1358 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING, 1975 
translations, 3 rotations, and scale), only the scale factor critically affects the measurement 
accuracy. 
In topographic applications, aerial photography are usually flown in a regular rectangular 
pattern with about 60 per cent forward overlap and 20 per cent sidelap. The camera axis 
usually is maintained within 3? of the vertical. In close-range applications, however, conver- 
gent photography is commonly used to maximize the mapping accuracy. The shape of the 
subject being mapped usually requires that photographs be taken from different angles and 
directions in order to provide stereo-coverage of the entire surface. As a consequence of the 
extensive stereoscopic coverage and extensive overlaps between stereo models, high internal 
accuracy can be achieved from the relative orientation ofthe photographs. Only scale controls 
in the object space would be needed for accurate mapping of the object. 
FULLY ANALYTICAL SOLUTION USING COLLINEARITY EQUATIONS 
The use of collinearity equations for a simultaneous least-squares adjustment of photo- 
grammetric blocks in analytical aerotriangulation has been well documented in the litera- 
ture?.19,30, 'he applicability of the method in close-range mapping problems has also been 
demonstrated!9,20,23, 
By using two collinearity equations to describe each ray of light in the photogrammetric 
block, a complete mathematical model of all the optical rays in the block can be constructed. 
Such a model can be represented by the matrix equation 
V + Bè + Bô = € (1) 
where V is a matrix of the residual errors in the image coordinates, À represents unknown 
corrections to the exterior orientation parameters of the photos, and à represents unknown 
corrections to the object-space coordinates. Observation equations can also be generated to 
describe other types of measurements such as: (1) exterior orientation of the camera during 
exposure (0i, 9;,x;, X;, Yi, and Zi); (2) ground coordinates of certain object points; (3) distances 
between points in the object space; (4) distances between the camera stations and some object 
points; (5) angles (horizontal, vertical, or phase) subtended by object points at the camera 
stations; (6) angles measured in the object space; (7) height of object points; and (8) azimuth of 
lines in the object space. Thus, a general mathematical solution may include the following 
types of observation equations: 
Observation Equations Type of Measurements 
V+Bö+BS = € (photo coordinates) 
V-§ „= (exterior orientation parameters) 
V A à. =E (ground control coordinates) 
V,+G,0+G,d =C, (angles) 
V.+G.9+G2A =C, (azimuths) 
V.+Gà+G0 eC. (distances) (2) 
The entire mathematical model may then be simply expressed as follows: 
à V + B6 = Ç (3) 
in which à = [2] . The normal equation for a least-squares solution is then given by the fol- 
lowing expression: 
(B'WB) 6 = B'WC (4) 
in which Wis the weight matrix. In actual computation, the d-matrix is first solved for from a set 
of reduced normal equations, and then the d;-matrices inside the Ó-matrix are solved for one 
at a time. An iterative procedure must be used in the solution. The iteration is stopped when 
the corrections in the à and & matrices become negligibly small. 
This formulation of the photogrammetric problem provides complete flexibility in the 
weighting of the measurements and on the type of controls to be used in the scaling and 
orientation of the model. Figures 2 to 6 illustrate five different cases which easily can be 
solved using the above formulation. Table 1 lists for each of the five cases the type of 
measurements made and the manner in which the reference coordinate system in the object 
space is defined.