The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008
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5.2. Modified collinearity equations for along track
sequence.
The well-known collinearity equations need modification
before they are applied to pushbroom images. They are
modified based on the above Kepler equations (1) in a way
where the ground coordinates and the rotations of the
perspective center are modelled as a function of time.
0
'x-x c (t)
T-To
= AM(t)
Y~Y c (t)
-c
_Z-Z e (i)_
Where:
c is the principal distance
X, Y, Z are the ground coordinates of a point
Xc(t), Yc(t), Zc(t) are the ground coordinates of the framelet
perspective center as a function of time, as described in
equations (1)
A, is a scale factor which varies from point to point
M(t) is a 3x3 rotation matrix which brings the ground
coordinate system parallel to the framelet coordinate system
as a function of time, where the rotation angles are simulated
with first order polynomials
y is the y-ffamelet coordinates of the corresponding point
yo is a small offset from the perspective center origin.
5.3. Modified coplanarity equation for along track
sequence.
5.3.1 Introduction. In general, coplanarity for the
perspective geometry is the condition that the two exposure
stations of a stereopair, any object point, and its
corresponding image points on the two photos all lie in the a
common plane (Wolf et all, 2000), as illustrated in figure 2.
In the figure the points Lj, L 2 , a h a 2 and A all lie in the same
plane. The coplanarity condition is
0 — B x ’{D i ‘F 2 D 2 -F x ) + B y -{E 2 -F x E X -F 2X ) ^
+ B z • (E x • D 2 -E 2 D x )
where
B x =x Li -x Li
B y = Y TT\
B 2 =z L -z^
D = m X2 ■ x + m 22 • y - m 32 ■ c
E = m xx -x + m 2X ■ y-m 3X -c
F = m x3 ■ x + m 23 • y - m 33 • c
In equation (2) subscripts 1 and 2 affixed to terms D, E and F
indicate that the terms apply to either photo 1 or photo 2. The
m’s are function of the three rotations angles omega, phi,
kappa as represented in matrix M(t). One coplanarity
equation may be written for each object point which appears
in the stereo photos. The coplanarity equation does not
contain object space coordinates as unknowns. It contains
only the elements of the exterior orientation parameters of the
two photos of the stereo pair.
Figure 2. The coplanarity condition (Wolf et al., 2000)
5.3.2. Modification of coplanarity equation for pushbroom
images. The development of the coplanarity equation for a
pushbroom sensor could be based on the perspective images
equation (2) where the fundamental characteristic that the
pushbroom image consist of one-dimensional images, should
be taken into consideration. In first instance the D, E and F in
equation (2) should be modified in case of pushbroom images
as follows, because of the one dimensional framelets:
D = m 22 • y ~m 32 c
E = m 2X ■y-m 3X c
F = m 23 -y-m 32 -c
(3)
In general the epipolar curves for linear pushbroom images are
not very easy to derive (Kim, 2000). The reasons are mainly
the pushbroom geometry itself, along with the selected sensor
model. Kim concludes that the problem is that the best sensor
model has not been developed yet.
In this paper, a slight different approach is followed. We do try
to use the model which is capable to describe the base vector
change as the scanner moves along its trajectory. This model
can be the UCL sensor model which simulates this as functions
of the exterior orientation parameters of the center point of the
center framelet of the first along track image.
The development of the coplanarity equation is based on the
equation of the Kepler model (1). Thus, the velocity of the
satellite during the acquisition time of the images is not
constant. Moreover at present, the rotation angles are not
constant and simulated with first order polynomials.
Based on equations (1) the X object space coordinates of the
points Li,L 2 a.re:
X Li =X 0+ u x -t x -A -X 0 -tf
X Ll = X 0 +u x ’(t 2 + dt)- A • X 0 • (t 2 + dt) 2
(4)
