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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008 
Hence, the density of C given the covariance matrix £ is defined 
by 
MC| S ) = OCr-«exp(-n. t r(S-C)) (4) 
| S |n )/2 Y\T(n-k+l) 
k= 1 
where | • | is the determinant and fr(-) is the trace of a matrix, 
r( ) is the standard gamma function and q is the dimensionality 
of s. 
2.2 Wishart-based distance measure 
An often used distance measure for polarimetric SAR data is 
based on the Wishart distribution and is defined in (Lee et al., 
1997): 
d w { C,£) = -ilnp(C|£) (5) 
= In (|£|) + ME" 1 -C)+c (6) 
where 
In 
Tj-gte- 1 )/ 2 ]TJ r(n — /c + 1) 
k=1 
(7) 
The constant term c in (6) is class independent and can be omit 
ted, if this distance is used as in (Lee et al., 1997) to decide if the 
data point C more probably belongs to class ci represented by 
the covariance matrix £i instead of belonging to class C2 repre 
sented by £2. The distance measure simplifies to: 
d w (C, £) = In ( I £ I ) + fr(£ _1 • C) 
(8) 
Although this distance measure is not a metric, because it is nei 
ther homogeneous, nor symmetric and does not fullfill the tri 
angle inequality, it is often used and has shown its effectiveness 
in practice. Because of this and its direct relation to the density, 
function it will be used in a slightly modified version in this wjork. 
3 WEIGHTED PYRAMID LINKING 
3.1 Pyramid construction and initialisation 
The basic structure used in this approach is a multiresolution im 
age pyramid. While the original data (fully-polarimetric SAR 
data, multi-look complex covariance matrices) is forming the base, 
the higher levels are versions of the image with subsequent re 
duced resolutions. The height of the pyramid (the number of lev 
els without the bottom level 1=0) shall be noted by L. Each pixel 
is represented by a node in this pyramid. The value v\(n l (x, y)) 
of the node n at position (x, y) at level l is simply an average of 
a window with certain size s at the previous level / — 1: 
VZ E [1, L\ : vAn ) = V2(n l ) = - • v\(n) 
s 
n'€des(n*) 
(9) 
The difference between the covariance matrices vi (n l ) and V2(n l ) 
will be explained in section 3.3. Just note, that they are set to 
the same value during the initialisation. Each node n at level l 
(0 < l < L) is therefore connected with a set of nodes at level 
l — 1, called descendents des{n) and a set of nodes in level l -f 1 
called parents par(n). Nodes at the bottom level Z = 0 have 
only parents, while nodes at the top level Z = L have only de 
scendents. Only vertical connections between nodes at adjacent 
levels and no horizontal relations between nodes of the same level 
are used. 
The windows overlap by a predefined amount o of pixels 
(0 < o < s). The size s of the window and the overlap o define 
the decrease in resolution of the next level. The parameters used 
in this paper are a quadratic window size s of 4x4 = 16 and an 
overlap o of two pixels in x- and y-direction. Given this setting 
of o and s there will not be enough pixels at the border of a level 
for a whole window, if the dimensions of this level are not even. 
In that case, the level is simply extended with as many pixels as 
needed. These additional pixels have the same value as the bor 
der pixels. Due to this manipulation the border pixels gain greater 
influence on the pixels at the next level. However, this effect is 
insignificant as experiments have shown. 
3.2 Weight adjustment 
The most important part of this approach is the introduction of 
link strengths w between nodes on adjacent levels of the pyra 
mid. Instead of using only the descendent with the largest degree 
of association, all descendents contribute accordingly to their link 
strength to the node value at the next level. The link strength 
w(n,n') between node n and its descendent n' E des(n) is 
based on proximity and similarity: 
w(n,n) = exp(-dspec(V2(,n),Vl(n))) 
■ exp (—var(n)) 
•exp (-dspat(ri' ,n) 
(10) 
The first factor has the most crucial role. It measures the spec 
tral distance between two nodes at adjacent levels in the image 
pyramid. Any proper distance measure can be used here. As 
mentioned above an often used distance measure for polarimet 
ric SAR data is (8), which is based on the Wishart distribution. 
Note, that the link strength is used to define the contribution of 
a descendent to the value of the current node in comparison to 
all other descendents of this node. Furthermore, the values of 
neighbouring nodes at one level are unlikely to be equal and the 
number of looks can be different, too. All nodes in the pyramid 
at the same level will have the same number of looks merely after 
the initialisation. That is why c in (6) cannot be omitted and (8) 
cannot be used here. Therefore d S p e c(v2(n'), vi(n)) is defined 
as: 
dspec (V2(n'),vi(n)) - ~\np(v2(n')\vi(n)) 
(ID 
where p(v2(n')\v\{ri)) is the density of the Wishart distribution 
defined in (4). 
Within the second factor the euclidian distance d S pat{n\n) be 
tween the spatial positions of the two nodes is used. The spatial 
distance within the 4 x 4 neighbourhood is defined as: 
/ y/9 V5 V5 V9 \ 
\/5 vT vT Vb 
s/h \/ï y/l \/5 
V V9 Vò Vò V9 ) 
(12) 
As the link strength now depends on geometric closeness the re 
gions tend to be more compact, whereas they would have more 
irregular shapes without this factor. 
The third factor represents the variability of the descendent of 
node n. Since the goal is to segment the image into homoge 
neous regions, nodes that represent segments with high variabil 
ity should get a lower link strength than nodes representing more