Bethel, James 
  
collinearity equation can be derived easily as in frame photography. The sensor coordinate system (SCS) is a three- 
dimensional coordinate system for the image vector in the collinearity equations. The origin of the SCS is determined as 
the instantaneous perspective center. The x-direction is coincident to line direction ("along track", in the direction of the 
platform motion) and the y-direction is the same as the column direction ("cross track"). The z-direction can be 
determined by requiring a right-handed coordinate system. The z-coordinate of the image vector is fixed to -/7, where fl 
is the “calibrated” focal length. The calibrated focal length can be considered a constant value over the entire image and 
needs to be estimated because only a nominal value (79mm) is available. The relationship between the ground 
coordinates and corresponding image coordinates can then be expressed as equation (1) — (3). 
U 
F.=x+f1—=0 (1) 
: A= 
V 
F,=y+A 7 =0 2) 
U X-X, 
y |2M|Y-Y, G) 
W Z-Z, 
where X,y  :x,y coordinates of image point in image coordinate system 
X, Y, Z : X, Y, Z coordinates of object point in ground coordinate system 
A; YZ 1 ‘X,Y, Z coordinates of instantaneous perspective center in ground coordinate system 
M : 3x3 orthogonal rotation matrix from the ground coordinate System to image coordinate system 
Jl : calibrated focal length 
From equation (1) — (3), notice that the six EO (Exterior Orientation) parameters, consisting of three coordinates 
(X,, Y. 1) of the instantaneous perspective center position, and three independent rotational angles («, ©, K), have 
different values for each scan line. Therefore, at least three control points are required for each scan line. 
This problem can be addressed by using a priori information describing the platform behavior. Even though the 
HYDICE sensor may encounter severe air stream turbulence during its flight, the FSP preserves the sensor optical axis 
within one degree of nadir when the aircraft pitch and roll angles are within 5 degrees of level flight. Aircraft crew 
performs aircraft operations to keep the nominal trajectory with a certain range. The time interval between two adjacent 
lines is very short (8.3 - 50 msec). Therefore each of the six exterior parameters may change slowly as the line number 
increases. Also each EO parameter in a given scan line will be highly correlated to that in a neighboring scan line. 
Based on these assumptions, many models have been proposed and subjected to experiment. The spline model has been 
often used to model platform trajectories in time-dependent imaging applications. This approach involves the recovery 
of spline coefficients, with time as the independent variable, for each of the six elements of exterior orientation. Since 
the time interval between exposure of adjacent lines of imagery is constant, as determined by time tags in the data file, 
the six time-dependent elements of exterior orientation may be written as a function of line number. The spline model, 
although acceptable for modeling the general trend of an aircraft trajectory, is too rigid and too restrictive of control 
point distribution to accurately rectify airborne pushbroom imagery. The Gauss-Markov model, which is based on 
Gauss Markov process, is flexible in the sense that it can accommodate abrupt changes in the position and orientation of 
the sensor aboard the aircraft. 
2.2 Gauss Markov Process 
The Gauss-Markov (GM) process is a very useful process in engineering applications because many physical processes 
are well described by its stochastic properties, and because its mathematical description is relatively simple. 
The criterion for a first order Markov process is that the conditional probability distribution of a random process be 
dependent only on the one most recent point in the past. For the first order GM, we can make fictitious observations for 
6 EO per each line starting the second line, 
Fa =e "-(Ap),,—(Ap), =0 (4) 
where Ap is correction of each 6 EO element 
i is line number in the image(i = 2,3,...,1280) 
s is coefficient for each EO element 
  
184 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B7. Amsterdam 2000.