.
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-
935
I
Yi/- èm
mmM
№P|pfll
¡ÉglliÉ
l, €
S|||j||8pg
1111 HÜ&
WÊÊ8M
flit
mBM