274 
2.3 Forward transform and inverse transform of RPC 
Model 
The geometry of image space and object space can be 
expressed through two forms of RFM, forward and inverse. 
Inverse form shows the direct transformation from image to 
object space, and forward form represents a more general type 
of RFM (the form provided by Space Imaging, Inc.). 
By the extended epipolarity model based on the projection 
track method, the transformation from the left image to object 
space is the forward form, and that from object space to the 
right image is the inverse form. 
2.3.1 Inverse RPC Model: The inverse transform is simple, 
RFCs are used in the RPC equations to calculate an image 
(sample, line) coordinate from an object (longitude, latitude, 
height) coordinate, and the specific processes are as follows: 
(1) Transform the flat coordinates and the corresponding 
height into the geodetic latitude and longitude, and the 
height above the WGS 84 ellipsoid. 
(2) Referring to Eq.(2), the object(longitude, latitude, height) 
coordinate can be normalized; 
(3) Referring to Eq.(l), the normalized coordinate in the image 
space can be solved; 
(4) Referring to Eq.(4), the (line, sample) coordinate can be 
solved; 
2.3.2 Forward RPC Model: The forward RPC is a 
transformation from the row and column indices of pixels in 
the image space and the corresponding height to the 
coordinates in the object space. 
Referring to Eq(l),the image point coordinates are measured on 
the image ,the corresponding elevation can be given artificially. 
Hence, the unknown quantities are the coordinates of ground 
point, that is to say two Equations two unknowns. In the RPC 
Model, the normalized coordinates (P, L)in the object-space are 
nonlinear equations, which need linearization and iterative 
least-square solution. For the strongly nonlinear problems, the 
solution obtained in the iterative process is always difficult, 
even divergent due to the numerical instability. 
2.4 Non-iterative forward transformation 
In view of the shortcomings, this paper proposes another 
conception: rounding out the forward transformation by the 
inverse form. 
The general distortion of remote sensing image can be seen as 
the compositive effect of the translation, scaling, rotation, 
affine, twisting, bending and higher distortion ,it is difficult to 
be described by a simple affine transformation. But in an 
infinitesimal local area, the distortion can be seen as an affine 
transformation including translation, scaling and rotation. The 
author exploits this theory to complete the non-iterative 
forward transformation by inverse form. 
Firstly, define the affine transformation between the ground 
point and image point: 
lat = ao + a\s + ail, Ion = bo + bis + bil (5) 
where lat,Ion = object coordinates in ground coordinate 
system 
s,l = image coordinates 
Then, the detailed process for non-iterative forward 
transformation is described in the following sections: 
(1) According to the assumed elevation value H and RFCs, 
calculate the image coordinates corresponding to these 
five ground points. 
{LAT _ OFF, LONG _ OFF)- 
{LAT _ OFF + LAT _ SCALE / 2.0, LONG _ OFF + LONG _ SCALE / 2.0); 
{LAT _ OFF - LAT _ SCALE / 2.0, LONG _ OFF + LONG _ SCALE / 2.0); 
{LAT _ OFF + LAT _ SCALE / 2.0, LONG _ OFF - LONG _ SCALE / 2.0); 
{LAT _ OFF - LAT _ SCALE / 2.0, LONG _ OFF - LONG _ SCALE / 2.0); 
They are the centre and comer points of the rectangle, 
whose centre is (LAT OFF, LONG OFF), and length 
and wideness are LAT SCALE and LONG SCALE 
distinguishingly. 
(2) Calculate the affine transformation coefficients of the 
general image according to the image and ground 
coordinates of step (1). 
(3) Calculate the approximate latitude and longitude 
(latp,lonp) coordinates corresponding to the chosen 
image point(s,/) with the affine transformation 
coefficients which are calculated in step (2). 
(4) Calculate the image coordinates (Sp,lp) corresponding 
to the ground point ( latp,lonp , H) by the inverse 
transformation. 
Cs — | S — Sp | 
(5) Calculate the residual error ,if 
ei=\l—lp\ 
e = (e s ) 2 + (el) 2 <0.01 pixel,the (lat P ,lon P ) is the 
solution ,and lat = latp ,lon = lonp ;else go to step 6; 
(6) Calculate the corresponding pixel size of the ground 
point according to the RFCs and the geodetic 
coordinates by Eq(6). Given the geodetic coordinates 
(lat,Ion, H) of the ground point, the corresponding 
image coordinates (s\,l\) can be derived from the 
inverse RPC model. Calculate the image coordinates 
( 52,/2 ) which is corresponding to the geodetic 
coordinates (latp + 0.01* LAT SCALE, lon P ,H) .And 
then the pixel size nearby the image point (52,/2) can 
be obtained by Eq(6). 
0.01*LAT _ SCALE 
PixelSize= (6) 
sqrt ((si - s2) * (si - s2) + (/1 -12) * (/1 - 12)) 
(7) According to the assumed elevation value H, calculate 
the image coordinates corresponding to these five points 
by using the inverse RPC Model. 
(latp, lonp, H), 
(latp + es* PixelSize, lonp + el* PixelSize, H), 
(latp+es*PixelSize,lonp-el*PixelSize,H), 
(latp-es* PixelSize, lonp+el* PixelSize,H), 
(latp-es* PixelSize,lonp-el* PixelSize,H), 
To calculate the affine transformation coefficients of this 
area, whose centre is (latp,lonp) , and length and 
wideness are 2es* PixelSize and lei * PixelSize .Then 
go to step(3),and circulate this iterative procedure until 
matching step(5).The result is the ground coordinates 
(latp,lonp, H) corresponding to the chosen point on the 
left image.