ACCURACY OF CLOSE-RANGE ANALYTICAL RESTITUTIONS 353 
photogrammetry by the means of homologous lines. Note that a natural linear or curved line is 
generally better defined than successive points belonging to it.); and 
€ the accuracy of comparator* measurements. 
REDUNDANCE OF MEASUREMENTS 
The redundance of measurements can take place in three ways: number of comparator 
measurements for each image point of each picture; number of frames at each station,** and 
number of stations; and number of targets for each object point (an object point is defined as 
the barycenter of n neighbouring targets). 
PREDICTION OF ACCURACY 
THE METHOD OF ACCURACY PREDICTORS 
Accuracy predictors are formulas or diagrams which give accuracy as a function of the 
principal parameters of a photogrammetric system. The aim of accuracy studies is precisely 
to provide simple and reliable accuracy predictors. Accuracy predictors can exist only for 
simple configurations of the data acquisition system (for example the symmetric case of the 
pair) which happily are the most frequent in practice. 
The method of mathematical models, though more complex, presents on the other hand 
more possibilities. 
THE METHOD OF MATHEMATICAL MODELS 
The method of mathematical models consists in building fictitious data, that is, the image 
coordinates of each object point, and then in simulating the image coordinate error for each 
image point. After computation, photogrammetric spatial coordinates are compared to the 
original “true” coordinates, and the accuracy is estimated and compared with the accuracy 
estimated from practical experiments. If the mathematical model is correct, it can be used to 
predict accuracy in any particular case. 
The essential part of the mathematical model is of course the simulation of the image 
coordinate error. This can be done in two ways. 
Analytic formulation of each kind of error. This is the case for different mathematical 
models considered in aerial photogrammetry. For example Meier,? considered the coordi- 
nate errors in the left and right hand photos as a function of film deformation ds, 
irregularities in the emulsion and the film support dr,, as well as the optical errors dr,, with 
the formulation 
ds1 = Fo + Fis 
dri = (Us + Urs) 
dra =k + ka (5) 
where s negative side, p principal distance, and r radial distance. 
To these systematic errors, it would be convenient to include the setting measurement 
error, but Meier? didn't (it should be noted that it was the purpose in Meier? to simulate 
analog plotting with the Planimat). 
Synthetic formulation. It is assumed as a first (and often, sufficient) approximation, that 
the distribution of the errors is normal .N(0, o), as are those for the image coordinates x and 
y which are otherwise considered independent. 
The second model is of course simpler than the first one. We will see that it seems very 
satisfactory for the case of a pair in analytic close-range photogrammetry; therefore, we need 
no longer consider the first model. 
Now, if the existence of a mathematical normal model can be proved for a given topology 
(that is for focal lengths, number of stations, and number of image measurements fixed), it is 
possible to use it for the following aims: 
(a) Determination of the optimal geometry for a given topology, that is, determination of 
the position of stations which provide the best accuracy. It should be noted that here the 
choice of the standard deviation o of the normal law is of no matter. As a matter of fact, 
* We suppose all the measurements (x and y) to have the same accuracy o. 
** We suppose that only one camera is used at each station.