Fraser, Clive
As it happens, it is unlikely that a standard EO of the 2-image (L and R) network could be achieved via a central perspective
model such as a modified collinearity approach (Sect.3) without the imposition of constraints on the EO parameters, notably
positional constraints based on precise ephemeris data. This is due to instability that results from over-parameterization. It
is well known that for the standard six EO parameters, the pitch angle is very highly correlated to position along the flight
line, and the roll angle displays strong projective coupling to cross-strip position. For the 2-image configuration represented
by L and R in Figure 1, correlation coefficients exceeding 0.99 can be anticipated with a bundle adjustment that does not
employ EO constraints. To some degree the addition of the central image removes this instability and allows for a bundle
adjustment to be carried out in the absence of EO constraints. Nevertheless, attention must still be paid to solution stability.
3 FUNDAMENTAL MATHEMATICAL MODEL
3.1 Modified Collinearity Equations
Prior to the discussion of alternative sensor orientation models for satellite imagery, a brief review is provided of what is
acknowledged to be the most rigorous restitution model for satellite image orientation and triangulation. The well-known
collinearity equations, which provide the fundamental mathematical model for restitution of photogrammetric frame
imagery, are equally applicable to satellite line scanner imagery, though in modified form. The modified model takes into
account the fact that the line scanner represents a perspective projection in the cross-track direction (y) only, and a parallel
projection in the x, or flight-line direction. This yields the following equations related to a particular scan line at time t:
0 - Xo =-cX!/z!
y-yo 7-cY'/Z (1)
and
XY ZY = RICE), YY ZZ WM.
where y, is the image coordinate within the scan line (the x coordinate is zero); xo, yo are the coordinates of the principal
point; c is the principal distance; X, Y, Z are the coordinates of the ground point; xe ye Z{ are the object space
coordinates of the sensor at time t; and R, is the sensor orientation matrix, again at time t.
In order to perform EO and subsequent ground point triangulation using Egs. 1, it is necessary to model the orientation
parameters (R,, XC ; YS 3 zc) as a function of time, otherwise the model is too over-parametrerised to support practical
implementation. The modelling of the sensor platform dynamics as a function of time or scan-line number is less
problematic for spaceborne sensors than for airborne linear array scanners due to the relatively smooth and quite well
described orbital trajectory of the satellite.
3.2 Bundle Adjustment Formulation
In the case where the orbital parameters of the satellite are known a priori, the positional elements of the EO can be
constrained to some degree. This incorporation of prior knowledge regarding satellite motion can range from the simple
assumption that the EO parameters vary either linearly or as a quadratic function over a short arc length, to the case where R,
and X? ; ys ; 2 are accurately known through the use of on-board GPS and star trackers which determine sensor attitude
angles. A common approach, lying somewhere between these two, is to enforce the platform motion to be in accordance
with a true Keplerian orbital trajectory. Thus, the ‘shape’ of the trajectory is assumed known a priori, but not the position.
With these considerations in mind, a combined mathematical model for satellite line scanner imagery can be written as
V = A, X; + AsX> + A3 X3 -L P (2)
Vo Cx * €5x; - £, ;P.
where x;, x; and x represent the EO, object point and additional parameters, respectively; A;, A» and A; are the related
design matrices; C, and C; are coefficient matrices of orbital constraint functions; v and v, are vectors of residuals; £ and &
454 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B7. Amsterdam 2000.
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