Multi-polarization reconstruction from compact polarimetry based on modified four-component...

Journal of Systems Engineering and Electronics

Vol. 25, No. 3, June 2014, pp.1–9

Multi-polarization reconstruction from compact polarimetrybased on modified four-component scattering decomposition

Junjun Yin∗ and Jian Yang

Department of Electronic Engineering, Tsinghua University, Beijing 100084, China

Abstract: An improved algorithm for multi-polarization reconstruc-tion from compact polarimetry (CP) is proposed in this paper. Ac-cording to two fundamental assumptions in compact polarimetricreconstruction, two improvements are proposed. Firstly, the four-component model-based decomposition algorithm is modified witha new volume scattering model. The decomposed helix scatteringcomponent is then used to deal with the non-reflection symmetrycondition in compact polarimetric measurements. Using the de-composed power and considering scattering mechanism of eachcomponent, an average relationship between the co-polarized andcross-polarized channels is developed over the original polariza-tion state extrapolation model. E-SAR polarimetric data acquiredover the Oberpfaffenhofen area and JPL/AIRSAR polarimetric dataacquired over San Francisco are used for verification and good re-construction results are obtained, demonstrating the effectivenessof the proposed method.

Keywords: polarimetric synthetic aperture radar (SAR), targetdecomposition, compact polarimetry (CP), multi-polarization re-construction.

DOI: 10.1109/JSEE.2014.000

1. Introduction

The polarimetric synthetic aperture radar (PolSAR) hasbeen widely used in many earth observing applications,such as terrain classification [1–4], land cover monitoring[5–7], and targets detection [8–10]. The space-borne fullypolarimetric SAR sensor has many advantages. However,it suffers from increment of the pulse repetition frequency,the power consumption, and the downloading data rate. Inaddition, the imaging coverage of full polarimetry is onlyhalf the width of a single-polarized or dual-polarized sys-tem. The dual polarization SAR system is a compromisingchoice between full polarization and single polarization. A

Manuscript received October 19, 2012.*Corresponding author.This work was supported by the National Natural Science Foundation

of China (41171317), the State Key Program of the Natural Science Foun-dation of China (61132008), and the Research Foundation of TsinghuaUniversity.

dual-polarized SAR, which transmits a single polariza-tion and receives two orthogonal polarizations, does notprovide complete information of targets pertaining to thequadrature polarization states, but it offers more informa-tion than a single-polarized system [11,12].

In order to obtain more information from the dual polar-ization, Souyris et al. proposed a dual polarization imag-ing mode, i.e., compact polarimetry (CP) [12–15], basedon one unique special transmitted polarization and two or-thogonal polarizations in reception. There are mainly twoways to cope with the CP measurements. One is to use CPdata directly without any assumptions; the other is to re-construct the multi-channel polarimetric information overextended/distributed targets from the CP design. In themulti-polarization reconstruction procedure, two assump-tions are very essential. One is the well-known reflectionsymmetry assumption, and the other is the polarizationstate extrapolation model. With both the assumptions, aniterative process was introduced by Souyris et al., and thereconstructed polarimetric data performed well to a certaindegree [12–15].

However, the reflection symmetry is not always validespecially in urban areas, where the reconstructed resultsare always far from the actual values. In order to derivemore target information from CP and accommodate thefully polarimetric (FP) information reconstruction schemewith the more general scattering cases, an improved polari-metric information reconstruction algorithm is proposedin this paper. This algorithm is based on modified four-component decomposition with a new volume scatteringmodel. By assuming a coherency matrix is totally decom-posed into four individual components, an average extrap-olated model relating to the different scattering mecha-nisms is proposed. Using the proposed reconstruction al-gorithm, the helix scattering power can be estimated fromthe 2 × 2 CP covariance matrix.

The outline of this paper is given as follows. A brief

2 Journal of Systems Engineering and Electronics Vol. 25, No. 3, June 2014

overview of the four-component decomposition [16] andthe modified decomposition method are summarized inSection 2. In Section 3, the linear π/4 compact polarimet-ric mode is briefly described, and then a multi-polarizationreconstruction algorithm is proposed. Two sets of polari-metric SAR (PolSAR) data are used for demonstrating theeffectiveness of the proposed reconstruction method. Thecorresponding experimental results and analysis are pre-sented in Section 4. Section 5 draws the conclusions.

2. Modified four-component modeldecomposition

In a linear horizontal (H) and vertical (V) polarization base,the Pauli target scattering vector is defined as follows underthe reciprocity principle for the monostatic backscatteringcase.

kp =1√2[SHH + SVV SHH − SVV 2SHV]T (1)

where SHV is the backscattered coefficient by V in trans-mission and H in reception. In the multi-look case, the scat-tering coherency matrix 〈T 〉 is usually used to deal withstatistical scattering effects. 〈T 〉 given in the following is anon-negative definite Hermitian matrix.

〈T 〉 = 〈kpkHp 〉 (2)

where 〈·〉 denotes the ensemble averaging, and the super-script H denotes the conjugate transpose.

2.1 Four-component decomposition

The Yamaguchi four-component decomposition modelsthe coherency/covariance matrix as the contribution offour scattering mechanisms, i.e., the surface scattering, thedouble-bounce scattering, the volume scattering, and thehelix scattering [16].

〈T 〉 = fs · Ts + fd · Td + fv · 〈T 〉v + fc · Tc (3)

where fs, fd, fv, and fc correspond to the coefficientsof the four scattering components, which are non-negativeand proportional to their powers, Ts, Td, and Tc denote thesurface scattering, the double-bounce scattering, and thehelix scattering models, respectively, and they are modeledas follows based on different physical scattering mecha-nisms.

Ts =

⎡⎣ 1 β∗ 0

β |β|2 00 0 0

⎤⎦ , Tc =

⎡⎣ 0 0 0

0 1 ±j0 ∓j 1

⎤⎦ ,

Td =

⎡⎣ |α|2 α 0

α∗ 1 00 0 0

⎤⎦ (4)

where α and β are unknown parameters to be de-termined. There are three volume scattering modelsfor 〈T 〉v, and the choice is based on the value of10 lg(|SVV|2/|SHH|2). Please refer to [16] for more de-tails.

When the original four-component decomposition is ap-plied to the real PolSAR data, some scattering componentpowers may become negative. To overcome this problem,an improved decomposition is proposed with a power con-straint [17]. The basic principle is that if the decomposedpower becomes negative, then the power is forced to zeroand let the sum of the decomposed powers be equal tospan.

span =12〈|SHH + SVV|2〉+

12〈|SHH − SVV|2〉 + 2〈|SHV|2〉

(5)

where span is the Frobenius norm of the scattering vec-tor. Four-component decomposition has been successfullyapplied to analyze the PolSAR data, especially for the areawith man-made targets (e.g., the urban area) where the re-flection symmetry condition does not hold.

2.2 Volume scattering model

The volume scattering can be regarded as an ensemble av-eraging of chaotic scattering states, and cannot be char-acterized as a deterministic scattering process. Instead ofthe original three volume models in [16], a special volumescattering model [18] 〈T 〉vol is adopted in this paper.

〈T 〉vol =

⎡⎣ 1 0 0

0 1 00 0 1

⎤⎦ . (6)

There are three reasons for us to choose this model forFP information reconstruction. First, it follows from thefully depolarized wave assumption. This model satisfiesthe polarization state extrapolation model [12] which isrelated to the intensity ratio between cross-polarized andco-polarized channels and the linear co-polarization cohe-rence |ρ|.⎧⎪⎨

⎪⎩〈|SHV|2〉

〈|SHH|2〉 + 〈|SVV|2〉 =14(1 − |ρ|)

ρ = 〈SHHS∗VV〉/

√〈|SHH|2〉〈|SVV|2〉

. (7)

This model is extrapolated from the case where thebackscattered wave is either fully polarized or fully de-polarized. For a fully polarized backscattered wave froma simple point target, |ρ| ≈ 1; for a fully depolarizedbackscattered wave, which means that the co-polarizedchannels are almost completely uncorrelated, and theaverage power received by the orthogonal antennas donot depend on their polarization states, |ρ| ≈ 0 and

Junjun Yin et al.: Multi-polarization reconstruction from compact polarimetry based on modified four-component scattering... 3

< |SHH|2 >≈< |SVV|2 >≈ 2 < |SHV|2 >. Tak-ing the Freeman volume scattering model [19] for exam-ple, we have 〈|SHV|2〉/(〈|SHH|2〉 + 〈|SVV|2〉) = 1/6 and|ρ| = 1/3. Though both the values follow from the factor1/4 in (7), the co-polarization coherence |ρ| �= 0, whichis not consistent with the fully depolarized backscatter-ing case [14, 15]. Using the volume scattering model in(6), we have 〈|SHV|2〉/(〈|SHH|2〉 + 〈|SVV|2〉) = 1/4 and|ρ| = 0. Both the values are consistent with the relation-ship model in (7) and the fully depolarized backscatteringwave assumption.

The second reason is that Tvol has the maximum polari-metric entropy H . H introduced by Cloude and Pottier rep-resents the randomness of backscattered waves. High en-tropy circumstance often occurs in the region with highlyanisotropic scattering elements, e.g., scattering from for-est canopies and scattering from vegetated surfaces. In ex-treme, H = 1 denotes totally random scattering. This sit-uation is expected to occur when a significant amount ofmultiple scattering is present such as in the case of scatter-ing from extremely rough surfaces. If some deterministicscattering process where H = 0 is added to the modelTvol, the backscattering will be less random, leading toH < 1. Thus, all the scattering models can be regarded asan addition of Tvol and deterministic scattering processes,and the balance among them determines H . Thus, Tvol isselected as a description of ideal random scattering.

The third reason is that Tvol is an azimuthally symmet-ric scattering model, which reduces the orientation angleeffect when the target decomposition is expanded. Fromthe above analysis, the model (6) is reasonably employedfor characterizing the most random scattering targets, thena simple relationship between the helix scattering and thevolume scattering components can be obtained by using

this model. This is the basic idea of the modified four-component decomposition. The derived relations betweendifferent scattering models will be used for the reconstruc-tion of pseudo FP information.

2.3 Modified four-component decomposition

From the scattering models defined in (4) and (6), the co-herency matrix can be decomposed into four scatteringcomponents. By comparing the measured data with boththe sides of (3), we can derive the helix scattering powerand the volume scattering power as follows:

{fc = Im|〈(SHH − SVV)S∗

HV〉|, Pc = 2fc

fv = 2〈|SHV|2〉 − fc, Pv = 3fv(8)

where Pv and Pc denote the powers of the volume scatter-ing and the helix scattering, respectively. The flow chart ofthe whole modified four-component model-based decom-position algorithm is given in Fig. 1, where Ps and Pd arethe decomposed powers of the surface scattering and thedouble-bounce scattering components, respectively. Theforemost reason for adopting model (6) to decompose thecoherency matrix is that we can obtain a simple relation-ship expressed in (8) to relate the elements in a 3 × 3FP matrix to a 2 × 2 CP matrix. In the original Yam-aguchi four-component decomposition, three volume scat-tering models derived from different probability densityfunctions are used for decomposition, which leads to mul-tiple relationships between the helix scattering componentand the volume scattering component. The point of this pa-per is not to analyze the target scattering characteristics butto reconstruct the FP information from the linear compactmode. Using the new model in (6), a simpler formula (8)is obtained to relate the helix and the volume scattering

Fig. 1 Flow chart of the modified four-component decomposition


components to the cross-polarized measurement< |SHV|2 >. In this way, the helicity parameter fc can alsobe estimated from the reconstruction procedure. Based onthe decomposed powers, an improved multi-polarizationreconstruction algorithm is presented in next section.

3. Multi-polarization reconstruction

3.1 π/4 mode compact polarimetry

The π/4 mode [12] features two linear receiving polariza-tions, i.e., the horizontal and vertical polarizations, andthe transmitting polarization which is linear and orientedat 45◦. The scattering vector for the π/4 mode is given by

kπ/4 =[SHH + SHV SVV + SHV]T√

2. (9)

For simplicity, the constant coefficient 1/√

2 is omittedhereafter. Then Cπ/4 = 〈kπ/4k

Hπ/4〉 given in (10) is the

corresponding covariance matrix, which can be regardedas contributions of three parts, i.e., the part associated withthe co-polarized channels, the cross-polarized channel in-formation, and the correlations between co-polarized andcross-polarized backscattering coefficients.

Cπ/4 =[

C11 C12

C∗12 C22

]=

⟨[ |SHH|2 SHHS∗VV

SVVS∗HH |SVV|2

]⟩+

〈|SHV|2〉[

1 11 1

]+

⟨[2Re(SHHS∗

HV) SHHS∗HV + SHVS∗

VV

SHVS∗HH + SVVS∗

HV 2Re(SVVS∗HV)

]⟩(10)

where Cπ/4 is a semi-definite Hermitian matrix. Underthe assumption of reflection symmetry, there is a com-plete de-correlation between the co-polarization and cross-polarization, i.e., 〈SHHS∗

HV〉 = 〈SVVS∗HV〉 = 0.

Regarding the third term in (10) as zero by assumingthe reflection symmetry, we have an underdetermined sys-tem of three equations and four unknown variables, i.e.,〈|SHH|2〉, 〈|SHV|2〉, 〈|SVV|2〉 and 〈SHHS∗

VV〉. In order toconstruct the reflection symmetric fully polarimetric infor-mation, the pseudo deterministic trend (7) is used to re-late the four unknowns. 〈|SHV|2〉 is the key parameter forthe solution and can be solved by iteration. Please refer to[12] for more details of the Souyris’ reconstruction algo-rithm. The iteration termination condition is that either theco-polarized coherence is |ρ| = 1 or 〈|SHV|2〉 is conver-gent. If a converged value of 〈|SHV|2〉 is obtained as X ,the reconstructed fully polarimetric covariance matrix for

extended targets is shown as follows:

Cπ/4−FP =

⎡⎣C11 − X 0 C12 − X

0 2X 0C∗

12 − X 0 C22 − X

⎤⎦ . (11)

3.2 New polarization state extrapolation model

Two assumptions are introduced to perform the FP recon-struction procedure. One is the reflection symmetry, andthe other is the polarization state model. The reflectionsymmetry applies reasonably well in general analysis ofnatural distributed scatterers. For urban areas, however, thereflection symmetry does not hold because of the strongpoint target reflection. In order to accommodate the recon-structed results for more general cases and extract morepolarization information, it is necessary to consider an-other physical scattering mechanism which corresponds to〈SHHS∗

HV〉 �= 0 and 〈SVVS∗HV〉 �= 0. Therefore the modi-

fied four-component decomposition is adopted here. Fromthe third part of (10), we have

Im〈SHHS∗HV + S∗

VVSHV〉 = Im〈(SHH − SVV)S∗HV〉

(12)where Im〈(SHH − SVV)S∗

HV〉 is the helix scattering. LetF = Im(SHHS∗

HV + SHVS∗VV). The covariance matrix of

the π/4 mode can be approximated by

Cπ/4 ≈⟨[ |SHH|2 SHHS∗

VV

S∗HHSVV |SVV|2

]⟩+〈|SHV|2〉

[1 11 1

]+

⟨[0 jF

−jF 0

]⟩. (13)

Next we consider the polarization state model. Consi-dering the helix scattering component, we propose an im-proved relationship based on the modified four-componentdecomposition, which assumes that the coherency matrixof a pixel is totally contributed by four scattering compo-nents. Thus, we establish an average model with conside-ring each separate scattering mechanism.

First we calculate 〈|SHV|2〉/(〈|SHH|2〉 + 〈|SVV|2〉) and|ρ| according to (4) and (6), respectively. For the surfacescattering and the double-bounce scattering models, wehave

〈|SHV|2〉〈|SHH|2〉 + 〈|SVV|2〉 = 0, |ρ| = 1. (14)

Following van Zyl [20], we decide the co-polarized co-herence coefficient ρ is equal to 1 for the surface scat-tering, which means the resulting backscattered waves ofSHH and SVV are in phase, and we decide ρ is equal to−1 for the double-bounce scattering, which means that


the co-polarized phase difference will in general be nearly180◦. For the helix scattering model, we have

〈|SHV|2〉〈|SHH|2〉 + 〈|SVV|2〉 =

12, ρ = −1. (15)

The backscattered waves of these three models are fullypolarized, i.e., |ρ| = 1, which characterize deterministicscattering processes. In the volume scattering case, the po-larization ratio and the coherence coefficient are 1/4 and 0,respectively. Thus the volume scattering 〈T 〉vol is a fullydepolarized scattering of backscattered waves in extreme.

In order to acquire the relationship between the co-polarized and cross-polarized channels, in [12] a morephysical model based on the polarization properties is alsoinvestigated. Accordimg to the lightened rotation symme-try assumption [21], we have

4〈|SHV|2〉 = 〈|SHH|2〉 + 〈|SVV|2〉 − 2Re(SHHS∗VV).

(16)Three basic scattering mechanisms of the four mod-

els except the double-bounce scattering are consistentwith this hypothesis. However, the even number reflec-tion is an important scattering behavior in the backscat-tering process. Thus, reconstructed results extrapolated bythis model are not desirable. Though the relationship de-scribed in (7) has a better result [12], it does not quite fit thereal PolSAR data especially for urban areas with complexstructures. Combining with the advantages of both models,we propose an improved model.

By four-component decomposition, we assume that theFP coherency matrix has been decomposed and the corre-sponding results are Ps, Pd, Pv and Pc, respectively. Thenthe average value of 〈|SHV|2〉/(〈|SHH|2〉 + 〈|SVV|2〉) canbe regarded as the contribution of the four components. Wehave

〈|SHV|2〉〈|SHH|2〉 + 〈|SVV|2〉 =

14

Pv

span+

12

Pc

span. (17)

Assuming the backscattering covariance matrix is aweighted sum of the four scattering processes, the correla-tion coefficient ρ should be real (positive or negative) andρ ranges from −1 to 1. Similarly, the average value of ρ is

ρ =Ps

span− Pd

span− Pc

span. (18)

(18) can be rewritten as 1−ρ=(2Pd+2Pc + Pv)/span.For naturally distributed targets, ρ is complex. Thus, amodified polarization relationship considering the coher-ence coefficient phase for a more general scattering caseshould be established with |ρ|. The improved model is

shown as follows:

〈|SHV|2〉〈|SHH|2〉 + 〈|SVV|2〉 =

1 − sgn(Re〈SHHS∗VV〉)|ρ|

4

(2Pc + Pv

2Pd + 2Pc + Pv

)(19)

where sgn(x) is a signum function; Re(x) denotes thereal part of x. The right-hand side of (19) resembles (7),but with a coefficient which is less than one. From pre-vious researches related to compact polarimetry, a phe-nomenon observed by many researchers is that the valueof 〈|SHV|2〉/(〈|SHH|2〉 + 〈|SVV|2〉) is usually far smallerthan (1 − |ρ|)/4. Therefore, adding a coefficient (smallerthan 1) to the right side of (7) may produce a better re-construction result. Using the fully polarimetric E-SAROberpfaffenhofen data shown in Fig. 2, scatter plots of thetwo sides of (7) and (19) are shown in Fig. 3 (a) and (b),respectively. For a better equality, the scatter points shouldlie along the diagonal line to support the validity of the po-larization state model. However, few points fall close to thediagonal line in Fig. 3 (a), indicating that the extrapolatedmodel (7) is not always valid to fit the real data, at leastfor this data set. Using the proposed model, most data lieclose to the eye line in Fig. 3 (b), which is evident that (19)shows a better fit to the real backscattering mechanisms.

Next we will present how to estimate both the values of2Pc + Pv and 2Pd + 2Pc + Pv by using a simple approxi-mation approach based on the π/4 CP measurements.

Fig. 2 Pauli-basis image of original E-SAR Oberfaffenhofen FP

data


Fig. 3 Scatter distribution of the Oberfaffenhofen area

3.3 Parameter estimation

From the coherency matrices shown in (4) and (6), accor-ding to (10), we synthesize the corresponding compactscattering models as follows:

Cπ/4−s =[ |β|2 β

β∗ 1

], Cπ/4−d =

[ |α|2 αα∗ 1

]

Cπ/4−c =[

1 ±j∓j 1

], 〈C〉π/4−v =

[1.5 0.50.5 1.5

](20)

where Cπ/4−s, Cπ/4−d, Cπ/4−c, Cπ/4−v are the covari-ance matrices of the surface scattering, double-bouncescattering, helix scattering, and volume scattering com-ponents, respectively; α and β are defined after Freeman[19]. One constraint is that the powers of mutually re-lated components should be equal, i.e., span(Cπ/4−c) =

span(Tc) and span(〈C〉π/4−v) = span(〈T 〉vol), whichcan guarantee the decomposed powers from the CP modeare equal to those from the FP mode measurements.

We expand the compact covariance matrix as

Cπ/4 =fs·Cπ/4−s+fd·Cπ/4−d+fc·Cπ/4−c+fv·〈C〉π/4−v

(21)where fs, fd, fc, and fv are the expansion coefficients asthose defined in (3). Let the corresponding scattering po-wers be Ps, Pd, Pc, and Pv, respectively. By comparingthe measured data of the two sides of (21), we have⎧⎨

⎩fs + fd + fc + 1.5fv = C11

|β|2fs + |α|2fd + fc + 1.5fv = C22

βfs + αfd ± jfc + 0.5fv = C12

. (22)

Since the relationship between fc and fv has beenknown, as shown in (8), we have the previous threeequations with four unknowns α, β, fs, and fd, whichcan be solved in a similar manner as that in [19]. IfRe(C12) is positive, we decide that the surface scatter-ing is dominant and let α = −1. If Re(C12) is nega-tive, we decide that the double-bounce scattering is domi-nant and let β = 1. Finally, the surface scattering power Ps

and the double-bounce scattering power Pd can be derivedas

A =(

C11 − fc − 32fv

) (C22 − fc − 3

2fv

)−

(Re(C12) − 1

2fv

)2

B = C11 + C22 − 2fc − 2fv (23)⎧⎨⎩

if β = 1

thenPs =2A

B − 2real(C12)or

⎧⎨⎩

if α = −1

thenPd =2A

B − 2fv + 2real(C12)In Freeman’s decomposition, the sign of the real part of

〈SHHS∗VV〉 is used to decide whether the double-bounce

scattering or the surface scattering is dominant. In thisstudy, the decision principle is replaced by Re(C12) forthe CP mode. Several fully polarimetric data sets includingthe two images shown in the following experiment sectionhave been used to verify this decision principle. We usePs −Pd > 0 for the FP mode and Re(C12) > 0 for the CPmode to determine the area dominated by single-bounce oreven-bounce scattering. By comparing the two determinedresults, it is found that the area differentiation is no morethan 5%. Thus, this principle to decide which scatteringmechanism is predominant is valid for the CP mode. Thenthe approximated values of 2Pc + Pv and 2Pd + 2Pc + Pv

could be estimated and updated from the iteration process.


3.4 Proposed multi-polarization reconstructionalgorithm

We also employ an iterative approach to solve the non-linear system.

Initializations, as we can get as follows:⎧⎪⎨⎪⎩

fc(0) = |F(0)| = 0

ρ(0) =C12√

C11C22

⇒

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

X(0) =C11 + C22

2

(1 − |ρ(0)|3 − |ρ(0)|

)〈SHHS∗

VV〉(0) = ρ(0)

√(C11 − X(0))(C22 − X(0))

fc(0) = |F(0)| = |Im(C12 − 〈SHHS∗VV〉(0))|

fv(0) = 2X − fc(0)

.

(24)Iterations, as we can get as follows:

w =

⎧⎪⎪⎨⎪⎪⎩

4fc + 3fv

2Pd + 4fc + 3fv, Re(C12) � 0

4fc + 3fv

2span − 3fv − 2Ps, Re(C12) < 0

ρ(i+1) =C12 − X(i) − jF(i)√

(C11 − X(i))(C22 − X(i))

X(i+1) =C11+C22

2(1−sgn〈Re(C12)〉|ρ(i+1)|) · w

2 + (1 − sgn〈Re(C12)〉|ρ(i+1)|) · w

〈SHHS∗VV〉(i+1) =ρ(i+1)

√(C11−X(i+1))(C22−X(i+1))

fc(i+1) = |F(i+1)| = |Im(C12 − 〈SHHS∗VV〉(i+1))|

fv(i+1) = 2X(i+1) − fc(i+1)

where X = 〈|SHV|2〉, Pv = 3fv, Pc = 2fc, span =C11 + C22, and i is the iteration number. Before tak-ing the reconstruction procedure, we should determinewhich scattering mechanism is dominant, and then se-lect the corresponding formula in (23) to calculate thevalue of Ps or Pd for iteration. Since the number of un-knowns exceeds the number of equations, we let fc(0) =0 at first. Then the initial value for fc(0) is updated bythe initialized 〈|SHV|2〉(0) and 〈SHHS∗

VV〉(0). The vol-ume scattering coefficient fv is assigned by the ithestimated 〈|SHV|2〉(i) and fc(i). Due to the violationsof the underlying assumption in iteration, |ρ|(i) maybecome larger than one, or the power of the volu-me scattering may become negative throughout the itera-tions. In both cases, we regularize the approximation va-lues to be the (i − 1)th iterative results and then halt theiteration. Suppose the nth order estimated values of Fn

and Xn are given, the reconstructed FP coherency matrixis shown as follows:

Tπ/4−FP =⎡⎢⎢⎢⎢⎣

γ1 + 2Re(C12) − 4Xn

2γ2−j2(Im(C12)−Fn)

20

γ2+j2(Im(C12)−Fn)2

γ1 − 2Re(C12)2

jFn

0 −jFn 2Xn

⎤⎥⎥⎥⎥⎦

(26)where γ1 = C11 + C22, γ2 = C11 − C22.

Due to the relationship between the Lexicographic targetscattering vector and the Pauli scattering vector, the unitarytransformation formula between the scattering covariancematrix C and the scattering coherency matrix T is

Cπ/4−FP =⟨⎡⎣ |SHH|2

√2SHHS∗

HV SHHS∗VV√

2SHVS∗HH 2|SHV|2

√2SHVS∗

VV

SVVS∗HH

√2SVVS∗

HV |SVV|2

⎤⎦⟩

=

DT3 Tπ/4−FPD3 (27)

where D3 =1√2

⎡⎣ 1 0 1

1 0 −10

√2 0

⎤⎦.

Thus, the linear basis multi-polarization information isreconstructed.

4. Experimental results

The proposed multi-polarization reconstruction algorithmis applied to two PolSAR data sets. One is the E-SARL-band data acquired over the Oberfaffenhofen area inGermany. The image has 1 300 pixel× 1 200 pixel. Fig. 2is the Pauli–basis image, from which we can see severalkinds of terrain types such as airports, urban areas, farm-lands, and forests. The other is the NASA/JPL AIRSAR L-band data acquired over San Francisco, shown in Fig. 4. It

Fig. 4 San Francisco test area


has 700 pixel × 900 pixel and consists of a variety of dis-tinctive scattering mechanisms. Both images are filtered bya 3 × 3 average sliding window.

4.1 Reconstruction performance

First, an experiment is used to illustrate and assess the va-lidity of the modified reconstruction algorithm. The π/4mode compact polarimetric data are generated from theoriginal fully polarimetric data. Fig. 5 shows the Pauli-basis reconstructed results by the two methods, i.e., theSouyris’ method and the proposed method. It is clear thatboth images indicate a good overall agreement with the FPimage over the whole area, but Fig. 5 (b) looks much bluerthan Fig. 5 (a) in the forest area blocked by the bold whiterectangle. This phenomenon is closer to the FP Pauli-basisimage shown in Fig. 2 and more consistent with the actualphysical scattering mechanism in forests.

Fig. 5 Pauli-basis images of the reconstructed data

Fig. 6 shows the scatter plots of the Oberpfaffenhofentest data. Fig. 6 (a)–Fig. 6 (c) shows the Souyris recon-structed results versus the actual radiometric values, andFig. 6 (d)–Fig. 6 (f) shows the modified reconstructed re-sults versus the actual radiometric values.

By comparison, we find that the reconstructed resultsof mutual related channels are similar but the Souyris’method is somewhat inferior. Furthermore, the helix scat-tering type which is omitted in the Souyris’ reconstructionby assuming that the reflection symmetry is retained bythe proposed method. The San Francisco test site presentsa better reconstruction performance because the pixels inthis test site are more coherent on average than the Oberp-faffenhofen test data. Table 1 gives the modes and the stan-dard deviations of the relative errors associated with thereconstructed results. The values assessed in the first fourcolumns are calculated by the method of Souyris’ et al.,the values assessed in the last four columns are of the pro-posed method in this paper. Dataset 1 and Dataset 2 denotethe data acquired over the Oberpfaffenhofen area and theSan-Francisco area, respectively. Unit of the phase error isdegree. For example, the relative error for HH polarizationis given by

(〈|SHH|2〉CP − 〈|SHH|2〉FP)/〈|SHH|2〉FP. (28)

The mode gives the most frequently occurred value,which shows the bias from a perfect reconstruction. LetDataset 1 denote the Oberpfaffenhofen data, and letDataset 2 denote the San Francisco data. All the pixelsof both data sets are used to evaluate the reconstructionperformance. From Table 1, we can see that both methodsoverestimate the cross-polarized term, and underestimatethe co-polarized terms. The proposed method resemblesthe Souyris’ method in estimating the magnitudes of thethree channels, but is superior for extracting the phase in-formation, i.e., Angle 〈SHHS∗

VV〉.The estimation of 〈|SHV|2〉 is the critical point for the

reconstruction algorithms, so we re-assess the estimatedperformance of 〈|SHV|2〉 and 〈SHHS∗

VV〉with several typi-cal regions, i.e., forests, ocean areas, urban regions, farm-lands, and bare soils. These typical regions are separatelyselected from the two data sets, and the selected areas out-lined by rectangles with indexes have been shown in Fig. 2and Fig. 4, respectively. The corresponding relative errorsare shown in Table 2. For magnitude reconstruction of theco-polarized channels, we can see that both the methodshave similar reconstruction accuracies. For magnitude re-construction of the cross-polarized channel, the proposedmethod has a better performance in forests and urban reg-


Fig. 6 Distribution of the reconstructed multi-polarization data

Table 1 Modes (Mode) and standard deviations (Std.) of the relative errors associated with the reconstruction algorithms

Souyris’ method Proposed methodRelative Errors 〈|SHH|2 〉〈|SVV|2〉〈|SHV|2〉 Angle〈SHHS∗

VV〉〈|SHH|2〉〈|SVV|2〉〈|SHV|2〉 Angle〈SHHS∗VV〉

Mode −0.176 9 −0.188 2 0.993 8 0.163 3 −0.160 1 −0.155 0 0.855 0 0.057 6Dataset 1 Std. 0.090 4 0.115 8 2.011 9 1.814 7 0.093 5 0.116 5 2.014 8 1.053 4

Mode 0.143 4 0.065 2 0.00 8 0.020 0 0.140 2 −0.010 8 0.004 0 0.029 8Dataset 2 Std. 0.247 4 0.219 0 1.270 6 1.056 3 0.233 0 0.205 4 1.351 2 0.990 6

Table 2 Modes (Mode) and standard deviations (Std.) of the relative errors associated with reconstruction algorithms for some typical areas

Forest Baresoil Farmland Urban area Ocean areaRelative Errors

Mode Std. Mode Std. Mode Std. Mode Std. Mode Std.

〈|SHH|2〉1 −0.144 2 0.059 7 −0.197 1 0.036 3 −0.194 5 0.039 6 0.206 9 0.210 5 0.095 0 0.088 5〈|SVV|2〉1 −0.260 6 0.086 3 −0.244 4 0.052 6 −0.162 5 0.054 0 −0.287 3 0.145 2 0.037 3 0.044 5〈|SHV|2〉1 1.061 8 0.414 2 6.535 0 1.327 5 3.840 2 0.958 5 −0.967 0 1.562 1 −0.072 1 1.065 5

Angle 〈SHHS∗VV〉1 0.342 2 2.049 8 0.605 5 3.856 0 0.295 9 0.652 7 −0.041 7 2.120 6 0.233 8 1.633 2

〈|SHH|2〉2 −0.111 7 0.070 6 −0.175 4 0.031 0 −0.170 8 0.044 5 0.196 5 0.199 6 0.098 2 0.079 7〈|SVV|2〉2 −0.230 7 0.116 5 −0.206 7 0.047 8 −0.158 7 0.053 1 −0.141 3 0.034 6 0.036 0 0.044 6〈|SHV|2〉2 0.702 1 0.412 0 5.693 8 1.179 4 2.929 9 0.911 3 0.526 2 1.361 0 −0.126 0 0.978 7

Angle 〈SHHS∗VV〉2 0.263 9 1.842 2 0.220 0 1.384 9 0.113 5 0.637 6 0.047 2 1.218 7 0.060 3 1.499 2


ions, and the reconstructed biases are smaller. For theocean area, the reconstruction of 〈|SHV|2〉 is not goodbecause of the low backscattering energy in the cross-polarized channel from sea surface. For the reconstruc-tion of 〈SHHS∗

VV〉, whose phase information is also thephase of ρ, the proposed method has a stable superiorityin the phase information extraction. This is because thatthe helix scattering component Im(SHHS∗

HV + SHVS∗VV)

in (13) is used to compensate for the phase informationdistortion. Im(SHHS∗

HV + SHVS∗VV) is generally regarded

as zero under the reflection symmetry assumption and infact affects the reconstructed accuracy of the phase of theco-polarized coherence. The modified method takes thiseffect into consideration, revising the imaginary part of〈SHHS∗

VV〉. Significant improvement can be found withthe selected baresoils and urban areas. But in the areawhere the volume scattering is dominant, the improvementis less because in this circumstance the phase informationis meaningless.

SAR calibration is to provide imagery in which the pixelvalues can be directly related to the radar backscatter of thescene. It is required in accurately estimating of geophysi-cal parameters. Conventional suggested requirements arethat the radiometric correction is within 0.5 dB and the es-timated phase error is known to be within 10◦. Thus, weemploy a similar method to quantitatively assess the recon-structions. The fully polarimetirc SAR data can be seen asthe true backscattered values, and the reconstructed datafrom CP are the values needed to be calibrated [14].

Comparisons of both methods have already been shownin Table 1 and Table 2, which listed the relative errors ofthe reconstructed values. Since compact polarimetry needshigher order statistics [12], the estimated errors are statisti-cally related to the fully polarimetric data. Take the recon-structed 〈|SHH|2〉π/4 for example. The radiometric bias isdefined as

10 log10(〈|SHH|2〉π/4/〈|SHH|2〉FP).

Compared with Dataset 2, Dataset 1 has a relativelylarger error variance, which indicates an expected poor re-construction performance, so we choose Dataset 1 for as-sessment. The estimated average reconstructed biases for〈|SHH|2〉 and 〈|SVV|2〉 are −0.845 5 dB and −0.905 5 dB,respectively, by Souyris’ method, compared with −0.757 7dB and −0.731 4 dB by the proposed method. Both the re-constructed magnitude errors of the co-polarized channelsare beyond the suggested limits of ±0.5 dB, but the errorsare all within ±0.8 dB by the proposed method accordingto another calibration requirement for some specific ap-plications [22]. But the reconstruction errors of 〈|SHV|2〉

by both methods tend to be larger. The reproduced val-ues are far away from the real radiometric values. Thisis because when taking the reconstruction procedure intoconsideration, two neglected parts, i.e., Re(SHHS∗

HV) andRe(SHVS∗

VV), which are ideally regarded as zero by as-suming the reflection symmetry, are actually added ontothe cross-polarized channel. We take Dataset 1 for exam-ple. These three parts of the E-SAR test data are compara-ble. We have

mean( 〈|SHV|2〉

Re(SHHS∗HV) + (Re(SHVS∗

VV))

)= −4.325 6,

which means that the omitted parts might cause 18% er-ror at least for this data set theoretically. If the two omittedparts are comparable with the cross-polarized channel in-tensity, it is not possible to reproduce 〈|SHV|2〉 within thelimit of ± 0.5 dB. This is the main reason why the cross-polarized intensity reconstructed data always tend to benot accurate. With respect to the phase information recon-struction, the average phase errors are 2.291◦ and 1.666◦

by the Souyris’ method and the proposed method, respec-tively. Both of them are within the required error margin of± 10◦.

For typical areas with different scattering mechanisms,the estimated phase error biases are −12◦, 4.54◦, 14◦,0.49◦, and 1.76◦ by the Souyris’ method, according tothe specific terrain type order listed in Table 2. Labels 1–5 in Fig. 2 and Fig. 4 present the forests, the baresoils,farmlands, urban areas, and ocean areas, separately. Thefirst four rows with subscript 1 are the estimations of theSouyris’ method, and the last four rows with subscript2 are the estimations of the proposed method. By theproposed method, the corresponding estimated phase er-rors are −9.2◦, 1.65◦, 5.7◦, −0.5◦, and −0.95◦, respec-tively. The Souyris’ method has a relatively larger phaseerror −12◦ in forests and 14◦ in farmlands. In forests,the double-bounce scattering and the helix scattering arein general in relatively larger proportion. The proposedmethod could compensate for the phase reconstruction dis-tortion to a certain degree by subtracting the imaginary partF . For farmlands, where the rough surface scattering oftenhappens, the backscattered waves might be dominated ei-ther by odd number or even number reflections [20]. Thusapplying the same extrapolation model (7) to the pixelswith different scattering properties could not yield goodresults. For the selected farmland test area, the mean errorof (7) (i.e., the subtraction between the two sides of (7))is −0.117 5, as compared with −0.007 5 of the improvedmodel in (19).

The reconstructed values from CP may be not accurateenough for geophysical parameter estimation. However,


it is sufficient for some applications, such as terrain typeclassification, metallic target detection, and oil-spill detec-tion.

4.2 Classification performance

To test the classification capability with the reproduceddata from compact polarimetry, classification accuracies ofseveral typical terrain types with reconstructed data are in-vestigated in this part. The AIRSAR San Francisco data isused firstly. The reflection symmetry assumption is validin some areas (e.g., the baresoil area and the ocean area),but not valid in other areas (e.g., the urban area and the for-est area). The complex Wishart maximum likelihood (ML)classifier is adopted for the supervised classification. Us-ing the ML classifier, the PolSAR image is classified intofour classes, i.e., ocean surfaces, urban areas, baresoil ar-eas, and forests.

For valid comparison, the three images (the original FPimage, the reconstructed image by the Souyris’ method,the reconstructed image by the proposed method) are clas-sified using the same training samples. To quantitativelycompare the classification accuracy, we regard the clas-sified results using the original FP data as the referencemap. Though the classification results of the FP data arenot completely consistent with the real ground truth mea-surements, they are reasonable for classification evaluationof compact polarimetry. The supervised classification ac-curacy of the ML classifier is somewhat related to the se-lected training samples, but an average experimental re-sult could reduce this influence. No matter what kind ofthe classifier is applied, it does not affect the assessmentof the multi-polarization reconstruction algorithm as longas the classifier is valid. The confusion matrices relatingto both the reconstruction algorithms are listed in Table 3,expressed in percentage. The classification rates are aver-age of several experiments, and the sum of each columnis 1. We can observe that the classification accuracy withthe data reproduced by the proposed algorithm is better,especially for the forest classification.

Table 3 Terrain type classification accuracies (presented in percent-age) of San Francisco data set(a) Data reproduced by the proposed method.

FPCP

Ocean area Urban area Baresoil Forest

Ocean 0.981 0 0.092 0.000

Urban 0 0.896 0.001 0.011

Bare-soil 0.019 0 0.648 0.033

Forest 0 0.104 0.259 0.956

(b) Data reproduced by the Souyris’ method

FPCP

Ocean area Urban area Baresoil Forest

Ocean 0.972 0 0.074 0.006Urban 0 0.871 0.000 0.070

Baresoil 0.027 0.020 0.611 0.030Forest 0 0.108 0.314 0.892

The classified result for forests with the reproduced databy the proposed method is close to the classified result withthe original fully polarimetric data, but the result by theSouyris’ method is not good enough. The standard devia-tion of the relative error, which is defined in (28), is a goodmeasure for precision. For forests, the relative mean squareerrors of 〈|SHV|2〉 estimated by the Souyris and proposedmethods are 1.817 and 0.432, respectively. Next, the totalclassification accuracy is considered, which is an importantparameter and could show the quality of the reconstructeddata. From Table 3, it is observed that the data reproducedby the proposed method give a total classification accuracyof 87.03%, which is the average of the diagonal values inTable 3 (a). It is higher than the average rate 83.71% relat-ing to the Souyris’ method shown in Table 3 (b).

The second data set used for assessment is the Oberp-faffenhofen data. The target classes over this area are com-plicated. Using the ML classifier, we simply classify thisimage into four classes, i.e., forests, buildings, the flat-ground, and farmlands. The confusion matrices are shownin Table 4. The classification results of the original fullyPolSAR data are also used as the reference for evalua-tion. From inspection of Table 3 and Table 4, the classifi-cation capability of the reconstructed data by the proposedmethod is acceptable, which shows the effectiveness of thedeveloped method for the application of terrain classifica-tion.

Table 4 Terrain type classification accuracies (expressed in percent)of Oberpfaffenhofen area(a) Classification with the data reproduced by the proposed method.

FPCP

Forest Flat-ground Building Farmland

Forest 0.892 0 0.335 0Flat-ground 0 0.939 0 0Building 0.037 0 0.665 0Farmland 0.071 0.061 0 1

(b) Classification with the data reproduced by the Souyris’ method

FPCP

Forest Flat-ground Building Farmland

Forest 0.876 0 0.329 0.004Flat-ground 0.004 0.936 0 0.003Building 0.048 0 0.671 0Farmland 0.072 0.064 0.0000 0.993

5. Conclusions

Based on a modified four-component model-based decom-


position, an improved multi-polarization reconstruction al-gorithm has been proposed in this paper for the π/4 com-pact polarimetry. In the proposed reconstruction algorithm,we consider the non-reflection symmetric scattering case,and recover more information. A special volume scatter-ing model, which is more consistent with the assumptionof fully depolarized backscattering scenario, is employedto decompose the target scattering coherency matrix. Inthis way, a simple relationship between the helix scatteringand the volume scattering components is derived. Then therelationship is used to compensate for the phase informa-tion distortion when implementing the reconstruction. Us-ing the decomposed powers of the four scattering com-ponents and considering the respective polarimetric prop-erties, we establish an average polarization state modeland then present the reconstruction procedure. The pro-posed algorithm takes account of the non-reflection sym-metry condition and the physical scattering mechanisms,and thus has a better reconstruction performance comparedwith the Souyris’ method, which is demonstrated by itsimplementations on two PolSAR data sets. Experimentalresults show that the proposed method is more suitable forterrain type analysis.

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Biographies

Junjun Yin was born in 1983. Now she is aPh.D. candidate in the Department of ElectronicEngineering, Tsinghua University, Beijing. Her re-search interests include compact polarimetry, shipand oil-spill detections in multi-polarization SARimagery, and terrain type clas sification and segmen-tation.E-mail:[email protected]

Jian Yang was born in 1965. He received hisB.S. and M.S. degrees from Northwestern Polytech-nical University, Xi’an, China, in 1985 and 1990,respectively, and Ph.D. degree from Niigata Univer-sity, Niigata, Japan, in 1999. In 1985, he joined theDepartment of Applied Mathematics, Northwest-ern Polytechnical University. From 1999 to 2000,he was an assistant professor with Niigata Univer-

sity. In April 2000, he joined the Department of Electronic Engineering,Tsinghua University, Beijing, China, where he is now a professor. His re-search interests include radar polarimetry, remote sensing, mathematicalmodeling, optimization in engineering, and fuzzy theory.E-mail: [email protected]

Multi-polarization reconstruction from compact polarimetry based on modified four-component...

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