ISPRS Commission II, Vol.34, Part 3A „Photogrammetric Computer Vision‘, Graz, 2002
coordinate systems to each other. It maps the coordinates from
the object space into the image space (Mikhail, Bethel and
McGlone, 2001). There are two types of parameters. First, the
exterior parameters which include the location and orientation
parameters. Lens distortion and focal length are examples of
the second type, which are called the interior parameters. The
model specifically covers and takes into account the lens
distortion through some parameters that model radial,
decentering, and affinity distortion.
X-x mry'ey + Ax
=—f FA AA "1, tn, 12 72.)
LXE YL {Y =F +7. (2-7)
(1)
y=y.=2y=y +Ay
ze EX-XerBOY- Y» (Z-—27)
LX ~X +r (TF =Y Yr. (Z=2)
Where: x, y and x', y: ideal and measured target coordinates
in image space
x,, y, : Principal point coordinates in image space
? x, ? y: distortion corrections in x, y directions
f: camera focal length
r ij: the ith row and jth element of the orientation
matrix R
X,Y,Z: target coordinates in object space
Xc,Yc,Zc: exposure station coordinate in object space
The rotation matrix R expresses the orientation of the image
coordinate system with respect to the object coordinate system.
The distortion effects including radial, lens decentering, and
affinity were computed through the equations below.
Ax =X(kr® + kt - ky) p(r? +232) +2p,X y
(2)
Ay zv Tk +kr°)+2p,x y+ pr +2ÿ")+a,x+a,ÿ
-— /, — / 2 —2 —2
where: ur 1 dt CUS vy
k, , P;» À; radial , decentering, affinity distortion
coefficients
The two condition equations for each target will be:
F,=X+X(kr’ +k,r* + kr’) + p,(r’ + 2X) + 2p, Xp
+f X ANZ ZZ.) x
LX Xn -Y)tuuz-2Z)
(3)
F,-yty(kr * kr! - kr) -2pxy* pr 423^?) +
LUX =X y+, (Y~Y +n ll~Z) =0
axo yd =
2 YET
From the equations above, each target observation will
generate two equations. Consequently, the number of equations
will be twice the number of targets in the image for each
camera.
4.2 Solution method
The unified least squares approach was used to solve this
system since some a priori knowledge is available for a
number of parameters (Mikhail and Ackerman, 1976). Using
the a priori knowledge of the parameters is the distinction
between ordinary least squares and unified least squares. This
knowledge is utilized to give those parameters initial values
and weights. In this sense, some of the parameters were treated
as observations with low precision by assigning large variances
to them. Since the system is non-linear, the parameter values
will be updated iteratively by adding the correction to them.
The system will converge when the correction vector values are
negligible. Then the final correction will be added to the
parameters to get the final estimated values. In our case here,
the system converges with few iterations since the precision of
the observations was very high.
5. DISTORTION ANALYSIS
5.1 Radial Distortion
The term used for the displacement of an imaged object
radially either towards or away from the principle point is
radial distortion (Atkinson, 1996). The magnitude of this
displacement is usually determined to micrometer precision
and it varies with the lens focusing. Radial Distortion is
included in the math model and its magnitude can be
calculated as follows:
Ar=kr°+kr+kr"
(4)
0, =Ar*x/r 0, =Arxylr
The radial distortion curve was constructed based on the
equation above as shown in figure 5. The resulting curves were
obtained for all four cameras and the maximum radial
distortion was around 30 micrometers.
The following step was done to level or balance the curve
based on equalizing the maximum and the minimum distortion
values. This procedure is done only to balance the positive and
negative excursions of the distortion function about zero. This
step has no effect on the final results of the corrected
coordinates; it is just cosmetic but accepted professional
practice. Mathematically, balancing the curve leads to a change
in the radial distortion parameters and consequently the focal
length and other related camera parameters. The aim of this
balancing procedure is to make Ka io as shown in
figure 6 and the condition equation will be:
tox — CELXtan(e )+r. —CFLxtan(e_ )=0 (5)
So the new focal length is:
ro +r.
max min
) + tan(e_. )
7 tan(o
max
(6)
After getting the revised focal length, the calibration
adjustment program is run again but with a fixed focal length
(CF
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