(XIX-B3, 2012 
SENSING 
sion uncertainty 
neters. This paper 
onent *m' as well 
le parameters for 
riterion has been 
1ese criterions are 
based on outputs 
opy is sensitive to 
ly. Soft classified 
ing SCM, overall 
accuracies, while 
iges for different 
ich gives highest 
7 noise clustering 
ta collection and 
any classification 
solution to this 
ild be one where 
1d removed from 
introduced such 
noise class. The 
ctional type (K- 
ability to detect 
tly demonstrated 
e to both fuzzy 
regression based 
plays a pivotal 
use without the 
eparate the good 
ccess the quality 
istering problem 
fore it computes 
m, 1997). 
ut entropy is not 
/ set theory and 
'ompare several 
ew of robustness 
this paper is to 
of FCM, PCM 
espect to fuzzy 
' as uncertainty 
; of parameters 
without entropy 
SESSMENT 
iced by Bezdek 
rique each data 
    
  
point belongs to a cluster to some degree that is specified by a 
membership grade, and the sum of the memberships for each 
pixel must be unity. This can be achieved by minimizing the 
generalized least - square error objective function in Eq. (3), 
Ni c = 2 3 
JU.) -EX(u) |x, =x, (3) 
Subject to constraints Eq (4), 
Yu, 21 for alli 
N 
LH, >0 for all j @ 
Osi <1 for all i, j 
where X; is the vector denoting spectral response of a pixel i, x 
is the collection of vector of cluster centers x,, pj are class 
membership values of a pixel, c and N are number of clusters 
and pixels respectively, m is a weighting exponent (1<m<co), 
which controls the degree of fuzziness, | X x is the squared 
  
distance (di) between X; and x,, and is given in Eq (5), 
à -|x,-x[, -(X,-x,) 4(x, -x;) (5) 
where A is the weight matrix. Amongst a number of A-norms, 
three namely Euclidean, Diagonal and Mahalonobis norm, each 
induced by specific weight matrix, are widely used. The 
formulations of each norm are given as (Bezdek, 1981) in Eq 
6), 
= I | Euclidean Norm 
Az D Diagonal Norm (6) 
Az c Mahalonbis Norm 
Where I is the identity matrix, D; is the diagonal matrix having 
diagonal elements as the eigen values of the variance covariance 
matrix, C; given in Eq (7), 
c Xxx) y e 
The class membership matrix pj is obtained in Eq (8) and (9); 
1 
c ( q? Von (8) 
s) 
k=1 
j 
where di-Y? (9) 
j=l 
2.2 Possibilistic c-Means Approach (PCM) 
In PCM, for a good classification is it expected that actual 
feature classes will have high membership value, while 
unrepresentative features will have low membership values 
(Krishnapuram and Keller, 1993). The objective function, 
which satisfies this requirement, may be formulated in Eq (10); 
N c » 2 c N o. (10) 
4U)- 3 Yu) X, -v| +21, 2 (1-4) 
  
  
Subject to constraints; 
max ,u, > 0 foralli 
N 
SN 20 for all j 
isl 
Osu, si for all i, j 
uj is calculated from Eq. (8). 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B3, 2012 
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia 
In Eq. (10) where n; is the suitable positive number, first term 
demands that the distances from the feature vectors to the 
prototypes be as low as possible, whereas the second term 
forces the ju; to be as large as possible, thus avoiding the trivial 
solution. Generally, n; depends on the shape and average size of 
the cluster j and its value may be computed in Eq (11); 
S 2 
> upd 
n, K-—— (11) 
m 
2, 
i=l 
Where K is a constant and is generally kept as one. After this, 
class memberships, uj are obtained in Eq (12); 
zu io y (12) 
y d? Ao 
1+| — 
N; 
2.3 Noise clustering without Entropy 
A concept of "Noise Cluster' was introduced such that noisy 
data points may be assigned to the noise class. The approach is 
developed for objective functional type (K-means or fuzzy K- 
means) algorithms, and its ability to detect 'good' clusters 
amongst noisy data is demonstrated (Dave and Krishnapuram, 
1997). Noise clustering, as a robust clustering method, 
performs partitioning of data sets reducing errors caused by 
outliers. In many situations outliers contain important 
information and their correct identification are crucial. NC is a 
method, which can be adapted to any prototype-based 
clustering algorithm like k-means and fuzzy c-means (FCM) 
(Frank, et al. 2007). The main concept of the NC algorithm is 
the introduction of a single noise cluster that will hopefully 
contain all noise data points . Data points whose distances to all 
clusters exceed a certain threshold are considered as outlier. 
This distance is called the noise distance. The presence of the 
noise , cluster allows outliers to have arbitrarily small 
memberships in good clusters (Dave, et al. 2007). In other 
classifiers where noise data points are not separate and present 
in information class, may lead to some kind of information 
scepticism. The objective function, which satisfies this 
requirement, may be formulated in EQ (13), (14) and (15); 
U (u, DEP 9 (13) 
i 
  
c d; Hom) d Jm) (14) 
i > ; HR 
k=1 di ö 
Where I=k=¢ 
Where 1=j=c¢ 
and 
1 -l 
e S Jom) (15) 
Hon = = TS +1 
j=1 d; 
O> 0, any float value greater than zero 
Where 8 >m>1, (any constant float value more than 1) 
N- row * column (image size) 
i = stands for pixel position at i^ location distance between X; 
and Vj