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TYPES OF DISTORTION
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The calibration of optical sensors in computer vision is
an important issue in autonomous navigation, stereo
vision and numerous other applications where accu-
rate positional observations are required. Various tech-
niques have been developed for the calibration of sen-
sors based on the traditional pinhole camera model.
Typically, the following types of geometrical distortion
have been recognized and dealt with [1]:

* RADIAL DISTORTION: This type of distortion is
point-symmetric at the optical center of the lens
and causes an inward or outward shift of image
points from their initial perspective projection.
About the optical center, radial distortion is ex-
pressed as

    r' = r + k1*r^3 + k2*r^5 + k3*r^7 +...   ; 

where ki are radial distortion coeficients, r is the
observed radial component of a projected point
and r', its predicted perspective projection [2].
The distortion is due to the camera lens so it
happens after perspective projection but before
scaling, skewing, and digitizing.


* DECENTERING DISTORTION: The misalignment of
the optical centers of various lens elements in the
sensor induces a decentering distortion which has
both a radial and a tangential component. They
are expressed as

r' = r + 3* (n1*r^2 + n2*r^4 + n3*r^6 +... ) sin(t-to);
t' = t + (n1*r^2 + n2*r^4 + n3*r^6 +... ) cos(t-t0);

where ni are the decentering distortion coeficients,
t is the observed angular component of a projected
point, t' is its predicted perspective projection and
t0 is the angle between the positive y-axis and the
axis of maximum tangential distortion due to de-
centering [2].


* THIN PRISM: Manufacturing imperfections of lens
elements and misalignment of CCD sensor arrays
from their ideal, perpendicular orientation to the
optical axis introduce additional radial and tan-
gential distortions which are given by

r' = r + (e1*r2 + e2*r4 + e3*r6 +... ) sin(t-t1);
t' = t + (e1*r2 + e2*r4 + e3*r6 +... ) cos(t-t1);

where ei are the thin prism distortion coeficients
and t1 is the angle between the positive y-axis and
the axis of maximum tangential distortion due to
thin prism [2].


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[1] J. Weng, P. Cohen, and M. Herniou. Camera cali-
bration with distortion models and accuracy eval-
uation. IEEE PAMI, 14(10): 965-980, 1992.

[2] American Society of Photogrammetry. Manual of
Photogrammetry, 4th edition. 1980.
