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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B4. Beijing 2008 
The RFM, as expressed in equation 1, provides a direct map 
ping from 3D object space coordinates (usually offset normal 
ized latitude, longitude, and height) to 2D image coordinates 
(usually offset normalized line and sample values). 
(1) 
P 2 (X,Y,Z) 
_PfX,Y,Z) 
L ~ P 4 (X,Y,Z) 
where r and c are row and column coordinates and 
P, ^ ¿ = 1,2,3,4 ^ are third-order polynomial functions of object 
space coordinates *' Y ’ Z that transforms a point from the 
objects space to the image space. There are 39 Rational 
Polynomial Coefficients (RPCs) in each equation, including 20 
in the numerator and 19 in the denominator, and they are 
usually provided by satellite image providers. 
Tao and Hu (2002) derived a least-squares solution to the RFM 
and comprehensively evaluated and analyzed the results of nu 
merous tests with different data sets. 
2.2 Updating RPCs with Additional GCPs 
GCPs are required for the elimination of biases, and a lot of 
research has been done on this respect. Hu and Tao (2002) used 
batch iterative least-squares method and an incremental discrete 
Kalman filtering method to update or improve the existing RFM 
solutions when additional GCPs are available. Robertson (2003) 
assessed the absolute geometric accuracy of a sample set of 
QuickBird products using a full photogrammetric block 
adjustment. 
According to Wang et al. (2005), basically, there are two kinds 
of schemes to improve the geo-positioning accuracy of RFM 
with additional GCPs, object space and image space. There are 
four different models defined in both spaces to refine the RPC- 
derived ground coordinates. They are translation model, scale 
and translation model, affine model, and second-order affine 
model. Here are some basic mathematical principles of the 
models. 
2.2.1 Geometric Correction in Object Space 
The four geometric correction models are shown as equation 2- 
5 as below, 
{P = a 0 + X 
L = b 0 + Y 
[H = c 0 +Z 
\P = a 0 +a,X 
\L = K+bJ 
[// = c 0 +c,Z 
I P = a 0 + a ] X + a 2 Y + a 3 Z 
L = b 0 +b x X + b 2 Y + b 2 Z 
H = c 0 + c,X + c 2 Y + cjZ 
P — Qq + X + a 2 Y + ajZ + a 4 XY + a^XZ + afYZ + a 2 X 2 + afY 2 + a 9 Z‘ 
■ L = b 0 + b x X + b 2 Y + bfZ + b 4 XY + b 5 XZ + b 6 YZ + b 7 X 2 + b s Y 2 + b 9 Z 2 
where (p,l,h) are the GCP coordinates, (x,y,Z) are the 
corresponding coordinates from RPC solution. ( a c )are the 
transformation coefficients. Geometric correction in object 
space is the coordinate conversion from coordinate system 
defined by RPCs-derived ground points to the coordinate 
system defined by GCPs. In translation model, the translation 
(a 0 ,6 0 ,c 0 ) are added to achieved the improved ground 
coordinates (p,L,H) and at least one GCP is needed for 
computation. Non-homogeneous scale distortions are corrected 
using additional scale factors (a^b^cf in the scale and 
translation model. Higher scale distortions can be estimated and 
eliminated in the affine transformation and second-order affine 
transformation models, respectively. In practice, the image 
points (r,c)of each GCP can be measured from the image, and 
the RFM triangulation is then applied for the derivation of 
ground coordinates (X,Y,Z) ■ The least square adjustment is 
then employed for the calculation of optimal estimates of the 
transformation parameters when the over-determined equations 
are established using the (X,Y,Z) anc * l h e corresponding 
(P,L,H) coordinates. The Check Points (CkPs) are used to 
estimate the root mean square error (RMSE) for each model by 
compare the differences of their known and calculated ground 
coordinates from the transformation parameters. 
2.2.2 Geometric Correction in Image Space 
The geometric correction in image space is also referred to as 
the bias-compensated RFM (Fraser and Hanley 2003; Fraser 
and Hanley 2005) Incorporation of image shift and drift terms 
into the basic model of Equation 6 yields a bias-compensated 
RFM, which takes the form: 
2 2 PfX,Y,Z) ... 
r + a 0 +a x r + a 2 c + a 2 rc + a 4 r +a f c = (6) 
r 2 (A,I 
» , » , ,2,2 P*(XJ 9 Z) 
c + L + hr + b.c + hrc + b.r +Lc = — 
P A (X,Y,Z) 
Within this formulation there are four choices of additional pa 
rameter sets: 1) which affect an image coordinate transla 
tion; 2) MM , which model shift and drift; 3) 
^o,a l ,a 2 ,b u ,b^b 2 , w |jj c jj d escr ib e an affine transformation^) 
a 0 ,a l ,...a 5 ,b 0 ,...,b 5 ^ w j 1 j c j 1 describe a second-order affine trans 
formation. The additional parameters can be solved using the 
multi-image, multi-point bundle adjustment developed by 
Fraser and Hanley (2003). For each GCP, the image coordinates 
(r,c) can be obtained by measurement, the parameters 
(a ¿>.) and ground points (X,Y,Z) of CkPs can be determined 
simultaneously by the bundle adjustment incorporating the 
GCPs and CkPs. The CkPs are then used for the accuracy esti 
mation. 
3. STUDY SITE AND DATA SET 
3.1 Study Area and QuickBird Across-track Stereo 
Imagery 
The study area is shanghai metropolitan area, China, latitude 
ranges from 31°08'52.8" to 31°17'59.6 ", longitude from 
121°25'28.9" to 121°36'49.0 ", the elevation range between 12 
and 14 m, very low relief within about 3 meters, total area is 
about 300km 2 .Two QuickBird basic images were collected in 
Feb and May 2004 by DigitalGlobe in Shanghai area, China, 
making a pair of across-track stereo imagery. The scan 
directions were both forward. The satellite azimuth and 
elevation angles for imagery were provided in the metadata 
files. The convergent angle was calculated according to the 
equation in Li et al. (2007): 
(2) 
(3) 
(4)