Radiosity
Dr. Scott Schaefer
1

Radiosity
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Radiosity



Physically based model for light interaction
View independent lighting
Accounts for indirect illumination
 Color bleeding
 Soft shadows
3/38

The Rendering Equation
L o
( p , v )

L e
( p , v )
 
 f ( l , v ) L o
( r ( p , l ),
 l ) cos
 i d
 i
4/38

The Rendering Equation
L o
( p , v )

L e
( p , v )
 
 f ( l , v ) L o
( r ( p , l ),
 l ) cos
 i d
 i
Outgoing radiance from surface at p in the direction v
5/38

The Rendering Equation
L o
( p , v )

L e
( p , v )
 
 f ( l , v ) L o
( r ( p , l ),
 l ) cos
 i d
 i
Emitted radiance from surface at p in the direction v
6/38

The Rendering Equation
L o
( p , v )

L e
( p , v )
 
 f ( l , v ) L o
( r ( p , l ),
 l ) cos
 i d
 i
BRDF of surface at p
7/38

The Rendering Equation
L o
( p , v )

L e
( p , v )
 
 f ( l , v ) L o
( r ( p , l ),
 l ) cos
 i d
 i
Ray cast from p in the direction of l
8/38

The Rendering Equation
L o
( p , v )

L e
( p , v )
 
 f ( l , v ) L o
( r ( p , l ),
 l ) cos
 i d
 i
Output radiance of intersected surface in the direction of l
9/38

The Rendering Equation
L o
( p , v )

L e
( p , v )
 
 f ( l , v ) L o
( r ( p , l ),
 l ) cos
 i d
 i
Angle between l and n
10/38

The Rendering Equation
L o
( p , v )

L e
( p , v )
 
 f ( l , v ) L o
( r ( p , l ),
 l ) cos
 i d
 i
Integral about hemisphere centered at p
11/38

BRDF’s
B idirectional R eflectance D istribution F unction
 Determines amount of incoming light from direction l reflected from surface in the direction of v v l
Image taken from “A Data-Driven Reflectance Model”
12/38

Discretizing the Rendering Equation
 Assume perfectly diffuse surfaces f ( l , v )
 c
 Discretize space into patches
 Color is constant per patch
Image taken from “Radiosity on Graphics Hardware”
13/38

Discretizing the Rendering Equation
L o
( p , v )

L e
( p , v )
 
 f ( l , v ) L o
( r ( p , l ),
 l ) cos
 i d
 i
14/38

Discretizing the Rendering Equation
L i

L i , e
 
 c i
L j h i , j cos
 i d
 i h i , j


1

 0 patch i is visible to patch j along l patch i is not visibl e to patch j along l
15/38

Geometric Computation of Form Factors
L i

L i , e
 
 c i
L j h i , j cos
 i d
 i
N
16/38

Geometric Computation of Form Factors
L i

L i , e
 
 c i
L j h i , j cos
 i d
 i
N
17/38

Geometric Computation of Form Factors
L i

L i , e

2 


1 c i
L j cos
 d


2

1
N
18/38

Geometric Computation of Form Factors
L i

L i , e
 c i
L j
 sin
 
2 sin

1


2

1
N
19/38

Geometric Computation of Form Factors
L i

L i , e
 c i
L j
 sin
 
2 sin

1


1

2 sin(

)

2 sin(

1
)
N
20/38

Geometric Computation of Form Factors
N
21/38

Geometric Computation of Form Factors
Project patches onto hemisphere
N
22/38

Geometric Computation of Form Factors
N
Project spherical patches onto tangent plane
23/38

Geometric Computation of Form Factors
N
Divide by area of disc in tangent plane ( for surfaces)
24/38

Geometric Computation of Form Factors
L i

L i , e
  j c i
L j
F i , j
Divide by area of disc in tangent plane ( for surfaces)
N
F i , j
= form factor
25/38

Matrix Computation of Radiosity
L i

L i , e
  j c i
L j
F i , j







1


 c c
2 c n

1
F
1 , 1
F
2 , 1
F n , 1

1
 c
1
F
1 , 2 c
2
F
2 , 2

 c n
F n , 2





 c c
1
F
1 , n
2

F
2 , n
1
 c n
F n , n




 







L
L
2

L n
1














L
L e e

, 1
, 2
L e , n







26/38

Matrix Computation of Radiosity
L i

L i , e
  j c i
L j
F i , j




L
L



L n
1
2














L
L e , 2
L e e

, 1
, n














 c c c n
1
2
F
1 , 1
F
2

F n
, 1
, 1 c
1
F
1 , 2 c
2
F
2 , 2
 c n
F n , 2



 c c
1
2
F
F
2

1 , n
, n c n
F n , n




 







L
L
2

L n
1






27/38

Matrix Computation of Radiosity
L i

L i , e
  j c i
L j
F i , j







L
L i i

L i n
1
2















L
L e e

,
, 1
L e , n
2














 c c c n
1
2
F
1 , 1
F
2

F n
, 1
, 1 c c c
2
1 n
F
1 , 2
F
2 , 2

F n , 2
Jacobi iteration



 c
1
F
1 , n c
2
F
2 , n
 c n
F n , n


 










L
L i

1
2
L i i


1
1

1 n







28/38

Matrix Computation of Radiosity
L i

L i , e
  j c i
L j
F i , j
L i 
L e

K L i

1
29/38

Matrix Computation of Radiosity
L e
30/38

Matrix Computation of Radiosity
L
 e
K L e
31/38

Matrix Computation of Radiosity
L e

K L e

K
2
L e
32/38

Matrix Computation of Radiosity
L e

K L e

K
2
L e

K
3
L e
33/38

Radiosity Examples
Image taken from http://www.9jcg.com/tutorials/jason_jacobs/radiosity_01.jpg
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Radiosity Examples
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Radiosity Examples
Image taken from www.cs.dartmouth.edu/~spl/Academic/ComputerGraphics/Fall2004/radiosityExample.jpg
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Advantages of Radiosity


Global illumination method: modeling diffuse inter-reflection
Color bleeding: a red wall next to a white one casts a reddish glow on the white wall



Soft shadows: an area light source casts a soft shadow from a polygon
No ambient hack
View independent: assigns a brightness to every surface
37/38

Disadvantages of Radiosity




Radiation is uniform in all directions
Radiosity is piecewise constant
No surface is transparent or translucent
Must determine how to subdivide shapes into small enough patches
38/38