International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences,
Vol XXXV, Part B3. Istanbul 2004
images, each is associated with different exposure stations. The
mathematical model that relates a point in the object space and
its corresponding point in the image space is the collinearity
equations, which uses EOP of the appropriate image (in which
the point appears).
In contrast, a scene is the recorded sensory data associated with
one (as in frame images) or more exposure stations (as in linear
array scanners) that maps near-continuous object space in a
short single trip of the sensor. Therefore, in frame images,
image and scene are identical terms, while in linear array
scanners, the scene is an array of consecutive 1D images.
Consequently, it is important to distinguish between the scene
and image coordinates. As shown in Figure 2b, i and y are the
scene coordinates, while in Figure 2a, x; and y; are the image
coordinates for image number i. Only x; and y; can be used in
the collinearity equations, while i indicates the image number or
the time of exposure.
| | , Column $8
DEL D —
12 i n or time
(b)
Figure 2. A sequence of ID images (a) constituting a scene (b)
2.2.2 Stereo Coverage
One of the main objectives of photogrammetry is to reconstruct
the three-dimensional object space from 2D images/scenes. This
is usually achieved by intersecting light rays of corresponding
points in different views. Therefore, different views or stereo
coverage is essential for deriving 3D information regarding the
object space. In linear array scanners, stereo coverage can be
achieved using one of the following ways:
e One scanner and across track stereo coverage using
roll angles: Stereo coverage can be achieved by tilting
the camera sideways across the flight direction
(different roll angles), Figure 3a. This has been
adopted in SPOT. A drawback is the large time gap
between images of the stereo pair, and consequently
changes may occur between the two scenes (Wang,
1999).
e One scanner and along track stereo coverage using
pitch angles: In this case the camera is tilted forward
and backward along the flight direction (different
pitch angles), Figure 3b. This type of stereo coverage
is used in IKONOS. This method has the advantage of
reducing the time gap between the scenes constituting
the stereo pair.
e Three scanners (three-line cameras): In this case, three
scanners are used to capture backward-looking, nadir,
and forward-looking scenes, Figure 3c. Continuous
stereo or triple coverage can be achieved along the
flight line with reduced time gaps. However, different
radiometric qualities exist between the scenes. This
method is implemented in MOMS and ADS40
(Heipke et al., 1996; Sandau et al., 2000; Fraser et al.,
2001).
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(a) (b) (c)
Figure 3. Stereo coverage in linear array scanners achieved by
roll angle rotation in two flying paths (a), pitch angle
rotation in the same flying path (b), and three-line
cameras (c).
2.2.3 Rigorous Modelling
Rigorous (exact) modelling of linear array scanners describes
the actual geometric formation of the scenes at the time of
photography, that requires the knowledge of IOP of the scanner
and the EOP of each image in the scene. Usually, EOP do not
abruptly change their values between consecutive images in the
scene, especially for space-based scenes. Therefore, most
rigorous modelling methods adapt polynomial representation of
EOP (Lee et al., 2000; Wang, 1999). The parameters, describing
the polynomial functions, are either given (directly) from the
navigation units such as GPS/INS mounted on the platform, or
indirectly estimated using ground control in bundle adjustment
(Lee and Habib, 2002; Habib et al., 2001; Habib and Beshah,
1998). Other methods (Lee et al., 2000; McGlone and Mikhail,
1981; Ethridge, 1977) use piecewise polynomial model for
representing the flight trajectory and the platform attitude. This
option is preferable if the scene time is large, and the variations
of EOP do not comply with one set of polynomial functions.
For indirect estimation of the polynomial coefficients using
ground control points, instability of the bundle adjustment
exists, especially for space-based scenes (Fraser et al., 2001;
Wang, 1999). This is attributed to the narrow Angular Field of
View (AFOV) of space scenes, which results in very narrow
bundles in the adjustment procedures. To avoid the polynomial
representation of EOP, Lee and Habib (2002) explicitly dealt
with all EOP associates with the scene. Linear feature
constraints were used to aid independent recovery of the EOP of
the images as well as to increase the geometric strength of the
bundle adjustment.
2.3 Epipolar Geometry of Linear Array Scanner Scenes
Kim (2000) modelled the changes of EOP as second order
polynomial functions in scanner position and heading, and first
order functions in pitch and roll angles. This model is called
“Orun and Natarajan” model as cited from (Orun and Natarajan,
1994). The author proved that the epipolar line is no longer a
straight line, rather has hyperbola-like shape. In addition, the
author proved that epipolar lines do not exist in conjugate pairs.
The next section investigates the epipolar geometry of linear
array scanners using constant-velocity-constant-attitude EOP
model, which is a subclass of *Orun and Natarajan" model. But
first, epipolar line determination in linear array scanners has to
be explained.
As discussed earlier, scanner at successive exposure stations has
different perspective centres and different attitude, therefore,
EOP will vary from one scan line to the other. Hence, the
epipolar lines should be clearly defined in such scenes before
studying their geometry. Figure 4 shows a schematic drawing of
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