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2. BACKGROUND
2.1 Epipolar Geometry of Frame Cameras
21.1 Definitions
It is important to list some terms with their definitions before
going into any detailed discussion (Cho et al., 1992). These
terms will be used throughout the analysis of the epipolar
geometry of frame images. Figure 1 shows two frame images,
relatively oriented similar to that at the time of exposure. O and
O' are the perspective centres of the left and right images at the
time of exposure, respectively.
o base line
fPipoy, lad epipolar plane er NY
n Y; : e
ES
Figure 1. Epipolar geometry in frame images
Epipolar plane: The epipolar plane for a given image point p in
one of the images is the plane that passes through the point p
and both perspective centres, O and O '.
Epipolar line: The epipolar line can be defined in two ways.
First, it can be defined as the intersection of the epipolar plane
with an image, which produces a straight line. Secondly, the
epipolar line can be represented by the locus of all possible
conjugate points of p on the other image (by changing the
height of the corresponding object point). The latter definition
will be used when dealing with linear array scanners.
It should be noted that no DEM is needed to determine the
epipolar line. Selecting several points along the ray (Op), i.e.,
choosing different height values of the object point, will yield
the same epipolar line (/,) in the other image, Figure 1.
Another important property of epipolar lines in frame images is
their existence in conjugate pairs. Consider Figure 1, where I,
is the epipolar line in the right image for point p in the left
image, and p, p^; are two different points in the right image
selected on /',. The epipolar lines of points p'; and p» will be
identical, denoted as 1,, and will pass through the point p. This
can be easily seen from the figure since all these points and
lines should be in the same plane (the epipolar plane).
21.2 Epipolar Line Determination in Frame Images
Method 1: Collinearity Equations Through the Object Space
The collinearity equations (Kraus, 1993), Equation 1, relate a
point in the object space and its corresponding point in the
Image space.
X k X, X =X,
1 (1)
5 [=| FG Y Yo
Zr La i "e
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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
Where:
Xy are the image coordinates of a point in the image;
Xo,Yo,C are the Interior Orientation Parameters, IOP, of the
frame camera;
Xo, Yo, Zy are the position of the exposure station;
R is the rotation matrix of the image;
Xi, Yi, Zp are the coordinates of the object space point;
À is the scale factor.
Considering Figure 1, two sets of collinearity equations can be
written for point p by setting the scale factor À; to two arbitrary
values. This results in two arbitrary object space points, P, and
P,, along the ray Op. The two object points are then re-
projected into the right image with known orientation
parameters. The resulting points, p’; and p,, form the epipolar
line.
Method 2: Coplanarity Condition Without Visiting the Object
Space
The coplanarity condition (Kraus, 1993), Equation 2 can be
directly used to determine the epipolar line equation.
(OO'&Op)e O' p'2 0 (2)
Where:
OO’ is the vector connecting the two perspective centres;
Op is the vector connecting the point of interest p in the left
image with its perspective centre, O;
Op' is the vector connecting the corresponding point in the
right image with its perspective centre; and
@,° symbolize vectors cross and dot products, respectively.
Since only the coordinates of the corresponding point in the
right image (i.e., x' and y) are unknown, Equation 2 becomes
the epipolar line equation. One has to note that in this method,
the object space point, or its possible location, has not been
dealt with. Next section deals with linear array scanners as
alternative to frame cameras.
2.2 Linear Array Scanners
2.2.1 Motivations for using Linear Array Scanners
Two-dimensional digital cameras capture the data using two-
dimensional CCD array. However, the limited number of pixels
in current digital imaging systems hinders their application
towards extended large scale mapping functions.
One-dimensional digital cameras (linear array scanners) can be
used to obtain large ground coverage and maintain a ground
resolution comparable with scanned analogue photographs.
However, they capture only one-dimensional image (narrow
strip) per snap shot. Ground coverage is achieved by moving the
scanner (airborne or space-borne) and capturing many. 1D
images. The scene of an area of interest is obtained by stitching
together the resulting 1D images. It is important to note that
every 1D image is associated with one-exposure station, and
therefore, each image has its own set of Exterior Orientation
Parameters (EOP).
A clear distinction is made between the two terms “scene” and
"image" throughout the analysis of linear array scanners.
An image is defined as the recorded sensory data associated
with one exposure station. In case of a frame image, it contains
only one exposure station, and consequently it is one complete
image. In case of linear array scanner, there are many 1D