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From now on, only the first term of the distortion function is
taken into account because it is, using video cameras, the only
significant parameter /Melen, 1994/. Nothing really prevents one
to include the other terms in the model, too.
The approximate radial distortion correction is
dx = k(x, "x E
A (4)
dy 7 ky, - 3907 15
and solving the equation (3) for x,, y,, it is also obtained
X. = Xa dx (5)
ye = Vi dy
which give approximate values of the radially corrected image
coordinates during iteration.
3.3 Solution of radial distortion
Let
//
Xc
x; y; 11 M |y - 0. (6)
1
be the singular correlation equation of two images which are
only linearly deformed. M is the 3x3-singular correlation matrix
containing seven independent parameters. See /Niini, 1994/ and
/Niini, 1995/ for details concerning the parameterization of M.
Now, expressing x,', y,', X,', y," with equations (5) and (4), and
replacing them into (6), the resulting equation is obtained
T
xx k@l | xx ka 1g)
/ uod (7)
Ye eye 9| M |y: -Gz-y9kGz -n)|
1 1
This is the singular correlation condition containing the radial
distortion parameters. Differentiating equation (7) with respect
to the unknown parameters and observations, a general system
of type Ax+Bv=l is obtained, which can be solved as explained
previously in /Niini, 1994 and 1995/. The only exception is that
in each iteration step, new radially corrected observations x, y,
have to be iterated, because they are also used in the
computation of the correlations.
Using two images only, it is possible to solve the common
radial distortion if the approximate interior orientation
parameters are known. Because radial distortion is a fixed part
of the interior orientation, like the camera constant, the
solvability of it follows the rules concerning the solution of
interior orientation from singular correlations.
For example, if the interior orientation is not known, at least
three images are required to solve the unknown interior
orientation and the radial distortion of the images. With two
different cameras, at least two images must have been taken
583
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
with both cameras. Using more images and thus more singular
correlations in the block, it is also possible to solve different
radial distortions of different cameras, provided that the
minimum requirements for each different camera are fulfilled,
and the block geometry constraints the solution sufficiently.
Special cases. Large radial distortion can be utilized to solve for
the approximate principal point in the very first step of the
adjustment. This is reasonable if the approximate image center,
for some reason, is too far from the true principal point.
Decentering terms of the radial distortion are derived from
equation (3), with one radial term only:
A ais à kx ke 9) (8)
Ya = ¥. * ky 021) ky (03-12)
and taking p,-k,x, p;-k,y, as additional decentering terms.
Then the principal point is computed from x,=p,/k and y =p./k.
Thus, if the radial distortion parameter k, is large enough, a
substantially better approximate center for the image is easily
obtained. In later iterations, these decentering terms can be
ignored, and the determination of the principal point can be left
in its original place in the block adjustment.
4. COMPENSATION OF LINEAR DISTORTION
After radial distortion and singular correlation parameters have
been determined, the interior orientation can be computed, using
the iterative way presented in /Niini, 1993/ or /Niini, 1994/.
The resulting linear transformation from linearly distorted
coordinates to the ideal image coordinates is the following.
x; = (xx) + BY)
n a, -y 6
Zi = pn
where Cp the camera constant, is now also dependent on the
choice of ry. Other terms are the same as in equation (2).
5. GENERAL BLOCK SOLUTION
Finally, a general and one step adjustment can be made for all
the block parameters based on the following factorization of the
singular correlation matrix, adopted from /Niini, 1994/.
M-C;R; (B,-B)R;C, (10)
This factorization expresses the singular correlation matrix
between two images in terms of the upper triangular interior
orientation matrices C,, C,, orthogonal rotation matrices R,, R,,
and skew symmetric base matrices B,, B,, both of which contain
the three projection center coordinates of the corresponding
image. All matrices are 3x3-matrices and their initial values are
obtained from the projective block adjustment.
