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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B5. Beijing 2008 
• An improvement of the adjustment results due to the utiliza 
tion of the complete information content of the observations. 
• An accuracy specification for each variance component and 
therewith for each group of measurement. 
The weights of the observations are given by the quotient of the 
variance of the unit weight <7q and the variances of the observa 
tions a\. At this, do is a constant (in general ero = 1) and cri are 
the variance components, which can be estimated. In the course 
of the VCE, the VC-matrix of the unknown parameters V(x) can 
be divided into additive components Vi(x) = 0f(A'f2 -1 A) -1 ; 
each component i represents a group of observations. See Koch 
(2004) for further information for the computation. 
Parametrization: Like in Baltsavias (1991), the radiometric gray 
value correction terms were determined prior to the actual LS ad 
justment. This approach yields a more robust solution for the 
remaining geometric parameters. 
The geometric transformation between both pairs of patches gv; 
and rvi with i = [1; 2] can be modeled by an affine transforma 
tion with two shifts ao,bo, two scales ai, 62 in row and column 
direction and two parameters a2,61 for rotation and shear. The 
gray values as well as the range values remain unaffected by the 
shift and rotation parameters and are resampled only, according 
to their new (non-integer) image positions. The remaining scale 
parameters cause geometric variations in both search patches, but 
effect a change in range values, too. Consequently, the range off 
set do depends on a 1,62 and can be integrated into the basic 2-D 
LSM approach, which allows a closed LST solution. Furthermore 
it was assumed that there is no depth scale variation in the range 
patches. Therefore, the depth scale parameter d\ is set to 1 in the 
following considerations. 
The relation between depth variations and - in a first instance - a 
consistent scale in row and column direction A := |(ai + 62) is 
given by 
rv i(*>y) (4) 
A = 
rv 2 (x',r/') 
Due to scale-invariant range value differences, Equation 4 is ex 
pressed for the two center pixels (x c ,y c ) and (x' c ,y ,c ). The 
range value adjustment of the center pixel becomes 
rvi(x c ,y c ) = A • rv 2 (x ,c ,y /c ) 
(5) 
Therewith, the range values of the remaining pixels in the neigh 
borhood (x n , y n ) and (x ,n ,y' n ) can be formulated: 
rvi(a: n ,y n ) = rvi(x c ,y c ) 
+ [rv 2 (x m ,y ,n ) - rv 2 (x /c ,y' c )] 
(6) 
Substituting rvi(x c ,y c ) in Equation 6 by Equation 5 yields a 
range value correction term according to scale variations for all 
pixels: 
d 0 = rvi (x,y) - rv 2 {x , y) = rv 2 (x /c ,y ,c ) • (A - 1) (7) 
Finally, the observation equation 2 can be expressed as 
ai + 62 
rvi(x,y) - v 2 (x,y) = rv 2 (x' c ,r/ c ) • 
-I-1 ■ rv 2 (x',y') 
- 1 
(8) 
Using this integrated model, all transformation parameters can 
be determined based on intensity and range observations. The 
GMM minimizes the sum of squares of the intensity and range 
value differences. The range offset do in dependency of ai, 62 is 
considered in the GMM of observation vector. 
4 MISSPECIFIED VC-MATRICES IN GENERALIZED 
MULTIPLE LINEAR REGRESSION MODELS AND 
THEIR CONSEQUENCES 
In statistics, a random variable is called heteroscedastic (HS), if 
at least two different observations do not have the same variance 
(Greene, 2007). Estimating the variances of each observation is 
impossible due to the lack of redundancy and is not desirable 
from a geodetic point of view. As a result, a HS pattern has to 
be introduced in the form of an assumed VC-matrix of a vector 
valued random variable (in general by giving equal weights to 
each vector component). This assumption may cause inaccurate 
parameter estimations and invalid statistical hypotheses tests. A 
(partial) correction for heteroscedasticity can be achieved by the 
application of a weighted LS estimation method. 
In this article, the weighting of the two different types of ob 
servations with a-priori unknown quality is firstly performed by 
VCE (Section 3). However, the variances of the two types of 
observations estimated by VCE may not correspond to the true 
variances of each individual observation. To fulfill the require 
ments of a robust VC-matrix estimation in statistical context, al 
ternative approaches for VC-matrix calculation are proposed and 
evaluated, which yield estimators that are consistent for the true 
VC-matrix. These include the heteroscedasticity consistent esti 
mator (HC; White, 1980) disclaiming any autocorrelation within 
the disturbances as well as the heteroscedasticity and autocorrela 
tion consistent estimator (HAC; Newey and West, 1986), assum 
ing a general dispersion pattern. 
The multiple linear regression model is a well documented tool in 
statistics (e.g. Seber, 2003). If the usual LS assumptions are true, 
the generalized least squares estimator (GLSE; Greene, 2007) is 
the best linear unbiased estimator of the GMM with known VC- 
matrix within the class of linear estimators: 
x= (A'n^A^A'n- 1 ! 
(9) 
The GLSE will still be unbiased assuming that (i) the known VC- 
matrix Ci 1 is the first approximation of the non-spherical behavior 
with respect to error term and (ii) the true dispersion pattern of 1 is 
given by fio, which should only belong to the set of all symmetric 
and positive definite matrices. This could be seen by using the 
expectation operator on x: 
£(x) = E 
= (A'ft^Aj^A'il^Ax 
(10) 
It is feasible to use the GLSE as long as the true parameter vector 
x has been estimated on the average, independent of any arbitrary 
approximation of the true VC-matrix of the population. Regard 
ing to the alternative GLSE expression 
x = x + (A'nr 1 A) -1 A'iir 1 v (11) 
the VC-matrix of x can be calculated as follows 
V(x) = E [(x-x)(x-x)'] 
= E [(A / nr 1 A) _1 A'nr 1 vv , nr 1 A(A / i2r 1 A) _1 ] 
= (A'fi^Aj^A'nr^on^AtA'ft^A) -1 
* (A'iir'A)- 1 
(12) 
As shown it Equation 12, the use of 171 as the true VC-matrix of 
the population results in a wrong computation of the estimated 
parameters. The consequences of this failure within the esti- 
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