s x, r is obtained when the ground backscattering coefficient
g x, r goes through a 2D linear system hx, r. The system
impulse response hx, r can be divided into the convolution of the impulse response in azimuth direction h
a
and in range direction h
r
as shown below:
, ,
,
a r
h x r h
x r h x r
, ,
,
c
g x r
r R x r
x x
A ,
3
, ,
, s
x r g
x r h x r
4 The impulse responses in azimuth direction h
a
and in range direction h
r
are given as follows:
4 ,
exp
a a
h x r
x j
r x
,
5
2
4 2
, exp
r r
r
K h x r
r j
r c
c
. 6
Our goal is to obtain an exact value of A. We make convolution between h
s
x, r andhx, r to get:
, ,
,
s
h x r h
x r x r
. 7
The filter that satisfies 7 above can have good performance on solving the convolution. In fact, it can be proved that hx,
r h
s
x, r in both azimuth and range directions are Sinc function instead of unit impulse response
δx, r. For Sinc function, there are peak sidelobes, which degrade the quality of
imaging. We can make a single point echo. When the point echo is a circle area function Somb, x
, y is the center of coordinate
2 2
2 2
, sin
x y
I x y H
c x
x y
y
2 2
2 2
2 2
2 2
sin
x y
x y
H x
x y
y x
x y
y
. 8
The simplified circle area function Somb is approximated by:
1
2 Somb
r r
r
J
9 J
1
is Bessel function of the first kind. After performing the Range-DopplerRD imaging processing, we obtain the image
in Fig. 1.
Figure 1. Point echo RD image
In Figure 1, the point target after imaging isn’t a circle area function Somb while it is a 2D Sinc function which is shown
below:
, sin
sin
x y
I x y H
c x
x c
y y
.
10
2.1 Parameter Estimation based on QE peak
Sinc function with Gaussian mask is equal to the Fourier transform of Gaussian function multiplying rectangular window
in frequency domain. Here, we transfer the result to frequency domain for solving the problem of parameter estimation.
Assume sincπxsincπy is unit 2D Synthetic Sinc Model. The
width of Sinc function in time domain is τ
x
, τ
y
. The width of Sinc function in frequency domain is
τ
u
, τ
v
. We can then get the scale variation of
τ
u
’, τ
v
’ at X and Y direction in frequency domain because of the change of
σ
x
, σ
y
in time domain. In time and frequency domains, the scaled transfers of the bottom of
peak are:
,
x x
x u
u x
11 ,
y y
y v
v y
12
τ
x
’, τ
y
’ denotes the radius of ellipse bottom at x axis and y axis in time domain respectively;
τ
u
’, τ
v
’ represents radius of rectangle bottom at x axis and y axis in frequency domain respectively.
The Fourier Transform of Gaussian Function and Sinc function can be expressed as:
2 2
2 2
2 2
, e
l y
t x
m x y
F
13
, ,
f
I x y G
u v
F
14 Adopting the form of 13, 14. A simple form of IFT can then
be obtained as
2 2 2 2
2
4 2
2 1
1
, ,
e
t l
f
u v
F u v G
u v
F
F
15 According to 1D inverse Fourier Transform, the result of 1D
IFT is given as,
sin
x x
c x
m
1
2 1
x x
F j
x u
x x
F
G u e
, 16
Then
2 2
1 2
2
2
t x
x
jx j x
x x
x
f x F
G e
e e
d
F
2 2
2 2
2
2 2
2 2
x t
x x
t t
x
x x
x x
j x
t t
e e
d
. 17
Since ω is independent of x, the integral can be expressed as
2 2
2 2
2
2 2
2 2
x t
x x
t t
x
x x
x x
j x
t t
e e
d
, 18
2 2
2
2
x t
x x
x x
x t
e
.
19 When σ
t
≥1, the Gaussian function
2 2
2
2
t
t
e d
is in the interval [ πσ
x
, πσ
x
]. The integral can be approximated as the area of G
2 πσ
x
σ
x
ω : SG
1 σ
x
σ
x
u , where
ω is angle frequency and f means frequency, and
ω=2πf. Then G
2 πσ
x
σ
x
ω
=
G
1 σ
x
σ
x
u . fx becomes
2 2
2 1
2
x t
x
x x
x x
t
f x S G
u e
20
2015 International Workshop on Image and Data Fusion, 21 – 23 July 2015, Kona, Hawaii, USA
This contribution has been peer-reviewed. doi:10.5194isprsarchives-XL-7-W4-45-2015
46
Therefore the 2D convolution of fx, y can be approximated as the multiple of volume VG
τ
f
’
σ
x
u, σ
y
v and Gaussian function,
as be described in Fig.3 and Fig.4.
2 2
2 2
2 2
, ,
2
y x
t l
f
y y
x x
x y
x y
t l
f x y V G
u v
e
. 21
Where,
,
f
x y
u v x
y
V G u
v H
. 22
-10 -5
5 10
-10 -5
5 10
-0.5 0.5
1
Range a3D Synthesized Sinc Peak without Gauss Mask
Azimuth A
m pl
iyud e
bFFT Contour Synthesized Sinc Peak without Gauss Mask Range
A zi
m ut
h
-1.5 -1
-0.5 0.5
1 1.5
-1.5 -1
-0.5 0.5
1 1.5
Figure 3 3D synthesized Sinc peak
-10 -5
5 10
-10 -5
5 10
-1 1
Range a3D Synthesized Sinc Peak with Gausst=l=1 Mask
Azimuth A
m pl
iyud e
-10 -5
5 10
-10 -5
5 10
-0.05 0.05
Range c3D Synthesized Sinc Peak with Gausst=l=4 Mask
Azimuth A
m pl
iyud e
-10 -5
5 10
-10 -5
5 10
0.01 0.02
Range e3D Synthesized Sinc Peak with Gausst=l=13 Mask
Azimuth A
m pl
iyud e
bFFT Contour Synthesized Sinc Peak with Gausst=l=1 Mask Range
A zi
m ut
h -1.5
-1 -0.5
0.5 1
1.5 -1
1
dFFT Contour Synthesized Sinc Peak with Gausst=l=4 Mask Range
A zi
m ut
h -1.5
-1 -0.5
0.5 1
1.5 -1
1
fFFT Contour Synthesized Sinc Peak with Gausst=l=13 Mask Range
A zi
m ut
h -1.5
-1 -0.5
0.5 1
1.5 -1
1
Figure 4 Compare Sinc Mask effect by different σ Guass
function In Figure 4, simulations show that the results of Sinc function
masked with Gauss function with different values of width in time and frequency domains. In Figure 3, the original peak
model is σ
x
=1, σ
y
=1, τ
x
=2, τ
y
=2， then τ
u
’=1, τ
v
’=1 in frequency domain. We find the value ofσ
t
,σ
l
is the more crossed the σ
x
, σ
y
, the result τ
u
’, τ
v
’ of frequency is the better than others in Figure 4. And, we can prove that the equations above are
correct because the simulation results are close to the Gaussian Function with window. But when σ
t
,σ
l
are over large, the peak will be disappeared.
For extracting peak parameters, we should make sure the position x, y.The first derivative of a local maximum curve is
zero. The peak position can be defined as the point which is both of first derivative zero-crossing points transverse and
longitudinal directions. Finding the interested point of two zero- gradient level line in x and y directions. So, f
x
, f
y
mean first- order partial derivative in x and y directions and f
xx
, f
yy
mean second-order partial derivative. When f
x
x , y
=0 f
y
x , y
=0 ,and f
xx
x , y
0, f
yy
x , y
0,we can find the position of peak value x
,y . The method same as the Sythtic Gauss Peak
modelSGP in [11].In Figure 5, We can see when mask is 4 pixel, the estimation effect is better than other mask widths with
10dB noise. So we should choose the fit mask width for different Sythtic Sinc Peak modelSSPσ width for extracting
exactly position information.
1 2
3 4
5 6
-0.2 -0.15
-0.1 -0.05
0.05 0.1
0.15 0.2
aEstimation Peak Position for Centeredx=0.6 y=0.1SSP By QE
Width of SSP with addnoise and multiplnoise E
st im
a te
d P
e a
k P
o si
ti o
n x-
x0 Mask width=1p
Mask width=2p Mask width=4p
Best Maskw line
1 2
3 4
5 6
1 2
3 4
5 6
bstd. of Estimation Peak Position for Centeredx=0.6 y=0.1SSP By QE
Width of SSP with addnoise and multiplnoise st
d .
o f
E st
im a
te d
P e
a k
P o
si ti
o n
x- x0
Mask width=1p Mask width=2p
Mask width=4p Best Maskw line
Figure 5 QE estimation of posion when Sinc peak is at x=0.6,
y=0.1 with 10dB noise We can then use QE algorithms to detect parameters of peak.
According to f
x
x, y=Ix, y m
x
x, y, f
xx
x, y=Ix, y m
xx
x, y, we have
2 4
2
, 1
,
xx x
t t
f x y
x x
f x y
23
2 4
2
, 1
,
y yy
l l
y y
f x y
f x y
24
, 2
2
f
x y
x y
u v t
l t
l
A V G u
v H
2
t l
u v
A H
25 where A means test amplitude,
τ
u
=1, τ
v
=1.Using 23-25, σ
x
, σ
y
, H can be solved.
1 2
3 4
5 6
1 2
3 4
5 6
a Estimation Peak W idth for Centeredx=0.6,y=0.1SSP By QE
Width of SSP with addnoise and multiplnoise E
st im
a te
d P
e a
k W
id th
si g
m a
x
Mask width=1p Mask width=2p
Mask width=4p Best Maskw line
1 2
3 4
5 6
0.002 0.004
0.006 0.008
0.01 0.012
0.014 0.016
0.018 0.02
bstd. of Estimation Peak Width for Centeredx=0.6,y=0.1SSP By QE
Width of SSP with addnoise and multiplnoise st
d .
o f
E st
im a
te d
P e
a k
W id
th si
g m
a x-
si g
m a
x0 Mask width=1p
Mask width=2p Mask width=4p
Best Maskw line
Figure 6 QE estimation of width when Sinc peak is at x=0.6,
y=0.1 with 10dB noise
2015 International Workshop on Image and Data Fusion, 21 – 23 July 2015, Kona, Hawaii, USA
This contribution has been peer-reviewed. doi:10.5194isprsarchives-XL-7-W4-45-2015
47
1 2
3 4
5 6
0.2 0.4
0.6 0.8
1 1.2
1.4 1.6
1.8 2
aEstimation Peak Amplitude for Centeredx=0.6,y=0.1SSP By QE
Width of SSP with addnoise and multiplnoise E
st im
a te
d P
e a
k A
m p
li tu
d e
A Mask width=1p
Mask width=2p Mask width=4p
Best Maskw line
1 2
3 4
5 6
0.02 0.04
0.06 0.08
0.1 0.12
0.14 0.16
0.18 0.2
bstd. of Estimation Peak Amplitude for Centeredx=0.6,y=0.1SSP By QE
Width of SSP with addnoise and multiplnoise st
d .
o f
E st
im a
te d
P e
a k
A m
p li
tu d
e A
-A Mask width=1p
Mask width=2p Mask width=4p
Figure 7. QE estimation of amplitude when Sinc peak is at x=0.6, y=0.1 with 10dB noise
In SSP proposed, its size of sample is 41 41
.Assure A=1, width changes from 1 to 5 pixel. We start QE estimation of
width and amplitude of peak in Fig. 6 and Fig. 7. The error of width estimation is 10
-2
. The error of amplitude estimation is 10
- 2
.This method is more effective than SGP
[11]
. 3. SYNTHETIC SINC MODEL FLUCTUATION
ANALYSIS 3.1 Position Fluctuation Analysis
Because noise causes the fluctuation of the estimation of peak and width estimates, we can use the variance of fluctuation to
evaluate different estimation algorithm.
, ,
h x y I x y
I
26 δ is the fluctuation of peak.
, ,
, ,
f x y
h x y m x y
I m
I m
f x y f
, 37
where
, f
I m x y
27 In the same way, we can obtain the equation below:
, ,
,
x x
f x y
I x y m
x y
28
, ,
,
xx xx
f x y
I x y m
x y
29
The first order and second order derivatives are derived as below:
,
x x
f I
m x y
30
,
xx xx
f I
m x y
31 In the simulation, we add the noise which observes Gamma
distribution where
2
1
N
n
. The first order and second order variance can be calculated as
2
1 4
N l
t
Var f
32
2 5
3 16
x N
l t
Var f
33
2 7
15 32
xx N
l t
Var f
34 In Figure 8,
α refers to the coefficient of fluctuation in one pixel. and
α＜1, α=f
x1
f
x1
﹣f
x2
，
x t
35 t means one pixel,
x
,
2 1
1 2
2 1
2
1
x x
x x
x x
x f
f f
f f
f
36
2 1
2
1
x x
x
Var x
Var f
f f
37
1- αt
αt X
1 x
f
2 x
f
x f
Figure 8 Peak position f Fluctuation
When x
1
and x
2
is very close, f
x1
﹣f
x2
is an approximate of the second order derivative of f. In the Sinc peak, we can obtain the
fluctuation of peak is
6 2
2 2
2 2
5
4 3
16
t l
N u
v l
t
Var x
H
38
3.2 Fluctuation Analysis of width and amplitude
