International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
Figure 3. Motivations for using parallel projection
Therefore, many space scenes, such as IKONOS, can be
assumed to comply with parallel projection. The next section
presents the different forms of the parallel projection model.
3.2 Forms of the Parallel Projection Model
Figure 4. Parallel projection parameters
The parallel projection model, as shown in Figure 4, involves
the following parameters:
« Two components of the unit projection vector (L, M);
e Orientation angles of the scene plane (w, @, K);
e Two shift values (Ax, Ay); and
e Scale factor (s).
The non-linear form of the parallel projection model relating an
object point, P, to its scene point, p, can be expressed as:
X I X Ax
y|2sA4R M |+sR"| Y |+|Ay B.
0 N | 7 0
where:
(X, Y, Z) are the object coordinates of the points of interest, P;
(x, v) | are the scene coordinates of the corresponding point,
Ps
R is the rotation matrix between the object and scene
coordinate systems;
N is the Z-component of the unit projection vector - 1.e.,
N 2 J1- D - M? ;and
À is the distance between the object and image points,
which can be computed from the third equation in (1).
The linear form of the parallel projection model is derived by
eliminating À from Equations 1. It can be expressed as:
x= À À + 4, ) + dal + 4,
yadAhA! HAL + A,
(2)
where A, to Ag are the linear parallel projection parameters, and
will be denoted as 2-D Affine parameters (Ono et al., 1999). It
is important to mention that Equations 1 are useful if the
scanner navigation data are available. On the other hand, if GCP
are available, the linear model in Equations 2 becomes easier to
use. The transformations between the linear and non-linear
forms and their relation to the scanner navigation data are
included in (Morgan et al., 2004b). One has to note that a
parallel projection between two planes becomes 6-parameters
standard Affine transformation. This can be easily seen if we
consider planar object space (i.e., considering Z as a linear
function of X and Y in Equations 2). The next subsection deals
with a pre-requisite transformation prior to handling scenes
according to parallel projection.
3.3 Perspective To Parallel (PTP) Transformation
Original scenes captured by linear array scanners conform to the
rigorous perspective geometry along the scan lines. Therefore,
before dealing with the parallel projection model, Perspective
To Parallel (PTP) transformation of the scene coordinates are
required. Such a transformation alters the scene coordinates
along the scan lines to make them conform to the parallel
projection (Ono et al., 1999), Equation 3.
| 1 (3)
y = y rt a EE
y
1— lan (y )
C
where:
C is the scanner principal distance;
y is the scanner roll angle; and
y,» are the coordinates along the scan line according to
parallel and perspective projection, respectively.
One has to note that Equation 3 assumes a flat terrain. In
addition, it requires the knowledge of the scanner roll angle,
which might (or might not) be available from the navigation
data. Therefore, it is preferred to estimate the roll angle together
with the parallel projection parameters using GCP. Combining
the linear form of parallel projection and PTP transformation
results in:
International Arch
xe4X-
tar
+ penes
The parameters in
estimated using a
considered as a 1
model, by takin;
transformation. Th
parallel projection
array scanner scent
4. EPIPOLAR
SCANNER S
The epipolar line
model, can be expr
Gx+6
where (x, y) and (
points in the left an
G, to G4, express
can be derived ba
point corresponde
corresponding poin
From Equation 5,
straight in the pare
lines, in any of the
Therefore, the ep
transform the origi
parallax values. Or
different angle, to
scene rows. Aften
obtain correspondi
transformation can
[E NECNON C
tz
Figure 5.
Xl] | cos (6
X1 s|-sin(
Y, S cos(0'
y, — sin (6
where:
Qs, ys) are t
transfo
Ey.) are t
transfo
0,0 are the
respect
-Ay/2, Ay/2 are the
respect
lS, § are the
respect
