An Analysis of Polarization Coherence Tomography Using Different Function Expansion
Ma Peifeng(1,2),Zhang Hong(1), Wang Chao(1),Zhang Bo(1), Wu Fan(1), Tang Yixian(1)
(1)Key
Laboratory of Digital Earth, Center for Earth Observation and Digital Earth, CAS,
No.9 Dengzhuangnan Road, Haidian District, Beijing, 100094, China
(2)Graduate
University of Chinese Academy of Sciences, Beijing, China
E-mail: peifengma1986@gmail.com ,
hzhang@ceode.ac.cn
employ
Abstract
multi-baseline
data.
In
addition,
while
In this paper we investigate the Polarization
dual-baseline data can be used to introduce another two
Coherence Tomography technique and propose a
F-L polynomials, its poor condition that defines the
different function expansion to reconstruct vertical
stability of inversion and sensitivity to noise confines the
profile function. Firstly we develop a new orthogonal
application. In [3] a regularization technique has been
family and then the coefficients are estimated by matrix
proposed at cost to loss of precision. F-L polynomials
inversion for the specific series. Finally we represent the
are orthogonal on [-1,1] by weight of 1, however, in
vertical profile function using these polynomials. Further
practice for various two-layer models mixed surface plus
we analyze the stability in new expansion compared to
volume case, scattering amplitude is stronger on the top
Fourier-Legendre approximation. In the end this method
of canopy corresponding to volume scattering and near
is validated using simulated dual-baseline data.
the ground corresponding to surface-canopy dihedral
Keywords: PCT, dual-baseline
response.
Therefore,
we
investigate
the
feasibility
of
reconstructing the vertical profile function in a new
I. Introduction
Polarization Coherence Tomography (PCT) is an
advanced approach used to reconstruct the vertical
orthogonal family in this paper. Firstly in sectionⅡwe
profile function in penetrable volume scattering [1],
summarize
which overcome the limitation of traditional methods in
tomography problem as a new orthogonal series
3-D SAR imaging that they suffer from an inherent
expansion. Then In section Ⅲ the validity of this
requirement for collection of a large number of
the
main
procedure
of
formulating
operational baselines. The vertical profile function can
expansion is demonstrated using simulated dual-baseline
be
data. Further
represented
by
the
Fourier-Legendre
(F-L)
we evaluate the stability of this
polynomials if only single- baseline data is available. As
approximation approach and compare it with the results
a new application technology in radar remote sensing,
in Fourier-Legendre series by Cloude in simulated
PCT is important for improved vegetation species
scenario. Finally in section Ⅳ
discrimination,
component
biomass
analysis
and
biodiversity studies [2].
we provide our
conclusions and discussions.
Assuming a priori knowledge of volume depth and
ground topography, Tomography reduces to solution of a
set of linear equations for the unknown coefficients
Ⅱ. PCT Using New Function Expansion
using PCT technology. For single-baseline data, only
PCT is a radar image processing technology that
two unknown coefficients can be obtained in F-L series,
employs function expansion to reconstruct normalized
which limits the approximation resolution. In order to
vertical profile of arbitrary polarization channel using
achieve higher accuracy of the estimation, we must
F-L series. Here instead of generating profile in F-L
series, we deduce orthogonal functions on [-1,1] by
2
weight of x , the first few polynomials are shown as:
P0 (z ) = 1
P1 ( z ) = z
1
(5 z 2 − 3)
2
1
P3 ( z ) = (7 z 3 − 5 z )
2
1
P 4 ( z ) = (6 3 z 4 − 7 0 z 2 + 1 5)
8
P2 ( z ) =
(1)
Now we turn to the basic definition of interferometric
complex coherence [1]:
Figure 1.
For the function a ( z ') , we can develop it with series:
a( z ') = ∑ an Pn ( z)
hv
∫
γ = e ik
z
z0
f ( z )e
ik
z
z
Plot of basis functions
dz
n
0
(5)
hv
∫
f ( z )d z
By a chain of the transformation as mentioned above
(2)
0
coherence can be written as:
Where Z0 is the position of the bottom of scattering layer,
Kz is vertical wave number. To retrieve the vertical
structure f ( z ) in functions we first normalize the range
of integral by a change of variable , results are shown as:
1
∫ (1+ a P (z ') + a P(z ') + a P(z ') +...)z ' e
2 ikv z '
0 0
~
γ=
1 1
1 1
dz '
−1
1
∫ (1+ a P (z '))z ' dz '
2
0 0
−1
hv
∫
0
h i k z hv
f ( z ) e ik z z dz = v e 2
2
1
∫
g ( z ')e
i
k z hv
z'
2
−1
1
1
−1
−1
(3)
=
Instead of expanding the function q ( z ') directly in
polynomials,
q ( z ') = a ( z ') z '2 .
we
make
an
assumption
−1
1
(1+ a0 ) ∫ z '2 dz '
We rescale the range g(z ') =1+q(z ') so that q ( z ') ≥ −1 .
Legendre
1
(1+ a0 ) ∫ z '2 eikvz 'dz ' + a1 ∫ P1(z ')z '2 eikvz'dz ' + a2 ∫ P2 (z ')z '2 eikvz 'dz ' +...
dz '
−1
=3
(1+ a0 ) f0 + a1 f1 + a2 f2 +...an fn
(1+ a0 )
= 3( f0 + a10 f1 + a20 f2 +...an0 fn )
(6)
Then the real polynomials used to
where
approximate can be written as:
unknown
Q0 (z) = z 2
Q1 ( z) = z3
1
Q2 ( z ) = z 2 (5z 2 − 3)
2
1 2 3
Q3 ( z ) = z (7 z − 5z)
2
1
Q4 ( z ) = z 2 (63z 4 − 70z 2 + 15)
8
normalized:
k
v
=
hvk
2
coefficients
an 0 =
~
z
,
by
γ = γ e − ik z e − ik , the
z 0
zero-order
v
term
are
an
1 + a 0 . So Evaluation of the each
component involves determination of the function f i ,
which is defined from the product of interferometric
(4)
wave number k z and vegetation height hv . The even
index functions are real while odd are pure imaginary.
If we can calculate the unknown coefficients an0 by
inverting this relation, the function of relative scattering
phase to develop the vertical structure, which is
density can be obtained as:
illustrated by the vertical scattering function of one
f (z) = (2z / hv −1)2(1+a10P1(2z / hv −1) +a20P2(2z / hv −1) +...an0Pn(2z / hv −1)), 0 ≤ z ≤ hv.
As
for
dual-baseline
data,
the
first
pixel P in Figure 4 and tomography along range line
AA’ in Figure 5.
five
polynomials will be used to approximate. Linear
formulation can be written as shown in equation (7) :
f

0
3 y
 f1
0
x
1
x
3
0
f
f
x
2
0
0
f3y
f2y
0
~


Im ag(γ x )


0  a10 
~



f4x  a20  Re al (γ x ) − 3 f0x 
.
=

~
0  a30  
y

  Im ag(γ )

f4y  a40  
~

y
y
Re al (γ ) − 3 f0  .(7)
Estimated tree height
Figure 3.
where x and y are identification of two baselines. Four
coefficients can be inverted using dual-baseline data.
Ⅲ. Tomography Reconstruction Using Dual-Baseline
Figure 4.
POLInSAR Data in New Function Expansion
Profile at Point P
In order to demonstrate the effectiveness of the new
function expansion in the generation of coherence
tomography,
we
POLInSAR
data
employ
L
simulated
band
by
dual-baseline
ESA
released
POLSARPro. The forest is initialized deciduous with the
height of 10m and the ground phase of 0.
Tomography of arbitrary polarization channel can be
developed. We mainly investigated two representative
Figure 5.
Polarimetric tomography along line AA’
polarization: cross polarized HV channel (volume
Note that in the HV polarization channel scattering
scattering dominated) and copolarized HH-VV channel
amplitude close to the crown is stronger, while in
(surface dihedral response dominated). The coherence of
HH-VV scattering amplitude close to the ground is
HV and HH-VV polarization is shown in Figure 2.
stronger, corresponding to the volume dominated and
dihedral
response
dominated
in
practice.
The
reconstruction of CT is inevitably subject to the effect
of noise, such as temporal decorrelation, statistical
fluctuations in coherence estimation and coherence bias
with limited data samples, incurring some ambiguity of
HV
Figure 2.
HH-VV
Complex coherence (L = 11)
the results such as the volume dominated in HH-VV
and dihedral response dominated in HV shown in
We retrieve forest height using the Three-Stage method,
Figure 5. So the sensitivity to noise is a key point we
the estimated height are as shown in Figure 3. Then we
should consider. From the matrix inversion formula
employ the estimated forest height and topography
[ F ] a = g we find that the stability of inversion is
attributed to the condition number (CN) of matrix [F].
As for single-baseline in Fourier-Legendre expansion
In this paper we have employed PCT technology to
CN can be expressed as:
CN = −
1
k v2
= −
sin ( k v )
f2
3 co s( k v ) − (3 − k v2 )
kv
(8)
reconstruct the vertical profile function of forest and
introduced a new function expansion approach. This
method
and in term of the new expansion,CN can be written as :
CN=-
Ⅳ.Conclusion
1
1
=−
3sin( k v ) 21cos( k v ) 81sin( k v ) 180 cos( k v ) 180 sin( k v )
f2
+
−
−
+
kv
k v2
k v3
k v4
k v5
has
been
validated
using
POLSARPro
simulated dual-baseline data. Since dual-baseline
provides two complex coherences, four coefficients can
be estimated and the highest polynomial order used to
(9)
approximate is four in F-L expansion. In the new
We draw the two function curves in the same
expansion, however, the highest order is six and the
coordinates in Figure 6 and find that values of CN in
results can exhibit the internal fine structure. In addition,
the approximation method of this paper are smaller than
due to better conditioning of the matrix inversion, the
in Fourier-Legendre by Cloude. That is to say, inversion
is better conditioned and system is less sensitive to
measurement vector a is less susceptible to noise,
which is always present in synthetic aperture radar
errors. In order to validate the advantage of this
interference processing.
approximation, we compare the standard variance of
coefficients using dual-baseline data above in two
expansion method in TableⅠ, from which it can be seen
that std in the new expansion is less than that in
Fourier-Legendre series.
V. Acknowledgment
This work is supported by National High-tech R&D
Program of china
National
Ⅰ
Table . Standard variance of profile coefficients for HV
channel in two function expansion
（
Natural
Grant No.2009AA12Z118）and
Science
Fund
Project(No.
40971198 and No. 40701106).
References
[1] S. R.Cloude,“Polarization coherence tomography”,
Fourier-Legendre
New Expansion
a10
0.42
0.23
Radio Sciences, 41:1-27,2006
a20
0.85
0.66
[2] Mette, T., “Forest Biomass Estimation from
a30
1.54
1.25
Polarimetric SAR Interferometry”, DLR Research
a40
13.20
0.003
Report 207-10, ISSN 1434-8454, 2007
[3] S. R. Cloude, “Dual-Baseline Coherence
Tomography”, IEEE Geoscience and Remote Sensing
Letters, 4(1):127-131, 2007
[4] Papathanassiou, K. P., and S. R. Cloude, “Single
baseline polarimetric SAR interferometry”, IEEE Trans.
Geosci. Remote Sensing., 39, 2352-2363, 2001
[5] S. R. Cloude, and Papathanassiou, K. P., “Forest
Figure 6.
Comparison of condition number: Red
(equation (8)),
Green(equation (9))
Vertical Structure Estimation Using Coherence
Tomography”, IGARSS2008,2008.