350 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING, 1976 
  
Check- point 
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3 
Incorrect estimation Correct estimation 
of accuracy inside 
the whole volume 
Fic. 3. Distribution of check points. 
(b) The distribution of the n points inside the volume is regular (Figure 3). In particular, 
any extrapolation about accuracy outside the check point volume is not valid. 
(c) The volume is not too deep. If the volume is too deep, then evaluation accuracy has to 
be estimated for successive slices. 
At last, we demonstrate the fundamental announced property of the RMS spatial 
residuals: An RMS spatial residual is proportional to the RMS error of the comparator 
measurements. 
It can be shown in the following way: for each check-object point M, 
  
  
  
  
Xiir 
(true coordinates X;; — || X4; || ; 
3iT 
Xiipu 
computed coordinates: X;»j = || Xoirr || ), 
Xsipn 
  
  
  
  
it is obvious that X;py = X;r plus a linear function of the measurement errors (measurement 
errors for the image points relating to M;, but also for the image points of any control point 
used in the computation). 
Then, if we name Ry; the RMS spatial residual at the point M,, we have (all the measure- 
ments are assumed to be independent) 
By = E (Xirn - Xa) (Xipn — Xir) 7 q?y o? 
and for the RMS spatial residual RXYZ worked out with all the n check points 
Roz-vil sS Rm, 
n 1 
RXYZ =q o (8) 
with q depending only on the object volume. 
ESTIMATION OF THE RMS SPATIAL RESIDUALS FROM CONTROL POINTS 
The previous criteria can be used without difficulty in laboratory experiments. However, 
it is more difficult in the field where it is often costly to provide enough control points for 
good determinations and enough check points for good estimation ofthe obtained accuracy. 
However, for lack of check points or check measurements one may use the RMS residuals 
computed with control points. We will name them R'XYZ, R'X, etc. Unfortunately, if the 
number n of control points is below 25 to 30, there is (statistically) a sensible overestimation 
of accuracy (See Appendix A), that is, 
R'XYZ« RXYZ. 
Nevertheless, it seems possible and reliable, at least in the case of the pair, to compute a 
corrective coefficient K so as to have 
RXYZ = K R'XYZ (statistically) (9) 
K depending on the number n of control points, the computational method, and the number 
r of unknowns estimated in the least-square adjustment. If from each control point we obtain 
p observation equations, we compute K from 
K=V EL 
pn-r (10)