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3. CORRECTIONS 
3.1 Geometric Corrections 
The geometric characteristics investigated in this 
calibration procedure include lens distortions (radial and 
decentering), offsets of the principal point and additional 
parameters which represented the effects of shear and tilt 
of the CCD sensor array. Mathematical models for these 
geometric characteristics are given below. 
3.1.1 Radial Lens Distortion: Radial lens distortion ,ór, 
is mathematically represented by a polynomial function 
of the radial distance r (Fryer & Brown, 1986 and Karara, 
1989) where 
óre Kp pi e Kor* SK pug is (1) 
where K; and K2 were the terms determined during the 
bundle adjustment. Ks was considered to be 
insignificant for the simple lenses used on inexpensive 
video cameras and assumed to be zero. Also 
r'm(x-x Yi (yy? (2) 
where x and y are image coordinates and x, and y, are 
offsets of the principal point. Figures 2 and 4 in Section 
5 graph the results of Equation 1 produced from the 
bundle adjustments. 
3.1.2 Decentering Lens Distortion: The decentering 
distortion equations follow the derivation in Fryer & 
Brown, 1986, where 
Ax 2 [Pi (r? + 2x? )+2P2xy][1 + P3 r° + .…] (3) 
Ay = [Pz (r? +2y2 )+2P;xyl[1+ Ps r? +..] (4) 
where Ax and Ay are the decentering distortion values in 
the x and y coordinates respectively, and r is the radial 
distance. A conventional representation of decentering 
distortion is as a Profile function P(r) where 
P(r) = r? (P12 « P2* )* (5) 
It is the profile function P(r) which has been graphed in 
Figures 3 and 5 later in this paper. P; and P2 may be 
easily determined from the plumbline method of lens 
calibration (Karara, 1989) or from a suitably rigorous 
bundle adjustment, as long as the correlation with offsets 
of the principal point have been duly acknowledged. 
3.1.3 Offsets of the Principal Point: This 
characteristic of cameras is due to the imperfect 
alignment of the line of autocollimation to the centre of 
the image plane. The parameters representing this 
characteristic are conventionally referred to as x, and y, 
and as previously mentioned, are highly correlated to the 
decentering distortion parameters of P; and P2,. The 
values of x, and y, for each camera, as determined by 
the bundle adjustment, are shown in Table 4 in Section 5 
of this paper. 
139 
3.1.4 Additional Parameters: The shear and tilt of 
sensor are represented by additional parameter terms 
from the following image coordinate correction equations 
(See Fraser, 1982). 
AX= AXı + AX2 (6) 
Ay = Ay: * Aye (7) 
where 
AX; = - Xp + (- X/0) de + K1 x 7 + K x 1° + Ks xP + 
Pi (3X + )+2P2xy (8) 
Ay; = -yp+(-y/c)dc+ Ki yP + Ke yr «Kay? 
2Pi x y+P: (X +3) (9) 
AX) m X yay 4 dpx^y- da x-y* (10) 
Aya = bi X * Da y * ba X y * Da x^ «bs x^ y 4 
bo X y* (11) 
where c is the initial value for the focal length and dc is 
the required adjustment to that value and 
X 2 X- Xp (12) 
y? y-yo (13) 
The additional parameters determined in the bundle 
adjustment were a; , b; , b» and bs. 
3.2 Radiometric Corrections 
The radiometric characteristics of digital cameras 
encompass aspects of the electronics and signal 
transmission involved in acquiring and converting the 
analog signal to the digital image. As these factors differ 
with the different type of camera, some characteristics 
are mainly found in analog video cameras with others 
mainly associated with digital still cameras. 
Phase patterns (as described by Ge, 1993), line jitter and 
noise are radiometric characteristics associated with 
analog video cameras when used in conjunction with a 
frame grabber. These characteristics are all due to the 
signal transmission and sampling of the signal by the 
frame grabber when producing the digital image. The 
effects on the image can be reduced without requiring 
new or expensive equipment, by taking a series of 
images and averaging them (see Hôflinger & Beyer, 
1993). This approach was incorporated into the 
procedure as described in Section 4 of this paper. 
Another consideration for analog video cameras is to 
ensure that they have been thoroughly warmed up before 
images are acquired (see Beyer, 1992). The time 
adopted for this procedure was 90 minutes. 
Digital still cameras do not have the above problems, as 
the image is converted to digital format immediately 
within the camera. Image compression within digital still 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B1. Vienna 1996