International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B2. Istanbul 2004
3. UTILIZING STRAIGHT-LINE FEATURES IN
PHOTOGRAMMETRIC ACTIVITIES
Incorporating linear features in photogrammetric applications
has to address two key issues. First, one should determine the
most convenient alternative for representing straight-line
features in the image and object space. Then, a mathematical
model has to be derived to incorporate the geometric attributes
of these primitives to solve the photogrammetric problem in
question.
3.1 Object Space Representation of Straight Line Features
The authors’ prior research showed that a straight line in the
object space is properly represented using two points (X,, Yı,
Z,) and (5, Y», Z;) along the line, which can be the beginning
and end points of that segment, Figure 1. Therefore, we will end
up with well-defined line segments in space, which can be
directly incorporated in GIS databases. In addition, such
representation will have no singularity (i.e., it is capable of
representing all line segments in space). Moreover, free-form
linear features can be represented with sufficient accuracy by a
set of connected finite straight-line segments. Finally, such
representation would lead to a straightforward perspective
transformation between the image and object space features.
Such transformation functions can be easily incorporated in
existing bundle adjustment procedures. Comparison between
this representation and other representations can be found in
(Habib and Morgan, 2003).
Figure 1. Object line representation and the mathematical model
- the coplanarity condition
3.2 Image Space Representation of Straight Line Features
After representing the object space line, one has to consider its
representation in the image space. As discussed in the previous
section, two points define the object line. These points can be
selected in one or two images within which this line appears,
but they need not be visible in other images, Figure 2. On the
other hand, corresponding linear features in overlapping images
will be defined by a sequence of intermediate points along these
features. The measured intermediate points need not to be
conjugate. Figure 2 illustrates two scenarios for the
measurement of the end and intermediate points. The end points
can be either identified in a single image (Figure 2-a) or in two
different images (Figure 2-b).
This representation can handle raw images captured by frame
cameras in the presence of distortions (i.e., it allows for possible
deviations from straightness in the image space features).
Therefore, such a representation can be incorporated in a bundle
adjustment with self-calibration procedure to solve for the IOP
612
of the implemented camera. Moreover, it is suitable for dealing
with linear features in scenes captured by linear array scanners
since perturbations in the flight trajectory would cause
deviations from straightness in the acquired images of these
features.
| | |
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Image À Image / Image & Image /
* Point defining the line in the object space # Point defining the line in the object space
X Intermediate Points
(a) : (b)
X Intermediate Points
Figure 2. End points can be measured in one image (a) or in two
different images (b) while the intermediate points
are measured in all images within which the line
appears
3.3 Perspective Transformation of Straight Line Features
In this section, we will discuss a method for image-to-object
space perspective transformation of straight-lines. The
relationship between the measured end points and the
corresponding ground coordinates, (X;, Yi, Z;) and (X5, Y», Z;),
is established through the collinearity equations. For every
intermediate point, a constraint has to be introduced to ensure
that the two ground points, the intermediate image point, and
the corresponding perspective centre are coplanar. This
constraint can be expressed as the triple product in Equation 1,
Figure 1.
(v, x v,)ev, 20 (1)
where:
vy, vo are the vectors connecting the perspective centre and the
two object points (the end points), and
V3 is the vector connecting the intermediate point and the
perspective centre, rotated into the ground coordinate
system.
The constraint in Equation 1 incorporates the ground
coordinates of the points defining the object line, the IOP of the
camera, the image coordinates of the intermediate point, and the
EOP of the image. Thus, it does not introduce new parameters.
Control lines can be also considered. In such a case, there is no
need to measure the end points in the imagery since the control
line already defines them. The depicted constraint in Equation 1
is suitable for the estimation of distortion parameters associated
with frame cameras. Moreover, for scenes captured by linear
array scanners, it can be used to estimate variations in the EOP
of the scanner along the flight trajectory. Such capability is
attributed to measuring numerous intermediate points along the
linear feature in the image space. Furthermore, the constraint in
Equation 1 can be easily incorporated in existing bundle
adjustment programs.
3.4 Straight-Lines in Other Photogrammetric and Medical
Applications
In this section, we will investigate the use of linear features in
other photogrammetric and medical applications; namely,
image-to-image registration (using 2-D line segments) and
surface-to-surface registration (i.e., absolute orientation using 3-
D line segments). For these applications, the line segments will
be represented by their end points, which need not be conjugate.
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