Image Formation Fundamentals
Basic Concepts (Continued…)
How are images represented
in the computer?
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Image digitization
•
•
Sampling means measuring the value of an image at a finite number of points.
Quantization is the representation of the measured value at the sampled point by an
integer.
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Image digitization (cont’d)
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Image quantization (Example)
256 gray levels (8bits/pixel) 32 gray levels (5 bits/pixel) 16 gray levels (4 bits/pixel)
8 gray levels (3 bits/pixel) 4 gray levels (2 bits/pixel)
2 gray levels (1 bit/pixel)
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Image sampling (example)
original image
sampled by a factor of 4
sampled by a factor of 2
sampled by a factor of 8
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Digital image
• An image is represented by a rectangular array of integers.
• An integer represents the brightness or darkness of the image
at that point.
• N: # of rows, M: # of columns, Q: # of gray levels
– N = 2 n , M = 2 m , Q = 2 q (q is the # of bits/pixel)
– Storage requirements: NxMxQ (e.g., N=M=1024, q=8,
1MB)
f (0,0)
f (1,0)
...
f ( N  1,0)
f (0,1)
f (1,1)
...
f ( N  1,1)
...
...
...
...
f (0, M  1)
f (1, M  1)
...
f ( N  1, M  1)
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Image formation
• There are two parts to the image formation
process:
– The geometry of image formation, which
determines where in the image plane the
projection of a point in the scene will be
located.
– The physics of light, which determines the
brightness of a point in the image plane as a
function of illumination and surface
properties.
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A Simple model of image formation
• The scene is illuminated
by a single source.
• The scene reflects
radiation towards the
camera.
• The camera senses it
via chemicals on film.
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Pinhole cameras
•
Abstract camera model - box
with a small hole in it
•
Pinhole cameras work in practice
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Real Pinhole Cameras
Pinhole too big many directions are
averaged, blurring the
image
Pinhole too smalldiffraction effects blur
the image
Generally, pinhole
cameras are dark, because
a very small set of rays
from a particular point
hits the screen.
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The reason for lenses
Lenses gather and
focus light, allowing
for brighter images.
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The thin lens
Thin Lens Properties:
1. A ray entering parallel to optical axis
goes through the focal point.
2. A ray emerging from focal point is parallel
to optical axis
3. A ray through the optical center is unaltered
1 1 1
 
z' z f
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The thin lens
1 1 1
 
z' z f
Note that, if the image plane is very
small and/or z >> z’, then z’ is
approximately equal to f
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Lens Realities
Real lenses have a finite depth of field, and usually
suffer from a variety of defects
Spherical Aberration
vignetting
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The equation of projection
•
Equating z’ and f
– We have, by similar triangles,
that (x, y, z) -> (-f x/z, -f y/z, -f)
– Ignore the third coordinate, and
flip the image around to get:
x y
(x, y, z)  ( f , f )
z z
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Distant objects are smaller
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Parallel lines meet
common to draw film plane
in front of the focal point
A Good Exercise: Show this is the case!
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Orthographic projection
Suppose I let f go to infinity; then
ux
v y
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The model for orthographic
projection
X 
U  1 0 0 0
Y 
V   0 1 0 0
  
Z
W  0 0 0 1 
T 
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Weak perspective
•
Issue
– perspective effects, but not over
the scale of individual objects
– collect points into a group at
about the same depth, then
divide each point by the depth of
its group
– Adv: easy
– Disadv: wrong
u  sx
v  sy
s  f /Z*
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The model for weak perspective
projection
X
0  
U  1 0 0
  
 Y 
0  
 V   0 1 0
Z
W   0 0 0 Z * / f  
  
 T 
 
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Model for perspective projection
U  1 0
V  0 1
  
W  0 0
0
0
1
f
X 

0
Y 

0 
Z 
0 
T 
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Intrinsic Parameters
Intrinsic Parameters describe the conversion from
unit focal length metric to pixel coordinates (and the reverse)
 x
  1 / sx
 

 y   0
 w
 0
  pix 
0
1/ sy
0
ox  x 
 
o y  y   K int p
1  w  mm
It is common to combine scale and focal length together
as the are both scaling factors; note projection is unitless in this case!
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Image formation - Recap
pixel coordinate
system
If we consider unit
focal length
x1
image coordinate
system
Scaling factor = depth of the
point X
camera coordinate
system
(R,T)
world coordinate
system
Taken from MASKS (invitation to 3D vision)
Camera parameters
• Summary:
–
–
–
–
points expressed in external frame
points are converted to canonical camera coordinates
points are projected
points are converted to pixel units
X
 U   Transforma tion
 Transforma tion  Transforma tion
 
  


 Y 
 V    representi ng
 representi ng
 representi ng
 
W   intrinsic parameters  projection model  extrinsic parameters  Z 
  


 T 
 
point in pixel
coords.
point in metric
image coords.
point in cam.
coords.
point in
world coords.
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Camera Calibration
The problem:
Compute the camera intrinsic and extrinsic
parameters using only observed camera data.
Calibration with a Rig
Use the fact that both 3-D and 2-D coordinates of feature
points on a pre-fabricated object (e.g., a cube) are known.
Calibration with Multiple Plane Images
Actually used in practice these days
Calibration Continued…