IEEE Geoscience and Remote Sensing Magazine - September 2015 - 17

In the literature, explicit estimation of class statistical
parameters has been addressed with the ExpectationMaximization (EM) algorithm which is an iterative approach to maximum-likelihood (ML) estimation for incomplete data problems [78]. The iterative equations that
characterize the EM algorithm are different according
to the statistical model adopted for the distributions of
the classes. The most suitable statistical model varies according to the kind of data. If optical passive sensor data
are considered the most common statistical models are:
i) Gaussian [7],[79]-[81]; ii) mixture of Gaussians [55];
iii) Rayleigh (for the magnitude of unchanged samples
computed according to CVA) [47]; vi) Rice (for the magnitude of changed samples computed according to CVA)
[47]; v) Uniform (for the direction of unchanged samples
computed according to CVA) [47]; vi) Non-uniform (for
the direction of changed samples computed according
to CVA) [47]. If SAR images are considered, it has been
shown that the Generalized Gaussian [2],[82], Weibull or
Nakagami-Gamma [83],[84] distributions allow to better
handle the complexity of the class distributions. The iterative equations needed for performing EM parameter
optimization under the Gaussian, mixture of Gaussian
and Generalized Gaussian class models can be found in
[2],[55] and [36], respectively, whereas more details on
the validity of the Rayleigh and Rice models can be found
in [47].
Once the statistical parameters are computed, pixelbased or context-based decision rules from the pattern
recognition literature can be applied. The review of such
methods is out of the scope of the present manuscript.
However, we recall the most widely used approaches in
the context of change detection. Concerning pixel-based
methods, we can mention: i) Bayes rule for minimum
error [2],[36],[47],[80]; ii) Bayes rule for minimum cost
[80]; iii) Neyman-Pearson criterion [80]. The Bayes rule
for minimum cost and the Neyman-Pearson criterion allow considering the costs of false and/or missed alarms
in the decision process. Bayesian decision theory can be
used also in multisensor change detection [109]. Here fusion is carried out according to the consensus theory by
integrating the estimates of statistical terms over different sensors. In the fusion step a weight is associated to
each source according to its expected reliability. Within
the Bayesian decision theory framework different techniques for reducing the effects of the residual registration
noise between multitemporal images have been integrated [48], [49],[85].
Another set of methods is based on the optimization
of objective (cost) functions. The fact that, generally, the
change index is one-dimensional makes this process easy.
The choice of the cost function plays a fundamental role in
the accuracy of the results. In the change detection literature, several objective functions have been employed based
on: i) discriminant analysis and inter- and intra-class measures [86]; ii) Bayes decision rule for minimum error [87]; iii)
september 2015

ieee Geoscience and remote sensing magazine

distribution free fuzzy entropy measure [88]. The optimization of objective functions leads to an implicit estimation
of the class statistical parameters [87],[89]-[96]. According
to the kind of data, different assumptions on the statistical distribution of classes can be made. As an example, the
Kittler and Illingworth criterion has been used under both
the Gaussian [89] and the Generalized Gaussian [2],[82]
assumption for the statistical distributions of classes. Also
methods based on Machine Learning and clustering that
minimize a cost function can be listed in this category. In
the literature examples can be found based on Support Vector Machine [90],[91], clustering and kernel-based clustering [65],[92],[93], neural networks [94],[98].
The use of fuzzy theory is another possibility. This kind
of techniques rely on the assumption that some ambiguity exists that arises from the overlapping nature of classes
or image properties [17],[18]. The ambiguity of an image
can be expressed in terms of radiometry (e.g., fuzzy entropy, hybrid entropy, correlation, etc.) or geometry (e.g.,
compactness, high and width, length and breadth, index
of area coverage, degree of adjacency, etc.). The decision
threshold is selected as the value where the membership
function shows a global minimum or maximum depending on the selected ambiguity measure [17],[18]. Fuzzy
clustering approaches belongs to this group as well [26].
Some approaches involve spatial-context information
in the decision process. This is justified by the reasonable
assumption that changes are large if compared with the
spatial resolution of the sensor. Thus a pixel is likely to be
surrounded by pixels of the
same class. The use of interpixel dependence may yield
more reliable and accurate
Spatial-context-baSed
change-detection results. A
methodS aSSumeS that
fully automatic approach to
changeS are large if
the unsupervised analysis
compared with the
of the change index, which
exploits the spatial contexSpatial reSolution of
tual information to reduce
the SenSor and that
the effect of noise in the depixelS are likely to be
tection procedure, has been
Surrounded by pixelS of
proposed in [2],[36]. The
the Same claSS.
solution is developed in the
context of Bayesian decision
theory, where the spatial
context of each pixel is modelled by the use of Markov Random Fields. Another effective technique capable to consider the spatial-contextual information is based on adaptive parcels, i.e., small
homogeneous regions shared by both original images
[29],[97]. The adaptive nature of parcels allows spatial-contextual information to be exploited so that noise may be
reduced without damaging the boundaries of the changed
areas. Spatial correlation between neighboring pixels has
been modelled by using Hopfield neural network [94],[98]
as well. This solution is fully automatic and distribution
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