IEEE Geoscience and Remote Sensing Magazine - March 2021 - 89
Speckle
Satisfies the
Central Limit
Theorem
Does Not
Satisfy the
Central Limit
Theorem
RCS obeying the beta distribution of the first kind and
the speckle obeying the CD [23] (the K, G 0, U, and W
distributions are all probability distributions belonging
to the Pearson system [42]-[44]).
The distribution functions and their related statistical
characteristics based on the product model are summarized in Table 2, which is introduced in the next section.
Because the U and W distributions can also be modeled
under the NR speckle model,
THE IMAGE GENERATED BY
they are introduced in detail
REPLACING n PIXELS WITH
in t he " Statistical Dist r ibutions of the NR Speckle
A NEW PIXEL BEARING
Model " section and are not
THEIR AVERAGE VALUE IS
covered here.
CALLED AN n-LOOK IMAGE.
CD
The square root CD is used
to describe the multilook amplitude images of homogenous
areas [1], which have been derived from the Rayleigh
speckle model in the " Statistical Distributions of the
Rayleigh Speckle Model " section. It can be remodeled as
the product of the RCS and the speckle under the product model. For the intensity image, if the RCS intensity
is a constant X v = o, the multilook speckle intensity Yv
obeys the CD Yv + Gamma ^n, n h ; then, the image intensity Z v obeys the CD according to the product model.
According to the statistical moment in Table 2, when the
Rayleigh Speckle Model [4]
Single Look:
Negative Exponential
Distribution
Multilook:
ΓD
×
×
×
Denotes " Product Model "
GC
Distribution [40]
Constant
GIG Distribution
Denotes " Generalization "
Single Look:
Negative Exponential
Distribution
Multilook: ΓD
GΓD
GΓD
RCS
Product Model (1981)
Rayleigh Speckle Model
satisfied. The speckle component can be modeled as generalized CD (GCD) when the central limit theorem is
not satisfied.
The RC S component a lso has mult iple dif ferent
modeling methods, depending on their different physical scenarios:
◗◗ The RCS is considered a constant for the homogenous
area. The Rayleigh speckle model can be obtained if the
central limit theorem is satisfied according to the product model displayed in Figure 6.
◗◗ For the inhomogeneous area, the RCS can be modeled
as a GCD, a generalized inverse Gaussian (GIG) distribution, and beta distributions of the first and second
kinds [23]. And the GIG distribution can be degenerated
into IG distribution, CD, and inverse CD under certain
conditions.
As depicted in Figure 6, under the product model,
1) generalized compound (GC) distributed SAR images can be obtained when the RCS obeys the GCD and
the speckle obeys the GCD [38]-[40]; 2) the SAR image
obeying the G distribution can be obtained by the RCS
obeying the GIG distribution and the speckle obeying
the CD (the G distribution can be degenerated into
G h, K, and G 0 distributions under certain conditions
[41]); 3) U-distributed SAR images can be obtained by
the RCS obeying a beta distribution of the second kind
and a gamma-distributed speckle [23]; 4) the SAR image obeying the W distribution can be obtained by the
IG Distribution
×
G Distribution [41]
×
G h Distribution [47]
×
K Distribution [9]
×
G 0 Distribution [41]
Beta Distribution
of the Second Kind
×
U Distribution [23]
Beta Distribution
of the First Kind
×
W Distribution [23]
ΓD
Inverse ΓD
FIGURE 6. The statistical distributions of scattered intensity based on the product model.
MARCH 2021
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE
89
IEEE Geoscience and Remote Sensing Magazine - March 2021
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