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9 MODELING APPROACH
2.1 Geometric Characteristics of High-resolution Satellite Imagery
1m high-resolution satellite imagery has a far narrower field view angle, one degrees, than mid resolution
(5m-10m on the ground) satellite imagery such as one of SPOT. This kind of projection of imagery is better
modeled by parallel rather than central one, since if conventional rotation angle based orientation parameters
are used, high couplings occur among them. In addition, a satellite moves smoothly along the Keplerian orbit
in space, and one scene covers a very small ground area. In such short term, the orientation angles are almost
regarded as constant and the flight path of satellite is approximately straight, if Gauss-Kriiger coordinate
system is adopted as a reference coordinate system (Konecny et al, 1987). These characteristics mean that the
collinearity equations between high-resolution satellite imagery and the ground points can be simplified as far
as the small area mapping purposes.
2.2 Geometry of Line Scanner Imagery
The projection of a CCD line scanner is one-dimensional central perspective, and each line image has different
exterior orientation, though it has high correlation to neighbors. Let exterior orientation parameters for line
number i be expressed by coordinates of the projection center, Xoi: Yoi: Zoi and angles ¢;,w;,k;. These
parameters are time variant. Many studies have indicated that attitude parameters can be expressed with
low-order polynomials in any Cartesian coordinate system. The collinearity equations are described as:
Oz an(X - Xa) t as(Y — Yoi) + 13 Z — Zei) (1)
Pi aX — Xoi) + a22(Y — Yoi) + a23(Z — Zoi) (2)
anse Mapa Y a pa
where (X, Y, Z) is the ground coordinates of an object point, c is principal distance, y is a coordinate of image
point in scanning direction, a;; (7=1,2,3: j=1,2,3) are elements of rotation matrix I5; R,; Rs.
Assuming the scene is projected to an image by parallel projection, c can be set to infinity. In this case the
second equations can be described as follows:
Ya = a21(X — Xoj) -F a22(Y — Yoi) - aza(Z — Zoi) (3)
Where y, is an affine projection image coordinate in scanning direction. Since affine projection image has actu-
ally small distortions, image coordinate y, must be transformed to a corresponding original image coordinate,
y. Now, let AZ indicate height difference between the average ground level and object height Z, the relation
between y, and y is given in the form
y(c+ AZ/(Acosw)) ;
i (4)
¢c—ytanw
Where À is a scale factor. If existing DTMs of observed area are available, this transformation can be previously
carried out before on-line mapping.
2.3 2D Affine Projection Model
In small area mapping using high-resolution satellite imagery, we can assume further that the sensor moves
linearly in space and the attitude is almost unchanged. The projection center in each line is described as
follows:
Ao = X, +A Ni (5)
With constant X, and AX. The similar expressions are defined likewise for Y,i and Z,i. Line number i is
expressed from Equation 1,
aii (X me Xo) + a12(Y = Yo) 3: a13(Z — Zo) (6)
zu )
a, A X + Q129AY + a3ÂZ
Here line number i is replaced by image coordinate x in flight direction. Assuming that the attitude does not
change, a,; are regarded as constant parameters. Equation 6 arranged for the constant coefficients leads to an
algebraic expression.
r= AX + AY + AZ + Ay (7)
Equation 3 is expressed by similarly
Ja Az X + AY + AZ + As (8)
where A; (j—1,* $8) are independent orientation parameters, which are not time-valiant. Equation 7 and 8 are
basic equations of 2D affine model.
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 673
