Edward M. Mikhail 
  
thereby improving rectification accuracy. The criterion for a first order Markov process is that the probability 
distribution function F[x(/)] of the random process x(f) is dependent only on the one most recent point in time. 
For each line of imagery in which a point is observed, two collinearity condition equations are written as in the case for 
the piecewise polynomial model. Assuming that the interior orientation of the sensor is known, there is a total of 6L 
parameters carried in the least squares adjustment, where L is the total number of lines in the image. Therefore, for 
each line in the image starting with the second line, six equations are written which reflect the constraints resulting form 
the Gauss-Markov process. 
Although these six equations per line are treated as observation equations in the least squares adjustment algorithm, 
they are essentially constraint equations effectively reducing the number of unknown parameters from 6L to 6. 
Therefore, a unique solution may be obtained if three control points are available. 
As the number of observed points corresponding to control points or linear features increases, the redundant 
"measurements can contribute significantly to the recovery of exterior orientation elements in the vicinity of the 
observation. This effect occurs if the weights assigned to the constraint equations are low enough to allow the 
parameters to vary significantly from one line to the next. When the platform model provides this flexibility in 
parameter recovery, there are greater than six independent parameters being estimated; therefore the redundancy is less 
than 2P - 6, where P is the number of control points. 
Similarly, the second order Gauss-Markov process, (x(f), can be defined as a Gaussian random process whose 
conditional probability distribution is dependent only on the two previous points. For each scan line in the image staring 
with the third line, six equations are written. These 6L-12 equations reduce the number of unknown parameters from 6L 
to 12. 
2.2.2 HyMap Modeling (Whisk-broom) 
The HyMap (Hyperspectral Mapping) sensor uses a whiskbroom scanner, unlike the HYDICE sensor that uses a 
pushbroom scanner. It sweeps from side to side as the platform moves forward. Therefore each image pixel, which is 
collected at a different time, requires its own set of six exterior orientation elements. To simplify this situation, it is 
assumed that the 
time to complete one scan line is small enough to consider one exposure station for each scan line. Then, each scan line 
of a whiskbroom image can be modeled as a panoramic image, instead of a framelet as used in the pushbroom model. 
Modeling from line to line remains the same for both imaging modes. 
2.2.3 Control Features 
The most common control feature used in the triangulation of multispectral imagery as well as traditional frame 
photography is the control point. Control point coordinates are commonly obtained from a field survey or a GPS 
survey. In our data set, however, the control point coordinates were easily and reliably obtained from the triangulation 
of pass points in frame photography that included the entire HYDICE coverage. The image coordinates (X,y) are line 
and sample values measured on the HYDICE imagery. 
Linear features offer some advantages over control points in their use as control features. Linear features are abundant 
in urban imagery and are often easier to extract using automated tools. When used in overlapping imagery, point to 
point correspondence is not required. Furthermore, without knowing their absolute location on the ground, linear 
features and their inherent constraints can contribute significantly to a solution by reducing the ground control 
requirements. 
Although the term linear feature encompasses any continuous feature with a negligible width and includes such 
parameterizations as splines, we limit consideration to straight line segments. Although there are several different 
parameterizations of a straight line in 3D space, we choose to model object space lines using two end points. At a given 
scan line, an image vector, which is rotated to the ground coordinate system, should be on a plane that is defined by 
three points: two end points of the line in the ground space and the position of the instantaneous perspective center. 
2.2.4 Experimental Results 
Two HYDICE images are used. The first data set was collected over the Washington DC mall in August 1995 (Figure 
7). Its ground sample distance is about 3.2 meters. Its flight height was about 6320m. From Figure 7, the straight line 
features like roads and building edges along the flight direction display a modest degree of roll-induced "waviness". 
  
596 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 
  
  
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