The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B4. Beijing 2008 
Since the 2D polynomial functions do not reflect the sources of 
distortion during the image formation and do not correct for 
terrain relief distortions, they are limited to images with few or 
small distortions, such as nadir-viewing images, systematically- 
corrected images and/or small images over flat terrain. These 
functions correct for local distortions at the ground control point 
(GCP) location. They are very sensitive to input errors and 
hence GCPs have to be numerous and regularly distributed. 
Consequently, these functions should not be used when precise 
geometric positioning is required for multi-source multi-format 
data integration and in high relief areas. If a second order 
polynomial function is used, the following is obtained: 
x — (Zq + a x u + a 2 v + a 3 uv + a 4 u 2 + a 5 v 2 
y = fî 0 +/3 ] u + j3 2 v + fî 3 uv + fi 4 u 2 +fi 5 v 2 
(2) 
From a mathematical point of view, a first order polynomial 
transformation requires a minimum of 3 points, a second order 
polynomial requires a minimum of 6 points and, a third order 
polynomial requires a minimum of 10 points. Generally, if the 
order of polynomial model is n, we must at least have a set of 
M=(n+l)(n+2)/2 GCPs to solve Equations (1). Suppose that {(Uj, 
Vj): i = 1, ..., n} and {(x i? y;): i=l, ..., n} are the tying GCP 
pairs from image and real world coordinate systems 
respectively, we can form n equations for both x and y as 
following: 
2 2 
= a 0 + a x u i + a 2 v i + a 3 u i v i + a A u i + a 5 v t + s x . 
£„ = 
£ 
£„ 
x \ 
y l 
£ 
£ 
*2 
yi 
, £.. — 
; 
’ y 
; 
£ 
£„ 
x n 
L r.J 
,A = 
«0 
Po 
a, 
Px 
a 2 
,B = 
Pi 
a 3 
Pi 
a 4 
A 
a, 
Ps. 
2 2 
1 u x V, WjV, Mj V, 
w = 
1 
u 2 v 2 
u 2 
1 
U n 
*1 
y\ 
x = 
Jt 2 
, Y = 
Y 2 
_ x "_ 
Xn_ 
This means that the two sets of coefficients aj and Pj could be 
estimated separately. The least square estimation can be used to 
determine the coefficients of Oj and (3j from GCPs. 
y, = Po+ P\ u >+ Pi v i+ Pi u i v i +/W + Ps v , 2 + e „ Min[e T x e x ] = (Wk-X) T (WA-X) 
( 3 ) Mm[e T y e v ] = {WB-Y) T (WB-Y) 
(6) 
By assuming that the e are independent and identically 
distributed (iid) observation errors, i.e., 
These are two standard least square problems. The estimation 
for cij and Pj are in Equation (7). 
£ x 
"O' 
<7 2 0 
£ = 
, £ ~ N< 
X 
£ y_ 
0 
i 
o 
Is 
1 
(4) A = (W‘Wy'W‘ X 
B = (W T Wy'W T Y 
(7) 
the equations constructed from the GCPs by (3) could be The c onding variance components estimation (Ou, 1989) 
separated into two sets of equations as shown in (5) below, afe ^ £q ua ti 0 n (8) 
X = WA + s x 
Y = WB + £ v 
where 
(5) & x 2 =^(WA-X) T (WA-X) 
ô y 2 =^(Wê-Y) T (WB-Y) 
(8) 
where r = rank(W). Thus, the transformation functions f and g 
are determined. The total variance could be estimated by: 
/r 9 * 2 * 2 
(9) 
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