CMRT09: Object Extraction for 3D City Models, Road Databases and Traffic Monitoring - Concepts, Algorithms, and Evaluation 
166 
In practice, the identity of Eq. (3) does not hold strictly, neither 
for ideal single-pass interferometers. Reasons therefore are 
varying bi-directional scattering, varying volume scattering, 
thermal noise, etc. However, a guess about the similarity of the 
two complex-valued SAR images u ] and li-, can be computed 
for each pixel \i,k\ by the coherence estimate | jj = | f\i, /c] | 
calculated in a predefined local neighbourhood W : 
X m i[ 
w 
\k]u* 2 
[a] 
p 
W, [/, A ] 
I 2 1 
w 
u 2 [i,k] 
I 2 
Based on the coherence estimate one can derive the probability 
density distribution (pdf) of the interferometric phase 
pdf (^;L) of the expectation (j) depending on the number 
of looks L, i.e. the amount of averaging independent pixels 
(Lee et al., 1994): 
pdf (^; l) = A{</>\ L)+ L) 
with 
A(<!>-,L) = 
r(L + l/2) I 
(H; 
>i 2 ) t M' 
cos! 
{</>-</>) 
2 Æ r(L) I 
^1-\Ÿ\ cos 2 ! 
|L + l/2 
(9) 
B(frL) 
L-m 2 ) 1 
2 F,(l,1;]/2; |jj 2 cos 2 (fi - i*)) 
and F(x) — (x — 1)! being the Gamma function 
and 2 F]{d,b‘,C’, z) being the hypergeometric Gaussian 
function. Figure 4 shows the shape of Eq. 9 for a fixed 
coherence and varying averaging while, in Figure 5, averaging 
is fixed and coherence varies. 
Although the pdf of the interferometric phase is not strictly 
Gaussian, it can be seen from the functions displayed in Figures 
4 and 5 that the pdf s first- and second-order moment ((j) and 
<7^) carry the most information of this distribution. 
Furthermore, assuming that (/>{z) is locally linear, one can 
write after Taylor expansion of (f){z) and omitting higher order 
terms: 
d(f) _ A0 
dz A z 
(10) 
Computing O 0 numerically from Eq. 9 and inserting 
— <7^ and AZ — (7- into Eq. 10 yields finally the 
standard deviation of the height estimates: 
<7 = 
d(/>/dz 
CD 
Figure 6 visualizes the behaviour of the standard deviation of 
the interferometric phase for varying coherence and number of 
looks. An evident feature of this function is the large influence 
of averaging for moderate coherence values. Only four looks, 
for instance, improve the standard deviation approximately by 
50% at a coherence of 0.65. Figure 7 shows typical height 
distributions for varying coherence and a specific fixed set of 
sensor parameters. 
Figure 4: pdf of interferometric phase for fixed coherence and varying 
Figure 5: pdf of interferometric phase for fixed averaging and varying 
coherence. 
Figure 6: Standard deviation of interferometric phase for varying 
coherence and number of looks.