point. This procedure will therefore rut out the 
variance due to common parameters, mainly the 
uncnspensated systematic errors in the two processes, 
leaving only the component due to the error of the 
approximation in the variante. fbvigusly, this 
should be of a lower order of magnitude, In 
photograæmetry, es control point information is 
usually available and as its accuracy i5 generally 
higher, it can be used as a control variate, This 
sage idea can be further extended to other known 
control points also, thus giving rise to the concept 
of a sulti-control variable (MOV) technique, An 
attempt is made to present this extended application 
into the popular area of rlose-range photogramsetry 
in this paper, 
2. THE MULTI-CONTROL VARIATE (NCV) METHOD 
The idea of a single-conirol variable Monte Carlo 
technique for reduction of variation of observations, 
which thereby increases the precision of the 
estimate, can be readily extended to a sultiple 
control variates technique. The basic conputationsl 
concept to be used in this aultiple case is explained 
in a nutshell in /Kobayashi, 1981), Following this 
concept, we proceed to define a new random variable 
I 
748) = VIR) - bi d OG CEDE D, del2546, 
R: randos nuaber (ros the sireas used. —..,12.1) 
14 © denotes the covariance matrix of 
lor Inbal. 00% & MH © denotes the cross 
covariance vector between Y and 1; 
ist DR tit, 0 hitdfseueg el 
soi LE Dov 8 po 3 wh. i28 
thes the optimal value Bo for H - [b1,b2,,..,bh3 is 
ke = & -1 re 0,4 
which leads to 
Var{l} = Vari) -C G -1 CT 5 VarlY3U -R2YX) 
© 
„en. 
Where RY) ic the multiple correlation coefficient 
between Y and X, The square of the correlation 
coefficient is often called the coefficient of 
determination, sz it represents the fraction of the 
total variation of Y explained by variation nf À, 
Here, as ELIJ-EIY), computed value of I is used for 
y, The idea behind the aultiple-control variate 
variance reduction is similar o regression analysis 
(special case nf analysis of covariance}, — However, 
in the regression analysis we usually wish to 
investigate the power of à set of predictive 
variables Y in explaining the variation of a responce 
variable Y, whereas in variante reduction by the 
sulli-tontrol —variste —sethod, we evaluate the 
additional reduction in the variante against the 
additional computation involved. We should bear in 
mind that it is possible to achieve any desired 
reduction of variante hy using the mean of a 
sufficiently long sisulation run, i.s,, we could use 
the arithaelic aean in place of each observation, 
The HCY method has been successfully applied in 
studying the queueing systes in industrial oper ations 
research. — Referring to Graver, Kobayashi, 1980) 
reports that gulti-control  variste method (three 
control variates only] cuts the variance to about 87 
{that is by a factor of 12,5! e£. the initial value, 
It ic interesting to note that the Expected value of 
7 and Y would still be ihe same when the negative 
value before the suamation in eq.2,1 is changed io a 
positive sign. This fact has been used to di by 
e9,2.1 a& follows; 
Zi = VIR) - — bid IR)-EINI3 for hi0 11.12.62] 
TIR) = VIR} ¢  hi{¥i GO-EDYME for bio ..,12.6b) 
i. A SIMULATION EXPERIMENT RASED ON THE NORMAL CASE 
In order to evaluate the polentislities of the MOY 
technique, il is necessary ta set up a framework for 
the simulation study, This aspect is covered in this 
section. In preparing a data set, the true object 
space Coordinates were assused and the torresponding 
photo coordinates in the left and right photographs 
were calculated. The calculated photo coordinates 
conform to the ‘normal case ‘ in terrestrial and 
close-range photogrammetry, Accordingly, the pf fect 
of the tilis and rotations is not included in the 
study and  hente it results in à certain 
approximation, Obviously, the advantage gained 35 
the simplicity of the model. Using three different 
representative object to caaera distantes, three dats 
sete were set up, 
400 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B5. Vienna 1996 
  
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