ACCURACY OF CLOSE-RANGE ANALYTICAL RESTITUTIONS 349 
  
  
Fic. 2. Estimating accuracy for an object volume. 
space (stations, orientation of the optical axis, object volume) in order to be able to compare 
results obtained from different working scales and from all available focal lengths. 
There is a solution in the symmetrical case of a pair (at least for identical left and right 
focal lengths). I do not know if it can be extended to other geometries. 
THE RMS SPATIAL RESIDUALS 
We consider n check points in the studied volume, that is, points whose true coordinates 
are known but not used in the photogrammetric computations. 
Then, if X;r, Y;r, and Zir are the true coordinates of the check point M; (i = 1, M), and 
Xipu, Yipn, and Zirn its photogrammetric coordinates, an estimation of the RMS spatial 
residual is 
  
<a 
1 7 
RXYZ = V [(Xipp — Xi? t (Yin — Yir? * (Zion — Zu] (5) 
(the true RMS residual RXYZ is in fact the limit of RXYZ for n infinite, with points in every 
part of the volume). 
It is interesting to determine the maximum spatial residual among the n check points: 
RMXYZ = Max V(Xipy — Xu?  (Yipn — Yir)? * (Zn — Za. (6) 
Of course, if necessary, analogous quantities can be estimated for the three axes. For 
example, in the X-direction. 
RX -A/ 1X (Xuu - Xa)? and RMX = max |Xien — Xor|- (7) 
In addition to its simplicity, such a criterion can be correctly estimated provided that: 
(a) The number n of check points is sufficient. It is presumably rather complex, and 
perhaps impossible, to give for the quantity RXYZ the exact confidence limits on a given per 
cent level. 
Then we proceed in the simplistic following way: We assume the variables (X;pu — Xir), 
(Yipy — Yi), and (Zipy — Zir) are normal and independent, and have the same standard devia- 
tion o. Under these conditions, and if we call s an estimation of e, we have 
j 
$2 — dn 2 [Xin — Xp? t (Yipy — Yir? * (Zipy — Zir)?] = T R? XYZ. 
  
Hence the confidence limits are the same for c and RXYZ. They are given by well known 
statistical tables, e.g., on the five per cent level we find 
n Confidence limits of RXYZ 
3 0.67 RXYZ 1.92 RXYZ 
6 0.75 RXYZ 1.50 RXYZ 
8 0.78 RXYZ 1.40 RXYZ 
10 0.80 RXYZ 1.34 RXYZ 
15 0.83 RXYZ 124 RXYZ 
25 0.86 RXYZ 1.90 RXYZ 
It is seen that, even for a small number of check points, the estimation of the RMS residual 
is not so bad, i.e., it is satisfactory when n >15.