# Full text: Mapping without the sun

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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.

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