IEEE Geoscience and Remote Sensing Magazine - March 2021 - 90

K DISTRIBUTION
The K v distribution is a classical statistical-distribution
model that describes inhomogeneous regions [2], [41].
The K v distribution can be derived based on several
different modeling methods. The modeling under the
product model is described here, and other modeling
methods are detailed in the " Statistical Distributions
of the NR Speckle Model " section. The K v distribution
can also be seen as a special case of the G v distribution under the condition that c " 0 [41]. At this time, the

number of looks n is known, the second-order sample mot 2 can be used to estimate the parameter, o = m
t 2.
ment m
G DISTRIBUTION
The G distribution was proposed by Frery et al. in 1997 [41]
using the product model to describe the statistical characteristics of inhomogeneous areas. In this model, the RCS intensity obeys the GIG distribution X v + GIG ^b, c, a h [46], and
the speckle obeys the CD. According to the product model,
the SAR image intensity Z v obeys the G v distribution.

TABLE 2. THE STATISTICAL DISTRIBUTION AND CHARACTERISTICS OF A PRODUCT MODEL FOR INTENSITY DATA.
DISTRIBUTION
TYPE

THE PDF OF
RCS P(Xv = x)

THE PDF OF
SPECKLE P(Yv = y)

CD [1]

pC

pC

G distribution
[43], [44]

p GIG

p Gammav ^ Z v = z; n, o h =
p G v ^ Z v = z ; b, c, a , n h

n ^b/ch z
c + nz
c
m
b
C ^ nh K a ^2 bc h

p Gamma

p K v ^ Z v = z ; b, a , n h =

G0 distribution
[43]

p Gamma-1

p G0v ^ Z v = z; c, - a, nh =

Gh distribution
[43], [41]

p IG

p Gvk ^ Z v = z; m, nh

p GC

p GC

2m
r e

m2
n

n z
c
C ^nh
n n-1

b1 v1

Notes and
supplements

z
a b2 v2

oj

nz

E " Z vr , =

K a- n _ 2 b ^c + nz h i

^m + 2nz h n

m

m

-1 - 2n
4

#0 3 x b v - b v - 1 # exp <- ` ax j
2 2

1 1

c 2 K a+ r ^2 bc h C ^ n + r h
m
n2 b
K a ^2 bc h C ^ nh
r

E " Z vr , = ^ nbh-r
E " Z vr , = ^c/nh r

# K n + 12 c

b1 b2
2z C ^ v 1 h C ^ v 2 h

C ^n + r h o r
`nj
^nh

E " Z vr , = c

n n C ^n - ahc - a z n - 1
C ^ nh C ^- ah ^c + nz hn - a

p GCv ^ Z v = z; a, b 1, v 1, b 2, v 2h =
#

a-n
2

exp ` -

a+n
2bn
^bnz h 2 -1 K a- n ^2 bnz h
C^ n h C ^ a h

K distribution
[2], [43]

=

1
n
` jz
C ^nh o

n n-1

a/2 ^ n - 1h

n

=

GC distribution
[38]

r-ORDER MOMENT E(Zvr )

THE PDF OF SAR IMAGES P(Zv = z)

b2

^m + 2nz h m

n

m

z
- c x m F dx
b1

C ^ n + r h C ^a + r h
C ^ nh C ^ah

C ^ n + r h C ^- a - r h
C ^ nh C ^- ah
r

2m n 2
r a n2 k e

E " Z vr , =
#

m2
n

1

n- 4

K r -1/2 ^m / n h C ^ n + r h
C ^nh

E " Z vr , = a 2r

Ca

2r
2r
+ v 1 k Ca
+ v2k
b2
b1
C ^v 1h C ^v 2h

The symbols in this table are as follows: n: the number of looks; K n ^ $ h: the second type of modified Bessel function; C ^ $ h:
gamma function; X v: RCS; Yv: speckle; Z v = X v $ Yv : SAR intensity image.
The detailed expressions in the second column, i.e., the PDF of the RCS, are
p C ^ X v = x; o h = (

1, x = o
; o 20,
0, others

a
2

" ^b, ch: b 2 0, c $ 0 ,
cbm
c
c
p GIG ^ X v = x; b, c, ah =
x^a - 1h # e - `bx + x j ; a ! R, ^b, ch ! H a ; H a = * " ^b, ch: b 2 0, c 2 0 ,
2K a ^2 bc h
" ^b, ch: b $ 0, c 2 0 ,

p Gamma ^ X v = x; b, ah =
p Gamma-1 ^ X v = x; a, ch =
p IG ^ X v = x ; n, mh = ;
p GC ^ X v = x; a, b 2, v 2h =

if a 2 0
if a = 0 ,
if a 1 0

b a ^a - 1h - bx
x
e ; b, a 2 0 ,
C ^ah
c- a
x a - 1 e - c/x ; - a, c 2 0,
C ^- ah

1
m 1/2
x
E # exp :- m a 2n + 2x kD; n $ 0, m 2 0,
2r x 3
b2 v2 - 1

b2
c xm
2 x aC ^v 2h a

x
# exp <- c a m F; a, b 2, v 2 2 0.
b2

The detailed expressions in the third column, i.e. PDF of speckle, are the gamma-distributed speckle Yv + Gamma ^ n, nh , which is
p C ^Yv = y h =

2n n 2n - 1
y
exp ^- ny 2h , y, n 2 0,
C ^nh

and the generalized gamma-distributed speckle Yv ~GC ^ b 1, v 1h , which is
p GC ^Yv = y h =

90

b1 v1
b1
b1
y 2 - 1 exp 8- y 2 B; b 1, v 1 2 0.
2C ^v 1h

IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

MARCH 2021
IEEE Geoscience and Remote Sensing Magazine - March 2021

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