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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part BI. Beijing 2008 
calibration area. Two-dimensional calibration is not only to be 
produced, but also provides high accuracy in measurement. 
Tsai.[l] proposed the two-step method based on the RAC 
(radial alignment constraint), which can get the calibration 
result by linear equations. Triggs[2] use the absolute quadratic 
curve principle to calibrate. Zhang zhengyou[3] calibrates the 
camera by the orthogonal rotation matrix of conditions and 
nonlinear optimization in China, Zhang Yongjun[4] proposed 
the 2-D DLT (direct linear transformation) with a bundle 
adjustment of the camera calibration algorithm, and so on. The 
purpose of all those methods is to calibrate non-metric camera. 
Thus, what are the influence factors in camera calibration? 
Thin prism distortion arises from imperfection in lens design 
and manufacturing as well as camera assembly. This type of 
distortion can be adequately amended by the adjunction of a 
thin prism to the optical system, causing additional amounts of 
radial and tangential distortions. It can be expressed as: 
5 up = SiC" 2 + v2 ) + 0[(u,v)*] (24) 
S vp =s 2 (u 2 + v 2 ) + 0[(u,v) 4 ] 
So, the total amount of lens distortion can be expressed as: 
2. THE ANALYSIS OF INFLUENCE FACTORS 
There are several influence factors in calibration. In this section, 
we try to analysis each influence factor in theoretic. 
2.1 Camera and lens 
In non-metric camera, the scales in u, v directions are often 
inconsistent, so focal length f is accustomed to be decomposed 
into two directions, expressed as f u , f v respectively. f u stands for 
the focal length in u axis and f v in v axis. And sometimes, we 
need to take the skewness of the two image axes into account. 
So the camera intrinsic matrix is often expressed by A: 
r 
fv 
0 
v 0 
1 
(2-1) 
Where, y is the skewness of the two image axes, and (U Q , Vq ) is the 
principle point coordinate in image plane. 
For ordinary digital camera, lens distortion can’t be ignored. In 
weng’s article, the lens distortion is departed into three parts: 
radial distortion, decentering distortion and thin prism distortion. 
Radial distortion causes an inward or outward displacement of a 
given image point from its ideal location. The radial distortion 
of a perfectly centred lens is governed by an expression of the 
following form: 
S u (u,v) = sfu 2 + v 2 ) + p x (3u 2 + v 2 ) 
+ 2 p 2 uv + k x u(u 2 + v 2 ) (2-5) 
S v (u,v) = s 2 (u 2 + v 2 ) + p 2 (u 2 + 3v 2 ) 
+ 2p x uv + k x v(u 2 + v 2 ) 
2.2 The shape of control points 
In Photogrammetry, the object space translates into image plane 
through perspective projection. After projection, line is still line, 
but other objects can’t keep the shape. For instance, the object 
is a circle before projection, but the shape is not a circle any 
more after projection. So, the centres of the shape are not 
consistence with each before and after projection. Many 
researchers ignore the error in their study. When we use the 
circular control points, it will cause error to extract point 
coordinates. So cross control points is better than circular 
control points for calibration. If we want to avoid the bias 
caused by extracting points, we can rectify the error through 
some method [5]. 
Let us assume that a circular control point R with radius r is 
located on the image plane so that its centre is at the origin of 
the planar coordinate frame H. circles are quadratic curves that 
can be expressed in the following manner: 
AX 2 H +2BX H Y H +CY 2 +2DX H +2EY H +F = 0 (2-6) 
Where A, B, C, D, E and F are coefficients that define the shape 
and location of the curve. In homogeneous coordinates, this 
curve can be written as: 
8 pr - k x p 3 + k 2 p 5 + k 2 p n +... (2-2) 
Where p is the radial distance from the principal point of the image 
plane, and k x , k 2 ,k 2 ... are the coefficients of radial distortion. 
The optical centres of lens elements are not strictly collinear. 
This defect introduces what is called decentering distortion, 
which can be described by the following expressions: 
Where 
For the circle R, 
X 
-1 t r 
H 
Q 
X 
H 
= 0 
Q = 
A B D 
BCE 
D E F 
0=0 
0 
0 
(2-7) 
S ud = A(3w 2 + V 2 ) + 2p 2 uv+ 0[(m,v) 4 ] 
s vd = 2p x uv + p 2 (u 2 + 3v 2 ) + 0[(w, v) 4 ] 
(2-3) 
0 
1