Homogeneous representation

Points

Vectors

Transformation

representation
Stefano Soatto (c)
UCLA Vision Lab
1
Lecture 4: Image formation
Stefano Soatto (c)
UCLA Vision Lab
2
Image Formation



Vision infers world properties form images.
How do images depend on these properties?
Two key elements



Geometry
Radiometry
We consider only simple models of these
Stefano Soatto (c)
UCLA Vision Lab
3
Image formation (Chapter 3)
Stefano Soatto (c)
UCLA Vision Lab
4
Representation of images
Stefano Soatto (c)
UCLA Vision Lab
5
(
Similar triangles <P’F’S’>,<ROF’> and <PSF><QOF> 
( z '  f )( z  f )  f 2
Stefano Soatto (c)
1
1
1


z'  z f
UCLA Vision Lab
6
Pinhole model
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UCLA Vision Lab
7
Forward pinhole
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UCLA Vision Lab
8
Distant objects are smaller
Stefano Soatto (c)
UCLA Vision
Lab
(Forsyth
& Ponce)
9
Parallel lines meet
Common to draw image plane in front of the focal point.
Moving the image plane merely scales the image.
Stefano Soatto (c)
UCLA Vision
Lab
(Forsyth
& Ponce)
10
Vanishing points
• Each set of parallel lines meets at a different
point
– The vanishing point for this direction
• Sets of parallel lines on the same plane lead to
collinear vanishing points.
– The line is called the horizon for that plane
Stefano Soatto (c)
UCLA Vision Lab
11
Properties of Projection





Points project to points
Lines project to lines
Planes project to the whole image or a half image
Angles are not preserved
Degenerate cases



Line through focal point projects to a point.
Plane through focal point projects to line
Plane perpendicular to image plane projects to part of
the image (with horizon).
Stefano Soatto (c)
UCLA Vision Lab
12
Orthographic projection
x x
'
y y
'
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UCLA Vision Lab
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Stefano Soatto (c)
UCLA Vision Lab
14
Cameras with Lenses
Stefano Soatto (c)
UCLA Vision
Lab
(Forsyth
& Ponce)
15
Stefano Soatto (c)
UCLA Vision Lab
16
Assumptions for thin lens equation




Lens surfaces are spherical
Incoming light rays make a small angle with
the optical axis
The lens thickness is small compared to the
radii of curvature
The refractive index is the same for the
media on both sides of the lens
Stefano Soatto (c)
UCLA Vision Lab
17
Stefano Soatto (c)
UCLA Vision Lab
18
Blur circle
Points a t distance  z are brought into focus at distance z '
A point at distance  z
z ' from the lens
is imaged at point
1 1
1


z'  z f
-z
z’
P
and so
Q
d
b
Q’
f
f
z ' z ' 
( z  z)
(z  f ) (z  f )
Thus points at distance z will give rise to a blur circle of diameter
d
b  | z ' z '|
z'
P’
z’
-z
with d the diameter of the lens
Stefano Soatto (c)
UCLA Vision Lab
19
Interaction of light with matter





Absorption
Scattering
Refraction
Reflection
Other effects:


Diffraction: deviation of straight propagation in the
presence of obstacles
Fluorescence:absorbtion of light of a given wavelength by a
fluorescent molecule causes reemission at another
wavelength
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UCLA Vision Lab
20
Refraction
n1, n2: indexes of refraction
Stefano Soatto (c)
UCLA Vision Lab
21
Solid Angle
hemisphere
radian
dq
2
2
2 2
 d    sin q ddq
 dq  2
q

0
0
0 0

2
 2  sin qdq
steradian (sr)
d
q

0
 2 [ cosq ]0

d  d sinq dq
Sphere:
Stefano Soatto (c)
UCLA Vision Lab
 2
4
22
Radiometric Terms
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UCLA Vision Lab
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Irradiance and Radiance
Irradiance Definition: power per unit area incident on a surface
d
E
dA
[W/m2 = lux]
Radiance Definition: power per unit area and projected solid angle
d 2
L  
cos q dA d
[W/m2sr]
Li 
qi
d 2
dE 
 Li  cos q i di
dA
E
 L cos q d
i
i
i
A
H2
Stefano Soatto (c)
UCLA Vision Lab
24
Radiant Intensity
Definition: flux per unit solid angle
d
I   
d
[W/sr = cd (candela)]
   I  d
S
: Radiant flux [W]

2
I  
[ ]
Stefano Soatto (c)
UCLA Vision Lab
25
Isotropic Point Source
Stefano Soatto (c)
UCLA Vision Lab
26
Isotropic Point Source
: Radiant flux [W]
All directions: solid angle 4
d : Radiant flux per
I   
d unit solid angle [W/sr]
r
dA  r 2 d
E

 I   
4
: Radiant intensity
d Id  d  1 dA





dA dA 4 dA 4 r 2 dA 4r 2
• Note inverse square law fall off.
Stefano Soatto (c)
UCLA Vision Lab
27
Isotropic Point Source
: Radiant flux [W]
All directions: solid angle 4
dA  r 2
1
d r
q
cos q
d : Radiant flux per
I   
d unit solid angle [W/sr]
h

 I   
4
: Radiant intensity
d Id  d  cos q dA  cos 3 q
E




2
dA dA 4 dA 4 r dA 4 h 2
Stefano Soatto (c)
 h  r cos q
• Note cosine dependency.
UCLA Vision Lab
28
Isotropic Point Source
Point source at a finite distance
r
q
h
 cos q
E
2
4 h
3
• Note inverse square law fall off.
• Note cosine dependency.
Stefano Soatto (c)
UCLA Vision Lab
29
Irradiance from Area Sources
Stefano Soatto (c)
UCLA Vision Lab
30
Hemispherical Source
L
E
2
0


2
0
L cosq i sin q i dq i di
[
]

L 2
2
  sin q i 02 di
2 0
L 2
  di
2 0
 L
Stefano Soatto (c)
UCLA Vision Lab
31
Reflectance
Li(x,i)
The surface becomes a light source
Lr(x,)
qi
Ei
Ei
dLr=fr dEi
dLr
fr 
dEi
Lr   f r dE i
Reflectance: ratio of radiance to irradiance
Stefano Soatto (c)
UCLA Vision Lab
32
BRDF
Stefano Soatto (c)
UCLA Vision Lab
33
BRDF
dL
r ,r;r E
dLrrq
; Eii 
i , ii;;q

q r r, r 
f rrq
i ,
i ;
i;
dEi iq
dE
i , ii 
dLr i ;r ; Ei   f r i ;r  dEi i 
d 2 r / d  r

d i
dEi i   Li i di  Li i cosqi di
Lr r    f r i ; r  dEi i 
i
Stefano Soatto (c)
  f r i ; r  Li i  cosqi di
i
UCLA Vision Lab
34
Reflection Equation
Stefano Soatto (c)
UCLA Vision Lab
35
qi
Perfectly Diffuse Reflection
Perfectly Diffuse Surface
•Appears equally bright from all viewing directions (qr, r)
•Reflects all incident light, i.e.,
 LB(qr, r) is constant for all directions (qr, r)
 E   Lr r  d r  
r
LB 1
fB 

Es 
Stefano Soatto (c)
 q
r
LB 
LB cos q r sin q r dq r dr   LB
r
L   cos q d

 
1
i
i
i
i
i
UCLA Vision Lab
36
qi
Common Diffuse Reflection
Normal Diffuse Surface
•Appears almost equally bright from most viewing
directions (qr, r), qr << 90°
•Reflects only a fraction of incident light, i.e.,
LB
fB 
 B
Es
LB   B  Li i cosqi di
i
Reflectance : Albedo
Stefano Soatto (c)
UCLA Vision Lab
37
Perfectly Diffuse Reflection
Distant point light source
d ( q i  q s )d ( i  s )
E( q i ,i )  E
sin q s
qi
Lr   f B Li (q i , i ) cosq i di
i

1
L (q ,  ) cosq sin q dq d


q


i


Stefano Soatto (c)
1

1

i
i
i
i
i
i
i
i
E
 d (q
i q i
i
 q s )d (i  s ) cosq i dq i di
E cosq s
Lambertian cosine Law
UCLA Vision Lab
38
Law of Reflection
Li ( qi ,i )
Lr ( q r ,r )
Lr (qr , r )  Li (qi , i   )
Stefano Soatto (c)
UCLA Vision Lab
39
Perfectly Specular Reflection
From the definition of BRDF, the surface radiance is:
Lr (q r , r )  
 q
To satisfy:
i
f s (q r , r ;qi , i ) Li (qi , i ) cosqi sin qi dqi di
i
Lr (qr , r )  Li (qi , i   )
d (qi  q r )d (i  r   )
f s (q r , r ;qi , i ) 
sin qi cosqi
Li (q r , r   )  
d (q

 q
Stefano Soatto (c)
i
i
 q r )d (i  r   ) Li (qi , i )dqi di
i
UCLA Vision Lab
40
Lambertian Examples
Lambertian sphere as the light
moves.
(Steve Seitz)
Scene
(Oren and Nayar)
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UCLA Vision Lab
41
Lambertian + Specular Model
L( P,  0 ,  0 )   d ( P)  L( P, i ,  i ) cosi d

  s ( P) L( P,  s ,  s ) cosn ( s  0 )
Stefano Soatto (c)
UCLA Vision Lab
42
Lambertian + specular
• Two parameters: how shiny, what kind of shiny.
• Advantages
– easy to manipulate
– very often quite close true
• Disadvantages
– some surfaces are not
• e.g. underside of CD’s, feathers of many birds,
blue spots on many marine crustaceans and fish,
most rough surfaces, oil films (skin!), wet surfaces
– Generally, very little advantage in modelling
behaviour of light at a surface in more detail -- it is
quite difficult to understand behaviour of L+S
surfaces (but in graphics???)
Stefano Soatto (c)
43
UCLA Vision Lab
Lambertian+Specular+Ambient
(http://graphics.cs.ucdavis.edu/GraphicsNotes/Shading/Shading.html)
Stefano Soatto (c)
UCLA Vision Lab
44