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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part Bl. Istanbul 2004
Equation | can be interpreted in several ways. It can be viewed,
for instance, as a 3D affine transformation followed by an
orthogonal projection, as a 3D similarity transformation
followed by a skew parallel projection, or as a skew parallel
projection followed by a similarity transformation. In the
application. of the affine projection model, it is useful to
consider its connection to the central-perspective model, i.e. a
departure from collinearity equation. In fact, the introduction
of the common scale factor is the key to converting the non-
linear collinearity equation to a form of simple linear
transformation. The affine model is a further generalised form,
which allows affinity (non-uniform scaling and skew distortion)
in the image to object space transformation.
In 3D scene reconstruction, a model formed by a stereo pair of
affine images can be created from four corresponding
(conjugate) points, and can be related to the object space by a
3D affine transformation (twelve degrees of freedom). On the
other hand, for the special case of central perspective projection
where the internal geometry is known, five points are required
to form a model which is then transformed to object space by a
similarity transformation (seven degrees of freedom). The
distinction between these two approaches lies in the inclusion or
omission of affinity and different scale factors in each axis of
the model space. To form a model of correct shape from affine
images, constraints describing orthogonality and uniform
scaling have to be imposed among the affine parameters. Also,
a third image has to be added to the network to resolve
ambiguity arising from the adoption of a common scale factor
(Ono & Hattori, 2003).
For satellite line scanner imagery, however, these constraints
can be neglected because of the geometry of pushbroom
scanners. With narrow field of view imaging systems,
uncertainties and perturbations of the sensor and other
parameters of affine distortion rarely assume significance. This
also allows constraint-free application of the affine model to
processed imagery (eg rectified) as well as raw scanner imagery.
3. MODEL VALIDITY
In the previous section, the affine model was presented as a 2D
camera model. However, the geometry of a line scanner is
based on a central-perspective projection. More precisely, it is
comprised of a one-dimensional central-perspective projection
in the scanning direction and an approximately parallel
projection in the satellite track direction. Therefore, in the
application of the affine model to line scanner imagery, we
have to be aware that two main conditions need to be preserved.
Those relate to the parallelism of the imaging planes, and to an
accounting for projective discrepancies between central-
perspective and affine projection. The following subsections
briefly address these issues.
3.1 Satellite trajectory and object coordinate system
The satellite's trajectory is based on Keplar's motion and is
non-linear in a Cartesian frame. This indicates that the
direction of the pointing angle of the sensor with respect to the
directions of the Z-axis (or the height direction) of the object
coordinate system is time variant. In other words, the imaging
planes are not parallel to each other. Furthermore, in a
Cartesian system the distance to the curved ground surface is
not constant, which in turn implies that the scale factor involved
in Equation | cannot be constant for the entire scene. Therefore,
the affine model could experience accuracy degradation when
employed in a Cartesian frame. The use of additional
parameters or the subdividing of an image strip into several
sections, with the discrepancy level below a given tolerance.
can offer solutions to this problem (Hattori et al., 2003).
As it happens, the parallelism of image planes is better
preserved in a map projection system (map grid coordinates and
ellipsoidal height). This is because the conformal map
projection can be viewed as a cylindrical projection. A
conformal map grid system such as UTM is a flat plane which
is obtained by unfolding a cylinder wrapped around the
ellipsoid at the central meridian. Considering that an orbital
ellipse for the imaging satellite has a focus at the centre of mass
of the earth, and has a small eccentricity, the view direction of
the sensor with respect to the normal to the Earth’s ellipsoid
does not change drastically. Hence, the constructed image
planes retain near-parallelism in a map projection reference
system.
There are other important concerns relating to the orbital
trajectory. Among these are the effects of perturbed motion of
the sensor during image acquisition. For instance, if the roll
angle © changes continuously, a skew distortion will appear on
the image. This type of skew distortion is also caused by earth
rotation. Although the combined effects of all possible orbital
perturbations can be complex, the affine model has shown itself
capable of absorbing the perturbations of a fixed (non-agile)
sensor to a considerable extent. On the other hand, if the sensor
is steerable (agile), care has to be taken with the application of
the affine model, which basically has the form of a simple
linear transformation. This issue will be further discussed in
Section 3.3.
It also follows that utilization of a geographic coordinate
system with the affine model is not desirable because of its non-
linear nature. From the standpoint that the affine model can be
regarded as a special case of the third-order rational function
model, it may well require higher-order terms when employed
in a geographic coordinate system. For a nominal scene of
HRSI, the author’s experience suggests that accuracy
degradation will be anticipated in both a geographic reference
system (latitude, longitude, height) or in a local tangential
system (X,Y,Z) (Hanley et al., 2002).
3.2 Projective discrepancies
For affine theory to be rigorously applicable to the orientation
of line scanner imagery, it is mandatory that the projection
discrepancy between a central-perspective and an affine
projection be compensated (Okamoto et al, 1992). The
relationship between a central perspective and an affine
coordinate is illustrated for the scan line direction in Figure 1,
where the projective relationship between a ground point and a
line scanner image is shown at unit scale. The ground point P is
projected onto the image point p by a central perspective
projection and the image point p, by an affine projection.
The central perspective image coordinate y can be converted to
the affine image coordinate y, by the following (Okamoto et al.,
1992; Hattori et al., 2000; Fraser & Yamakawa, 2004):
1 (2)
y-tan w
f=
/
