DIGITAL ANALYSIS OF ERROR ELLIPSOIDS IN ANALYTICAL PHOTOGRAMMETRIC STEREOTRIANGULATION 
Dr. IULIAN C. BARBALATA 
Professor of Photogrammetry 
Université de Moncton - CUSLM 
165, Boulevard Hébert, Edmundston 
New Brunswick E3V 2S8, CANADA 
Commission III 
ABSTRACT : 
By means of error propagation and error ellipsoids, researche was conducted to perform analyses on the 
accuracy in three dimensional space of photogrammetric stereotriangulated point positions. 
A mathematical model was developped and investigated for error ellipsoids, the main aim being to esta- 
tablish an operational procedure for the evaluation of quality of least squares adjustment. Theoretical 
error study based on fictious photography was also performed, without resorting to tedious sampling 
techniques (e.g. Monte Carlo method). 
The determination of spatial accuracy for selected discrete points requires several steps, involving the 
evaluation of the characteristic equation of covariance matrix, as well as the computation of their ei- 
genvalues and eigenvectors by an iterative procedure. The semiaxes of rotated ellipsoid were then deter- 
mined together with their three rotation parameters. 
A computer program named EROELIPS was developped and tested with real data to compute the eigenvalues and 
eigenvectors of each covariance matrix corresponding to selected points, as well as the parameters of 
standard error ellipsoid in three dimensional space. The program also provides the coordinates of each 
selected triangulated point in the shifted coordinate system and the parameters of constant probability 
ellipsoid for different confidence levels. 
KEY WORDS: Accuracy, Aerotriangulation, Algorithm, Analytical, Photogrammetry, Theory. 
1. INTRODUCTION 
The mathematical model developped by ‘the author where : 
(Barbalata,1979,1980-a) for multiple station ana- 
  
  
  
  
lytical stereotriangulation is based on projecti- Xi. 9,4 are image coordinates reduced to the ap- 
ve methods (DLT) and the end product of the pho- proximate principal point and corrected approxi- 
togrammetric process is a list of coordinates mately for radial and decentering distortion. 
which define the spatial position of a finite but 
large number of discrete points. Since the un- c. is calibrated principal distance, 
known coordinates of the control points were car- i 
ried in the present solution, the error propaga- A,B,...F are the elements of orthogonal orienta- 
tion associated with these also emerges as a by- tion matrix. 
product of the solution  (Barbalata,  1980-b; 
Brown D.C.,1955, 1958, 1968, 1976; El Hakim and AX. Xo—XO ,^Y.— Y. —-YO, A2, - 2.- 20 
Faig,1980; El Hakim, 1986; Wong and Elphingstone 13 3 4 13^. 3 i 13.57 i 
1972). in which x9,Y9 23 are space perspective center 
coordinates. 
In this context, the method of propagation of 
variance and covariance has been used to eva- If one consider the Direct Linear Transformation, 
luate the accuracy of the quantities determined then the solution is based on the following pair 
in a least squares solution (Anderson, 1973; of collinearity equations (Barbalata,1979,1980-a) 
Barbalata, 1978, 1988-a, 1988-b, 1990; Forester, 
1975; Gyer and Kenefick, 1970; Kilpela and all, a, X. t 85 Y; + aj 2j + a, 
1981; Kratky and El Hakim, 1983; Li Deren and x; +0X,; = J = 0, 
Shanjie, 1989; Mikhail, 1970; Salsig, 1990). J 5 ag Xqgt a, Y. +a, 2. +1 
J 1 (1-b) 
2. MATHEMATICAL MODEL OF ANALYTICAL TRIANGULATION as X * ac Y; * aj 2, + ag s 
y.tAy.. = 
If one consider the Bundle Adjustment Method,then tj 1J ag X. * &40Y: * 8442. *1 
the solution is based on the following pair of J J J 
collinearity equations which express the relation where: 
ship between the image coordinates (x,y) of point 
j on photo i and the corresponding ground coordi- Xijt^Xij ; y14*2y,.;are image coordinates reduced 
nates Xj, Yj, 2j of point j: to the approximate principal points and corrected 
approximately for radial and decentering distor- 
. A E + Bi DY; 5 + Ci AZ; tion. 
X. FC, 
13 i Dj OX; ; +E; mE + F; AZ; 3 üt x E 2 are the ground coordinates of point j, 
A, DX, oo B. OY.. t€. 62.. : (a1, a2 , ....a11J are the Projective Parameters 
$1: 7.53 rmbt denn of Transformation (PTP). 
B OX; CE, OY... t0. 42. 
D; ex: + E; Yij + Fi Zij 
639