yortant issue in close 
n applying Modern 
| designed to operate 
analysis and adjust- 
co-ordinate system 
ıry co-ordinate sys- 
‘the point P, in the 
ction is extended by 
rors (e.g. distortion 
quire high accuracy. 
-ordinate measure- 
(geodetic) observa- 
  
rojection 
M 
Image Plane I Oo 5. 
   
Object Space 
Figure. 1b: Geometry of the central projection 
2. ADJUSTMENT TECHNIQUES WITHIN PROMPT 
Within photogrammetric adjustment and analysis of data 
there are different tasks the numerical procedures have to 
serve. Besides the necessary analysis of the data (amount 
of redundancy, rank deficiency of the design matrix within 
the adjustment of indirect observations, detection of 
multiple observations, number of independent normal 
equations within the observations and more) the determi- 
nation of the parameters from the redundant observations is 
most important. 
Within the mathematical idea of the Lp-Norm estimators 
the method of least squares is central for the determination 
of the parameters and their related variances. 
Let there be an (n,1) size vector v of residuals then the 
norm of this vector 
( Mn 
Nyv)- IM, = > bi) G) 
can be minimized due to its value p. For p-1 the least 
absolute value estimation (LAVE) is obtained, for p-2 the 
method of least squares (LS) and for p-infinite the 
so-called MINIMAX method (sometimes called 
Tschebyscheff-norm). From the statistical point of view the 
LAVE is a maximum likelihood estimate for random errors 
following a Laplacian distribution, corresponding the 
method of least squares for random errors following a 
normal distribution and the MINIMAX method for random 
errors following a rectangular distribution. 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
Within the idea of norm estimation techniques the LAVE 
technique may be regarded as a so called robust adjustment 
technique. These numerical procedures are introduced to 
adjustment and analysis tools to ensure blundered data 
(so-called outliers) to be detected by the size of their 
corresponding residual. LS is well known to obtain statisti- 
cally best results, this means the corresponding parameter's 
variances become minimal within all Lp-norm estimators. 
MINIMAX might be applied to reduce the absolute size of 
the maximum residual to its minimum (among all Lp- 
Norm estimators). Obviously different adjustment tech- 
niques might obtain different results for the parameters. 
From the statistical point of view these techniques obtain 
almost the same numerical results for a large amount of 
data with the random errors following a normal distribution 
[KAMPMANN, KRAUSE, 1995]. 
PROMPT applies different numerical strategies for the 
computations of the parameters and carries out some 
important numerical checks. Besides the well known 
normal equation formulations from least squares technique, 
additional techniques from Linear Programming are intro- 
duced. To reduce the amount of computation so called 
sparse techniques are applied for several purposes. Espe- 
cially when operating nonregular normal equations within 
the sparse technology, the so-called datum transformation 
(S-transformation) of the parameters had to be applied 
[GREPEL, 1987]. Numerical checks for the accuracy of 
the computation are carried out. 
PROMPT was designed to operate different adjustment 
models other then the adjustment of indirect observations. 
The introduction of additional equality constraints for the 
parameters x may be introduced for several Lp-Norm 
estimators applying the transformation of them into the 
adjustment model of indirect observations [KAMPMANN, 
1992]. 
3. ADJUSTMENT WITH RANK DEFICIENCY 
DESIGN MATRICES 
Consider the (n,u) size design matrix A with n denoting the 
number of observations and u denoting the number of 
parameters to determine and rank A = q = u-r. The integer 
value r denotes the rank deficiency of the design matrix. If 
r>0 the matrix of normal equations (ATA J! is a nonregular 
(u,u) size matrix with rank (ATA y! =u-r. 
To obtain a unique determination of the (u,1) size vector x 
of the parameters r equality constraints may be introduced 
to the least squares adjustment. These constraints are 
formulated within the (ru) size matrix E and may be 
numerically derived from the necessary condition A E- 0 
where 0 denotes a (n,r) size zero matrix. 
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