Bethel, James 
  
parallel and vertical. Therefore the elevation of a constrained line couldn't be estimated reliably. To relieve this 
situation, we need to assume that linear features are horizontal in the ground space with known elevation. Then the 
constrained line model can be made based on the fact that three consecutive points should be on a straight line in the 
ground space. 
1 1 1 
Fey do dep (9) 
Y 5h Y, 
U, 
rU d 
V, 
Y zi Zh 
i ( i Jy. 
where [U, V W,| -M'lx, VN; - f| 
2.3.3 Semi-automated Line Extraction. As mentioned previously, the sampling rate is an important factor for the 
Gauss-Markov model. To increase the sampling rate, it is desirable to extract each point on every scan line for each 
linear feature. Digitizing points manually on the line is time consuming and error-prone work. Fortunately, this time 
consuming work can be replaced by a semi-automatic method, which extracts lines automatically given initial 
approximate delineation. The method used in this paper is based on time-delayed discrete dynamic programming for 
energy minimization of active contours [Amini, et. al., 1990]. In this method, we start with an initial line that is 
determined approximately. The position of the line is updated by the influence of image gradients near the edge, and by 
internal smoothness of the line. The update continues until no change is produced in the estimated line position. 
An energy minimization model has been often used to extract linear features. A typical objective function of such an 
energy minimizing model can be expressed by equation (10). 
Min. E = YE, (p,)+ EP ) (10) 
i=l 
where, Ex(p)=lelp,— pl" + Blpu 22; + Pal?) (11) 
E dise ( D; ) : Energy function related to image gradient 
p; : position (/,c) of i" point on the line 
C, D: coefficients 
This function consists of two energy functions, internal energy and edge energy. The internal energy serves as the 
forces that make the line to be smooth. The first term of internal energy is used to make consecutive points closer to one 
another and the second term makes the shape of line smooth. The edge energy represents the forces that make the line 
take the shape of salient features present in the image. Thus, the line is attracted to image points with high gradient 
values. 
Dynamic programming is a very useful method for the solution of many optimization problems where a sequence of 
inter-related decisions is required. For many optimization problems, it is often possible to decompose a problem into a 
sequence of smaller sub-problems. Then, these problems can be solved recursively one at a time. Therefore, 
computational effort can be reduced significantly. 
2.4 Experiments 
The purpose of this section is to compare the accuracy obtained from the restitution of HYDICE imagery using different 
platform models. In order to quantify the performance of each model, we examine the residuals of check points. Since 
we have only single image coverage, the Z-coordinate must be fixed to its known value. This does not imply a flat 
terrain assumption, only that the various Z coordinates of points and features are indeterminate using a single image. 
The estimated coordinates, X | and Y... , are computed as follows: 
est est ? 
  
186 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B7. Amsterdam 2000.