The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B5. Beijing 2008 
937 
&aQ 
Obo 
^al 
&b2 
<?d0 
_0 
a gv 
CTgv 
0"rv 
CTgv 
ârv 
Comments 
[pixel] 
[mm] 
[16bit] 
[16bit] 
[16bit] 
0.037 
0.026 
0.008 
0.007 
4.160 
1.0 
1.0 
X 
X 
65.112 
37.669 
Equal weights 
0.036 
0.026 
0.010 
0.007 
5.742 
1.0 
l-E+7 
X 
X 
69.930 
51.810 
Overemphasis intensity measurement 
0.069 
0.054 
0.007 
0.007 
0.475 
l-E+7 
1.0 
X 
X 
103.232 
7.616 
Overemphasis range measurement 
0.041 
0.034 
0.006 
0.006 
0.641 
1.0 
1.0 
1616.916 
120.719 
69.108 
7.499 
VCE (two groups of observation) 
Table 1: RIM sequence analysis using different stochastic models. Experimental trials performed with 2.5-D LST using intensity and 
range channel. 
tr(V(Sc)vcE) tr{V(x) HC ) tr(V(x)HAc) tr(. ( '.')*** R Homogeneity Comments 
0.00349* 
0.00492** 
0.00550*** 
0.709 
0.634 
[0.10; 9.60] 
tzl 
tzl 
Static scene 
Measuring mark 
Images taken form different point of views 
0.00688* 
0.01696*** 
0.01234** 
0.557 
0.405 
[0.10; 9.60] 
tzl 
tzl 
tzl 
tzl 
tzl 
tzl 
Kinematic scene 
0.00260*** 
0.05099** 
0.00257** 
0.04448* 
0.00256* 
0.12381*** 
0.986 
0.872 
0.999 
0.359 
[0.14; 7.15] 
[0.14; 7.15] 
Natural gray value gradient 
Images taken form one point of view over time 
Table 2: Homogeneity of variance covariance matrices. 
The experiment shows that the use of complementary information 
improve accuracy and reliability for RIM matching tasks. 2.5-D 
LST is most impressive when dealing with significant range off 
sets between template and search patches. In particular the range 
channel supports gray value observations in scale adjustment and 
provides additional information in the case of low contrast within 
intensity patches. 
5.2 Experiment 2: Stochastic Model 
When processing heterogeneous data, an adaption of the stochas 
tic model is necessary. A splitting of a single heterogeneous ob 
servation group in several ones allows the consideration of mul 
tiple variance factors (Section 3.2). Thus, it is possible to tap 
the full information potential of the available observations. The 
results of some experiments on a RIM data set with high inten 
sity and range contrast as well as a range offset of about 30 cm 
between template and search patch accentuate the need for an 
adapted stochastic model (Table 1). The following experimental 
setup was used: 2.5-D LST with (i) equal weights for intensity 
and range observations (ag VtT . v = 1 bit), an overemphasis of (ii) 
the intensity (a® v = 1 • E + 7 bit) or (iii) the distance measure 
ment (cTg V = 1 • E + 7 bit) and (iv) a stochastic model, which was 
estimated by VCE with a® g — 1 bit and a^ g = 1 bit as initial 
a-priori SD for a VCE. 
In Table 1 (Row 1-3) a negative influence on parameter accura 
cies is obvious for a deficient weighting of the measurements. 
The specified a-posteriori SD of the adjusted observations have 
higher variances, compared to a well-balanced weighting (Row 
4). Those balanced weighting could be achieved by VCE with 
two groups of observations. The precision of the shift parame 
ters is within the order of */30 pixel. The scales can be determined 
with a precision cr a i,b2 of 0.006, and the corresponding range 
offset SD ado (derived from Equation 7 by the law of the prop 
agation of errors) becomes 0.6 mm (based on do = 294.1 mm, 
corn 0.2 %). 
Applying a VCE, precision information of the original observa 
tions becomes available: In this case, the a-priori SD of the inten 
sity measurement a gv is 1600 gray value, which corresponds to 
16 bit resp. 6 gray values referring to 8 bit. This magnitude is re 
alistic and comprehensible due to a poorer signal-to-noise ratio of 
CMOS sensors (Lange, 2000), background illumination and vari 
ations within charge-to-voltage relation. Furthermore, a gv aligns 
with previous results empirically determined by Hempel (2006). 
The range values have been measured with an SD of 121 counts 
resp. 1.4 cm (it = 20.2 ms; mf = 20.0 MHz), which corre 
sponds to Hempel’s results as well. Free of systematic errors, the 
averaged a-posteriori SD of the adjusted observations è gv ,rv can 
be specified with 69 counts (16 bit-range) resp. 0.3 gray values 
(8 bit-range) for the intensity channel and 8 counts resp. 1 mm for 
the range channel. Those values reflect that the raw data match 
well with the established model. 
The experiment shows that 2.5-D LST in combination with a 
VCE improves the parameter accuracies. Especially if no ade 
quate a-priori precision information is available, an optimal uti 
lization of the whole content of information becomes possible. 
Furthermore, the procedure delivers valuable information on the 
sensor and data quality. 
5.3 Experiment 3: Robust VC-Matrix Estimation 
The following experiments have been performed to show, whether 
an enhanced stochastic model in the form of a robust VC-matrix 
estimation (Section 4) is useful for the presented functional model 
(Section 3.2) and to quantify differences in the precision of the 
underlying VC-estimators (VCE, HC and HAC). Testing the de 
terminants of the estimated VC-matrices of the GMM parameter 
vector x would be of a great benefit since this approach will in 
corporate the covariances between the components of x as well as 
their variances. However, this test procedure will only work for 
orthogonal regressions or stochastic processes (Rounault, 2007), 
which both do not fit into the presented framework. In order to 
still derive a valid value for the homogeneity of the several VC- 
matrices, the traces tr(V(St)vcE), fr(V(x)nc) and 
tr(V(it)HAC) of the estimated matrices are tested against each 
other by usual F-test procedures: 
• Null hypothesis: 
H 0 : al = al (16) 
• Test Statistics: 
_ ¿r(V(x)«) tr(V(St)„) 
£r(V(x)**) ' ir(V(x)***) 
with tr(V(x)„) < tr(V(x)»„) < fr (V(x)»»«) 
• Acceptance region: 
R — ^-^n— l,n— 1, ^ ) F n — l,n—1,1— ?£ j 
with F: Quantile of F-distribution 
n: Number of unknown parameters 
a: Significance level 
(17) 
(18)