IEEE Geoscience and Remote Sensing Magazine - March 2021 - 103

the GCD becomes the exponential distribution. If v = 2,
the GCD degrades into the Nakagami distribution. If
v = 1, the GCD degenerates to the CD. If v = - 1, the GCD
becomes the inverse CD. If l " ∞ , the GCD degrades
into a log-normal distribution. If l = 1 , the GCD is a
Weibull distribution.
FINITE MIXTURES
All of the statistical distributions mentioned in the previous section belong to the parametric method [1], [26], and
each is usually appropriate for a specific applicable scene.
However, usually there are multiple types of targets in a remote sensing image. Also, with the acquisition of high- and
ultrahigh-resolution SAR images, more details are highlighted, making it difficult to model SAR images with only
one kind of statistical distribution [1].
To solve this problem, the finite-mixture model
(FMM) [29], [73] has attracted the attention of scholars.
The high flexibility of the FMM model makes it widely
used in the fields of pattern recognition [74], [75], signal and image analysis [76]-[78], machine learning [79],
[80], and remote sensing [30], [78], [81], [83]. The FMM
is a semiparametric method that models the unknown
probability distribution as a linear combination of parameterized mixture components, with each mixture
component belonging to a dictionary of SAR specific distributions [29].
The FMM models the PDF of the SAR amplitude image z
as a linear combination of k mixture components [29], [30]:
	

fz ^ z|M h =

k

/ a i fi (z|m i),(105)

i=1

where fi ^$| m ih, i = 1, 2, f, k represents the probability distribution of the mixture component with parameters m i,
and " a 1, f, a k , is a set of mixing proportions such that
	

k

/ a i = 1, 0 # a i # 1,(106)

i=1

where M is a set collecting all the parameters of FMM,
which is
M = " m 1, f, m k, a 1, f, a k , .
The distribution of each mixture component can be
selected from the statistical distributions introduced previously. The selected R types of mixture components f j ^$| p jh,
j = 1, 2, f, R make up a dictionary D = " f1, f2, f, fk ,, where
p j represents the parameter of each mixture-component
distribution [30].
Consider the k-independent random variables " A 1, f, A k ,
with distributions characterized by the densities " f1, f, fk ,.
Assume that the observation of the experiment consists
of first choosing one of these random variables with
probability " a 1, f, a k , and then sampling from it. In
other words,
MARCH 2021

IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

	

A=*

A 1 with probability a 1,
h
A k with probability a k .

The resulting random variable A obeys the mixture distribution characterized by the k laws " f1, f, fk , with proportions " a 1, f, a k , .
Given an SAR image and a dictionary, it is necessary
to use an optimization method that determines the optimal number k of mixed components, the mixing proportion a k, and the distribution parameter m i of each
mixture component.
Blake et al. [14], [15] experimentally verified that, using
only a single-probability distribution, such as the K, lognormal, Weibull, gamma, exponential distribution, and so
forth, it is not possible to accurately model high-resolution SAR images. The authors'
experiments show that the
mixed distribution composed
THE FMM IS A
of two K distributions or two
SEMIPARAMETRIC METHOD
log-normal distributions can
THAT MODELS THE
be used to describe high-resoUNKNOWN PROBABILITY
lution SAR clutter. This is beDISTRIBUTION AS A LINEAR
cause the model is expanded
COMBINATION OF
from a single two-parameter
PARAMETERIZED MIXTURE
probability distribution to the
double five-parameter probaCOMPONENTS, WITH EACH
bility distribution, which can
MIXTURE COMPONENT
better describe the complexBELONGING TO A
ity of high-resolution images.
DICTIONARY OF SAR SPECIFIC
Blacknell [84], [85] proposed
DISTRIBUTIONS.
a correlated Gaussian-mixture
distribution (GMD) model to
approximate the correlated
K-clutter image in the logarithmic domain, and the model
is applied to target detection, which shows that the GMD
model realizes better detection results compared with a single Gaussian distribution. At the same time, it is easier for
the GMD model to introduce correlation information than
it is for other distributions. The introduction of correlation
information can also significantly improve the performance of target detection.
Moser et al. [30] used an FMM to model medium-resolution SAR amplitude images. Each mixture component was
selected from a dictionary of six specific distributions: lognormal, Nakagami, GGR, SaSGR, Weibull, and K. The authors proposed a dictionary-based stochastic expectation
maximization (DSEM) that can automatically estimate the
parameters of the optimal mixture model combined with
the method of log-cumulants (MoLC), and the parameters
include the number and type as well as the distribution
parameters of the mixed components. Furthermore, enhanced DSEM (EDSEM) is proposed in [83], which can
more quickly estimate the number of mixed components.
EDSEM further extended the DSEM algorithm for medium-resolution SAR images to very-high-resolution images,
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IEEE Geoscience and Remote Sensing Magazine - March 2021

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