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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