International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B2. Istanbul 2004 
  
3.1 Nonparametric algorithm of density estimation 
The widespread approaches to  nonparametric density 
estimation are approaches using RP and k-NN methods, defined 
by (2) and (3) respectively: 
= Hn, 
P s 
=] € 
x X 
1 
Y 
IE 
= | Er i 
p(X |œ;)= n; Ta 
v=] 3 
s=l v= 
where n. — the number of elements in the sample V, of the 
class (9,; P - the number of image feature bands; A = 
smoothing parameters of the class (9, ; d(u) — kernel function 
of density distribution function. 
—]1 
24-1] M (3) 
] £X 
m X Q. — n 
p(X |œ;) N.X) 
N V(k 
n? 
where K, — distance parameter, N — the sample size. At that in 
case of Euclidean distance: 
5 
ZR" 
A °T{(n+2)/2] 
V(k,,N,X)= (4) 
where V(k,,N,x) — in common case is the amount of all 
points for that the distance to the point x less or equal R, , A — 
K 
unit matrix, /' — gamma-function. 
Direct using of algorithm on the basis of (2) u (3) leads to 
significantly low computational performance of density 
estimation. 
The proposed original density estimation algorithm is based 
upon modifications of the mentioned RP and k-NN algorithms. 
Let's consider the issues of these modifications. 
To increase the computational performance of conditional 
density estimation algorithms the advance kernel function 
calculation algorithm is applied. The idea of the algorithm is 
based upon the exclusion of the periodic low computational 
operations of kernel functions d(z) during density estimation 
in each component v-th of multidimensional feature vector 
X={x,v=1P} at (2) by buffering (caching) once 
calculated function values. 
To increase the computational performance of conditional 
density estimation of k-NN algorithm it is proposed and 
developed the modification of the algorithm. The main issue of 
the modification is in the following. According to (3) the 
parameter defined computational performance of density 
estimation is V'(k , N, x), which represents the distance in a 
current metrics. Therefore the acceleration of calculation of this 
parameter will lead to increase the computational efficiency of 
density estimation as a whole. 
To perform a faster search of nearest neighbors the application 
of the algorithm of a spatial indexing is proposed. The spatial 
indexing allows to find easier and faster a necessary point in 
multidimensional space according to (3) u (4) and to calculate 
the density probability value. 
The spatial indexing of data is a process of reflection a 
multidimensional space to the one-dimensional space by the 
indexes structure where each index corresponds to the point of 
multidimensional space. There are some approaches to the 
spatial indexing with different curves, such as Zet, Hilbert etc. 
The developed algorithm of density estimation is based upon 
Zet-curves, because the investigation results of the proposed 
density estimation algorithm with two different types of the 
index curves (Zet and Hilbert) demonstrated that the density 
estimation by algorithm with Zet-curves gives more robust 
results and index structure creation in this case is performed 
faster. 
Some conducted research shows that more effective is a 
combination of mentioned algorithms of density estimation 
defined by (2) n (3). At that what specific algorithm should be 
applied in each specific case is defined upon data dimension 
P.Incase P €3 the modification of RP algorithm is applied, 
in case P x4 the modification of k-NN algorithm is applied. 
The obtained algorithm, combined possibilities of couple 
nonparametric density estimation algorithms, allows the 
computational performance increase in dozens times compared 
to traditional nonparametric algorithms. 
3.2 Feature space forming 
Statistical: Recently in the tasks of interpretation the additional 
(texture) information in some way usually is used (Haralick 
R.M. & Joo H. A, 1986). More widespread method for 
considering the texture information is statistical approach to 
forming Haralick texture characteristics. The texture features 
for each pixel are computed over a moving box of a defined 
size. In this study, first moment textures have been used, which 
are defined by first-order histograms representing the rate of 
occurrence of each grey level within the moving box. Further 
descriptions of the textures used can be found in (Haralick R.M. 
& Joo H. A, 1986). The following texture characteristics have 
been computed: variance, entropy, energy, skewness, kurtosis, 
coefficient of variation. However to apply this hopeful 
approach to RS images interpretation, difficulties of 
informative feature selection should be overcome. 
The algorithm of forming feature space by the authors used 
might be represented in some sequential steps. In the first step 
the texture characteristics for all bands of RS image with 
neighborhood size 3x3, 7x7, 10x10 are calculated. In the 
second step the selection of 5 more informative features are 
carried out. The selection is performed according to algorithm 
of informative feature selection based upon criterion of 
pairwise separability Jeffries-Matusita (JM-distance). In 
common case JM-distance is: 
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