Lectures 9: The camera

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• Lectures 9 The camera
• Contents
• The imaging device
• The 3D-to-2D transformation of the pin hole
  model
• Camera calibration
• Radial distortion
• Telemetry by an uncalibrated camera

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Rays in a simplified digital camera

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• Thin-lens formula a -1 b -1 f -1
• The image at right shows a camera lens with
• f 12 mm (fixed)
• Distance b set so that a bf/(b-f) 3 ft 0.9
  m
• aperture set to 2.8. Thus the aperture angle
  0.36 radians. (Exact definition aperture
  f/ddiaphragm)

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Depth of focus

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Camera geometry Global frame X-Y-Z
Image coordinates (u,v), Camera frame x-y-z
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The camera transformation using the pin hole
model The object coordinate (x,y,z) in the
camera frame related to the image coordinate (u,v)
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Transformation from the global frame to the
camera frame
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The full camera transformation expressed by a 4x3
camera matrix
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Using relations between 3D points and their images
Sonka et al. p. 455
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Task 2
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Task 3
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Conversion between internal and external camera
parameters
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Deviations from the pin hole model of a camera
Image distortion
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Telemetry by an uncalibrated camera Consider the
same object taken by a camera from many different
directions
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How to measure the 3D coordinates of the land
marks? Number of land marks 10 Number of
images 5 There is a relation between the image
position of the landmark centers, the camera
parameters and the 3D landmark coordinates. The
image positions can be obtained by image
analysis. There are 50-x coordinates and 50
y-coordinates. We get a total of 100
equations. How many unknowns 6 camera
parameters 5 x 6 30 pose parameters, 30 3D
coordinates of the landmarks. A total of 66
unknowns.
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The best we can do is to search for the smallest
square error. The set of equations is non-linear
so the standard methods for solving
overdetermined equations do not work. If you have
a start guess you can linearize the equations and
apply Newtons method. Among the 30 3D
coordinates, 7 can be chosen to be fixed. At
Maersk we choose all three coordinates of two
land marks (the two on the plate surface) and the
height of a third to be fixed. Thus there are 59
unknowns in the example
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The method gives not only the landmark
coordinates, but also the 5 linear camera
parameters and the distortion parameter. Typical
result Mean error between the observed image
positions and the projected image positions (mean
over 100 coordinates) is 0.2 pixels. The image
size of the 200 x 150 mm plate was 800 x 600
pixels. This means that 0.2 pixels corresponds to
50 micrometer! After telemetric measurement, the
calibration template used in a simple calibration
lead to a mean error of 0.1 pixels.