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```									Camera Simulation

Effect           Cause

Field of view    Film size,
stops and pupils
Depth of field   Aperture,
focal length
Motion blur      Shutter
Exposure Film speed,
aperture, shutter

References
Photography, B. London and J. Upton
Optics in Photography, R. Kingslake
The Camera, The Negative, The Print, A. Adams
CS348B Lecture 7                                   Pat Hanrahan, 2005
Topics

Ray tracing lenses
Focus
Field of view
Depth of focus / depth of field
Exposure

CS348B Lecture 7                  Pat Hanrahan, 2005
Lenses
Refraction

Snell’s Law

n sin I   n sin I   N
I

I
n   n

CS348B Lecture 7                                Pat Hanrahan, 2005
Paraxial Approximation

e0
sinU  u

tanU  u

U

z

Rays deviate only
slightly from the axis
CS348B Lecture 7                 Pat Hanrahan, 2005
Incident Ray

Angles: ccw is positive; cw is negative

I

U                    

I U 

The sum of the interior angles is equal to the exterior angle.
CS348B Lecture 7                                      Pat Hanrahan, 2005
Refracted Ray

    I
U 

( )  I   (U )

I   U  

CS348B Lecture 7                        Pat Hanrahan, 2005
Derivation

Paraxial approximation

I U   i  u
I   U     i   u  

CS348B Lecture 7                       Pat Hanrahan, 2005
Derivation

Paraxial approximation

I U   i  u
I   U     i   u  

Snell’s Law
n sin I   n sin I  ni   ni
n(u  )  n(u  )

CS348B Lecture 7                           Pat Hanrahan, 2005
Ray Coordinates
h                h         h
u           u         
z               z        R

h
u                 R         u 

z                       z

CS348B Lecture 7                                    Pat Hanrahan, 2005
Gauss’ Formula

Paraxial approximation to Snell’s Law
n(u  )  n(u  )
Ray coordinates
h           h         h
u                  u
z          R         z
Thin lens equation
h h       h h
n(  )  n(  )
z R      z R
n n ( n  n )
                   Holds for any height, any ray!
z z     R
CS348B Lecture 7                                Pat Hanrahan, 2005
Vergence
Diverging                       Converging

V 0       V 0                V 0

n n 1                
Vergence                 V         m  diopters 
r z                 
Thin lens equation       V  V  P
1
Surface Power equation   P  ( n   n)
R
CS348B Lecture 7                               Pat Hanrahan, 2005
Lens-makers Formula

Refractive Power
 1  1 1
P  ( n   n)    
 R1 R2  f

Converging       Diverging

CS348B Lecture 7                                Pat Hanrahan, 2005
Conjugate Points

1 1 1
 
z z f

To focus: move lens relative to backplane
Horizontal rays converge on focal point in the focal plane
CS348B Lecture 7                                             Pat Hanrahan, 2005
Gauss’ Ray Tracing Construction

Parallel Ray

Focal Ray
Chief Ray

Object                                          Image

CS348B Lecture 7                                  Pat Hanrahan, 2005
Ray Tracing: Finite Aperture

Focal Plane   Aperture Plane      Back Plane

CS348B Lecture 7                     Pat Hanrahan, 2005
Real Lens

Cutaway section of a Vivitar Series 1 90mm f/2.5 lens
Cover photo, Kingslake, Optics in Photography
CS348B Lecture 7                                           Pat Hanrahan, 2005
Double Gauss

Data from W. Smith,
Modern Lens Design, p 312

(mm)        (mm)
58.950         7.520   1.670    47.1        50.4
169.660         0.240                        50.4
38.550         8.050   1.670    47.1        46.0
81.540         6.550   1.699    30.1        46.0
25.500     11.410                           36.0
9.000                        34.2
-28.990         2.360   1.603    38.0        34.0
81.540     12.130      1.658    57.3        40.0
-40.770         0.380                        40.0
874.130         6.440   1.717    48.0        40.0
-79.460     72.228                           40.0

CS348B Lecture 7                                       Pat Hanrahan, 2005
Ray Tracing Through Lenses

200 mm telephoto               35 mm wide-angle

50 mm double-gauss                16 mm fisheye

From Kolb, Mitchell and Hanrahan (1995)

CS348B Lecture 7                                  Pat Hanrahan, 2005
Thick Lenses

Equivalent Lens

Refraction occurs at the principal planes
CS348B Lecture 7                              Pat Hanrahan, 2005
Field of View
Field of View

From London and Upton
CS348B Lecture 7                           Pat Hanrahan, 2005
Field of View

From London and Upton
CS348B Lecture 7                           Pat Hanrahan, 2005
Field of View
Redrawn from Kingslake,
Field of view                           Optics in Photography

fov filmsize
tan      
2      f

Types of lenses
    Normal                26º
Film diagonal  focal length
    Wide-angle 75-90º
    Narrow-angle          10º

CS348B Lecture 7                                      Pat Hanrahan, 2005
Perspective Transformation

Thin lens equation

1 1 1          fz
   z 
z z f        z f
fx
 x 
z f
fy
 y 
z f

Represent transformation as a 4x4 matrix
CS348B Lecture 7                           Pat Hanrahan, 2005
Depth of Field
Depth of Field

From London and Upton
CS348B Lecture 7                           Pat Hanrahan, 2005
Circle of Confusion

s            s
z            z        d

a                  c

Focal Plane                Back Plane

Circle of confusion proportional                 c d  s  z
 
to the size of the aperture                      a z     z

CS348B Lecture 7                                        Pat Hanrahan, 2005
Depth of Focus [Image Space]
s           s
Depth of focus 
a
Equal circles of confusion
Two planes: near and far
zf             z f

c d  s  z f
f
c
     
a z f   z f
d
f
zn                
zn
c dn zn  s 
  
                                               c

a zn     
zn


dn
CS348B Lecture 7                               Pat Hanrahan, 2005
Depth of Focus [Image Space]
s           s
Depth of focus 
a
Equal circles of confusion

zf             z f

c d  s  z f
f              1 1  c
                     1                                      c
a z f   z f       z f s  a 
d
f
zn                
zn
c dn zn  s
          1 1  c                                            c
               1  

a zn     
zn     zn s   a 


dn
CS348B Lecture 7                               Pat Hanrahan, 2005
Depth of Focus [Image Space]

Depth of focus 
Equal circles of confusion

1 1  c              1 1  c            z f
 1                1                       s
z f s  a          zn s   a 

c
1 1      1
 2

z f zn  s                        
zn

1 1 2c 1
 

z f zn a s

CS348B Lecture 7                         Pat Hanrahan, 2005
Depth of Field [Object Space]

Depth of field 
Equal circles of confusion
zf
1 1 1         1 1 1     1 1 1
                       
s s f         
zn zn f   z f z f f
c
1 1    1
 2
zn z f s                           zn

1 1 2c  1 1  2 c 1                                    c
    
zn z f a  f s a f

CS348B Lecture 7                               Pat Hanrahan, 2005
Hyperfocal Distance
1 1    1                   a
 2                  N     zf
zn z f s                   f

1 1 2c 1     cN    1                                     c
       2 2 2
zn z f a f    f    H
zn
When
c
H
s  H  zn  , z f  
2

H is the hyperfocal distance

CS348B Lecture 7                              Pat Hanrahan, 2005
Depth of Field Scale

CS348B Lecture 7       Pat Hanrahan, 2005
Factors Affecting DOF
From http://www.kodak.com/global/en/consumer/pictureTaking/cameraCare/cameCar6.shtml

1 cN
 2
H f

CS348B Lecture 7                                                              Pat Hanrahan, 2005
Resolving Power

 Diffraction limit
f
c  1.22
a
     1.22  64  .500m=0.040 mm 
 35mm film (Leica standard)

c  0.025mm

 CCD/CMOS pixel aperture

c  0.0116 mm (Nikon D1)

CS348B Lecture 7                                       Pat Hanrahan, 2005
Exposure

a

f

2
a
E   L cos  d  L  sin   L
2
 

4 f 

CS348B Lecture 7                               Pat Hanrahan, 2005
Relative Aperture or F-Stop

f
a
a
f
N

 1
F-Number and exposure:          EL        2
4N
Fstops: 1.4 2 2.8 4.0 5.6 8 11 16 22 32 45 64
1 stop doubles exposure
CS348B Lecture 7                                Pat Hanrahan, 2005
Camera Exposure

Exposure           H  E T

Exposure overdetermined
Aperture: f-stop - 1 stop doubles H
Decreases depth of field
Shutter: Doubling the open time doubles H
Increases motion blur

CS348B Lecture 7                             Pat Hanrahan, 2005
Aperture vs Shutter

f/16               f/4               f/2
1/8s             1/125s            1/500s

From London and Upton
CS348B Lecture 7                           Pat Hanrahan, 2005
High Dynamic Range

Sixteen photographs of the Stanford Memorial Church
taken at 1-stop increments from 30s to 1/1000s.
From Debevec and Malik, High dynamic range
photographs.

CS348B Lecture 7                        Pat Hanrahan, 2005
Simulated Photograph

Adaptive histogram   With glare, contrast, blur
CS348B Lecture 7                              Pat Hanrahan, 2005
Camera Simulation

A
L( x, , t,  )
L( x ,  , t,  )


R   P( x ,  ) S( x ,  , t ) L(T ( x ,  ,  ), t,  ) dA( x )  d  dt d
AT 

Sensor response                             P( x ,  )
Lens                                        ( x,  )  T ( x ,  ,  )
Shutter                                     S( x ,  , t )
Scene radiance                              L( x, , t,  )
CS348B Lecture 7                                                              Pat Hanrahan, 2005

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