In: Wagner W., Szekely, B. (eds.): ISPRS TC VII Symposium - 100 Years ISPRS, Vienna, Austria, July 5-7, 2010, IAPRS, Vol. XXXVIII, Part 7B 
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Figure 1. Class transition diagram with four classes. 
The vector 'a = ['a/,...,'a n ] with 0 < 'a, < 1 represents, for a 
particular image object, the fuzzy classification defined over ft 
at time t, where 'a, denotes the membership value of the object 
to class co,- (for all co,- e il) at time t. It is further assumed that 'a, 
is a function of attribute values of the image object at time t. 
Based on the fuzzy label vector 'a and on the transition matrix 
T, the Fuzzy Markov Chain Model estimates the class 
membership values, represented by the vector 
,+/ p = [ i+7 p 7 ,.../ +/ p„ ] for the same object one time unit later by 
applying the following formula: 
(l) 
for i,j = 1The symbols T and ± represent respectively a t- 
norm and a s-norm. 
The transition law introduced in Equation (1) can be expressed 
in a more compact form by the equation below. 
1+1 P = 'a°T (2) 
In the expression “A°B”, the symbol denotes a special type 
of matrix multiplication, analogous to conventional matrix 
multiplication, where the product is replaced by a t-norm and 
the summation is replaced by a s-norm operator. 
The symbols r+y p and l+l a denote different distributions 
although both refer to the same object at the date t+1. While 
,+I a has been computed based on feature values at time t+1 
without any temporal transition transformation, ,+/ p is the result 
of applying the FMC transition law, 
Equation (2), upon the membership grades in 'a computed on 
the feature values of the image object corresponding to the same 
geographical object at time t. 
3. MULTITEMPORAL CLASSIFICATION MODEL 
3.1 Problem statement 
Let 'I and t+, l denote two co-registered images of the same 
geographical area acquired respectively at dates t and t+1. 
Accordingly, 'x and t+I \ stand for the feature vectors composed 
of spectral and spatial feature values describing the same 
geographical object respectively in '1 and ,+I I. We further 
denote with 'w and ,+y w the crisp label vectors for the object 
being analyzed at times t and t+1. Both 'w and ,+l w are n- 
dimensional unitary vectors of the form [0 ... 1 ... 0] having 
“1” in its i-th component and “0” otherwise, indicating that the 
object belongs to the class to,- at a particular time. Formally, 'w 
and ,+y w belong to a n dimensional space £2", where: 
£2" = { w = [ W],...,w n ] I w,- e {0,1} 3 
for all i = l,...,n and llwll = 1} 
The multitemporal classification problem treated in the present 
work consists of identifying the vector ,+/ w for each image 
object, based on the feature vectors 'x and ,+/ x, in other words, 
it is about finding a function M of the form: 
,+y w = M ('x,' +/ x) (4) 
3.2 General classification model 
The terms monotemporal and multitemporal will be used 
hereafter to designate classifiers whose inputs refer respectively 
to a single date or to multiple dates. 
The outcome of the multitemporal classifier can be viewed as 
the fusion of the outcome of two monotemporal classifiers. Let 
the first monotemporal classifier be represented by a function 
y C that computes membership values for the object being 
classified at the later time t+1 based exclusively on the feature 
values at time t+1, extracted from image <+/ I. The 
monotemporal classifier L C produces a «-dimensional fuzzy 
label vector denoted by ,+I a = [' +y cq, ' +y a 2 ,.../ +/ a„ ], where ,+y a, 
stands for the membership of the image object assigned by L C to 
the class co,-, for all to,- e £2 and for at least one i, ,+ 'a, f 0. So, 
L C can be viewed as a function of the form: 
1+1 a = L C ( ,+y x) (5) 
A second monotemporal E C is applied to the object feature 
vector 'x at time /. Analogously to the first monotemporal 
classifier, it generates a fuzzy label vector 'a, formally: 
'a = E C ( f x) (6) 
Since 'a refers to the membership distribution at the earlier time 
t and our interest is in the classification at the later time t+1, the 
FCM transition law is applied to infer the membership values at 
time t+1 based on the membership values at t. Thus, if T is the 
class transitions matrix representing the class transitions in two 
consecutive instants, we may estimate the classification at time 
t+1 by combining equations (2) and (6), yielding: 
,+y p = E C Ox) o T (7) 
The two fuzzy label vectors 1+1 a and ,+/ p are then combined in 
the next step by an aggregation function F to form a 
multitemporal fuzzy label vector ,+/ p = [' +/ p/,..., ,+ V« ] given 
by: 
' + V = F(' +y a, ,+/ p)= F[ l C C'x) , E C Ox) • T] (8) 
The final step is the defuzzification, performed by a function of 
the form: 
H: [0, 1]"—> £2” (9) 
that transforms the fuzzy label vector ' +/ p into a crisp one. 
Putting it all together, the multitemporal classifier M is given 
by Equation (10). 
,+y w = M(' +y x, 'x)= H{ F[ l C ( ,+/ x) , E C Ox) o T] } (10)