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CS348B Lecture 16 Pat Hanrahan, Spring 2005
Participating Media & Vol. Scattering
Applications
Clouds, smoke, water, …
Subsurface scattering: paint, skin, …
Scientific/medical visualization: CT, MRI, …
Topics
Absorption and emission
Scattering and phase functions
Volume rendering equation
Homogeneous media
Ray tracing volumes
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CS348B Lecture 16 Pat Hanrahan, Spring 2005
Absorption
L dL( , )L x
ds
( )a x
( , ) ( ) ( , )adL x x L x ds
Absorption cross-section:
Probability of being absorbed per unit length
( )a x
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CS348B Lecture 16 Pat Hanrahan, Spring 2005
Transmittance
0
ln ( , ) ( ) ( )s
aL x s x s ds s
0
( ) ( )s
as x s ds
Optical distance or depth
( , )( )
( , ) a
dL xx ds
L x
( , ) ( ) ( , )adL x x L x ds
Homogenous media: constant
( )a as s a
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CS348B Lecture 16 Pat Hanrahan, Spring 2005
Transmittance and Opacity
0
ln ( , ) ( ) ( )s
aL x s x s ds s
Transmittance
( , )( )
( , ) a
dL xx ds
L x
( , ) ( ) ( , )adL x x L x ds
( )( ) sT s e
( )( , ) ( , ) ( ) ( , )sL x s e L x T s L x
Opacity( ) 1 ( )s T s
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CS348B Lecture 16 Pat Hanrahan, Spring 2005
Out-Scatter
L dL( , )L x
ds
( )s x
( , ) ( ) ( , )sdL x x L x ds
sScattering cross-section:
Probability of being scattered per unit length
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CS348B Lecture 16 Pat Hanrahan, Spring 2005
Extinction
L dL( , )L x
ds
( )t x
( , ) ( ) ( , )tdL x x L x ds
t a s Total cross-section
0
( ) ( )s
ts x s ds Attenuation due to both absorption and scattering
s s
t a s
W
Albedo
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CS348B Lecture 16 Pat Hanrahan, Spring 2005
Black Clouds
From Greenler, Rainbows, halos and glories
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CS348B Lecture 16 Pat Hanrahan, Spring 2005
In-Scatter
2
( , ) ( ) ( ) ( , )s
S
S x x p L x d
L dL( , )L x
ds
( )s x
( ) ( )p p Reciprocity
2
( ) 1S
p d Energy conserving
Phase function ( )p
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CS348B Lecture 16 Pat Hanrahan, Spring 2005
Phase Functions
Phase angle
Phase functions
(from the phase of the moon)
1. Isotropic
-simple
2. Rayleigh
-molecules
3. Mie scattering
- small spheres
... Huge literature ...
cos
1(cos )
4p
2
4
3 1 cos(cos )
4p
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CS348B Lecture 16 Pat Hanrahan, Spring 2005
Blue Sky = Red Sunset
From Greenler, Rainbows, halos and glories
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CS348B Lecture 16 Pat Hanrahan, Spring 2005
Coronas and Halos
Moon Corona Sun Halos
From Greenler, Rainbows, halos and glories
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CS348B Lecture 16 Pat Hanrahan, Spring 2005
Henyey-Greenstein Phase Function
2
32 2
1 1(cos )
4 1 2 cos
gp
g g
Empirical phase function
0
2 (cos )cosp d g
g: average phase angle
g = -0.3
g = 0.6
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CS348B Lecture 16 Pat Hanrahan, Spring 2005
The Volume Rendering Equation
Integro-differential equation
Integro-integral equation
Attenuation: Absorption and scattering
Source: Scatter (+ emission)
( , )( ) ( , ) ( , )t
L xx L x S x
s
0
( )
0
( , ) ( )
s
t x s ds
L x e S x s ds
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CS348B Lecture 16 Pat Hanrahan, Spring 2005
Simple Atmosphere Model
Assumptions
Homogenous media
Constant source term (airlight)
Fog
Haze
( )( )t
L sL s S
s
( ) 1 t ts sL s e S e C S C
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CS348B Lecture 16 Pat Hanrahan, Spring 2005
The Sky
From Greenler, Rainbows, halos and glories
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CS348B Lecture 16 Pat Hanrahan, Spring 2005
Atmospheric Perspective
From Greenler, Rainbows, halos and glories
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CS348B Lecture 16 Pat Hanrahan, Spring 2005
Atmospheric Perspective
Aerial Perspective: loss of contrast and change in color
From Musgrave
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CS348B Lecture 16 Pat Hanrahan, Spring 2005
Semi-Infinite Homogenous Media
Reduced Intensity
Effective source term
Volume rendering equation
Integrating over depths
( , )cos ( , ) ( , )o
o t o o
L zL z S z
z
( , )( , ) (0, )izi iL z e L
( , )( , ) ( ) (0, )izo s i o iS z p e L
/ cos / cos
0
cos ( ) ( , ) ( )t o t iz zo o s i o iL e p e L dz
cos i cos o
cosz s cosdz ds
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CS348B Lecture 16 Pat Hanrahan, Spring 2005
Semi-Infinite Homogenous Media
Integrating over depths
/ cos / cos
0
1 1
cos cos
0
cos ( ) ( , ) ( )
( , ) ( )
1( , ) ( )
1 1cos cos
cos cos( , ) ( )
cos cos
t o t i
ti o
z zo o s i o i
z
s i o i
s i o i
ti o
i oi o i
i o
L e p e L dz
p L e dz
p L
W p L
cos i cos o
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CS348B Lecture 16 Pat Hanrahan, Spring 2005
Semi-Infinite Homogenous Media
BRDF
Seeliger’s Law or The Law of Diffuse Reflection
( , )( , )
( )cos
1( , )
cos cos
i or i o
i i
i oi o
LdLf
dE L
W p
cos i cos o
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CS348B Lecture 16 Pat Hanrahan, Spring 2005
Subsurface Scattering
Skin
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CS348B Lecture 16 Pat Hanrahan, Spring 2005
Volume Representations
3D arrays (uniform rectangular)
CT data
3D meshes
CFD, mechanical simulation
Simple shapes with solid texture
Ellipsoidal clouds with sum-of-sines densities
Hypertexture
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CS348B Lecture 16 Pat Hanrahan, Spring 2005
Scalar Volumes
Interpolation
Map scalars to optical properties( ), ( )s av v
1ls
ls
1ls
( ) trilinear( , , , , ( ))l lv s v i j k x s
Voxel
( , , )i j k
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CS348B Lecture 16 Pat Hanrahan, Spring 2005
Scalar Volumes
Scatter
s
s ds
s dsVoxel
( , , )i j k
( ( ), ) ( ) ( , ( ( ), )) ( , ( , ( )))s L s L LS x s s p x s x L x x x s
( , ( , ( )))s L LL x x x s
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CS348B Lecture 16 Pat Hanrahan, Spring 2005
Ray Marching
Primary ray
Direct Lighting
1
0
for( 0; 1; )
( ) ( , ( ( ), )) ( , ( , ( )))
1 ( ( ))
s L s L L
t
T
L
s s s ds
S s p x s x L x x x s
L L TS s
T T x s s
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CS348B Lecture 16 Pat Hanrahan, Spring 2005
Ray Marching
Shadow ray
Direct Lighting
1
for( 0; 1; )
1 ( ( ))
( ( )) ( ) ( , ( ( ), )) ( , ( , ( )))t
s L s L L
T
t t t dt
T T x t t
S x s s p x s x T L x x x s
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CS348B Lecture 16 Pat Hanrahan, Spring 2005
Beams of Light
From Greenler, Rainbows, halos and glories
From Minneart, Color and light in the open air
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CS348B Lecture 16 Pat Hanrahan, Spring 2005
Color and Opacity Volumes
M. Levoy, Ray tracing volume densities
( , , ) ( , , )
( , , )
C i j k R G B
A i j k
1ls
ls
1ls
( ( )) trilinear( , , , , ( ))l lc x s c i j k x s
Voxel
( , , )
( , , )* ( , , ), ( , , )
c i j k
C i j k A i j k A i j k
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CS348B Lecture 16 Pat Hanrahan, Spring 2005
Ray Marching
Primary ray
Direct Lighting
(0,0,0,0)
for( 0; 1; )
(1 ( )) ( )
C
s s s ds
C C C c s
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CS348B Lecture 16 Pat Hanrahan, Spring 2005
Volume Rendering Examples
From Marc Levoy
From Karl Heinz Hoehne