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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
two linear array scanner scenes. For a 1D image in the left scene
with O as its perspective centre, point p can correspond to many
epipolar planes (unlike the case of frame images - compare
Figures 1 and 4). In this case, there are as many epipolar planes
as the perspective centres in the right scene. Therefore, the
epipolar line cannot be defined as the intersection of planes.
Instead, the second definition used in frame images is adopted,
where the epipolar line is defined as the locus of all possible
conjugate points of p on the other scene based on the orientation
parameters.
:Epipolar
line
Figure 4. Epipolar line in linear array scanner scenes
In order to determine the epipolar line, EOP of the scan lines
together with the IOP of the scanner must be available. The
epipolar line can be determined in a similar way as discussed in
Section 2.1.2 by repeating the procedure for each scan line since
each scan line has different EOP. The change in EOP from one
scan line to the next is one of the factors that determine the
shape of the epipolar line.
3. UTILIZING CONSTANT VELOCITY AND
CONSTANT ATTITUDE MODEL
The motivation of investigating the constant-velocity-constant-
attitude model is that many space scenes are acquired in very
short time (e.g., about one second for IKONOS scene). In
addition, similar assumptions were made for EOP when
deriving the SDLT model for linear array scanners (Wang,
1999), The scanner, therefore, can be assumed to travel with
constant velocity and constant attitude during the scene capture,
Equations 3.
Ë E. Va , "1
as = X 0 = AX J- X = Xt AX i
7 — 7 7. ; r .
Y eYQt^AY Y nl .tóolu
PA n. Jg 5 : (3)
Ze Z5ALI zemazudt
= 2 ' '
K,-K KODAK
Q,-—(0 w=
(215 5 1
(2, =0 | 9-9
where:
J is the scan line number on the left scene;
: is the scan line number on the right scene;
(Xo, Yo, Zp) is the position of the first exposure station in the
left scene;
1027
(AX, AY, AZ) is the constant velocity vector of the scanner
while capturing the left scene;
(X, Ys, Z'y) is the position of the first exposure station in the
right scene;
(AX, AY', AZ’) is the constant velocity vector of the scanner
while capturing the right scene;
(©, @, K) are the rotation angles of the left scanner; and
(9. ov. x) are the rotation angles of the right scanner.
For a point (/, y;) on the left scene, the collinearity equations can
be written as:
A = À À + À,
Vis A Z +A,
(4)
where A, to A, are functions of the IOP and EOP of the left
scene. Equation 4 represents two planes parallel to Y and X
axes, respectively. Their intersection is a straight-line (a light
ray in space). On the other hand, for the right scene, the
collinearity equations are:
al p 7 FA
Xx xa XX S
T (5
Yio [=A R iow] Y -Y's
T€ ZZ
ZT
Using EOP from Equations 3, and substituting for X and Y from
Equation 4, Equations 5 can be written as:
VE, + WE, = iE, + E, (6)
where F, to E4 are functions of the IOP and EOP of the scanner.
Detailed derivation of the above equation can be found in
(Morgan, 2004).
4. ANALYSIS OF EPIPOLAR LINES
Equation 6 represents the locus of potential conjugate points in
the right scene (i.e., the equation of the epipolar line in the right
scene). It is important to note that y; is unknown (at image
number 7). The epipolar curve is a straight line if E50, hence y ;
is a linear function of i. Therefore, the term FE, has to be
analyzed. Setting £5 to 0, the condition for having the epipolar
curve as a straight line can be expressed as triple product of
three vectors vy, v; and vs, Figure 4, as:
(v, 8 v,)ov, 20 (7)
Equation 7 indicates that these three vectors, shown in Figure 4,
must be coplanar in order to have the epipolar curve as a
straight line. In reality, it is hard to find the above condition
valid and equals to absolute zero. In order to quantify the
straightness of the epipolar curve, the value of Æ, has to be
compared to the value of Æ,. The smaller the ratio of (E;/Ej), the
epipolar curve will become closer to being straight.
E, (rn ®v)or, (8)
E, T (v, ® B)ov,
where B is the airbase vector, connecting the perspective point
of the point under consideration in the left scene and the
perspective centre of the first image in the right scene. Recall
that epipolar lines in frame images are straight lines in both the
