The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B4. Beijing 2008
Kim (2000) formulated exact epipolar curve for the linear array
scanner scenes based on Orun and Natarajan’s orientation
model and derived some useful properties of the epipolar curve.
The shape of such epipolar curve is not a straight line but
hyperbola-like non-linear curves. It can be approximated by a
straight line for a small range but not for the entire image.
Conjugate epipolar pairs do not exist for this kind of imagery,
but exist “locally”.
The epipolarity equation derived by Kim is useful in analysing
the property, but it is difficulty to apply this equation to the
process of linear push-broom scenes due to the need of
orientation information. Morgan (2004) derived the epipolarity
of this kind of scenes based on two-dimensional affine model,
and obtained a linear model which can be used to resample the
entire scene. This paper proposes a method to generate the
approximate epipolar lines based on RFM without deriving
epipolarity equation.
2. DESCRIPTIONS OF RFM
RFM relates object point coordinates (X, Y, Z) to image pixel
coordinates (/, s') or vice visa, as physical sensor models, but in
the form of rational functions that are ratios of polynomials
(Yong, 2005). The RFM is essentially a generic form of the
rigorous collinearity equations. The RFM is divided into
forward RFM and inverse RFM according to the relationship
between object space and image space.
The forward RFM is a transformation from the coordinates in
the object space to the row and column indices of pixel in the
image space. The defined ratios have the form as follows:
l n = P\( X n’Y n ,Z n )/p 2 (X„,Y n ,Z n )
s n =Pi(X n ,Y n ,Z n )/p<(X n ,Y n ,Z n )
The inverse RFM is to transform the image coordinates to the
object coordinates, which can be presented as follows:
X n=Ps(Ls n ,Z n )/p 6 (l„,s n ,Z n ) (2)
Y n = pAL^n’ZJ/p s (i n ,s n ,z n )
where /„, s n are normalized image coordinates;
X n , Y n , Z„ are normalized object space coordinates;
Pi is a polynomial with the following form:
p(X. Y, Z) = £ tt • X "‘ ■ Y "' Z * <3)
;'=0 j=0 k=0
Where N is the polynomial order, c ijk called rational polynomial
coefficients. In order to improve the numerical stability and
minimize the introduction errors during computation, both
image coordinates and object space coordinates are normalized
in the range of [-1 +1] by applying offsetting and scaling
factors(NIMA, 2000). When N is three, Eq. (3) becomes a three
dimensional polynomial with 20 coefficients, which is the most
commonly used form by data and software vendors. The model
used in this paper is in the form of three order polynomial.
3. EPIPOLAR LINE GENERATION
The epipolar geometry of linear array scanner scenes can be
understood more easily with Figure 1. The p is a point on the
left scene; S is the perspective centre of point p. The light ray
pass through S and p hit the ground at point P. Every point on
the light ray can be projected to the right scene, and the
combination of these projected points can form a curve on the
right scene, which is called the epipolar curve of point p.
Figure 1. Epipolar geometry of linear array scanner scene
According to the definition of the epipolarity for linear array
scanner scene described above and the property summarized by
Kim (2000), a method to generate the approximate epipolar line
of point p based on RFM can be developed.
The procedure of epipolar line generation is as follows. For a
point p on the left scene give an elevation value Z and calculate
the corresponding object space coordinate use the inverse RPCs
of the left scene. Then project the object space point into the
right scene use the forward RPCs of the right scene. Repeat this
process for a quantity of times, each time the elevation of the
object point should change with equal interval along the light
ray connecting the perspective centre and image point on the
left scene, a series of image point on the right scene can be
obtained, fit them to be a line 1’. The line 1’ should be the
approximate epipolar line of point p and the conjugate point of
p should be located on the line 1’ or near from it. During this
process, the actual elevation of the object space point is not
required. In the experiment section some details such as how to
change the elevation and if the elevation range and the times
elevation changed have influence on the accuracy of the
epipolar line will be discussed.
4. EXPERIMENTS
4.1 Data Description
The dataset used in the experiment involves a stereo-pair
captured by IKONOS-2 over the south of Australia with 1
meter resolution. The specifications of these scenes are listed in
Table 1. Residential areas, water areas and mountains are
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