According to Eqs. 13 and 17, we can write Eq. 20 as follow
2 2
2, 2
3 1 exp
3 1
1 exp
c m
g
h h
r h
r
γ αβ
ω ω
⎛ ⎛
⎞ ⎛
⎞ =
− −
⎜ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎝ ⎠
⎝ ⎠
⎝ ⎞
⎛ ⎞
⎛ ⎞
⎟ +
− −
− ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎟ ⎝
⎠ ⎝
⎠ ⎠
21
2.3.2 First-order variogram: As described above, the tessellation process of Poisson lines partitions the domain into
non-overlapping cells within the image. The cell values are realizations of independent Gamma random variables. Two
pixels x and x + h separated by a distance ||h|| belong to the same cell, denoted event A. Its probability is related to the
covariance function of Z
m
x and thus to the second-order variogram of Z
m
x Garrigues et al., 2007
2 2,
m m
P A C
h h
σ γ
= =
−
22 The probability of
A
is equal to
2 2,
1 1
m
P A P A
h σ
γ = −
= − +
23 Conditioning on the event A and
A
, the theoretical first- order variogram can be decomposed as follow,
γ
1
h = 0.5P AE | Z x + h − Zx | A
⎡ ⎣
⎤ ⎦
+0.5PAE | Zx + h − Zx | A
⎡ ⎣
⎤ ⎦
24 where the first expectation value can be calculated from Eq. 19
E | Zx + h − Zx | A
⎡ ⎣
⎤ ⎦
= E σ ωZ
m
x + h − Z
m
x
⎡ ⎣
+ 1 −ω
2
Z
g
x + h − Z
g
x A
⎤ ⎦⎥
25
In the case of event A, the locations x and x + h are in the
same cell. As a result, Z
m
x + h - Z
m
x = 0, Eq. 25 reduce as
follows,
E | Zx + h − Zx | A
⎡ ⎣
⎤ ⎦
=
σω
E Z
g
x + h − Z
g
x
⎡ ⎣
⎤ ⎦
26 from Eq. 14 the expectation value of |Z
g
x - Z
g
x + h| can be
figured out. By the same token, the second one will be
E | Zx + h − Zx | A
⎡ ⎣
⎤ ⎦
= E σ |ωZ
m
x + h − Z
m
x
⎡⎣ + 1
−ω
2
Z
g
x + h − Z
g
x | A
⎤ ⎦⎥
27
In the case of the event
A
, the pixels x and x + h do not
belong to the same cell. Z
m
x + h, Z
m
x, Z
g
x and Z
g
x + h
are independent random variables with the same Gamma distribution. Using Eq. 14 the probability density function of
the difference Z
g
x + h - Z
g
x, as same as Z
m
x + h - Z
m
x,
can be figured out.
ω
and
2
1
ω
−
are weight values of the two random variables, then the probability density functions of
ω
Z and
Z
2
1
ω
−
can be derived. Let
ω
Z +
Z
2
1
ω
−
be a new random variable Z
c
, Eq. 27 becomes
σ
E[Z
c
]. Using the probability density function of Z, the expectation of Z
c
can be calculated. Through
β
the E[Z
c
] can be represented as a function of
γ
2, g
h. Here also gives the expression of
γ
1
h when
α
= 2 as an example.
1 2,
2 2,
2, 2
2, 3
2 2
3 2 ;
, 2
4 2
; 1
, 1
1 8
1
g g
m m
g g
m m
h h r
h r
h r h r
γ γ
σ ω σ
γ
γ σ
γ ω
ω ω
= ⎡
⎢ −
⎢⎣ ⎤
⎥ ⎥
+ −
+ ⎥
⎛ ⎞
⎥ −
⎜ ⎟
⎥ −
⎝ ⎠
⎦
28
From Eq. 28, for the multi-Gamma and mosaic models, the first- and second-order variograms are related quadratically and
linearly, respectively.
3. EXPERIMENT AND RESULTS 3.1 Parameter estimation by least-squares criterion
In this paper, least-squares criterion is employed to estimate the mixture model parameters, including first- and second-order
variograms, by fitting theoretical variograms to experimental ones Schabenberger Gotway, 2005. Let H = [h
1
, ..., h
n
] and
Θ
=
α
,
β
,
ω
, r
g
, r
m
. In real SAR images,
α
is a constant equal to the number of looks of SAR sensors. These parameters are
used to characterize the spatial structure in a SAR intensity image. Theoretical variogram family
γ
H,
Θ
can be calculated from Eqs. 20 and 24.
Let
{ }
H H
∈ =
h h
h :
ˆ ,
ˆ ˆ
2 1
γ γ
γ
be the
experimental variogram family, where the first- and second- order experimental variograms are computed according to
[ ]
∑ ∑
− +
= −
+ =
2 2
1
| |
2 1
ˆ |
| 2
1 ˆ
h N
h N
z h
z h
N h
z h
z h
N h
x x
x x
γ γ
29
where Nh is the number of the pairs of pixels separated by the distance h. For fitting
γ
H,
Θ
to
ˆ H γ
, assume
, ˆ
H ε
Θ H
H +
=
γ γ
30 where
ε
H is the n × 1 error vector with zero mean and variance-covariance matrix V
Θ
= Var[
ε
H].
By weighted least squares, the parameter vector
Θ
can be estimated so as to minimize the weighted sum of squares
[ ]
[ ]
2 2
2 2
2 1
2
ˆ ˆ
, ,
, ,
| |
ˆ ,
2 ,
T T
n j
j j
j j
N h h
h h
γ γ
γ γ
γ γ
γ
=
− −
= ⎡
⎤ =
− ⎣
⎦
∑
H H Θ
V Θ H
H Θ ε H Θ
V Θ ε H Θ Θ
Θ
31
3.2 Comparison of theoretical variograms with different values of parameters
To test the feasibility of the mixture model, we compare the theoretical variograms with different values of parameters.
First is for the second-order variogram Eq. 21. We set
σ
2
= 1,
ω
2
from 0 to 1 with interval 0.125, that is,
ω
2
= 0,
ω
1 2
= 0.125, ...,
ω
8 2
= 1. Then we set three examples: a r
g
= r
m
= 30 pixels. b r
g
= 10 pixels, r
m
= 50 pixels. c r
g
= 50 pixels, r
m
= ISPRS Annals of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume III-2, 2016
This contribution has been peer-reviewed. The double-blind peer-review was conducted on the basis of the full paper. doi:10.5194isprsannals-III-2-99-2016
102
