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Mapping without the sun
Zhang, Jixian

2.1 RFM
The RFM is a model that relates a ground point expressed
ground point ( P ’ L ’ H ) in object space coordinates to a
corresponding point in an image expressed image point in
image space coordinates (C.V. Tao, 2001). Four polynomials all
compute functions of three ground coordinates: longitude,
latitude and height. Each polynomial uses 20 coefficients at
least, and is essentially the generic form of the collinearity
equation, with the collinearity equation expanded by using
Taylor polynomials and represented in the rational form.
Wherein the various imaging parameters are used to determine
four polynomials that are used as follows:
x p t (P,L,H)
Pl (p,l,h)
' y p 3 (P,L,H)
P,{P,L,H) (1)
Where i. p < L > H ) is a set of normalized coordinates of latitude,
longitude and height in object space. The normalized
coordinates are obtained by applying a linear scaling factor and
a linear translation factor to the corresponding actual
coordinates of latitude, longitude and height in object space to
limit the magnitude of each coordinate of the normalized set to
a predetermined range. Each of the four polynomials Pi , Pk ,
p y , p* used in the RFM may be expressed generically as
p{P,L,H)="f M
/=0 y=0 hO
=a 0 +a i H+a 2 L+a } P+a t HL+a 5 HP+a t LP+a J H 2 +aj} +0^* +a t0 HLP
+a n H 2 L+a t2 H 1 P+a ty I?H+a u l}P+a ts HP l +a l6 LP l +o lr P J +a ll( Z, 3 +a t jf
where a iJk is the coefficients of the rational polynomial
function. In every item of the polynomial, the power of each
coordinate component P > L ’ H is no more than 3, and the sum of
the powers of coordinate components in an item is also no more
than 3, typically 1, 2 or 3. RFM based on RFC which is utilized
for HRSI provides a transformation from image to object space
coordinates in a geographic reference system and achieve a
quite high geometric accuracy. However, one of key problem at
RFM is that it cannot accommodate local image space
distortions and high frequency perturbations in exterior
orientation (C.S. Fraser, 2006), so a fined RFM which can
eliminate the distortions and improve the geometric accuracy of
the image space coordinates should be taken into account.
2.2 RFC
According to the different source of RFC, there are two
approaches to generate RFCs and build RFM. One method is
the terrain-dependent, the other is the terrain-independent (Liu
Jun, 2006). The former is directly computed from GCPs which
are measured in the field, while the latter is computed by least-
squares fitting from 3 dimensional control grid (see Figured).
In principle, the terrain-independent RPCs are a relatively
straightforward process. The interested area of image is then
surrounded by a fictitious array of object points, which are
positioned on horizontal planes at multiple elevations, and
number several hundred or even thousands.
Figurel Illustration of 3 dimensional control grid
Usually, the RPCs construct a reparameterization of the
rigorous sensor orientation model which is set up by the time
functions from satellite ephemeris and attitude which is directly
observed using on-board GPS receivers, gyros and star trackers.
However, both the satellite ephemeris and attitude are subject to
error no matter under what circumstances because standard
stereo image products are produced using satellite position and
attitude information measured by on-board GPS and the stellar
camera which have inevitably a certain range of error in the
measurement process. Hence, errors in sensor interior and
exterior orientation thus give rise to errors in the RPCs. Without
using ground control, there are obvious systematic errors. C. S.
Fraser proved by experiments: although the positioning errors
of RFM in image space reach 29 pixels and 17 pixels, RFM has
strong systematicness, for the size of error on each point is
similar and the direction is basically the same (C.S. Fraser,
2002; C.S. Fraser, 2003). In order to improve the accuracy of
stereo positioning, GCPs are needed to compensate the
systematic error of RFM. It can provide the transformation
either from object space to image space or from image space to
object space. In RFM, first-order items of the function can
eliminate the twisted distortion caused by optical projection,
second-order items can eliminate the errors caused by Earth
curvature, atmospheric refraction and lens distortion, and errors
caused by other unknown elements can be eliminated by cubic-
order items (C.V. Tao, 2000).
To improve the geometric accuracy of the image space
coordinates determined by the RPM, it may be required to
estimate and remove residual errors or biases either in the RPCs
or in the imaging parameters from which the coefficients were
derived. The refinement work may be accomplished by
adjusting the coefficients of the RFM such that when the
adjusted RPM is applied to the GCP object space coordinates,
the resulting image space coordinates more closely approximate
the GCP image space coordinates (Guo Zhang, 2005; Kwoh et
al. 2005).Therefore, without removing the biases of imaging
parameters, it can still improve the geometric accuracy of
images produced using a given camera model by performing a
preprocessing function to adjust the image space coordinates
using adjustable functions ^ and AT to obtain adjusted
image space coordinates x and Y (see Figure.2). Hence,
according to the analysis above, a new object space coordinate
system determined from the set of original object space
coordinates to improve geometric accuracy in the resulting
image space coordinates defines as follows: