Finally, the block parameters can further be enhanced in a
general, simultaneous adjustment, presented in chapter 5.
3. COMPENSATION OF THE RADIAL DISTORTION
The solution of the radial distortion is based on the fact that the
singular correlation between two images should produce zero
when only linearly deformed images are used. Any deviation
from zero, exceeding of course, the random noise due to the
observational errors, can then be interpreted as the effect of
nonlinear errors /Niini, 1990/. For example, using video
cameras, the radial distortion is usually significant. In practice,
it has also been shown to be sufficient to model the nonlinear
distortions with one parameter only, namely with the parameter
corresponding to the third power of the radius /Melen, 1994/.
Decentering or tangential distortion need not to be considered
here, because they are practically too dependent on the principal
point which, in turn, is allowed to change in this block
adjustment.
3.1 The camera model
A physical camera model is assumed. The properties of the
camera are: first, the lens may cause nonlinear (radially
symmetric) distortion to the ray of light when it passes through
the lens; second, the distortion caused by a poorly known sensor
geometry is linear. Thus, all linear distortions happens only after
the nonlinear ones. The compensation of the distortion should
then take place in the reverse order, not simultaneously.
The physical model used here is not exactly the same as the
traditional "physical model" in /Kilpelá et al, 1981/. The
traditional model treats linear and nonlinear distortions
simultaneously, which works well using aerial images with small
linear distortions, but it may be considered erroneous using
video images with apparent linear distortion /Melen, 1994/.
3.2 Elliptic radial distortion
When computing the singular correlations, it is not necessary to
compensate the linear distortions in advance since the interior
orientation can always be computed after the singular
correlations have been computed but not before. Instead, it is
preferable just to find a suitable expression for the nonlinear
distortion in terms of the linearly distorted observations.
To obtain the required expression, it is first studied how the
final, radially and linearly distorted observations x,, y,, are
formed from the ideal, undistorted image observations x,, y;:
2 2 4 4
x, =x, +k x,(r; -19) *kx(r, IQ) tas (1)
Y,-y,* ky; 9) kx -19) ^...
and
X X, +X," By ?
% (2)
Yır = Yo or
582
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
Above, rx; *yl is the radius computed from the principal
point, and k,, k,,... are the radial distortion parameters, r, is the
radial distance where radial distortion effect is wanted to be
Zero. X, y, are the principal point coordinates, o is affinity
(scale ratio between the x- and y-axes of the image coordinate
system), and B corresponds to the non-orthogonality angle
between the image coordinate-axes. The radial distortion model
is adopted from /Karara, 1989/.
The observations x,, y, are measured and they are subject to the
random observational errors. The term r, with a certain nonzero
value is useful in the adopted model because it prevents the
system to collapse into a trivial solution where the magnitude of
the radial distortion correction becomes as large as the image
coordinates themselves. Fixing r, also fixes the image coordinate
scale since, for a certain radial distortion profile, the camera
constant depends on r,, and of course, on the radial distortion
coefficients k; /Brown, 1968/. A suitable value for r, is, say 200
pixels, in a 512 x 512 pixel? image.
By replacing the inverse of (2) into (1), and after some
manipulations it is obtained
Xx, =X, kx. x a2... (3)
2 2
ES 2 yc * kıy Ye +19) +",
which express radially distorted observations as functions of the
radially corrected (but linearly distorted) observations x,, y, and
of the interior orientation parameters. Especially,
r7 (xx)? «(a By, -y "2B (x. x (y, -y,)
is the elliptical radius computed from the point x,, y.. Equation
(3) presents the elliptical radial distortion model, because for all
equidistant image points the corresponding radial distortion
pattern is elliptically symmetric, as seen in figure 1. It also
expresses the radial distortion effect directly in the linearly
deformed image coordinate system as required.
T
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