Rendering & Reconstructing Under Complex BRDF ’ s
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Transcript of Rendering & Reconstructing Under Complex BRDF ’ s
Rendering & Reconstructing Under Complex BRDF’s
AGENDA Introduction
Motivation Basic concepts
Photometric Stereo
Image Based Rendering (IBR)
Summary
IntroductionIn this lecture we will discuss the way
lightinteracts with matter and how to improverealism in CV and in other related areas
such ascomputer graphics, using this knowledge.
Motivation (1) – Constructing Geometry of an Object
Left: no light .
Right: A spot light is pointing down on the object from above and behind, reflecting off the surface of the sphere.
This simple highlight gives the viewer a completely different reading of the scene.
Motivation (2) – Giving Clues To an Object's Material
The objects reflects highlights differently.
Left: soft - as though the object were made of chalk.
Right: glossy - creates the perception of very shiny plastic.
Motivation (3) – Image Based Rendering - Changing View Point
Motivation (4) – Image Based Rendering - Changing Lights Direction
Given 1 object image (face) taken under single light source (unknown direction).
Motivation (4) – Image Based Rendering - changing lights
render same face under new lighting direction:
Basic Concepts Radiometry Solid angle Radiance Irradiance Radiosity & Exitance BRDF “Helmholtz” Reciprocity Rule Isotropic vs. Anisotropic Special BRDF
Diffuse surface & Lambertian Low Specular surface & Fong Model
Local vs. Global Illumination
Radiometry
Deals with the following Questions: How do we measure light? How “bright” will surfaces be? How does light interacts with surfaces?
Surface material
Radiometry – Some Answers..
How much light the surface receives
LN
How much of the received light is reflected
Brightness of a surface
Example
Same light source hitting two different surfaces:
Light hits the surface directly Light hits the surface at an angle
As a result the right surface receives less light per square inch !
Light Behavior
Absorbed transmitted
reflected
Combination
Fluorescence
Absorbing light at one
wavelength, and radiate
light at different
wavelength.
Simplifying Assumptions The light leaving a point on a surface is
due only to light arriving at this point.
No Fluorescence
Surfaces do not generate light internally - treating sources separately.
Radiometry – Formalization
(i,i)
(o, o)
LN
R
V
Radiometry – Formalization
Around any point there is a hemisphere ofdirections:
Spherical Coordinates
x = r sincos y= r sin sinz= r cos
Basic Concepts Radiometry Solid angle Radiance Irradiance Radiosity & Exitance BRDF “Helmholtz” Reciprocity Rule Isotropic vs. Anisotropic Special BRDF
Diffuse surface & Lambertian Low Specular surfac & Fong Model
Local vs. Global Illumination
Intro. To Solid Angle
Light is form of energy
light is measured in terms of flow through an area
light coming from a single direction
light coming from a small region
Solid Angle - Definition
The solid angle is the area of the projection of the object onto the unit sphere.
Units : steradians, abbreviated sr.
Solid Angle of a Small Patch
The solid angle subtended by a small patch area dA is:
2
cos
r
dAd
ddd sin
dA
Basic Concepts Radiometry Solid angle Radiance & Irradiance Radiosity & Exitance BRDF “Helmholtz” Reciprocity Rule Isotropic vs. Anisotropic Special BRDF
Diffuse surface & Lambertian Low Specular surfac & Fong Model
Local vs. Global Illumination
Radiance
Denoted: L(x,,) Units: Wm-2sr-1 .
Amount of energy traveling at some point in a specified direction, per unit time, per unit area perpendicular to the direction of travel, per unit solid angle
Radiance is Constant Along a Straight Line
Assuming light does not interact with the medium through which it travels – i.e. that we are in .
How much light is arriving at a surface.
A surface experiencing radiance L(x) coming in from d experiences irradiance:
Note: While the radiance is per area perpendicular to the direction of travel, the Irradiance is not.
Units: W*m -2
Irradiance
dxL cos,, ,,xEi
d cos
d
Basic Concepts Radiometry Solid angle Radiance Irradiance Radiosity & Exitance BRDF “Helmholtz” Reciprocity Rule Isotropic vs. Anisotropic Special BRDF
Diffuse surface & Lambertian Low Specular surfac & Fong Model
Local vs. Global Illumination
dLB cos),,(xx
Radiosity
Total power leaving a surface, per unit area on the surface.
To get it, integrate radiance over the hemisphere of outgoing directions: X
Exitance Light sources emit light, they are
sources of radiance
Exitance is the equivalent of radiosity for emitters:
dLE e cos),,(xx
Basic Concepts Radiometry Solid angle Radiance Irradiance Radiosity & Exitance BRDF “Helmholtz” Reciprocity Rule Isotropic vs. Anisotropic Special BRDF
Diffuse surface & Lambertian Low Specular surfac & Fong Model
Local vs. Global Illumination
Intuition: BRDF is a function that specifies the ratio between the incident light in one direction and the emitted light in a second direction.
The function defines properties of the surface (shininess,..)
BRDF – more formally
the ratio of the radiance in the outgoing direction to the incident irradiance at a point on the surface
Range: [0,infinity] (surprising?)Units: inverse steradians = sr -1
dL
L
Ei
L
iiii
ooo
ii
oooiioobrdf cos,
,
,
,,,,,
Outgoing radiance
irradiance
Basic Concepts Radiometry Solid angle Radiance Irradiance Radiosity & Exitance BRDF “Helmholtz” Reciprocity Rule Isotropic vs. Anisotropic Special BRDF
Diffuse surface & Lambertian Low Specular surfac & Fong Model
Local vs. Global Illumination
Helmholtz Reciprocity Rule
brdf is symmetric:
ooiiiioo brdfbrdf ,,,,,,,
=(i,i) (i,i)
(r,r) (r,r)
Basic Concepts Radiometry Solid angle Radiance Irradiance Radiosity & Exitance BRDF “Helmholtz” Reciprocity Rule Isotropic vs. Anisotropic Special BRDF
Diffuse surface & Lambertian Low Specular surfac & Fong Model
Local vs. Global Illumination
Isotropic vs. Anisotropic
Isotropic reflection - reflection that does not vary as the surface is rotated about the normal (the angle).
Isotropic – useful assumption.
Basic Concepts Radiometry Solid angle Radiance Irradiance Radiosity & Exitance BRDF “Helmholtz” Reciprocity Rule Isotropic vs. Anisotropic Special BRDF
Diffuse surface & Lambertian Low Specular surfac & Fong Model
Local vs. Global Illumination
Special BRDFs
Diffuse Light Illumination that a surface reflects
equally in all directions.
BRDF is constant:
The brightness is independent of the observer position.
Also called “Lambertian” Reflection.
xx brdfiioobrdf ,,,,
Ideal Diffuse Surfaces – ALBEDO definition
Albedo - The fraction of the incident radiance in a given direction that is reflected by a point on diffuse surface (in all possible directions).
Denoted d.
Also called diffuse reflectance.
xx
xx
brdf
oobrdf
oobrdfd
d
d
cos
cos
The radiant energy I from a diffuse surface:
Lambert’s Law
Unit normal
intensity of light source
Light unit vector
radiant
albedo
NLNLII Lˆ*ˆˆ**
L
N
Specular Surface
Light reflected from the surface unequally to all directions.
These are the bright spots on objects (polished metal, apple ...).
Phong Model – Specular Light
• How much reflection light you can see depends on where
you are
Different BRDF
PerfectlyPerfectly SpecularSpecular “ “Mirror”Mirror”
n n ∞ ∞
Different BRDFIncidentIncidentLightLightRayRay
SurfaceSurfaceNormalNormalReflectedReflected
LightLight
SlightlySlightly scatteredscattered Specular:Specular:
Different BRDF
perfectly Diffuse
IncidentIncidentLightLightRayRay
SurfaceSurfaceNormalNormal
Different BRDF
Combination ofDiffuse and Specular
IncidentIncidentLightLightRayRay
SurfaceSurfaceNormalNormal
Basic Concepts Radiometry Solid angle Radiance Irradiance Radiosity & Exitance BRDF “Helmholtz” Reciprocity Rule Isotropic vs. Anisotropic Special BRDF
Diffuse surface & Lambertian Low Specular surfac & Fong Model
Local vs. Global Illumination
Local vs. Global Illumination
Local illumination Everything is lit only by light sources
Global illumination Everything is lit by everything else
Local illumination
global illumination
O.K. So Now What?!
Photometric Stereo
The Problem
Given a set of images of the same object, from the same view point, under different given light sources…
Can We Recover The 3D Shape of The Object?
General Schema
Recover surface normal
Recover shape out of normals
Overview
Classic approach Recover surface normal when the
light is known Recover surface normal when the
light is unknown
New idea – “shape by example”
Since we keep the camera and the scene intact, each image pixel of the three images correspond to the same 3D point :
Classic Approach - Basic Idea
Classic Approach
Assumptions: “Lambertian” surfaces Point light sources that are distant
Lambert’s law:
normal
intensity of light source
light vector
Image intensity
albedo
NLNLII Lˆ*ˆˆ**
N
L
Vector Form
For each pixel p, the normals are the same and we
get 3 conditions respectively:
i = 1,2,3For each pixel p we get a vector :
pNLpN
L
L
L
I
I
I
Ip p
p
p
p
p
p
p
ˆ**ˆ**
3
2
1
3
2
1
Lp = 3*3 matrix
NpiLIipp **
Simple Case – Light is Known
In that case we get for each pixel:
pNLIp p ˆ**
IpLpN p1ˆ*
IpL
IpLpN
p
p
1
1
ˆ
N is a unit vector
More Complex – Light is Not Known - Factorization
For each pixel:
For f frames and p pixels, we get:
pNLIp p
*
pLp LIL ˆ*
NpN ˆ*
LNI f*p intensity matrix F*3 light
matrix3*p normals matrix
Factorization
If there is no noise, then rank (I) = 3.
By Singular Value Decomposition (SVD):
But, there are many solutions, since:
NLVUI T ~*
~
NLNAALNLI~~
*~~~
**~~
*~ 1
Shape and Materials by Example:A Photometric Stereo Approach
Aaron Hertzmann, University of Toronto
& Steve Seitz, University of Washington
The Idea
Consider the simple case of two objects photographed together
Orientation-Consistency Cue
Suppose, we would like to determine the shape of the bottle.
Under the right conditions it holds that:
“Two points with the same surface orientation reflect the same light toward the viewer”
Orientation-Consistency Cue
For example, if a point is in highlight on the bottle, then it must have the same surface normal as the region in highlight on the sphere.
Ambiguous But, what happens when:
There are multiple highlights on the sphere? Multiple points on the sphere with the same intensity.
Solution: taking pictures under more lighting conditions.
More Lighting Conditions…
General Assumptions
• At least one reference object of the same or similar material must be imaged under the same illumination.
• The shape of the reference object (sphere) is known.
• Lighting is distant.
• The camera is orthographic.
• Local illumination only – shadows, intereflection and so on are ignored.
Formalization
Given multiple images of reference and target objects - same viewpoint, different illuminations :
Ir1 , . . . , Irn - the reference images.
It1, . . . , Itn - target images.
Corresponding reference Iri and target Iti images are
captured under the same illumination.
Let Ir1,p be the intensity of pixel p in
reference image #1.
Define vector V to be the intensities at a same pixel over the n images. Vr
p = [Ir1,p , . . . , Irn,p]T
Vtp = [It1,p , . . . , Itn,p]T
Formalization – cont.
Basic Algorithm
Given Pixel p on the target object look for pixel q on thereference object s.t. ||Vp – Vq|| is minimized.
Pixels p and q have the same normal if
|| Vp – Vq|| is minimized.
Determining Normal of a Point
- =
Determining Normal of a Point
- =
- =
Reference objectTarget object
Distant light
Limitations of Basic Algorithm (1)
Must have uniform BRDF for each point on target object
Limitations of Basic Algorithm (2)
Reference object Target object
Distant light
Reference and target object are made of the same material
Target Object Has Different BRDF’s To overcome it, target object must be
either pure diffuse or pure specular. For diffuse object use lambert’s low:
light
ppt
pt lnI **
albedo normal
light
ppt
pt lnI **
Light source (direction & intencity)
light
pr
pr lnI **
Target object has different BRDF’s – The Trick
p and q have the same normal if
is minimized
qr
q
pt
pt
V
V
V
V r
Target Object Made of Multiple Materials
Assume every material can be represented as linear combination of k (base) materials .
Use k (independent) reference objects.
Each pixel in target material can be represented as a linear combination of the k reference materials.
Find material coefficient and pixel q for best corresponding with pixel p.
Advantages
The BRDF may be arbitrary.
BRDF may vary over the surface.
The illumination may be unknown.
Any number of light source.
Result – Uniform BRDFBottle Reconstruction
8 in total
Result – Unifrom BRDFVelvet Reconstruction
reference target
14 in total
Result – Multiple materialsCat Reconstruction
Gray, diffuse sphere Ceramic cat
Shiny, black sphere
13 in total
Image Based Rendering
Image Based Rendering (IBR) Input: Dense set of images from different
viewpoints or different illumination.
Goal: Create pictures of synthetic scenes under new illumination conditions or from new viewpoints.
The picture should be undistinguishable from photographs of real environments.
Rendering Algorithms
Differ in the assumptions made regarding lighting and reflectance in the scene and in the solution space.
local vs. global illumination algorithms.
view dependent vs. view independent solutions.
Agenda Local illumination, view dependent
algorithm for rendering a human face
Global illumination, view independent algorithm for acquiring the reflectance properties of complete scenes
Summary
Local IlluminationView Dependent
Acquiring the Reflectance Field of a Human Face
Paul Debevec, Tim Hawkins, Chris Tchou, Haarm-Pieter Duiker, Westley Sarokin, Mark
SagarSIGGRAPH 2000
Goals Acquire images of the face from 2
viewpoints under a dense sampling of incident illumination directions.
Construct a reflectance function for each pixel.
Render the face under new illumination conditions.
Challenges• Complex and individual shape of the face.
• Subtle and spatially varying reflectance properties of the skin.
• Complex deformation of the face during movement.
• Viewers are extremely sensitive to the appearance of other people’s faces.
Light Stage
Constructing Reflectance Functions
For each pixel location (x, y) in each camera, that location on the face is illuminated for 64 x 32 directions of and .
For each pixel we keep all radiance values under 2000 different illumination direction (reflectance function).
Rxy(, ) corresponding to the ray through the pixel (x,y) with illumination direction (, ) .
Novel Form of Illumination Rxy(, ) represents how much light is
reflected towards the camera by pixel (x,y) as a result of the illumination from direction (, ).
Solid angle covered by each of the illumination
directions
Illumination Map
One can capture illumination at a point in the real world with a single spherical “photograph” or environment map.
Two different projections of the same spherical image
Novel Form of Illumination
Grace Cathedral in San Francisco ,St. Peter's Basilica, The Uffizi Gallery in Florence ,the UC Berkeley Eucalyptus Grove and a synthetic test environment.
Results
Watching a movie…
Render a human face - summary
2000 images taken from a fixed viewpoint under different illumination conditions
A reflectance function for each pixel was created using these images
A linear combination for each pixel together with the illumination map enable rendering the face from natural illumination conditions
Global Illumination View Independent
Global illumination , view independent
Some basis The Global illumination equation Basic Radiosity methods
New idea – Inverse Global Illumination
Summary
Recall
Radiance – Amount of light.
BRDF – Ratio between out going radiance and coming irradiance.
Radiosity - Total power leaving a surface.
Exitance – Total power leaving a point on a light source.
Global Illumination Equation
Total power leaving a point in a specified direction:
Radiance Exitance BRDF Irradiance
dLLL ioobrdfooeoo cos),,(),,,,(),,(),,( xxxx
Total radiance leaving the point x on the surface
in direction (o,o)
Radiance emitted from the surace at point x in direction
(o,o) , equal zero for non light sources
The fraction of the incoming irradiance at point x, in direction (,) which is
reflected by the surface in direction (o,o)
The incoming irradiance at point x, in
direction (,)
Total light reflected by the surface
Basic Radiosity Methods
Originally introduced in 1950s as a method for computing radiant heat exchange between surfaces
Radiosity Algorithms
Solve the global illumination equation under a restrictive set of assumptions All surfaces are
perfectly diffuse Surfaces can be
broken into patches with constant radiosity
Assumptions allow us to simplify the global illumination equation
The Radiosity Algorithm For Image Synthesis
Form Factor Calculation
Solution to the system of equations
VisualizationRadiosity solution
Radiosity image
Input of scene geometry
Input of reflectance properties(albedo for each patch)
Viewing direction
Radiosity algorithm
The Form Factor
The form factor Fij is the fraction of the total radiance leaving a patch i which is received by patch j
A function of the scene geometry only
Sum to unity1,
1
N
j ijFi
The Discrete Radiosity Equation
N
jjijiii BFEB
1
dLLL ioobrdfooeoo cos),,(),,,,(),,(),,( xxxx
From Total radiance leaving point x in a
specific direction to the radiosity leaving a patch
i
From the radiance emitted by point x
in a specific direction to the exitance leaving
patch j
From BRDF to albedo
From integration of irradiance
over the hemisphere to
the sum over all the patches
The Discrete Radiosity Equation
B = E + F x B
E = MB where M = (IN - F)
N
jjijiii BFEB
1
NNNNNNNN
N
NN B
B
B
FFF
FF
FFF
E
E
E
B
B
B
2
1
21
222212
11121111
2
1
2
1
The Discrete Radiosity Equation (cont)
E = M x B
Dimension of M is given by the number of patches in the scene: N xN It’s a big system Iterative solution
NNNNNNNN
N
N B
B
B
FFF
FF
FFF
E
E
E
2
1
21
222212
11121111
2
1
1
1
1
Radiosity Algorithm – Pro & Cons Needs only be calculated once for
different viewing conditions
when geometry changes there is a need to recalculate the form factors
If lighting changes then only the equation needs resolving
Radiosity Algorithm - Results
Walking through the scene
Inverse Global IlluminationRecovering Reflectance Models
of Real Scenes from Photographs
Computer Science DivisionUniversity of California at
Berkeley
Yizhou Yu, Paul Debevec, Jitendra Malik & Tim Hawkins
Global Illumination
Reflectance Properties
Radiance Images
Geometry Illumination
Inverse Global Illumination
Reflectance Properties
Radiance Images
Geometry Illumination
Inverse Global illumination Outline
Motivation Goal Partial solutions
Inverse Radiosity Specular Parameters
Mutual Illumination Results Conclusion
Motivation Most Image Based Rendering
methods allow novel viewpoints, but not changes in lighting.
This paper shows recovery of reflectance parameters of a scene.
Can then relight scene.
Motivation - cont
Many authors have previously recovered reflectance parameters. e.g., Specular and diffuse parameters Spatially varying BRDFs
However, this is done in laboratory with controlled illumination
Good for individual objects, but not for an entire scene
Goal Estimation of the reflectance
properties of all surfaces in the scene at once.
Surfaces are illuminated in situ rather than as isolated samples.
Perform all of this from a relatively sparse set of photographs.
Simplifying Assumptions
No transmission Known geometry Known light source positions Known cameras positions Radiance maps Specular reflectance parameters
constant over large surface regions
Simplifying Assumptions - cont Each surface point captured in at
least one image
Each light source captured in at least one image
Image of highlight in each specular surface region in at least one image
First Step Toward The Full Solution
Inverse radiosity
Pure diffuse scenes
The environment is broken into patches with constant diffuse albedo
Inverse Radiosity
Input: Scene Geometry Lighting conditions Radiance distribution
Output: Diffuse albedo at each patch in the
environment
Input -Geometry and Camera Positions
Input - Light Sources
j
ijjFB
ijjFB
Inverse Radiosity
j
ijjiii FBEB iB
? ii EB ?iii EB
jB
)/()( j
ijjiii FBEB
Second Step Toward The Full Solution
Local illumination
Single surface
Single known light source.
Uniform BRDF’s – allows both diffuse and specular reflection
Local Illumination
Radiance Li obtained by a
measurement of each
Irradiance Ei obtained by known light source
Goal BRDF estimation using (Li , Ei)
Ei Li
Ward Reflectance Model Variant of Phong model. Using Ward’s model, the radiance of a patch is
given by:
d - albedo
s K(, ) - specular term
K - nonlinear function of , the incident and viewing directions .
- surface roughness (blur) vector.
iisd
i IKL
),(
Ward Reflectance Model
isotropic specular highlight
( is scalar)
anisotropic specular highlight( is 3-component vector)
Local Illumination
Li is radiance at Pi
iisd
i IKL
),(
Ei is irradiance at Pi
i is light & camera position 3 or 5 unknown parameters: d, s
and to be estimated
iE
i
Li
Local Illumination
One equation for each pixel of surface in image
Can be solved using nonlinear optimisation
2
,,)),(( min arg iisi
i
di IKIL
sd
2
,,)),(( min arg iisi
i
di IKIL
sd
Ready For The Real thing ….
jk ACL
jiji APAP FL
Mutual Illumination Very similar to
inverse radiosity
j
vC kC
With specular surface, no longer true
Before, radiance towards Pi from Aj was same as radiance towards Ck
j j
iAPCAPsAPAPdPCPC jivjijijiiv
ivKLFLEL ),(
j j
iAPCAPsAPAPdPCPC jivjijijiiv
ivKLFLEL ),(
Mutual Illumination
We can express the difference between the two as S
This is purely due to specularity e.g. Aj might look diffuse from Pi’s
viewpoint, but have a specular highlight from camera’s viewpoint
jikjkji APCACAP SLL
Mutual Illumination To recover all BRDF parameters for
all the surfaces we need: Radiance images covering the whole
scene Each surface patch needs to be assigned
a camera from which its radiance image is selected
At least one specular highlight on each surface needs to be visible in the set of images
Each sample point gives an equation
Mutual Illumination
Idea for iterative algorithm: assume zero S initially do
calculate L radiances from S estimates using global illumination
update all d, s, using L radiances re-estimate S using d, s, and L
loop until convergence
Mutual Illumination Highlight regions need special
treatment: detect in advance.
No guarantees for convergence.
No error bound on the recovered BRDF parameter values.
In practice work well.
Results
Results
Inverse Global Illumination - Summary
Inverse radiosity Recovering specular reflectance
properties from direct illumination The reflected light was divided into diffuse
and specular components Specular component was modeled using
Ward’s model A new technique for determining
reflectance properties of entire scenes taking into account mutual illumination.
Rendering & Reconstructing Under Complex BRDF’s - Summary
Few basic concepts
Photometric Stereo
Image Based Rendering (IBR)
The End
From Normals to Shape Given pixel (x,y) and its normal n = ,
we wish to find the z coordinate.
The corresponding surface point is(x,y,Z(x,y))
The x component of n: (1,0,Zx)
The y component of n: (0,1,Zy)
z
y
x
n
n
n
Factorization – cont.
Since the normal is orthogonal to its x and y components we get:
(1,0,Zx) x (0,1,Zy) = (-Zx, -Zy, 1)
And after normalization:
11
1ˆ
22 y
x
yxz
y
x
Z
Z
ZZn
n
n
n
Now, we can integrate Z over x and y to find out Z(x,y).
z
xx n
nZ
z
yy n
nZ
Factorization – cont.
11
1ˆ
22 y
x
yxz
y
x
Z
Z
ZZn
n
n
n
Radiosity & Exitance for diffuse surfaces
dLB cos),,()( xx
Diffuse surfaces, by definition, have outgoing radiance that does not depend on direction
)(cos)()( xxx oo LdLB
dLE e cos),,()( xx
)(cos)()( xxx ee LdLE
Radiance to Radiosity
Recall:
Simplifying the global illumination equation gives:
)(),,,,(
)()(
)()( 0
xx
xx
xx
diioobrdf
eLE
LB
dLLL ioobrdfooeoo cos,,,,,,,,,, xxxx
dLLL id
e cos,,xx
xx
dLEB id cos,,xxxx
Switching the Domain
We still have annoying radiance terms inside the integral
Radiance is constant along lines
The radiance arriving is coming from a diffuse surface, y :
,,,, yx LL
yBLL ,,,, yx
Switching the Domain (cont)
We can convert the integral over the hemisphere of solid angles into one over all the surfaces in a scene:
otherwise
isiblemutually v arey and x if
0
1,
cos2
yxV
r
dyd
Sd dyr
VyB
EBy
yxxxx2
coscos,
)(
dLEB id cos,,xxxx
If x and y are mutually visible
Discrete Formulation
Assume world is broken into N disjoint patches, Pj, j=1..N, each with area Aj
Define:
i
i
Pi
i
Pi
i
dxEA
E
dxBA
B
x
x
x
x
)(1
)(1
Discrete Formulation (cont)
Change the integral over surfaces to a sum over patches:
N
jPd
j
dyVr
BEB1
2,
coscos)(
yyxyxxx
ij
i P
N
jPd
iPi
dxdyVr
BEA
dxBA x
yx
yxyxxx1
2,
coscos)(
11
N
j P Pijiii
i j
dydxVrA
BEB1
2,
coscos1
x y
yx
Sum over all patches in the
scene
Sum all points x in patch i
The Form Factor
dydxyxVrA
Fi jPx Py
iij ),(
coscos12
Fij is the proportion of the total power leaving patch Pi that is received by patch Pj
N
j P Pijiii
i j
dydxVrA
BEB1
2,
coscos1
x y
yx
N
jjijiii BFEB
1
Form Factor Properties
Depends only on geometry
Reciprocity: AiFij=AjFji
Additivity: Fi(jk)=Fij +Fik
•Reverse additivity is not true (F (jk)i Fji +Fki , it’s the area weighted average of the individual form factors)
1,1
N
j ijFi
Sum to unity (all the power leaving patch i must get somewhere):
Solving the Linear System
The matrix is very large – iterative methods are preferred
Start by expressing each radiosity in terms of the others:
ijiijijij
N
jij FMNiEBM
,1 ,1
NiM
EB
M
MB
ii
ij
N
ijj ii
iji
1 ,1
Relaxation Methods
Jacobi relaxation: Start with a guess for Bi, then (at iteration m):
NiM
EB
M
MB
ii
imj
N
ijj ii
ijmi
1 ,)1(
1
)(
NiM
EB
M
MB
M
MB
ii
imj
N
ij ii
ijmj
i
j ii
ijmi
1 ,)1(
1
)(1
1
)(
Gauss-Siedel relaxation: Use values already computed in this iteration: