MATHEMATICAL FORMULATION & DIGITAL ANALYSIS 1363 
The innovation in the DLT approach is the concept of direct transformation from com- 
parator coordinates into object-space coordinates, thus by-passing the intermediate step of 
transforming image coordinates from a comparator system to a photo coordinate system. The 
concept is particularly useful for mapping with non-metric cameras which do not have fiducial 
marks in the focal plane to define the axes of the photo coordinate system. However, it is 
important to note that, since the orientation of the comparator axes with respect to the image 
plane vary from photo to photo in the measurement process, the coordinates of the principal 
point differ from photo to photo even though the actual position of the principal point may 
remain fixed in the image plane as the focus setting is changed. Consequently, the coordinates 
ofthe principal point must be calibrated for each photograph. Moreover, the rotation parame- 
ters (w, q, and k) represent rotations about the axes of the comparator coordinate system. 
For metric cameras which are equipped with fiducial marks to define the photo coordinate 
system and graduation marks to measure the rotation angles, there is no justifiable reason for 
using the DLT approach in data reduction. 
The concept of direct transformation can also be implemented by the use ofthe traditional 
collinearity equations. There is no modification needed in the basic collinearity equations, 
the only change being that the principal point coordinates (x,, y,) and the rotation parameters 
(w, @, and x) for each photo would be referred to the comparator axes rather than to the photo 
coordinate axes. 
FULLY ANALYTICAL SOLUTION USING COPLANARITY EQUATION 
The photogrammetric solution may be divided into two major phases: (1) relative orienta- 
tion of individual stereoscopic pair of photos, and (2) model linking and absolute orientation to 
the object-space coordinate system. 
The coplanarity equation is well suited for solving the problem of relative orientation. The 
equation simply states that the conjugate rays from the left and the right photos (A; and A; 
respectively) must intersect in space, i.e., 
(A; X An) :B=0 (10) 
where B is the vector joining the two exposure centers. The complete set of coplanarity 
equations for a stereoscopic pair of photos may be expressed in matrix notation as follows: 
AV +Bô =C (11) 
where V is a matrix ofthe residual errors in the image coordinates, Ó is a matrix of the unknown 
relative orientation parameters (4o, Ag, Ax, b,, and b,), A and B are coefficient matrices, and C 
is a matrix of the constant terms. After having solved for the ó-matrix, the spatial coordinates 
(with respect to an arbitrarily defined model coordinate system) of any image point in the 
stereoscopic coverage area may be computed by the method of intersection. The absolute 
orientation parameters (three rotations, three translations, and a scale factor) which relate the 
arbitrarily defined model coordinate system to the object-space coordinate system can be 
computed by the use of the fundamental projective transformation equation: 
Xj 31 1135 M13 Xi X, 
Yi | = À | M2, M29 M23 Y; — Y, (12) 
Aj 11131 11135 TI133 Zi = Za 
where xj, yj, and z; are model coordinates of point j; Xj, Y;, and Z; are the corresponding 
object-space coordinates of point j; A is the scale factor; X,, Y,, and Z, are the three translation 
parameters; and m,,, m;5, . . ., and mag are all functions of the three rotation parameters (o, g, 
and x). 
The above approach is particularly suited for the application cases which involve only a 
single pair of photos. The use of the coplanarity equation for relative orientation eliminates 
the need for computing approximate object-space coordinates of all model points before 
starting the solution. The computation process is so simple that it can even be programmed for 
use in a desk-top programmable calculator. 
For the application cases which require the use of multi-photo configurations, the photos 
must be sorted into stereoscopic pairs. After the relative orientation parameters of each pair 
have been determined, the individual models are then linked together and oriented to the 
object-space coordinate system in a simultaneous least-squares adjustment using equation 12. 
The second step is basically a triangulation process, and approximate object-space coordi-