In: Wagner W., Szekely, B. (eds.): ISPRS TC VII Symposium - 100 Years ISPRS, Vienna, Austria, July 5-7, 2010, IAPRS, Vol. XXXVIII, Part 7B 
classes of interest at the two dates against all those identified as 
changed pixels may provide some degree of information on the 
type of land-cover transition, but not with the accuracy and 
reliability provided by fully-supervised approaches). 
In this paper, we formulate this complex issue in terms of a 
compound decision problem (Duda et al., 2000) and propose a 
novel partially-supervised change-detection (PSCD) technique 
capable of exploiting the only prior knowledge available for the 
specific land-cover classes of interest at the two times (thus 
avoiding the need to rely on exhaustive ground-truth informa 
tion for all the classes), while providing accuracies comparable 
with those of fully-supervised methods. Moreover, it has the 
great advantage of being sensor-independent, which allows 
selecting at each date the set of sensors and features most suit 
able for characterizing the targeted class(es) of interest. 
The proposed method aims at estimating at each date the prob 
ability density function (PDF) and the prior of both the class(es) 
of interest and the remaining unknown land-cover classes (for 
which no ground truth is available) represented as a single un 
known information class. In particular, PDFs are approximated 
by a mixture of suitable basis functions whose free parameters 
are determined employing the iterative Expectation- 
Maximization (EM) algorithm (Dempster et al., 1977). Changed 
pixels are then identified using an iterative labelling strategy 
based on Markov random fields (MRF) (Solberg et al., 1996) 
which allows taking into account both spatial and temporal 
correlation between the two images, as well as properly con 
straining the probability estimates. 
For demonstrating the capabilities of the proposed method, 
extensive experimental trials have been carried out with differ 
ent combinations of multispectral, hyperspectral and SAR data. 
Obtained results confirmed the effectiveness and the reliability 
of the proposed technique, which provided very promising re 
sults. In particular, accuracies are comparable to those achieved 
with the PCC method in the presence of an exhaustive ground 
truth for each image both considering the maximum likelihood 
(ML) classifier (Richards and Jia, 2006), as well as support 
vector machines (SYM) (Cristianini and Shawe-Taylor, 2000). 
2. PROBLEM FORMULATION AND ASSUMPTIONS 
For the sake of simplicity we will describe the problem and the 
proposed method under the assumption of a single targeted 
land-cover transition of interest. The extension to the case of 
multiple transitions is straightforward. 
Let us consider two IxJ co-registered remote-sensing images 
*'={*#}&, and X 2 = {xl}';U, 4 e R 02 , referring 
the same geographical area at times t\ and h , respectively, 
where x!y, xj represent corresponding feature vectors (even 
derived from different sets of sensors at each date, respectively, 
and merged using a stacked vector approach (Richards and Jia, 
2006)) associated with the pixel at position (i,j) , and D\, Eh. 
define respective dimensionalities. 
Let £1'={cd ,...,co.L} and Q 2 ={ed ,...,ceL) be the set of land- 
cover classes characterizing X' and X 1 , respectively. In the 
following, we will denote as cdt e £2' and cdt e Q 2 the infor 
mation classes of interest at t\ and h , for which Ni and N2 
labelled training patters are available, respectively. Elence, 
coU ={Q‘-ceD and aLk - {Q. 1 -cd*.} will represent corre 
sponding unknown classes (each consisting of the merger of 
remaining classes, for which no ground truth is available). 
Let C and C 2 = {Cy }';/=, denote two sets of labels for 
X' and X 2 , respectively, where Cle {cd M ,coU} and 
C|e {ainx,ocLi} are associated with the pixel at position (i,j). 
In this framework, our aim is to identify the two sets C, C 2 
maximizing the posterior probability given the two images X 1 
X 2 and, finally, to draw pixels experiencing the targeted land- 
cover transition from edit to cdt. This can be formalized as a 
compound decision problem (Duda et al., 2000): 
{C',C 2 }=argmax{P(C l ,C 2 \X\X 2 )} (i) 
c‘,c 2 
According with the Bayes theory, finding a solution to (1) is 
equal to determine the sets of labels maximizing the likelihood 
C(X',X 2 \C',C 2 ) = P(C',C 2 )-p(X',X 2 \C',C 2 ). 
In the reasonable hypothesis of time-conditional independence, 
the problem can be written as: 
{C',C 2 }=argmax{C{X ,?d \ C ,C 2 )-P(C',C 2 )p(X\C')-p(X 2 \C 2 )) (2) 
where p(X'\C') and p(X 2 \C 2 ) represent the conditional 
PDFs at t\ and ti respectively. 
3. PROPOSED PARTIALLY-SUPERVISED CHANGE- 
DETECTION TECHNIQUE 
For addressing the complex task described in Section 2, we 
propose a novel partially-supervised technique aimed at ap 
proximating the class-conditional densities p(X' \C'), 
p(X 2 \C 2 ) as mixtures of suitable basis kernel functions and 
estimating the joint prior probability P(C',C 2 ) properly taking 
into consideration the spatio-temporal context. 
The rationale is based on the observation that the PDF of an 
image can be always approximated by a mixture of suitable 
kernels (i.e., Parzen density estimation (Duda et al., 2000)). 
Accordingly, similarly to what is commonly done in the context 
of Radial Basis Function Neural Networks (RBF-NN) (Bruz- 
zone and Femandez-Prieto, 1999), we model for each pixel of 
both images the PDFs p(x\), p(xl) as a mixture of K circu 
larly symmetric multivariate Gaussian functions. Kernel pa 
rameters (i.e., centres and variances) are initialized using the k- 
mean clustering algorithm (Bruzzone and Femandez-Prieto, 
1999), whereas final estimates are obtained by using the EM 
algorithm (Dempster et al., 1977). Then, class-conditional den 
sities of the interest class p(x)j\cdt), p(x 2 j \ cdt) are modelled 
by properly weighting the resulting set of kernels using again 
the EM algorithm over the training samples available for (dt 
and cdt. This is somewhat analogous to the training phase of 
RBF-NN which is generally carried out in two steps: i) selec 
tion of centres and variances of the kernel functions associated 
with hidden units on the basis of clustering techniques; and ii) 
computation of weights associated with the connections be 
tween the hidden and output layers on the basis of available 
training patterns. 
The PDF of the entire image is itself a mixture of the interest 
and unknown class-conditional densities, weighted by corre 
sponding prior probabilities. Accordingly, we obtain a first 
rough approximation for p(x\ |coU), p{x] \ ednk) initializing 
priors to 0.5. Afterwards, estimates are refined using a novel 
MRF-based iterative labelling strategy accounting for spatio- 
temporal correlation, which permits to model P(C!/,Cij) mutu 
ally considering the local neighbourhood of each pixel in the 
two images. Finally, changed pixels are identified and associ 
ated with the targeted land-cover transition by minimizing a