computer graphics (spring 2008) coms 4160, lecture 22: global illumination cs4160
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Computer Graphics (Spring 2008)Computer Graphics (Spring 2008)
COMS 4160, Lecture 22: Global Illumination
http://www.cs.columbia.edu/~cs4160
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Illumination ModelsIllumination Models
So far considered mainly local illumination Light directly from light sources to surface
Global Illumination: multiple bounces Already ray tracing: reflections/refractions
Some images courtesy Henrik Wann Jensen
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Global IlluminationGlobal Illumination
Diffuse interreflection, color bleeding [Cornell Box]
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Global IlluminationGlobal IlluminationCaustics: Focusing through specular surface
Major research effort in 80s, 90s till today
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Overview of lectureOverview of lecture
Theory for all methods (ray trace, radiosity)
We derive Rendering Equation [Kajiya 86] Major theoretical development in field Unifying framework for all global illumination
Discuss existing approaches as special cases
Fairly theoretical lecture (but important). Not well covered in any of the textbooks. Closest are 2.6.2 in Cohen and Wallace handout (but uses slightly different notation, argument [swaps x, x’ among other things]) and 19.2 in Shirley (different notation, omits emission, but has a reasonably good intuitive discussion that we somewhat follow).
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OutlineOutline
Reflectance Equation (review)
Global Illumination
Rendering Equation
As a general Integral Equation and Operator
Approximations (Ray Tracing, Radiosity)
Surface Parameterization (Standard Form)
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Reflectance Equation (review)
ir
x
( , ) ( , ) ( , ) ( , , )( )r r e r i i i r iL x L x L x f x n Reflected Light(Output Image)
Emission Incident Light (fromlight source)
BRDF Cosine of Incident angle
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Reflectance Equation (review)
ir
x
( , ) ( , ) ( , ) ( , , )( )r r e r i i i r iL x L x L x f x n Reflected Light(Output Image)
Emission Incident Light (fromlight source)
BRDF Cosine of Incident angle
Sum over all light sources
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Reflectance Equation (review)
ir
x
( , ) ( , ) ( , ) ( , , ) cosr r e r i i i r iiL x L x L x df x
Reflected Light(Output Image)
Emission Incident Light (fromlight source)
BRDF Cosine of Incident angle
Replace sum with integral
id
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Global Illumination
ir
x
( , ) ( , ) ( , , ) cos( , )r r e r i r ir iiL x dL x L x f x
Reflected Light(Output Image)
Emission ReflectedLight (fromsurface)
BRDF Cosine of Incident angle
id
Surfaces (interreflection)
dAx
i x x
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Rendering Equation
ir
x
( , ) ( , , ) c( , ) ( , ) ose r i rr r i ir iL x L xL x f x d
Reflected Light(Output Image)
Emission ReflectedLight
BRDF Cosine of Incident angle
id
Surfaces (interreflection)
dAx
UNKNOWN UNKNOWNKNOWN KNOWN KNOWN
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Rendering Equation (Kajiya 86)Rendering Equation (Kajiya 86)
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OutlineOutline
Reflectance Equation (review)
Global Illumination
Rendering Equation
As a general Integral Equation and Operator
Approximations (Ray Tracing, Radiosity)
Surface Parameterization (Standard Form)The material in this part of the lecture is fairly advanced andnot covered in any of the texts. The slides should be fairlycomplete. This section is fairly short, and I hope some of youwill get some insight into solutions for general global illumination
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Rendering Equation as Integral Equation
Reflected Light(Output Image)
Emission ReflectedLight
BRDF Cosine of Incident angle
UNKNOWN UNKNOWNKNOWN KNOWN KNOWN
( ) ( )( ) ( , )l u e u K u dvl v v
Is a Fredholm Integral Equation of second kind [extensively studied numerically] with canonical form
( , ) ( , , ) c( , ) ( , ) ose r i rr r i ir iL x L xL x f x d
Kernel of equation
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Linear Operator Equation
( ) ( )( ) ( , )l u e u K u dvl v v Kernel of equationLight Transport Operator
L E KL Can be discretized to a simple matrix equation[or system of simultaneous linear equations] (L, E are vectors, K is the light transport matrix)
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Solution Techniques
All global illumination methods try to solve (approximations of) the rendering equation– Too hard for analytic solution: numerical methods– General theory of solving integral equations
Radiosity (next lecture; usually diffuse surfaces)– General class numerical finite element methods
(divide surfaces in scene into a finite set elements or patches)
– Set up linear system (matrix) of simultaneous equations
– Solve iteratively
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Ray Tracing and extensions– General class numerical Monte Carlo methods– Approximate set of all paths of light in scene
L E KL IL K EL
( )I K EL 1( )I KL E
Binomial Theorem2 3( ...)I KL K K E
2 3 ...E KE K E K EL
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Ray Tracing2 3 ...E KE K E K EL
Emission directlyFrom light sources
Direct Illuminationon surfaces
Global Illumination(One bounce indirect)[Mirrors, Refraction]
(Two bounce indirect) [Caustics etc]
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Ray Tracing2 3 ...K EE K K EEL
Emission directlyFrom light sources
Direct Illuminationon surfaces
Global Illumination(One bounce indirect)[Mirrors, Refraction]
(Two bounce indirect) [Caustics etc]
OpenGL Shading
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OutlineOutline
Reflectance Equation (review)
Global Illumination
Rendering Equation
As a general Integral Equation and Operator
Approximations (Ray Tracing, Radiosity)
Surface Parameterization (Standard Form)Page 461 of Shirley is reasonably close to this part of lecture, although it uses different notation. See also pages 38 and 39 in handout, which may have a clearer explanation of the ideas.
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Rendering Equation
ir
x
( , ) ( , , ) c( , ) ( , ) ose r i rr r i ir iL x L xL x f x d
Reflected Light(Output Image)
Emission ReflectedLight
BRDF Cosine of Incident angle
id
Surfaces (interreflection)
dAx
UNKNOWN UNKNOWNKNOWN KNOWN KNOWN
i x x
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Change of Variables
Integral over angles sometimes insufficient. Write integral in terms of surface radiance only (change of variables)
( , ) ( , ) ( , ) ( , , ) cosr r e r r i i r i iL x L x L x df x
x
x
dA
i
i
i
o
id2
cos
| |o
i
dAd
x x
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Change of Variables
Integral over angles sometimes insufficient. Write integral in terms of surface radiance only (change of variables)
( , ) ( , ) ( , ) ( , , ) cosr r e r r i i r i iL x L x L x df x
2
cos
| |o
i
dAd
x x
all visible2
to
cos cos( , ) ( , ) ( , ) ( , , )
| |i o
r r e r r i i r
x x
L x L x L x f xx
dx
A
2
cos cos( , ) ( , )
| |i oG x x G x x
x x
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Rendering Equation: Standard Form
Integral over angles sometimes insufficient. Write integral in terms of surface radiance only (change of variables)
Domain integral awkward. Introduce binary visibility fn V
( , ) ( , ) ( , ) ( , , ) cosr r e r r i i r i iL x L x L x df x
2
cos
| |o
i
dAd
x x
all visible2
to
cos cos( , ) ( , ) ( , ) ( , , )
| |i o
r r e r r i i r
x x
L x L x L x f xx
dx
A
2
cos cos( , ) ( , )
| |i oG x x G x x
x x
all surfaces
( , ) ( , ) ( , ) ( , , ) ( , ) ( , )r r e r r
x
i i rL x L x L x f x G x dAx x V x
Same as equation 2.52 Cohen Wallace. It swaps primedAnd unprimed, omits angular args of BRDF, - sign.Same as equation above 19.3 in Shirley, except he has no emission, slightly diff. notation
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OverviewOverview
Theory for all methods (ray trace, radiosity)
We derive Rendering Equation [Kajiya 86] Major theoretical development in field Unifying framework for all global illumination
Discuss existing approaches as special cases