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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