Image Formation
Light can change the image and appearances (images from D. Jacobs)
What is the relation between pixel brightness and scene radiance?
Later: what is the relation between pixel brightness and scene reflectance ?
© 2002 by Davi Geiger
Computer Vision
September 2002
L1.1
Image Formation
 
A cos
solid angle subtended by a small patch of area A.
R2
n̂

R

A
L - radiance is the amount of light radiated from a surface per solid angle
(power per unit area per unit solid angle emitted from a surface. W m 2 sr 1)
E - irradiance is the amount of light falling in a surface
(power per unit area incident in a surface. W m 2)
© 2002 by Davi Geiger
Computer Vision
September 2002
L1.2
Surface Radiance and Image Irradiance
A
Pinhole
Camera
Model
n̂


f

z
I
Same solid angle
A cos
I cos 

( z / cos  ) 2 ( f / cos  ) 2
© 2002 by Davi Geiger
A cos  


I cos  
Computer Vision
z
f



2
September 2002
L1.3
Surface Radiance and Image Irradiance
A
n̂


d
f

z
Solid angle subtended by the lens,
as seen by the patch A
I
 d 2 cos 
 d
3


cos



2
4 ( z / cos  )
4z
2
Power from patch A through
the lens
2
P  L  A cos   LA
Thus, we conclude
3
  cos  cos 
4z
2
P
A   d 
 d
3
  cos 4 
L
cos

cos


L
 
I
I 4  z 
4 f 
2
E
 d
© 2002 by Davi Geiger
Computer Vision
September 2002
L1.4
Conclusions
EL
 d
2
  cos 4 
4 f 
•The irradiance at the image pixel is converted into the
brightness of the pixel
•Image Irradiance is proportional to Scene Radiance
•Scene distance, z, does not affect/reduce image brightness (the
model is too simplified, since in practice it does.)
•The angle of the scene patch with respect to the view ()
reduces the brightness by the cos 4  . In practice the effect is even
stronger.
© 2002 by Davi Geiger
Computer Vision
September 2002
L1.5
The Bidirectional Reflectance Distribution
Function (BRDF)
n̂
n̂
i
( i , i )
( e , e )
i
 L( e , e )
f ( i , i , e , e ) 
 E ( i , i )
BRDF - How bright a surface appears when
viewed from one direction while light
falls on it from another.
Usually f depends only on e  i ,  i , e : true for matte surfaces and specularly
reflecting surfaces.
© 2002 by Davi Geiger
Computer Vision
September 2002
L1.6
Extended Light Sources and BRDF
   sin  i i i
i
Light source radiance arriving through solid angle 
A
E ( i , i )  E ( i , i ) sin  i i i
i
Power arriving at patch A from 
P  A cos i E ( i , i )  A E ( i , i ) cos i sin  i i i
Foreshortening
E0
thus the irradiance arriving at patch A is
P




A

 2

0
 
E ( i , i ) cos i sin  i i i
The radiance of a patch A at direction ( e , e ) is thus, given by

L( e , e )  
 2

 0
© 2002 by Davi Geiger
f ( i , i , e , e ) E ( i , i ) cos i sin  i i i
Computer Vision
September 2002
L1.7
Special Cases of BRDF
1.
Lambertian Surfaces (matte)- appears equally bright from all viewing
directions and reflects all incident light, absorbing none, i.e. the
BRDF is constant and L  E0. . What constant f ?
L( e , e )  f

 2

0
 
E ( i , i ) cos i sin  i i i  f E0
Thus, the total “reflected power” from patch A becomes

 2

0
p  
since


 2

0
 
Using that L 
L( e , e ) A cos e sin  e e e  f E0 A 
cos e sin  e e e  
Foreshortening
p
 f E0  and for Lambertian surfaces L  E0 ,
A
we finally obtain
© 2002 by Davi Geiger
Computer Vision
f 
1

September 2002
L1.8
1.
2.
Special Cases of BRDF
S
Specular Surfaces (mirrors) – reflects all light arriving from the
direction ( i , i ) into the direction ( i , i   ). The BRDF is in this case
proportional to the product of two impulses,  ( e   i ) and
 (e  i   ).What is the factor of proportionality k ( i , i ) ?

 2
L( e , e )  

 0
k ( i , i )  ( e   i )  (e  i   ) E ( i , i ) cos i sin  i i i
 k ( e , e ) E ( e , e ) cos e sin  e

E0  
 2

 
L  

 0

0
2
E ( i , i ) cos i sin  i i i
L( e , e ) cos e sin  e e e
and for specular surfaces L  E0  k ( i , i ) 
1
cos  i sin  i
 ( e   i )  (e  i   )
we finally obtain f ( i , i , e , e ) 
cos i sin  i
© 2002 by Davi Geiger
Computer Vision
September 2002
L1.9
Lambertian Surface Brightness
How bright will a Lambertian surface be
when it is illuminated by a point source
of radiance E? and by a “sky” of uniform
radiance E?
n̂
For a point source the irradiance at the
surface is E0  E cos i and the radiance
must then be
L( e , e )  f E0 
1

( i , i )
( e , e )
E cos  i
Familiar cosine or “Lambert’s law” of reflection from matte surfaces
(surfaces covered with finely powdered transparent materials such as
barium sulfate or magnesium carbonate), and can approximate paper,
snow and matte paint.
Finally, for a “sky” of uniform radiance E we obtain

L( e , e )  
 2

 0
© 2002 by Davi Geiger
1
E cos  i sin  i i i  E !

Computer Vision
September 2002
L1.10