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Mapping without the sun
Zhang, Jixian

1.3 Sensor Model
Choosing a suitable sensor model is also very important to
resample CCD push-broom image. For such image sensor
models, including collinear equation, DLT and so on, are yet to
be determined which is the best one. During the past years the
photogrammetric community at large has become aware of the
use of rational functions for photogrammetric restitution. This
has been largely due to the need to use these for setting up
IKONOS data, as Space Imaging does not provide a
physicalsensor model with the image data, but offers a
parameter called a rational polynomial coefficient (RPC) in
order to use a rational function model (RFM) as a mathematical
sensor model.
The RPC model has recently raised considerable interest in the
photogrammetry and remote sensing community. The RPC
model is a generalized sensor model that is capable of
achieving high approximation accuracy. As the number of
sensors increases along with greater complexity, and as the
need for standard sensor model has become important, the
applicability of the RPC model is also increasing. A RPC
model can be generated from the physical sensor model and
substituted for all sensor models, such as the projective, the
linear push-broom and the SAR. This paper proposes extended
RPC model to describe the object-to-image space
2.1 Definition of RPC Model
The RPC Model is divided into forward RPC and inverse RPC
according to the relationship between object space and image
space (Yang, 2000; Tao and Hu, 2001b). In the case of the
forward RFM, the ratios of polynomials of object-space
coordinates define the image-space coordinates. General forms
of the RPC Model can be written as
Num L (P,L,H)
Den L (P, L, H)
Num s (P,L,H)
Den s (P,L,H)
where X , Y = normalized image coordinates
P = normalized geodetic latitude
L= normalized geodetic longitude
H= normalized height above the WGS 84 ellipsoid
Num L (P,L,H), Den L (P,L,Hy Num s (P,L,H)> Den s (P,L,H)
can be written as
Num L (P, L,H) -a x + a 2 L + a 3 P + a 4 H + a s LP + a k LH + a 7 PH + a % l}
+a 9 P 2 + a x0 H 2 + a x X PLH + a xl E + a x3 LP 2 + a X4 LH 2
+a xs L 2 P + a x6 P 3 + a xl PH 2 + a x% L 2 H + a X9 P 2 H + a 20 H 3
Den L (P, L, H) = b x + b 2 L + b 3 P + b 4 H + b 5 LP + b 6 LH + b 7 PH + b,L 2
+b g P 2 + b X0 H 2 + b xx PLH + b x2 P + b n LP 2 + b X4 LH 2
+b xs L 2 P + b i6 P 3 + b xl PH 2 + b Xi L 2 H + b l9 P 2 H + b 20 H 3
Num (P, L, H) = c + c L + c P + c H + c LP + c LH + c PH + cL
+c P' +c w H~ + c PLH + c t +c LP 2 + c LH
1 +c ¿P + c P’ + c PH' + c LH + c P H + cji
Den (P,L,H) = d+d 2 L + dP + dH + dLP + dLH + dPH + dt
+dP 2 +dH~ +d PLH + d t +d LP 2 +d LH 2
9 10 II 12 13 14
+d ÊP + d P 1 +dPH +d EH + d FH + d rf
15 16 17 18 19 20
where ai,bi,ci,di (i= 1,2,3 20) = polynomial coefficients,
(called RFCs) ,and usually, ^ and d x equal to 1.
In order to improve the numerical stability of the computation
and minimize the errors during the matrix computation, both
image-space coordinates and object-space coordinates are
normalized in the range of -1.0 to +1.0 by applying offsetting
and scaling factors .The normalization of the coordinates is
computed using the following equations. (OGC, 1999; Tao et
al., 2000)
Latitude - LAT OFF
p =
Longitude - LONG OFF
L = (2)
H =
Sample - SAMP __ OFF
X =
Y =
In these equations, LAT OFF, LONG OFF, HEIGHT OFF,
SAMP OFF and LINE OFF are offsetting factors, and LAT -
LINE SCALE are scaling factors. The RFCs supplied for the
IKONOS satellite of Space Imaging include 10 normalizing
parameters and 80 coefficients.
2.2 Improve the RPC accuracy with GCPs
There are two methods to improve the RPC accuracy using
Ground Control Points (GCPs).The one is to rectify the RFCs
using GCPs directly. This method is a little difficult, which
needs a large number of GCPs to calculate the eighty RFCs,
and the calculated coefficients may be correlativity. The other
uses few GCPs, which is not to rectify the RFCs but to
calculate the image’s parameters of affine transformation. It
needs for transforming relation between the measured
coordinates and the calculated coordinates by using RPC model.
There are two types of errors need to be corrected through
analyzing the impact of satellite parameters on the accuracy of
the geometric correction. One type of parameters is used to
rectify the errors in the row direction, and the other rectify in
the column direction. The author defines an affine
transformation to rectify the error of the image coordinates as
y = eo + e\ * sample + ei * line
x = /o + f \ * sample + /2 * line
where x and y are the measured coordinates of the GCPs on the
image plane to calculate the affine coefficients .Then, the affine
coefficients obtained are used in Eq 4 to calculate an rectified
(x,y) coordinate from an image(line, sample) coordinate.