368 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING, 1976 
where Xipy, Yipy, . . . , are the photogrammetrically computed spatial coordinates of n check 
points well acounting for the object volume. It should be recalled that a check point is a 
point whose true coordinates are not used in the computations. 
If control points are used instead of check points Equation A-1 gives a biased estimation 
R'XYZ of RXYZ with (in a statistical way) R'XYZ « RXYZ. The accuracy estimated from 
control points is, then, statistically overestimated. 
However, if only control points are available, it can be seen that in many computational 
procedures it is possible to compute a corrective coefficient K so that E(K x R'XYZ) = 
E(RXYZ). 
FIRST CASE: RESECTIONS IN SPACE OF THE TWO BUNDLES WITH THE RIGOROUS LEAST-SQUARES 
METHOD, AND INTERSECTION OF HOMOLOGOUS RAYS. 
Let us call m;, the observed image point corresponding to the control point M;. The com- 
pensated image point, M;«, is the intersection of SM; with the plane of the picture (S deter- 
mined by compensation) (Figure A-1). 
The residuals of the observation equations are the two components v;, and v;, of the vector 
22 
mi, Mic. As a consequence of the least-squares theory, we have 
E(vix) = E(vw) = 0 
1 n 
f= Y (0? + 04?) 
nr T (A-2) 
where s? is the unbiased estimation of the variance o? of the comparator measurements and 
r is the number of unknown parameters. 
The following assumption is considered verified in a practical sense though not 
theoretically correct, i.e., all the residuals v;, and v;, (i = 1, n) obey the same law, with a 
standard deviation o,, and are independent. Under these conditions, an unbiased estimate 
of 0,2 is 
] 5 2n — r 
2 = 25 Ye tv,2) 2n 82, 
  
(A-3) 
Then it is possible to define the corrective coefficient K. If RXYZ is the true RMS spatial 
residual, we know that RXYZ = q o where o is the RMS error of comparator measurements 
and q depends only on the object volume. It is, as a rule, the value obtained from a sufficient 
check-point population. 
If we consider an n control-point population, we will have 
R'3XYZ = qg'2 3,2. 
(For a given control point, M;, the photogrammetric determination, M;py, is obtained as an 
intersection of two rays, S1 m;,! and S2 m;,2, and subsequently the spatial residual M;M,py, is a 
function only of the left and right residuals v;, and v;,). 
q' is a stochastic variable depending only on the choice of the n control points and is in- 
dependent of s, such that E q?' — q?. 
Then, from Equation A-3, 
E(R'?XYZ) = Eq?' Es,? = q? A g? 
TL 
  
E(R'2XYZ) - 2^ — Rexyz 
2n 
  
  
S 
Fic. A-1. Geometry of the resection in space.