incomplete, given the presence of residual systematic errors and, therefore, 
biases in the distribution of adjustment residuals. However, this fact is 
of only limited consequence for the present discussion. 
PROJECTIVE INFLUENCES OF FILM UNFLATNESS 
A mechanism of projective compensation appears to be at work in the net- 
work adjustment such that the effective film topography, as modelled by the 
photo-invariant APs, is determined in such a way that maximum internal con- 
sistency of the relatively oriented bundles of rays is ensured. This feature 
is illustrated in Fig. 1 which shows a hypothetical imaging configuration of 
three convergent photographs. For each exposure, the point p; represents the 
image position of the object point P on the true focal plane, p' is the 
actual image position on the film surface and p'! is the image point on the 
mathematically modelled film surface. In an adjustment without film deforma- 
tion APs (the parameters a,, b, in Eqs. 6), the intersected object point will 
fall at a position P' within the shaded triangle shown in the figure. This 
occurs as a result of the departure of the photographic surface from the 
focal plane. The incorporation of photo-invariant APs in the self-calibration 
adjustment makes it possible for the film surface at each exposure to assume 
the positions indicated by the image points pi in Fig. 1, thus yielding the 
intersected object point position P". Note in the figure that, whereas the 
self-calibration adjustment produces a solution of higher internal consis- 
tency (smaller triangle of ray misclosure and therefore smaller photo coordi- 
nate residuals), the accuracy of the object point determination is poorer 
than that obtained in a solution with no APs, i.e. P" is further from P than 
is P'. This feature is entirely consistent with the least-squares principle, 
i.e. the quadratic form of the residuals should be a minimum. The quadratric 
form minimized in the solution of the photogrammetric normal equations is the 
following: 
$5 yw or y e yo + VOW (8) 
where V is the vector of image coordinate residuals; W is the weight matrix 
of the photo coordinate observations; y. V, V are the residual vectors and 
W,W,W are the weight matrices of the exterior orientation, object space and 
additional parameter prior means, respectively. For a minimal constraint 
adjustment, with loose a priori weights on the APs, viwy »VIWV 4 VIWV + 
VIWV. It is only in the presence of redundant object space control that the 
residuals V, which express the departure of points P' and P' from a control 
point P, form a significant component of à. 
From Fig. l it is apparent that if optimum external accuracy is to be 
attained, the photo-invariant APs must sufficiently model the actual film 
topography, rather than defining a mathematical surface which achieves mini- 
mum VIWV. Intuitively, there are three approaches that can be adopted in 
order to improve the adjustment accuracy, though not necessarily the precision: 
(i) the use of redundant object space control, which is often less than 
practicable in many close-range photogrammetric applications; 
(ii) restricting the order of the polynomial interpolation function; and 
(iii) increasing the degrees of freedom in the adjustment, by either using a 
high density target array, or by increasing the number of exposure 
stations. The former is a more effective and often more practicable 
approach. 
As regards increasing the degrees of freedom, provision of more image points 
aids in "controlling" the interpolation surface. Conversely, additional ex- 
posures tend to constrain the film topography modelling by facilitating multi- 
ray intersections for the object points. The order of the polynomial adopted 
is dependent, to a large degree,on the object space control configuration. 
160