ACCURACY OF CLOSE-RANGE ANALYTICAL RESTITUTIONS 371 
In the case of a one-dimensioned quantity estimated from n independent measurements, 
the mean-square-error VEe? of the mean of n measurements can be written as 
Ee? - E f? + o%n, 
c being the standard deviation of replicated measurements and VE f? being the RMS bias 
of the measuring system, which includes the estimated quantity. If systematic errors are 
well corrected, it can be assumed* that 
VE 8? =0 
and subsequently 
Ee? =(1 +X) 02 = (1 + 2) (9. (B-1) 
In the case of an r-dimensioned quantity P, estimated from n > r measurements, we can 
generalize Equation B-1 as follows: 
E ep ep? =(1+ 2r, =(1+ A) (B-2) 
where e, = P — P(P = true value of P), T; — variance of P, and T, ^ n/r T», the mean variance 
matrix of all elementary determinations of P (r measurements instead of n and control nets 
with neighboring geometrical characteristics). I» is of course proportional to 02, o being the 
RMS error of comparator measurements. Thus we can rewrite Equation B-2 as 
E ep ep" = T', [0 9 £] o? (B-3) 
with I", independent ofc. This can be adopted as the accuracy ofthe resection. Thus, one can 
conclude that the simplest way to account for the bias due to the resections operations is to 
substitute for all accuracy questions the value V1 +# c for the value o. 
As a consequence, the RMS spatial residual R,XYZ (resections with n control points), which 
can be computed from check measurements after intersection of homologous rays and which 
we know to be proportional to the RMS error of the comparator measurements, is proportional 
to V1 + fo, ie, 
RXYZ =k Vit Lo 
RXYZ = V1+ LR, qn 
THE TESTS 
The raw results of the tests are recorded in Tables B-1, B-2, and B-3. Six different photo- 
pairs and three different base-to-object distance ratios (0.86 for 101 and 102, 0.33 for 106 and 
107, 0.14 for 11 and 12) were employed. Table B-1 includes results for the direct linear trans- 
formation method, the principle of which was given earlier. Table B-2 presents results for the 
method of relative orientation followed by a least-squares adjustment to a control net. Table 
B-3 concerns the exclusive problem of relative orientation (number and definition of image 
points). The accuracy criterion is the RMS spatial residual rXYZ (um) referred to the image 
plane in Tables B-1 and B-2, and the RMS residual parallax (um) for Table B-3, computed 
from more than 100 distinct check targets. For each value of n (number of control points for 
* This can be justified as follows: It can be thought that the heart of the measuring process is the 
realization of a coincidence between two “spots”, one of the measuring device and one of the 
measured quantity. In fact, a spot never appears to the observer as a point (mathematical concept) but as 
a brilliance gaussian distribution with a standard deviation c,; and two different spots are known as 
distinct only when the sum ofthe two curves present a central minimum, that is, if the distance between 
the two spots is greater than 2-30, (Figure B-1). In other words, the definition of the spot of the measured 
quantity is defined with an RMS bias of o.. In addition, for the same reasons, the observation error for 
the coincidence also has a standard deviation o,, which we can assume for a well elaborated, precise, 
and calibrated measuring device to be equal to o (standard deviation of the elementary replicated 
observations). Finally the RMS bias is equal to a. 
*—» dz toc *——9 dz $0c 
Ne central minimum 17 antral minimum neal central minimum 
Fic. B-1. Gaussian distribution of a measurement.