The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B5. Beijing 2008
936
mation process will lead to invalidated statistical joint or single
hypotheses tests. The degree of complexity is further enhanced
when both Oi and fio are unknown (this is the common situa
tion in statistics) and have to be estimated from the data. The
estimated GLSE (or two stage Aitken estimator) is given by:
x = (A'ftr'A^A (13)
This estimator is neither linear due the fact that fix and v are cor
related, nor does it have known finite sample properties in gen
eral. At least it can be said that x is a consistent estimator of
x whenever the matrix series (f2i)£° converges in probability to
fio due Slutsky’s theorem (Greene, 2007). For sure it is impos
sible to estimate fio directly because of the 0.5N(N + 1) free
parameters within that VC-matrix (N: Number of observations).
The authors adopt two well known robust covariance estimation
procedures from econometrics to circumvent the direct estima
tion problem. The idea is quite simple: Seeking a consistent es
timator for A, instead of estimating a restricted
version of the true VC-matrix of the population. This matrix con
tains only 0.5n(n + 1) free parameters whereas n is referred the
number of components in x (n: Number of unknown parame
ters). Assuming that fio is a positive definite diagonal matrix
(without an autocorrelation pattern) and fix = I is the best ini
tial approximation over the underlying HS pattern, the approach
will coincide with White’s heteroscedasticity consistent estimator
(White, 1980). Furthermore, the only restrictions on fio are given
by symmetry and positive definiteness, which lead to the het
eroscedasticity and autocorrelation consistent estimator of Newey
and West (1986). The estimation process is carried out in the fol
lowing way:
1. Using Equation 13 whereas fix represents the stochastic model
and the best approximation of fio (fii: VC-matrix estimated
by VCE).
2. Estimating the true VC-matrix of x by
V(x)hc - (A'fi^Ar'iVA'nr'A)- 1 (14)
with E 0 = A'n^ 1 'i 0 nr 1 A
* 0 = Diag(v 2 )
and assuming a HC pattern as well as abstaining from any
autocorrelation within the disturbances only.
3. Estimating the true VC-matrix of x by
V(X)hac = (A / nr 1 A)~ 1 Ei(A / iir 1 A)- 1 (15)
with E i = [e 0 + f ij
f > =£?-.(' -iti)
■ E,=j+i( a iV.Vi-ja'_ j + ai-jVi-jVia'i)
p — floor ^4(iV/100) 2 / 9 j
and assuming a general dispersion pattern. The row vector a t
is the z th row of A := WA where W contains the recipro
cals of y/\7 whereas y/Xi is the z* eigenvalue of fix. This is
due to a spectral decomposition of fix (extension of the HAC
estimator on heterogeneous observations).
Since both VC-estimators are consistent for the true f2 of the pop
ulation (Newey and West, 1986; White, 1980) all single and joint
hypotheses tests are valid for the asymptotic case.
5.1 Experiment 1: Functional Model
To show the improved parameter accuracies and reliability of the
estimation, simulated and real data (Figure 3) with (i) high inten
sity contrast, (ii) high range contrast and (iii) balanced contrast
between both channels have been used. Furthermore, trials with
and without a range offset were performed. This configuration
should show the influence of different functional models on the
shift and scale parameters.
Figure 3: Some 3-D camera intensity (top) and range (down) im
ages. (a): Synthetic data, (b): Real data with measuring
marks, (c): Human hand, (d): Human head.
A synthetic data set has been created in order to determine the in
fluences of different functional models accurately. These data are
simply noisy (G gv — 1000 bit; o rv = 100 bit), but no system
atic errors occur (e.g. measurement uncertainties due to surface
conditions, background illumination or multipath effects). Real
data have been captured by the SwissRanger SR-3000 with con
stant integration time (it = 20.2 ms) and modulation frequency
(mf — 20.0 MHz). After exploring the potential of 2.5-D LST
with synthetic data, those experiments reflect the results, which
can be expected in practical use.
For single channel estimation it is obvious that insufficient con
trast in intensity or range observations will result in non
convergence of the solution vector. This is a general problem
of LSM: As the covariance matrix is generated from observations
with stochastic properties, the estimated standard deviations (SD)
of the transformation shift components will generally be too op
timistic, and correlations between parameters, indicating singu
larities caused by insufficient patch-gradients will often not be
detected (Maas, 2002).
Consequently the adjustment is stabilized, if one of the channels
provides sufficient contrast in both coordinate directions. Even
though it contains low information, the corresponding intensity
or range signal is not discarded but has some influence on the
adjustment in the form of a slight improvement of shift and scale
parameters against single channel estimation with high contrast
(approx. 5.0% for <7ao,bo and 20.0% for cr a x,62)- Therewith, the
entire available'information is used.
An improvement in scale adjustment can be achieved for signif
icant depth variations between template and search patch, inde
pendent of even or uneven range patches (approx. 50.0% for
CTal,62)-
An optimal 2.5-D LST solution can be achieved, if intensity and
range channel provide sufficient contrast. The SD for shift param
eters (j a0 ,6o are within a range of '/so to '/25 pixel for real data. A
range offset can be determined with a relative accuracy of 0.25 %
of the whole distance do.
5 RESULTS
The following section presents results of several experiments and
shows the effects of different functional and stochastic LST mod
els.
A measure of the quality of the introduced functional model pro
vide the SD of the adjusted observations d gv ^ rv . Those values
range around 100 bit for the intensity channel and 10 bit for the
range one. Specific details on the order of magnitude (16 bit-
range) will be given in Section 5.2.
