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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)