Stephan Scholze 
  
different flight lines and slightly different operating conditions while scanning the pictures may cause significant changes 
in the RGB-values of pixels induced from the same roof surface. To overcome this problem it was proposed in (Henricsson, 
1996) to use the CIE(L*a*b*) color space for computing the chromatic attributes of the flanking regions. 
Since differences in chroma may only be derived if the stimuli have the same lightness, in a first step pixels with outlier 
L*-values have to be removed from the region. To identify and discard the L*-outlier pixels we apply a Minimum Volume 
Ellipsoid (MVE) estimator (Rousseeuw and Leroy, 1987). It is defined as the ellipsoid of minimal volume that covers at 
least half of the data points. For increasing sample size, the breakdown point of the MVE estimator converges to 50 % 
(Rousseeuw, 1985), yielding robust estimates. Next, we compute the MVE estimates for the mean vector and covariance 
matrix of the a*-b*-distribution, only considering those pixels which are belonging to the adaptive region and also are 
L*-inliers. The obtained values robustly represent the chromatic properties of the flanking regions on both sides of a 
given line segment. 
3.3 Forming initial match hypotheses 
For a given line segment 1 in the first view, the preliminary set of possible correspondences is given by all geometrically 
possible line segments {1} in the second view. To prune this set further we exploit the chromatic attributes derived in the 
previous step. We will only keep correspondence candidates, which have similar mean color vector and covariance matrix 
in at least one flanking region. We do not require chromatic similarity on both sides of the line segments in order to be 
robust against frequent cases of occlusion. 
Let us now define our tests for similar mean color vector and covariance matrix. Both are applications of the likelihood 
ratio test for multivariate distributions (Krzanowski, 1993). The color distribution in the flanking regions associated with 
a straight line segment has a two-variate population in the variables (a*, 5"). Under the assumption of normality, the 
estimates for mean 4 and covariance matrix X are the sample mean X and the sample covariance matrix S. 
e Testing the sample mean x. 
Consider the sample mean color vector X4; — (a*,5*) of a given flanking region in the first view and its considered 
counterpart X^ , in the second view. Here, a* and b* denote the robustly estimated mean values of the color distribu- 
tion in the flanking regions in the CIE(L*a*b*) color space. We want to prove the null hypothesis: Ho : X4p = Xap 
against the alternative H7, : X/; Z X. The likelihood ratio test statistics leads to: 
len ESSI e + Ty) 5 
with S denoting the sample covariance matrix of the a*-b*-distribution in the first view and n being the number of 
data-points (pixels). If Hp is true, then £; has a distribution with a number of degrees of freedom p — 2. See 
below for choosing the appropriate significance level a of the test. 
e Testing the covariance matrix S. 
Let us denote the sample covariance matrix of a given flanking region in the first view as S and its counterpart in 
the second view S’. Analogous to testing the mean, we want to prove the null hypothesis: Hy : S’ = 8 against the 
alternative H, : S' Z S. Again the likelihood ratio test statistics leads to: 
ts = n trace(U) — nln |U| — np (6) 
with U 2 S^! S, n being the number of pixels and p the number of degrees of freedom. With a number of degrees of 
freedom p = 2, specifying the (symmetric and positiv-semidefinite) covariance matrix requires 1m = i p(p-41)23 
separate, independent entries, thus ts has a X2, distribution if Hp is true. 
e Choosing the significance level a. 
Conventionally one would require a significance level a of five percent. Thus, if we would observe tx > X2 0r 
ts > X3, we could disprove the respective null hypothesis with an error-probability of at most o. Simply setting 
9 
a = 5% and using the corresponding values X? turns out to be too restrictive. Although the CIE(L*a*b” ) color space 
used in conjunction with our robust estimation scheme results in stable estimates for region attributes over different 
views, still due to different illumination conditions, different surface orientations relative to the camera position and 
due to varying scanning results, regions belonging to corresponding line segments in different views do not show 
exactly the same color distribution in a*-b*-color space. Instead of using a hard coded value for a we therefore 
provide an automatic initialization step: our algorithm randomly picks a small number (~ 10) of line segments in 
one view. For these line segments all geometrically possible match partners in the other view are determined. For the 
resulting combinations we compute the quantities £& and ts as given in equations (5) and (6) respectively. We now 
have to discard obviously wrong matches, stemming from chromatically incompatible but geometrically possible 
combinations. A match is discarded if its significance level in the corresponding x?-test would fall below 0.001. 
From the remaining values we determine the median values tx and tg as estimates for the test values on the left hand 
side of equations (5) and (6). 
  
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