In: Wagner W., Szekely, B. (eds.): ISPRS TC VII Symposium - 100 Years ISPRS, Vienna, Austria, July 5-7, 2010, IAPRS, Vol. XXXVIII, Part 7B 
2>- 
[w4t*] (/) =—— 
[mUF'"-^) 
N, 
if x'ij 6 T‘ 
if x'gT' 
(12) 
The corresponding log-likelihood is given by: 
where j3> 0 tunes the influence of the context and S is the 
Kronecker delta function defined as: 
d[(Cl,Oj),(C' gh ,C 2 gh j\ 
|i if (a,a)=(c'.,o 
|o if (a,a)^(c^,o 
(18) 
in C(d' = m)=faj ■ ln[p(x'j I alt)] 
¡j=i 
3.3 Iterative Labelling 
(13) 
Solving Eq. (2) is equivalent to maximize the log-likelihood 
\rvL(X\X 2 \C ,C 2 ), which can be written as: 
Even in this case we assume that convergence is reached if the 
relative increase in the log-likelihood is lower than e . 
3.1.3 Estimation of p(x\j\aLk) 
It is worth noting that Eq. (3) can be re-written as: 
K K K 
P(ajU)-^MLkk-(/i(x'o)=^Mi-</)k(x'ij)-P(cdn t )-^Wma-<pk(x'j) (14) 
\nC(X\X 2 \C x ,C 2 )=-YJU^(X',C)-U^(C',C 2 )-\n(Z) (19) 
where ULxa(X',C) = -\n[p(X‘\C)], represents the class- 
conditional energy function at date /=1,2, while is 
given by (17). Since Z solely depends on the selected type of 
neighbourhood, the final problem becomes solving: 
Hence, \/k=\,...,K it holds P(ai^)-wUk = wi-P(tu'„,)-u4<*. [C ,C 2 )^^gmm{U^(C\C 2 )+Ulx,(X\C)+Ul,(X 2 ,C 2 )} (20) 
Since P(coin)+P(oink)=\, we have: c ’. c2 
, Wk-P(Oim)-Wmxk Wk-P(COLx)-wLk 
Wink k — = 
P(ccU) 1 -P(oim) 
(15) 
Then, as M', £', W' and WL have been determined, we can 
compute p(x'j\coU) substituting (15) into (6), upon it is possi 
ble to obtain a reliable estimate P(oinx) for the prior probability 
of the class of interest. This can be properly accomplished 
throughout the iterative labelling phase presented below. 
3.2 Joint Prior Modelling 
For modelling the joint prior P(C\C 2 ), we propose an ap 
proach based on MRF (Solberg et al., 1996). In particular, we 
assume that the couple of labels Cl, Cl associated with pixel 
at position (i,j) at times /1 and /2 depends on the couples of 
labels associated with pixels belonging to the spatial 
neighbourhood Go of (i,j) at the two dates (we always con 
sidered first order neighbourhoods). 
In other words, the higher the number of its spatial neighbours 
experiencing a certain land-cover transition is, the higher the 
probability for a given pixel of experiencing the same transition 
is. In this hypothesis, according with the MRF theory, it holds 
the equivalence: 
P(C'j,Cy \ C' g h,dh;(g,h)e 0 i ,) = Z- 1 -exp[-[/ № (a,C^)] (16) 
where Z = Z(Qij) is a normalizing constant called partition 
function, while C/cmuex is a Gibbs energy function (accounting 
for the spatio-temporal context) of the form: 
To this aim, we propose a strategy based on the Iterated Condi 
tional Modes (ICM) algorithm (Besag, 1996) which allows 
maximizing local conditional probabilities sequentially. In par 
ticular, at each iteration / we update the estimated prior prob 
abilities for the class of interest at each date P(cdnx) and, ac 
cordingly, also the class-conditional densities of the unknown 
classes p(x\j \ coU). The algorithm works as follows: 
Step 1. After estimating ^(x!> \cii M ), VxiyeA", / = 1,2 follow 
ing the approach described in the previous paragraphs, set 
P(oixxx)=0.5 (no prior knowledge is assumed to be available 
about the true prior P(coLx)) and compute the conditional den 
sity of the unknown classes p(x'ij\ccL ! .) accordingly; 
Step 2. Derive the initial sets of labels C, C 2 by solely mini 
mizing the non-contextual terms of Eq. (20), i.e. 
{C',C 2 } = argmm{UL(X',C')+UL(X 2 ,C 2 )}; 
Step 3. On the basis of current C' and C 2 , compute the new 
estimated prior probabilities ratioing the number of pixels asso 
ciated with the class of interest over the whole number of pix 
els, i.e. P(oinx) = \ax\/(I-J), CL = {G i \Gj£C , ,Gj=aL} I i f\ 
then, update the class-conditional densities for the unknown 
classes p(x'j \ccL±) accordingly; 
Step 4. Update C ] and C 2 according with Eq. (20); 
Step 5. Repeat Step 3 and Step 4 until no changes occur be 
tween successive iterations. 
At the end of the process, the final targeted change-detection 
map C' is defined as: 
c'={c;j\’:U, 
if Cl = cdnx and Cl = cdxx 
otherwise 
(21) 
4. EXPERIMENTAL RESULTS 
U^(Cl,Cl)=- Y, p-S[(Cl,C 2 j),(C gh ,Cl h j\ 
(g^Sn 
(17) In order to assess the effectiveness of the proposed technique, 
we carried out several experiments with different combinations