IEEE Geoscience and Remote Sensing Magazine - March 2021 - 92

p Z v ^ z; a, v h =

v

2
c z m Kv - 1 c 2 z m,
a
a z C^ v h a

(14)

and the corresponding r-order moment is
E " Z rv , = a r

C^r + 1h C^r + v h
.
C^ v h

(15)

If b 1 = b 2 = 2c, v 1 = 1, v 2 = 1/2, the GC distribution becomes
a Weibull distribution,
p Z v ^ z; a , c h =

3c

2
2c
c zm
a
a z C ^1/2 h

-1

z
K -1/2 d 2 c a m n . (16)
c

If b = a # 2 -1/c , (16) can be transformed into p WB ^ z; c, bh ,
as listed in Table 3.
VALIDITY ASSESSMENT OF THE PRODUCT MODEL
Although the product model greatly simplifies the statistical modeling of SAR images, few works have dealt
with the verification of their theoretical and physical
foundations. In particular, whether the speckle of highresolution SAR images is still multiplicative noise has
always been controversial. Some scholars believe that
the speckle should be seen as a scattering phenomenon
rather than multiplicative noise [1]. This section summarizes the current assessment of the validity of the
product model.
Early on, Tur et al. [49] studied the conditions applicable
to the multiplicative speckle from the theoretical model of
optical coherent imaging. The random intensity I ts received
at a certain point P = ^ x p, y ph can be expressed as
I ts ^ x p, y ph
=

##R

dxldyl # h ^ x p - xl, y p - ylh t ^ xl, ylh exp 6i{ ^ xl, ylh@ ,
2

cell

(17)
where h ^ x, y h represents the amplitude response of the system, { ^ xl, ylh is the random phase at the point ^ xl, ylh , and
t ^ xl, y lh indicates the scattering contribution at the point
^ xl, y lh . It should be noted that the meanings of t ^ xl, y lh and
{ ^ xl, ylh here correspond to the scattering amplitude and
phase of a single scatterer in a one-resolution cell in radar
imaging, respectively.
If and only if t ^ xl, y lh does not change within a resolution cell, according to the Cauchy-Schwarz inequality, (17)
can be expressed as [49]
I ts ^ x p, y ph = t ^ x p, y ph 2
#

##R

cell

dxldyl # h ^ x p - xl, y p - ylh exp[i{ ^ xl, ylh]

= a $ I inc ^ x p, y ph $ I s ^ x p, y ph ,

2

(18)

1
I inc ^ x p, y ph = a t ^ x p, y ph 2
I s ^ x p, y ph =

##R

dxldyl # h ^ x p - xl, y p - ylh exp[i{ ^ xl, ylh] ,
2

cell

(19)
92

where I inc ^ x, y h represents the incoherent image intensity
of t, I s ^ x, y h is the intensity value at t ^ x, y h = 1, and a is a
constant depending on the system parameters. Equation
(18) gives the product model in a coherent imaging system.
Based on (17)-(19), the literature [49] concludes that the
product model is established when t ^ xl, ylh is constant in a
resolution cell; the product model is no longer valid when
t ^ xl, ylh changes and there is detailed information within a
resolution cell.
In addition, Lee et al. studied the product model from
the perspective that the CV of the Rayleigh speckle model
should be constant for homogenous areas [2]. Suppose a
product model
z ^k, l h = x ^k, l h y ^k, l h ,

(20)

where z ^k, l h is the intensity or amplitude of the ^k, l h th
pixel in the SAR image, x ^k, l h is the reflection coefficient,
and y ^k, l h is the noise with mean of 1 and a standard deviation of v y .
Suppose x ^k, l h and y ^k, l h are statistically independent;
according to the product model and E 6y@ = 1, it can be derived that
E 6 z @ = E 6 x@ .

(21)

Furthermore, the variance of z can be derived as
Var ^ z h = E 6^ z - zr h2@ = ^ Var ^ x h + xr 2 h v 2y + Var ^ x h ,

(22)

and then the CV of z can be obtained as
Var ^ z h
=
zr

Var ^ x h ^1 + v 2y h
+ v 2y .
zr 2

(23)

There is no variation of the reflection coefficient over
the homogenous areas; therefore, Var ^ x h = 0, and then
Var ^ z h
= vy .
zr

(24)

Thus, the CV is constant in the multiplicative noise
model of the homogenous regions. In the literature
[2], it is explained that the homogenous SAR image
satisfies the product model by testing the CV of the
single-/multilook SAR amplitude/intensity image as
a constant.
It can be seen that the product model is theoretically
strictly established for only homogenous regions but not in
the inhomogeneous case. That is, the product model based
on the Rayleigh speckle model is theoretically strictly established, but there is no theoretical support for the product
model to be extended to inhomogeneous regions. However, the statistical modeling based on the product model
is still widely used because it is applicable to several typical
distribution models for inhomogeneous regions, such as
the K distribution, and it simplifies the derivation process
of statistical modeling [50].
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

MARCH 2021
IEEE Geoscience and Remote Sensing Magazine - March 2021

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