Full text: Mapping without the sun

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) 
Height - HEIGHT OFF 
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. 

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