
A=ω
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July 20, 1999
Radiosity -- Form Factors
✔Radiance
θω cos
2
dAd
dL
Φ=
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July 20, 1999
Radiosity -- Form Factor
✔Flux leaving Ei in direction of Ej
ωθθ ddAL iiiij cos)(radiance321
=Φ
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July 20, 1999
Radiosity - Form Factors
✔Solid angle of ray as seen by Ei
2/cos rdAd jjθω =
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July 20, 1999
Radiosity - Form Factor
✔Flux leaving Ei towards Ej
2
coscos
r
dAdAL jijiiij
θθ=Φ
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July 20, 1999
Radiosity - Form Factor
✔Recall: Form Factor is the fraction oftotal flux given off by one patch thatarrives at another.
✔We have the amount of flux from i to j(Φij)
✔What we need is total amount of fluxemitted by i (Φi)
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July 20, 1999
Radiosity - Form Factor
✔Recall: Patch i is Lambertian
iiiii dALdAM π==Φ
Note: it can be shown by integration of Lover a hemisphere that M = πL for aLambertian surface
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July 20, 1999
Radiosity - Form Factor
✔Calculating the fraction
2
coscos
rdAL
dAdALF
ii
jijiiijij π
θθ=
ΦΦ
=
2
coscos
r
dAF jji
ij πθθ
=
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July 20, 1999
Radiosity - Form Factor
✔Calculating the fraction
Depends ONLY on geometry!2
coscos
r
dAF jji
ij πθθ
=
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July 20, 1999
Radiosity - Form Factor
✔But wait, there’s more….– Patches are not points nor are they
differential areas– Must integrate over each patch.
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July 20, 1999
Radiosity - Form Factor
✔Integration over patches
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July 20, 1999
Radiosity - Form Factors
✔The final expression!
ji
A A
ji
iij dAdA
rAF
i j
∫ ∫=2
coscos1
πθθ
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July 20, 1999
Radiosity - Form Factors
✔Fun Facts about form factors– Solution first developed in 1760 by
Lambert– AiFij = AjFji
– Form Factor assumes a nonparticipatingmedium (like air or vacuum)
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July 20, 1999
Radiosity - Form Factors
✔So how do we solve this thing?– There is an analytic solution. But it is very
ugly and impractical (note: discovered in1993)
– Classic solution: The Hemicube method
– Any questions before we move on?
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July 20, 1999
Radiosity - Form Factors
✔Nusselt’s Analogy
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July 20, 1999
Radiosity - Form Factors
✔Nusselt’s Analogy– Project Ej onto plane of surface
– Form factor is fraction of projected surfacearea with respect to projected area of ahemisphere.
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July 20, 1999
Radiosity - Form Factors
✔Result from Nusselt’s Analogy
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July 20, 1999
Radiosity - Form Factors
✔Hemicube– Replace hemisphere with hemicube.
– Hemicube is divided into cells.– Can determine form factor for each cell
(delta form factors).
– Form factor for a patch is the sum of deltaform factors for cells that the projection ofthe patch covers.
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July 20, 1999
Radiosity - Form Factors
✔Hemicube
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July 20, 1999
Radiosity - Form Factors
✔Hemicube - Delta Form Factors– Gives an approximation of form factor for a
cell on the hemicube.
jji
EdE Ar
Fji
∆≈− 2
coscos
πθθ
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July 20, 1999
Radiosity - Form Factors
✔Hemicube - delta form factor (top side)
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July 20, 1999
Radiosity - Form Factors
✔Hemicube - Delta Form Factors– If we conveniently choose r to be 1 then
– and
122 ++= vur
rji
1coscos == θθ
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July 20, 1999
Radiosity - Form Factors
✔So...
2222 )1(
coscos
++∆
=∆≈∆vu
AA
rF top
topji
top ππθθ
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July 20, 1999
Radiosity - Form Factors
✔Similarly, it can be shown
2222 )1(
coscos
++∆=∆≈∆
nu
AnA
rF side
sideji
side ππθθ
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July 20, 1999
Radiosity - Form Factors
✔Fun Facts about Hemicube– Delta Form Factors can be precomputed
once then applied to each patch– Accuracy depends on hemicube resolution.– Most popular means of calculating form
factors
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July 20, 1999
Radiosity
✔Let’s recap– Divided our scene into n patches
– Set up this system of n equations:
−−−
−−−−−−
=
nnnnnnnn
n
n
on
o
o
M
M
M
FFF
FFF
FFF
M
M
M
L
L
LLLL
L
L
L2
1
1221
2212221212
11121111
2
1
1
1
1
ρρρ
ρρρρρρ
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July 20, 1999
Radiosity
✔Recap continued– For each i,j combination, calculate the form
factor Fij
– Now need to solve the radiosity equation
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July 20, 1999
Radiosity
✔Recap
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July 20, 1999
Radiosity
✔Solving the Radiosity Equation– Use Numerical Techniques (Jacobi /
Gauss-Sidel method)• Calculates full radiosity solution• O (n2)
– Progressive Refinement• Approximate solution• Better solution with each iteration.
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July 20, 1999
Radiosity - Progressive Refinement
✔Progressive Refinement– Based on Flux left to be distributed in
scene.– At each iteration only consider patch with
most flux to give
– Continue until flux left to be distributed fallsbelow a threshold
– Limit…threshold = 0 is equivalent to fullsolution.
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July 20, 1999
Radiosity - Progressive Refinement
✔Shooting flux into environment
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July 20, 1999
Radiosity - Progressive Refinement
✔For each patch I we keep track of:– Mi - current amount of radiant exitance
– ∆Mi - current amount of radiant exitancegained since last iteration
– ∆Miunsent - amount of unsent radiant
exitance since last iteration
✔Munsent - Total amount of unsent flux =∆Mi
unsent Ai
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July 20, 1999
Radiosity - Progressive Refinement
✔ Initialize Mi, ∆Mi, ∆Miunsent = Moi
✔ While Munsent > Threshold– Find element i with greatest ∆Mi
unsentAi
– For each other element j• Calculate form factor Fij
• Calculate ∆M, change of M due to ∆MiunsentAi
• Update ∆Munsent for element j• Update Total M for element j
– Set ∆Munsent for element I to 0
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July 20, 1999
Radiosity - Progressive Refinement
✔Solution gets progressively better witheach iteration
✔Later iterations contribute less to totalsolution than earlier iterations
✔Limit is equivalent to complete solution.
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July 20, 1999
Radiosity
✔Okay, so where are we?
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July 20, 1999
Radiosity
✔Finally! Rendering– Radiosity Solution gives us a view
independent solution– “Made to fit” texture mapping– Use traditional rendering to get view
dependent image• flat / Gouraud shading• Non-recursive ray tracing
– We are done!
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July 20, 1999
Radiosity - examples
✔Cornell Box
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July 20, 1999
Radiosity Examples
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July 20, 1999
Radiosity
✔Based on physically based energytransfer
✔Calculates steady state solution✔Assumes diffuse surfaces✔Computationally intensive
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July 20, 1999
Radiosity
✔Summary
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July 20, 1999
Radiosity
✔One thing we didn’t discuss– Meshing strategies
• How do we efficiently divide scene into patches• Adaptive Subdivision
– see Ashdown, Chapter 7
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July 20, 1999
Radiosity
✔Further Reading– Ashdown, Radiosity: A Programmer’s
Perspective– Cohen/Wallace, Radiosity and Realistic
Image Synthesis– papers on reserve
✔Any questions
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July 20, 1999
Next Class
✔Advanced Rendering– Two pass method
• Radiosity + ray tracing - Best of both worlds
– REYES
![Page 72: Rendering III : Radiosity](https://reader033.vdocuments.mx/reader033/viewer/2022042107/6256c17c16ae05576a460d25/html5/thumbnails/72.jpg)
July 20, 1999
Remember
✔Class Web Site:– http://www.cs.rit.edu/~jmg/virtualPhoto