Point Target Classification via Fast Lossless and Sufficient $\Omega$–$\Psi$–$\Phi$ Invariant Decomposition of High-Resolution and Fully Polarimetric SAR/ISAR Data

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CONTRIBUTEDP A P E R

Point Target Classificationvia Fast Lossless andSufficient �–�–�Invariant Decompositionof High-Resolution and FullyPolarimetric SAR/ISAR DataClassification of high-resolution SAR/ISAR data through decomposing the

radar target Sinclair matrix is discussed in this paper dispensing

full-resolution and lossless analysis.

By Riccardo Paladini, Member IEEE, Laurent Ferro Famil, Member IEEE,

Eric Pottier, Fellow IEEE, Marco Martorella, Senior Member IEEE,

Fabrizio Berizzi, Senior Member IEEE, and Enzo Dalle Mese, Life Fellow IEEE

ABSTRACT | The classification of high-resolution and fully

polarimetric SAR/ISAR data has gained a lot of attention in

remote sensing and surveillance problems and is addressed by

decomposing the radar target Sinclair matrix. In this paper, the

Sinclair matrix has been projected onto the circular polariza-

tion basis and is decomposed into five parameters that are

invariant to the relative phase �, the Faraday rotation �, and

the target orientation � without any information loss. The

physical interpretation of these parameters, useful for target

classification studies, is found in the wave-particle nature of

radar scattering phenomenon given the circular polarization of

elemental packets of energy. The proposed deterministic

target decomposition is based on the left-orthogonal special

unitary SU(2) basis, decomposing the signal backscattered by

point targets, represented by the target vector, via six special

unitary SU(4) rotation matrices, and by providing full reso-

lution and lossless analysis. Comparisons between the pro-

posed deterministic target decomposition and the Cameron,

Kennaugh, Krogager, and Touzi decompositions are also

pointed out. Generally, the proposed decomposition provides

simpler interpretation, faster parameter extraction, and better

generalization properties for the analysis of nonreciprocal or

random targets. Several polarimetric SAR/ISAR data sets of

UWB data, airborne fully polarimetric EMISAR data, and

spaceborne RADARSAT2 are used for illustrating the effective-

ness and the usefulness of this decomposition for the classi-

fication of point targets. Results are very promising for

application use in the next generation of high-resolution

spaceborne and airborne Pol-SAR and Pol-ISAR systems.

KEYWORDS | Automatic target classification; automatic target

recognition; classification algorithm; data mining; decomposi-

tion theorem; depolarization effect; deterministic processes;

Earth surface; eigenvalues and eigenfunctions; Einstein photon

Manuscript received June 22, 2011; revised March 24, 2012 and August 1, 2012;

accepted October 20, 2012. Date of current version February 14, 2013.

R. Paladini, M. Martorella, F. Berizzi, and E. Dalle Mese are with the

Information Engineering Department, University of Pisa, Pisa 56126, Italy

(e-mail: [email protected]).

L. Ferro Famil and E. Pottier are with the Institute of Electronics and

Telecommunications of Rennes (IETR), University of Rennes-1, Rennes 35000, France.

Digital Object Identifier: 10.1109/JPROC.2012.2227894

798 Proceedings of the IEEE | Vol. 101, No. 3, March 2013 0018-9219/$31.00 �2013 IEEE

circular polarization; Faraday rotation; geophysics computing;

invariant decomposition; lunar surface; matrix decomposition;

orientation invariant parameters; particle characterization of

radio scattering theory; polarimetry; polarization; polarization

transformation properties; radar; radar cross section (RCS);

radar polarimetry; radio scattering models; remote sensing by

radar; Sinclair matrix; synthetic aperture radar; target decom-

position; target scattering characterization; vectors

NOMENCLATURE AND ABBREVIATIONS

PolSAR Polarimetric synthetic aperture radar.

CTD Coherent target decomposition.

ITD Incoherent target decomposition.

UWB Ultrawideband.

LOS Line of sight.

SU Special unitary matrices.FSA Forward scattering alignment.

BSA Backward scattering alignment.

SVD Singular value decomposition.

DOF Degrees of freedom.

RCS Radar cross section.

SDH Sphere, diplane, helix decomposition.

RGB Red, green, blue.

HSV Hue, saturation, value.CFAR Constant false alarm rate.

�;�;� Physical distortions generated by: Faraday

rotation, target orientation around LOS,

target vector phase shift.

j Jones vector.

S Sinclair radar scattering matrix.

V Radar network voltage.

�i Kennaugh–Huynen con-eigenvalues.B Basis matrices of the Sinclair matrix.

U;V SU(2) generators for change of basis.

k; c Scattering vectors 2 C4.

v Feature vector.

T;D;H Sphere, diplane, and helix Sinclair

matrices.

Snonrec;Srec; Cameron additive decomposition

components.Smaxsym ;S

minsym

Rð!iÞ SU(4) generators for target vector

decomposition.

I . INTRODUCTION

Airborne and spaceborne fully polarimetric and dual co-

herent synthetic aperture radars (PolSARs) are emergent

technologies developed for remote sensing of terrestrialand planetary surfaces, providing full day–night coverage at

great nominal resolution and good image quality [1]–[3].

Compared to single-polarized SAR, fully polarimetric SARs

provide four images of the sensed scenario with different

characteristics depending on the scattering material and

shape. The informative contents of the four polarimetric

SAR channels are enlarged and can be used for the devel-

opment of several applications of remote sensing by creatinga mapping between the observed scattering matrix and the

scattering matrices of elemental scattering objects.

The radar target scattering matrix models the scatter-

ing process characterizing the radar target polarization

transformation properties as has been shown by Kennaugh

and Huynen since 1948 [4]–[6]. The radar target scatter-

ing matrix can be projected in a vector form as shown by

Cloude [7], providing the maximum information availablefor the remote sensing of terrestrial and planetary surfaces

[1]–[3]. Each pixel of high-resolution PolSAR data is

represented by a complex four-element vector character-

izing the polarization transformation properties of the

observed target.

The radar target scattering matrix and the averaged

target coherency matrix are the main mathematical tools

useful for characterizing the scattering process. The classi-cal theory developed for the radar target signature analysis

is based on linear polarization measurements, whereas

Kennaugh, Graves, Huynen, Boerner, Cloude, Van Zyl,

Krogager, Cameron, Pottier, Touzi, and others have shown

that the development of target decomposition theorems is

necessary for obtaining a physical interpretation of the

observed scattering mechanism [4], [8]–[15]. Following

the definition provided by Touzi, Boerner, and Luneburg,target decomposition theorems are divided into two main

families: CTDs, which are applied to single observation

scattering matrix for representing point targets, and

ITDs, which are applied to averaged measurements for

representing the statistics of distributed targets [16].

From a mathematical point of view, CTD theorems deal

with the representation of a deterministic signal out of a

single measurement and ITDs deal with the representa-tion of stochastic vector processes with large numbers of

observations.

The ITD, introduced by Huynen, Cloude, Van Zyl, and

Pottier, has matured considerably in the last two decades,

due to its application for the problem of filtering the

speckle interference, and currently, it is the primary source

of investigation in PolSAR classification and parameter

retrieval [1], [2], [5], [10], [13]–[15], [17]–[19]. The mul-tiplicative speckle noise is a fading phenomenon generated

by multiple scatterers superimposed in the same resolution

cell, and their importance is related to the range-azimuth

resolution of the SAR system. It is well known that radar

echo response is dominated by the contribution of scat-

terers having a dimension comparable or larger than the

radio wavelength (Mie and optical scattering), where

smaller objects (Rayleigh scattering) have reduced RCS.Next-generation PolSAR systems are increasing the reso-

lution considerably reaching the �=4 physical limit ob-

tained when the fractional bandwidth occupation of the

signal is unitary [i.e., ultrawideband (UWB) imaging] and

the synthetic antenna aperture is an arc of two radians. The

increased system resolution reduces the number of

significant scattering objects superimposed and the need

Paladini et al. : Point Target Classification via Fast Lossless and Sufficient �� Invariant Decomposition

Vol. 101, No. 3, March 2013 | Proceedings of the IEEE 799

for incoherent averaging would therefore vanish comingback to the single looked SAR analysis.

Recently, a new concept in radar polarimetry has been

introduced called ‘‘lossless and sufficient target decom-

position theorems.’’ This is to be applied to both target

vectors and covariance matrices [20]. Lossless and suffi-

cient decompositions are a special set of functions which

model all the N DOFs of a complex signal in terms of a

minimum number of useful parameters. Among losslessand sufficient decompositions, there is a special set of in-

variant decompositions which model the DOF of different

signals in terms of SUs, which have also been found [20].

This recent development is useful for obtaining the de-

composition of the circular polarization target coherency

matrix in terms of the eight orientation ð�Þ-invariant pa-

rameters and a meaningful estimator of the dominant

eigentarget orientation �1 [21].The extraction of �-invariant parameters, proposed by

Huynen, is useful for reducing the impact of the signal

distortions that are generated by geometrical rotation of

the scatterers around the radar LOS [5], [15]. More re-

cently, the development of �-invariant decomposition and

ground-target slope estimation algorithms has also ma-

tured considerably [1], [6], [14], [15], [21]–[27].

Next-generation low-frequency spaceborne PolSAR(L–P band) will also be affected by the Faraday rotation,

where the Earth’s magnetic field causes a rotation of the

polarization plane during the two-way propagation mea-

suring 2 � radians also generating a nonreciprocal scat-

tering [28]–[32]. The removal of the Faraday effect has

been proven effective by Bickel and Bates in 1965 by

analyzing the diversity of the cross-polarization channels

via the circular polarization scattering matrix and has beensuccessfully applied for the calibration of L-band PALSAR

[29]–[31]. Wang et al., in particular, have assessed the cross

relationships between the Faraday rotation and other

distortion sources such as crosstalk and channel imbalance.

The strong impact of small � rotations, on the order of a few

degrees, is destructive regarding image quality and target

decomposition parameters [32]. For these reasons, the anal-

ysis of nonsymmetric scattering matrices, neglected in mostof the works dealing with backscatter geometry, is funda-

mental for developing a new decomposition of the Sinclair

matrix invariant to target orientation and Faraday rotation.

In this paper, the fast lossless and sufficient �–�–�invariant decomposition of the target vector c 2 C4 is

proposed for the classification of high-resolution and fully

polarimetric SAR data. The proposed CTD, based on the

left–left orthogonal ðll?Þ circular polarization special uni-tary SU(2) basis, is represented in a target vector form for

the classification of point targets. The modulus of the

proposed scattering vector is invariant to the phase shift

expðj�0Þ, Faraday �, and target orientation � rotations.

The circular polarization scattering vector is also sufficient

for representing all the DOFs of the radar target scattering

matrix in a convenient way without any information loss.

The decomposition is also classified as computationally‘‘fast,’’ for using a reduced number of operations, and a

smaller memory use compared with other approaches.

The objectives of this paper are as follows: review all

the CTD theorems, present a new decomposition maxi-

mizing the advantages of the most efficient CTDs and

overcoming their limitations, provide an invariant charac-

terization of the scattering phenomena based on a physical

basis, derive the relationships between most of the CTDparameters currently used, and assess the proposed de-

composition theorem and classification algorithm with

high-resolution and fully polarimetric remote sensing SAR

data. The structure of the paper is organized as follows. In

Section II, the principal CTD theorems are reviewed

pointing out the main features of each one with a sim-

plified formalism [1], [2], [13]. In Section III, the proposed

CTD based on circular polarization scattering vector isdetailed underlining the physical meaning of the proposed

target �–�–�-invariant parameters. A new classification

metric for both symmetric and nonsymmetric targets

based on the distance in a Hilbert space is also presented.

In Section IV, the relationships between parameters from

CTD theorems are shown. Specifically, the fundamental

equivalence of different sets of characteristic parameters

for the representation of symmetric targets is postulated,where the proposed decomposition is found to be more

suitable for characterizing partially symmetric and par-

tially reciprocal targets. The experimental assessment is

performed in Section V, through UWB data of some ele-

mental targets measured in an anechoic chamber, EMISAR

airborne Pol-SAR images of a ship, and fully polarimetric

RADARSAT2 data of the Gibraltar strait [33], [34]. The

proposed decomposition is fully lossless, exploiting the sameDOF of single looked PolSAR data, where the ‘‘lossless and

sufficient �-invariant decomposition of random reciprocal

target’’ is lossless only by considering compressed three-by-

three complex coherency matrices data sets [21].

II . REVIEW OF DETERMINISTICTARGET DECOMPOSITION

A. The Sinclair Radar Target Scattering MatrixThe backward scattering alignment (BSA) is used as a

standard reference system of coordinates in radar polar-

imetry for backscatter geometry, as shown by Sinclair,

Kennaugh, Boerner, Van Zyl, Cloude, Pottier, Lee, and

Touzi in some contribution papers, review papers, and re-

cent monographs [1], [2], [4], [9], [10], [13], [14], [16]. TheBSA represents a unique reference system of coordinates

iz ¼ ix � iy, where ix; iy are the horizontal and vertical

components of the Jones vectors, representing the polari-

zation state of the electromagnetic fields transmitted and

received by the radar antenna, and iz is the direction of

propagation in transmission. Fig. 1 represents the geometry

of the scattering process according to BSA where the radar


800 Proceedings of the IEEE | Vol. 101, No. 3, March 2013

target reference system iz00 ¼ ix00 � iy00 is aligned to the BSA

via three rotations, namely, azimuth Rð�Þ, depression Rð�Þ,and orientation Rð�Þ, defining the target aspect

ix

iy

iz

24 35 ¼ cosð�Þ � sinð�Þ 0

sinð�Þ cosð�Þ 0

0 0 1

24 35�

cosð�Þ 0 � sinð�Þ0 1 0

sinð�Þ 0 cosð�Þ

24 35�

1 0 0

0 cosð�Þ � sinð�Þ0 sinð�Þ cosð�Þ

24 35 ix00

iy00

iz00

24 35: (1)

According to the Sinclair formalism used in this work, the

complex envelopes of the received electromagnetic fields jrx,

collected in the Jones vector, are modeled via the multi-

plication of the complex valued Sinclair matrix S by the com-

plex envelopes of the transmitted electromagnetic fields jtx

jrx ¼ Sjtx: (2)

The Sinclair matrix characterizes the polarization

transformation properties of radar targets, and it is the

primary source of investigation for a radar target signatureanalysis [4]. Nevertheless, the target nature is embedded

in the Sinclair matrix and considerable work has been done

in order to give a physical interpretation of its coefficients

for reconstructing the shape and the features of the ob-

served target [13]. The object of this investigation is known

as target decomposition, and it is introduced in this section

in the case of a deterministic target [1], [2], [13], [16].

B. Kennaugh–Huynen Decomposition of the RadarTarget Sinclair Scattering Matrix

In 1952, Kennaugh developed the first systematic studyof the radar target scattering phenomenon, by deeply ana-

lyzing the mathematical properties of the scattered echo

varying the antenna polarization [4]. The Kennaugh theory

has contributed to the optimum design of the antenna

polarization for single polarization radar and has proposed

the first radar target classification theory based on the

polarization transformation properties of the radar target

[5], [8], [9], [15], [35]–[40].Kennaugh proposed the search of the polarization state

jmax maximizing the received voltage measured at the ter-

minals of radar transceiver antenna Vmax

Vmax ¼ jtmaxSHVjmax: (3)

The maximum voltage is obtained through a matching

antenna condition and, for the Jones vector, it is formal-

ized via the following characteristic con-eigenvalues prob-lem having two solutions �1�2 if and only if SHV is

symmetric:

jmax ¼ SHV jmax ¼ �j�max: (4)

The maximum con-eigenvalue �1 represents the opti-

mum transceiver polarization state called ‘‘co-pol maxi-

mum,’’ where the entire con-eigenvalues spectrum is a

diagonal form of the radar scattering matrix obtained via

cosimilarity change of basis, as shown by Kennaugh [4] andGraves [41]

SGraves�1956D ¼ �1 0

0 �2

� �¼ UtSHVU (5)

where the columns of U represent the coordinate of a new

polarization basis and �1;2 are called con-eigenvalues and

can be complex valued [41]. In 1965, Huynen, in a

Proceedings of the IEEE issue dedicated to the radar

target scattering matrix, modeled the diagonal form of

S in terms of six characteristic parameters useful for clas-

sification studies. This model was extensively revised in

Fig. 1. BSA and the target aspect angle. The target reference system

½ix00 iy 00 iz00 � can be aligned to the radar antenna one ½ix iy iz� through

three rotations, namely: 1) azimuth rotation around target vertical iy00 ;

2) depression rotation around new cross-range axis ix0 ; and

3) orientation rotation around the radar LOS iz. The three Euler’s

rotations are a sufficient set for characterizing the target position

in space.



1978, and it is reported in this form in the followingequation [5], [6], [8]:

SHuy�1978D ¼mH exp j2ð�H þ �HÞð Þ

�1 0

0 tan2ð�HÞ expð�j4�HÞ

" #¼UtSHVU

U ¼ jmax jmax?½ �

¼cosð HÞ � sinð HÞsinð HÞ cosð HÞ

" #

�cosðHÞ j sinðHÞj sinðHÞ cosðHÞ

" #: (6)

The angles H and H are, respectively, the orientation

and the ellipticity of the optimal special unitary SU(2)

maximum copolarization basis U; mH is the amplitude of

the maximum return; �H, the con-eigenvalues phase dif-

ference, is called the ‘‘skip angle’’ characterizing the num-

ber of signal reflection; and �H is called the characteristic

or ‘‘polarizability’’ angle, which represents target sensitiv-ity to the incidence polarization [1], [2], [5], [7], [9], [37],

[38]. Huynen classified ðm; �H; �H; HÞ as invariant param-

eters being independent of the target � angle (but being

dependent on azimut � and elevation �), and ð H; �HÞ as

dynamical parameters being dependent on the orientation

and local position in range [8].

1) Con-Eigenvalues Phase Ambiguity and the Huynen Fork:In 1978, Huynen provided the first revision of the con-

eigenvalues diagonalization pointing out that the con-

eigenvalue (4) is ambiguous in terms of a complex phase

factor expðj�Þ

if Sj1 ¼�1j�1 ! Sj1 expð�j�Þ

¼�1 expð�2j�Þ j1 expð�j�Þð Þ� 8� 2 R: (7)

This problem known as the con-eigenvalues phaseambiguity has been discussed recently by Touzi [15],

Luneburg [39], and Touzi et al. [40]. The con-eigenvalues

phase ambiguity can evoke two side effects: the first is a

coherent shift of the two con-eigenvalues that is meaning-

less from a target identification point of view, affecting theabsolute phase �H; the second one, detailed in (8), shown

at the bottom of the page, is to generate the phase differ-

ence between the two con-eigenvalues creating ambiguity

in the interpretation of the skip angle �H ! �0 ¼ �H þ �00.Nevertheless, between the infinite choices of �0, a

‘‘special solution’’ of the con-eigenvalues problem provid-

ing ‘‘real-valued’’ con-eigenvalues exists. Such decomposi-

tion is called Takagi decomposition, as is also reported byHorn and Johnson [42].

Huynen [6] and Boerner et al. [35] have applied the

Takagi con-eigenvalues diagonalization procedure to the

Sinclair radar scattering matrices modeling the polariza-

tion basis U via the product of three SU(2) matrices, as

generated by the complex exponential form of the Pauli set

BP ¼ I J K Lf g

¼1 0

0 1

� �;

1 0

0 �1

� �;

0 1

1 0

� �;

0 j

�j 0

� �� (9)

thus obtaining

SHuy�1987D ¼

1 0

0 tan2ð�HÞ

" #

¼ 1

mH expðj2�HÞUtSHVU

U ¼ jmax jmax?½ �¼ expðj HK þ jHLþ j�HJÞ

¼cosð HÞ � sinð HÞsinð HÞ cosð HÞ

" #

�cosðHÞ j sinðHÞj sinðHÞ cosðHÞ

" #

�expð��HÞ 0

0 expð�HÞ

" #: (10)

The new parametrization in (10) defines a unique skip

angle �H, which is formally equivalent to the forms of (6)and (8) by substituting in (8) �00 ¼ ��H ! �0 ¼ 0.

The revision of the Huynen CTD from 1987 [see (10)]

is also useful for providing a more elegant group-theoretic

representation of the characteristic copolarization states

SAmbD ¼m expðj2�HÞ

exp j2ð�H þ �00Þð Þ 0

0 tan2ð�HÞ exp �j2ð�H þ �00Þð Þ

� �¼ UtSHVU

U ¼ jmax jmax?½ � ¼cosð HÞ � sinð HÞsinð HÞ cosð HÞ

� �cosðHÞ j sinðHÞj sinðHÞ cosðHÞ

� �expð��00Þ 0

0 expð�00Þ

� �(8)



[6], [35]. According to Huynen’s optimal polarizationtheory, three characteristic copolarization states exist for

the symmetric Sinclair matrix if j�1j 6¼ j�2j: the copolari-

zation maximum and two copolarization nulls constructing

the so-called polarization fork in a great circle of the

Poincare sphere. The two copolarization nulls are sepa-

rated angularly by the angle 4�H and are both separated

angularly � 2�H by the copolarization maximum gener-

ated by jmax [35], [37], [43]. It is worth noting thatcopolarization nulls are not defined for isotropic targets

like dihedrals, trihedrals, and quarter waves that scatter

the same power independently of antenna polarization.

Copolarization nulls also collapse in a needle configuration

for nonisotropic targets like dipoles and helices ð�H ¼ 0Þ,as shown by Boerner et al. [35] and Cameron and Rais [43].

The characteristic Huynen fork of a reciprocal target is

positioned on a great circle of the Poincare sphere via fourtransformations, the fork aperture of an angle 4�H, and

three SU(2) rotations formally equivalent to real Euler

rotations around the Cartesian axes of the Poincare sphere,

that are generated by angles H; H; �H, as shown in Fig. 2

[6], [35]. The homomorphism between the SU(2) matrices

in (10) and the real rotation group O(3) of rotations in the

3-D space has been discussed by Cloude in [2] and [7].

2) Symmetric Targets Theory: A fundamental milestone

of the Kennaugh theory, recently revised by Cameron, is

the definition of a set of targets termed symmetric [4],[12]. If a target response is diagonalized through a pair of

linear polarized antennas, by a rigid rotation around LOS,

then it is symmetric. It means that the newly rotated po-

larization basis is aligned with the target axes of symmetry.

By substituting H ¼ � in (5)

SCam�1990D ¼ �1 0

0 �2

� �¼: Uð�; H ¼ 0ÞtSsymUð�; H ¼ 0Þ (11)

where �1�2 can be in general complex quantities. Sym-

metric targets have five real DOFs: the target size, two

symmetric target shape parameters, relative phase expðj�Þ,and relative orientation �. Nevertheless, some symmetric

targets are also diagonalized by elliptical polarization, asdiscussed by Cameron and Leung [12] and Cameron and

Rais [43]: for this reason, this condition is not necessary

but it is only sufficient. Kennaugh’s report has also intro-

duced the polarization charts representing the received

voltage in decibel as a function of the antenna polarization

used [4]. Measurements of real radar targets have shown a

high degree of correlation between the voltages obtained

using similar orientation and the opposite sense of ellip-tical polarization antennas since 1952 [4]. This feature has

been related to the symmetry of most radar targets and is

discussed in Section IV introducing a novel definition of

target symmetry based on the energetic difference between

circular copolarized returns. In other words, for a symmetric

target, the equator of the Poincare sphere plays as a mirror

of symmetry in Kennaugh’s copolarization constant echo

loci [4].

3) Generalization of the Kennaugh–Huynen CTD for Non-symmetric Sinclair Matrix: In 1986, Davidovitz and Boerner

[44] as well as Cloude [7] and Czyz [45] proposed the

extension of the Kennaugh theory for nonreciprocal

scattering where the Sinclair matrix is nonsymmetric.

In this case, the solution of the optimization problem

called singular value decomposition (SVD) is found bydecoupling the maximum voltage problem through the

use of two different pairs of transmitting and receiving

antennas U;V

SDav�1986D ¼ UtSHVV: (12)

This solution underlines the complexity of the problem

in the general case, where U and V are two independent

SU(2) change of basis matrices performed in transmission

and reception. In order to maximize the voltage backscat-

tered by a nonreciprocal target, two independent polariza-

tion states should be used in transmission and in reception

Fig. 2. Huynen fork of a generic target defined in terms of four steps:

(a) fork opening angle 4�H ¼ 60�; (b) phase shift, rotation of the

H-fork of an angle �H ¼ 30� around the iy-axis of the Poincare sphere;

(c) ellipticity shift, rotation of the H-fork of an angle H ¼ 11� around

the ix-axis of the Poincare sphere; and (d) orientation shift, rotation

of the H-fork of an angle H ¼ 45� around the iz-axis of the

Poincare sphere.



(rx 6¼ tx and rx 6¼ tx). For this reason, the Kennaugh–Huynen decomposition has been difficult to apply in many

real-world problems, where the scattering matrix is nonre-

ciprocal given the presence of thermal and speckle noises,

propagation distortions, or, in the more general case, of

bistatic geometry.

More recently, Karnychev et al. [46] have reconsidered

the generalization of the Kennaugh–Huynen approach to

nonsymmetric scattering matrix, using the Pauli projectionand introducing a complex parameter �K

�K ¼ tanð�KÞ expðj KÞ (13)

where

• 0� � �K � 45� describes the degree of nonreci-

procity; the value �K ¼ 0 is for a reciprocal target

where �K ¼ 45� for a fully nonreciprocal target;

• �180� � K � 180� is the phase difference be-

tween symmetric and nonsymmetric scattering.

The Karnychev approach has been developed to obtain alossless representation of S in terms of eight parameters

½mH; �H; �H; �H; H; H; �K; K� [46]. The SPAN of the

scattering matrix has also been decomposed into the fol-

lowing polarization basis invariant scalar quantities [46]:

SPANðSÞ ¼ SPANðSrecÞ þ SPANðSnonrecÞ (14)

where

SPANðSÞ ¼ jS11j2 þ jS12j2 þ jS21j2 þ jS22j2

SPANðSrecÞ ¼ jS11j2 þ jS22j2 þ 12jS21 þ S21j2

SPANðSnonrecÞ ¼ 12jS21 � S12j2

8>><>>: (15)

and it follows

tanð�KÞ2 ¼SPANðSrecÞ

SPANðSÞ : (16)

4) Touzi TSVM and Its Generalization: In order to cir-

cumvent the Huynen skip angle ambiguity, Touzi in 2007

[15] projected the Kennaugh–Huynen con-eigenvalues in the

form of (6) on the Pauli basis: then, he computed a complex

ratio in order to extract two novel parameters �S;��S

tanð�sÞ expðj��sÞ ¼�1 � �2

�1 þ �2: (17)

The Touzi TSVM has decomposed the scattering vectorof a reciprocal target into the following �-invariant set of

features ½mH H H �S ��S �H�, where the new quanti-

ties �S;��S are functions of the Huynen parameters

ð�H; �HÞ, as shown in Section IV-A [15], [40].

The generalization of the Touzi’s TSVM to nonrecip-

rocal scattering has been proposed recently by Bombrum

for the decomposition of the nonsymmetric scattering ma-

trix [47]. The Bombrum decomposition recalls theDavidovitz and Boerner generalization of the Huynen de-

composition via SVD, where �H; �H are substituted by

cosimilarity Touzi’s parameters �S;��S, obtaining the fol-

lowing vector ½mH E E �S ��S �H R R�, where the

lower script E;R are for emission and reception antennas

[5], [6], [15], [47].

C. Graves Polar DecompositionIn 1956, Graves proposed the analysis and decomposi-

tion of the radar polarization power scattering matrix [41].

According to Graves, the Sinclair matrix can be written in

terms of the product of a nonnegative matrix P ¼ffiffiffiffiffiffiffiffiffiSt�Sp

and a unitary matrix U

S ¼ UP: (18)

This is called polar decomposition of a matrix, and it

is analogous to the polar form of a complex number [41].

In particular, P2 specifies the total power backscattered

from a deterministic target for any transmitted polariza-tion. An interesting property of the power scattering

matrix is that the same unitary matrix which diagonalizes

P2 by a similarity transformation diagonalizes the Sinclair

matrix by a congruent (cosimilarity) transformation [41].

This property has been used by Kotinsky and Boerner for

computing the Kennaugh–Huynen parameters of S [38].

The application of the polar decomposition has been

reconsidered in recent works of Carrea and Wanielik [48]and Souyris and Tison [49] by fully exploiting the eight

DOFs of the Sinclair matrix.

D. The Bickel Invariant Decomposition ParametersIn 1965, in a series of visionary papers about the radar

target scattering matrix, Bickel and Ormsby [28], Bickel

and Bates [29], and Bickel [50] modeled the propagation

distortions given by the Faraday rotation �B and birefrin-gence, introduced the problem of polarimetric calibration,

and extracted some important target invariants using the

circular polarization scattering matrix. In particular,

Bickel proposed the following characteristic invariants:

the amount of energy diffused SPAN, the matrix determi-

nant detðSÞ, the depolarization parameter DB, the ellipti-

city B, and orientation �B of the maximum return,



formally equivalent to the Huynen optimum copo-larization state jmax in (4) [50]

SPAN ¼ jS11j þ jS22j þ jS12j þ jS21jdetðSÞ ¼ S11S22 � S12S21

DB ¼ jSllj þ jSrrjð ÞSPAN

B ¼ 12tan�1 jSllj � jSrrj

2jSlrjjSll þ S�rrj

� ��B ¼ 1

4arg SllS

�rr

�B ¼ 1

4arg SlrS

�rl

:

8>>>>>>>>><>>>>>>>>>:(19)

The contribution of Bickel et al. in radar polarimetry hasbeen forgotten for many years and has been reformulated

in the target vector form in Section III. It is also worth

noting that the circular polarization scattering matrix can

be used for simplifying the computation of the Kennaugh–

Huynen con-eigenvectors of S.

E. Cloude’s Scattering Representation Basedon the Pauli Basis

In 1986, Cloude presented a Ph.D. dissertation about

the polarization transformation properties of radar targets.

Cloude’s work introduced the mathematics of group theory

and developed the basis for a radar target classificationtheory based on the vectorization of the radar target scat-

tering matrix [1], [2], [7], [13]

k ¼ 1

2trðSHV BiÞ ¼ ðk0 k1 k2 k3Þt 2 C4 (20)

where Bi is a orthonormal basis set under a Hermitian

inner product. The set of Pauli matrices in (9) represents,

respectively, the scattering of a trihedral, two oriented

(�2 ¼ 0, �3 ¼ =4) dihedrals, and one nonreciprocal

target, providing the well-known Pauli unitary scattering

vector [13], [17]

kP ¼1ffiffiffi2p ½Shh þ Svv Shh � Svv Shv þ Svh jShv � jSvh�t: (21)

The set of lexicographic basis matrices

BL ¼ 2 00 0

h i; 0 2

0 0

h i; 0 0

2 0

h i; 0 0

0 2

h in o(22)

produces instead a lexicographic ordering of the scattering

matrix elements

kL ¼ ½Shh Shv Svh Svv�t: (23)

The use of orthonormal components is sufficient forrepresenting the target vector in terms of uncorrelated

components, providing an optimal metric in Hilbert vector

space, based on the Hermitian inner product. Another

useful property of the Pauli basis is the discrimination

between the odd bounce, the even bounce, and the nonre-

ciprocal type of scattering mechanisms [13].

F. Cameron Decomposition and Touzi’s SSCM

1) Ambiguities Related to Kennaugh’s Approach for TargetClassification: In 1990, Cameron and Leung pointed out

that the Kennaugh decomposition of a reciprocal target is

an ambiguous description for certain types of radar targets

[12]. The dihedral, for example, is diagonalized given any

value of helicity H and the trihedral is diagonalized givenany value of orientation H [12], [23]. Nevertheless, for

the matching antenna design problem, multiple solutions

provide increased stability, and the Kennaugh–Huynen

decomposition is ambiguous when using H; H for target

classification. This problem arises during copol-max com-

putation, as suggested by Mieras [37] and Hubbert and

Biringi [51], whereas the solution degenerates for identical

con-eigenvalues, for example, for the trihedral and thedihedral. From a mathematical point of view, this condi-

tion is sufficient to demonstrate that the Kennaugh CTD is

not a function of the scattering coefficients. Another point

raised about the Kennaugh CTD is the complicated and

involved computation of con-eigenvalues solutions and the

complex generalization of the problem for the nonrecip-

rocal case [7], [44], [51].

2) Cameron Decomposition: To overcome certain ambi-

guities of the Kennaugh–Huynen approach, Cameron and

Leung [12], Cameron et al. [23], and Cameron and Rais

[52], [53] proposed a new decomposition of the Sinclair

matrix, into a nonreciprocal component Snonrec and a re-

ciprocal target Srec forming a subspace of S, being a ¼ffiffiffiffiffiffiffiffiffiffiffiffiSPANp

expðj�Þ a complex number, and �rec the distance

between S and Srec

S ¼ a cosð�recÞSrec þ sinð�recÞSnonrecf g; 0 � �rec �

2:

(24)

The reciprocal target subspace Srec according to the Touzi

and Charbonneau representation [54]

Srec ¼ cosðCÞSmaxsym þ sinðCÞSmin

sym

n o; 0 � C �

4(25)



is decomposed into a maximized symmetric Smaxsym and a

minimum symmetric Sminsym unit target components, where

(26), shown at the bottom of the page, is a unit three real

parameter model ðjzCj; ffzC;�Þ, and

Sminsym ¼ exp j�min

sym

� �� U �þ

4; H ¼ 0

� �t 1ffiffiffi2p BPJU �þ

4; H ¼ 0

� ��(27)

is a unit dihedral oriented of � =4, multiplied by acomplex phase �min

sym, and

Snonrec ¼exp j �

2þ �nonrec

ffiffiffi2p BPL (28)

is a unit one parameter ð�nonrecÞ nonreciprocal target, and

BPi are the elements of the Pauli basis in (9) and �C; �Ccomplex numbers [12], [23].

The degree to which S deviates from Smaxsym is measured

by angle C

cosðCÞ ¼S;Smax

sym

� � kSk Smax

sym

(29)

where ð:; :Þ is the inner Hermitian matrix product, and k:kis the matrix norm [54].

The main contributions of [12], [23], [52], and [53] are

the introduction of a complex unit disc zC for the repre-

sentation of symmetric targets and the development of an

unsupervised classification scheme of the radar target

scattering matrix into 11 classes. For symmetric targets,

the classification scheme proposed by Cameron is based on

a distance in the Hilbert space evaluated by using a metric

based on the zC parameter [23], [52], [53]. Nevertheless,after the introduction of the lossless and sufficient decom-

position theorems, we find that the Cameron CTD pro-

duces a suboptimal interpretation for partially symmetric

targets C > 0 that are decomposed into orthogonal maxi-

mum and minimum symmetric components. By the ana-

lysis of (25), it is evident that by considering only Smaxsym

some information is lost, where Smaxsym is extracted through

a maximization procedure [2], [12]. It is worth noting thatthe symmetric target cannot be considered as a subspace of

a reciprocal target, being a reciprocal target decomposed in

the coherent sum of two symmetric targets. This limit is

overcome in Section III introducing a special set of sym-

metric targets being a subspace of the reciprocal target

subspace.

3) Touzi SSCM Representation of Deterministic Target: In2002, Touzi and Charbonneau [54] revised the Cameron

decomposition introducing (25), and proposed a pair of

parameters ð T; ð�Sb � �SaÞTÞ, formally equivalent to zC

from the representation of the maximum symmetric scat-

tering component in the Pauli basis

kmaxP�sym ¼ expðj�SaÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij�Cj2 þ j"Cj2

q

�

cosð TÞ

sinð TÞ exp jð�Sb � �SaÞT

0

0

266664377775: (30)

Touzi and Charbonneau have also proposed the sphere

and diplane BPI�J Pauli vector components to represent

Smaxsym in the Poincare sphere ð�C;�CÞ using the four-

vector Stokes formalism [54]

� ¼j�j2 þ j"j2j�j2 � j"j2�"� þ ��"�jð�"� � ��"Þ

26643775¼

1

cosð2�CÞ cosð2�CÞcosð2�CÞ sinð2�CÞ

sinð2�CÞ

26643775: (31)

G. Krogager Sphere–Diplane–Helix Decomposition

1) Original Formulation: After the development of the

NASA–AIRSAR sensor, the research in radar polarimetry

has matured considerably. In 1990, Krogager [11] proposed

a physical decomposition of the radar target scattering

matrix into sphere kSej�s , �-oriented dihedral component

kD, and into a helix residual kH expðj2�Þ that can haveboth left or right helicity

SHV ¼ expðj’SÞkSTfþ expðj’RÞ kDDð8Þ þ kH expðj2�ÞH

� ��

Smaxsym ¼

Uð�; H ¼ 0Þt 1ffiffiffiffiffiffiffiffiffiffiffi1þjzCj2p 1 0

0 zC

� �Uð�; H ¼ 0Þ

1ffiffi2p �CBPI þ �C cosð�2�CÞBPJ þ sinð�2�CÞBPK½ �f g

8<: (26)



T ¼1 0

0 1

� �Dð�Þ ¼

cosð2�Þ sinð2�Þsinð2�Þ � cosð2�Þ

� �H ¼

1 j

j �1

� �: (32)

The Krogager sphere–diplane–helix (SDH) decomposi-

tion is meaningful for the classification of man-made

structures, due to the well-known fact that larger RCSs are

produced, for backscattering, by dihedral (even bounce)

and trihedral (odd bounce) reflectors [55]. Nevertheless,

the SDH decomposition is based on nonorthogonal com-ponents providing a nonoptimal information representa-

tion, and, being written in a vector form, it does not

conserve the signal energy [15]. For this reason, the

Krogager CTD is not suitable for developing an optimum

radar target classification theory.

2) Invariant Krogager Decomposition and Corr–Rodriguez�–�–�–� ModelVThe Sphere–Helices Basis Set: In 1995,Krogager and Czyz showed that the computation of the

‘‘sphere–diplane–helix decomposition’’ parameters is sim-

plified on the right–left circular basis, representing the

scattering of a sphere and left and right helices. The de-

velopment of the Krogager decomposition in the new basis

extracts a new set of features fkS; kD; kH;�K; ’S;�Kginvariant of �K, the target rotation around LOS [22]

SRL ¼ expðj�KÞ�

kS expðj’SÞ0 j

j 0

� �þ kD

expðj�KÞ 0

0 � expð�j�KÞ

� �þ kH expðj�KÞH

�(33)

where kHH is a residual and H can be both left1 00 0

� �or

right wound0 0

0 � 1

� �helix ðLL G9 RRÞ.

In 2002, Corr and Rodriguez [56] proposed an interest-

ing vector representation based on the orthogonal sphere

and left and right helices basis set for decomposing de-

terministic and random targets in terms of �-invariantparameters.

The ‘‘sphere–helix–helix decomposition’’ proposed by

Krogager, Czyz, Corr, and Rodrigues is written in a spinor

unit vector form providing some advantages compared to

the conventional Pauli basis, for a �-characterization of

reciprocal radar targets in terms of spinor phases. Gener-

alization of this concept to the nonsymmetric scattering

matrix case is shown in Section III using the circular po-larization basis representing the idealized scattering of

elemental packets of energy (left or right helicity photons).

H. Concluding Remarks About Existing CTDsIn this section, six different approaches have been dis-

cussed for the Sinclair matrix decomposition: 1) theKennaugh–Huynen cosimilarity [4], [5], [15], [35], [44],

[46]; 2) the polar decomposition [41], [48], [49]; c) the

Bickel circular polarization model [28], [29], [50]; d) the

Pauli basis model [13], [14]; e) the Cameron maximum-

symmetric decomposition [23], [43], [54]; and f) the maxi-

mum dihedral ‘‘sphere–diplane–helix decomposition [11],

[22], [55].

The properties of the CTDs revised in this section havebeen summarized in Table 1, namely: orthonormal vector

form, lossless and sufficient representation for: 1) symme-

tric, 2) reciprocal, and 3) nonreicprocal scattering, �-

invariance, and �-invariance.

The Kennaugh decomposition has, since the beginning,

proven to be an effective physical theory for radar target

scattering classification. Nevertheless, the solution of the

nonconventional eigenvalues problem has pointed outsome weak points [12], [39], [51], [57]. Kennaugh–Huynen

parameters are not written in a vector form, and this is less

robust from a classification viewpoint not being defined as a

proper strategy to compute optimum distances by using the

Huynen fork of two targets. Cloude’s contribution has been

fundamental in the introduction of target vectors and

spinor algebra, nevertheless angles �P; �P; �P;�Pi are not

�-invariant whereas the invariant �P parameter has showna very powerful feature for the characterization of random

targets [1], [2], [16].

The Krogager decomposition provides a simpler inter-

pretation of the symmetric scattering matrix in terms of

the superposition of dihedral and trihedral reflectors. The

Table 1 Properties of the CTD Parameters (amb. ¼ Ambiguous, l&s ¼ Lossless and Sufficient, and overab. ¼ Overabundant



interpretation of SDH for nonisotropic targets � � =4 isnot entirely clear, resulting in a superposition of sphere

and diplane, and the coefficients of the decomposition

being correlated for nonsymmetric scattering do not con-

serve the signal energy [11], [15], [23].

The Cameron decomposition has provided two contri-

butions: the development of an efficient target classifica-

tion algorithm based on the inner Hermitian product and a

very detailed work for the characterization of a symmetrictarget set [53]. Nevertheless, the Cameron CTD for a non-

symmetric target gives a suboptimal interpretation and the

generalization of the description of random targets is not

straightforward [52], whereas the circular polarization

scattering vector proposed in Section III has shown re-

cently a very good generalization for the random reciprocal

target [20], [21], [58]. On the other hand, the circular po-

larization approach of Bickel and Lee is effective in extract-ing some important �–�-invariants for both symmetric or

nonsymmetric targets and for the estimation of the

�-effects induced from the ground slope [25], [29], [50]. In

Section III, the scattering vector model proposed by Cloude

is developed in the circular polarization basis representing

the polarization of the elemental particles of the radiation

and is decomposed in terms of five physically meaningful

�–�–�-invariant parameters. This target vector is used forcompensating the Faraday distortion, for extracting

important target invariants, and for developing an optimum

maximum-likelihood classifier in a single shot.

III . FAST LOSSLESS AND SUFFICIENT�–�–�- INVARIANT SCATTERINGVECTOR DECOMPOSITION BASED ONTHE PARTICLE CHARACTERIZATIONOF RADIO SCATTERING

In this section, some novel contributions are presented,

namely: the fast decomposition of the scattering vector in

terms of a set of invariant parameters (Section III-B and

C), the introduction of the particle characterization of

radio scattering (Section III-D), the introduction of the

novel invariant target vector subspaces (Section III-E), andthe development of a generalized point target scattering

classification algorithm (Section III-F).

A. Modeling �-Dependence of the ScatteringVector Parameters

Unitary, four-element scattering vectors are canoni-

cally modeled in terms of seven angles, as shown by Cloude

and Pottier [14]

kP ¼

cosð�PÞ expðj�1Þsinð�PÞ cosð�PÞ expðj�2Þ

sinð�PÞ sinð�PÞ cosð�Þ expðj�3Þsinð�PÞ sinð�PÞ sinð�Þ expðj�4Þ

26643775: (34)

By using the Pauli basis, Cloude and Pottier modeled

the target orientation effect � in terms of the multipli-

cation with a special unitary SU(4) matrix R8P [2]

k0Pð�Þ ¼ RPð8ÞkP ¼

1 0 0 0

0 cosð2�Þ � sinð2�Þ 0

0 sinð2�Þ cosð2�Þ 0

0 0 0 1

26643775kP

(35)

thus obtaining (36), shown at the bottom of the page.

By comparing the angles of (36) and (34) it is clear thatmost of these quantities are strongly �-dependent.

By computing the eigenvectors of RPð8Þ, a new spe-

cial unitary SU(4) polarization basis Q modeling the �rotation in phasor is found

RCð�Þ ¼

expð�j2�Þ 0 0 0

0 1 0 0

0 0 1 0

0 0 0 expðj2�Þ

26643775

¼Q�1RPð8ÞQ

Q ¼ 1ffiffiffi2p

0 j j 0

1 0 0 �1

j 0 0 j

0 1 �1 0

26643775: (37)

k0Pð�Þ ¼

cos �0P

exp j�01

sin �0P

cos �0P

exp j�02

sin �0P

sin �0P

cosð�0Þ exp j�03

sin �0P

sin �0P

sinð�0Þ exp j�04

2666437775

¼

cosð�PÞ expðj�1Þsinð�PÞ cosð�PÞ cosð2�Þ expðj�2Þ � sinð�PÞ sinð�PÞ cosð�Þ sinð2�Þ expðj�3Þsinð�PÞ sinð�PÞ cosð�Þ cosð2�Þ expðj�2Þ þ sinð�PÞ cosð�PÞ sinð2�Þ expðj�3Þ

sinð�Þ sinð�Þ sinð�Þ expðj�4Þ

2666437775 (36)



The phasor modeling of RCð�Þ allows for generating asimple scattering vector model in terms of �-invariant

parameters. The search of �-independent features is use-

ful for reducing by one order the complexity of polarimet-

ric feature vector fluctuations as a function of the target

aspect angle vð�; �;�Þ ! vð�; �Þ [4], [5], [33]. The four

eigenvectors represented by the columns of Q perform a

polarimetric change of basis into the left–left orthogonal

( ¼ =4, � ¼ 0) special unitary SU(2) polarization basisof the Sinclair matrix

SC¼Sll Sll?

Sl?l Sl?l?

� �¼ 1ffiffiffi

2p

1 j

j 1

� �Shh Shv

Svh Svv

� �1ffiffiffi2p

1 j

j 1

� �¼ 1

2

Shh þ jShv þ jSvh � Svv jShh þ Shv � Svh þ jSvv

jShh � Shv þ Svh þ jSvv �Shh þ jShv þ jSvh þ Svv

� �:

(38)

The circular polarization scattering matrix of a recip-rocal target (38) is equivalent to the polarimetric change of

basis provided by Q to RPð8Þ in (37). The direction of the

electric field phasors of a left and right circular polarized

plane wave rotates in a plane transverse to the LOS, with

constant angular speed, describing a circle. For this reason,

� variation is equivalent to a phase shift on the circularlybased copolarized returns. The modulus of the circular

polarization scattering vector is also independent of �,

whereas by using linear polarization, the fluctuations

generated by � are experienced in order of 15 dB or more [5].

It is also worth noting that the scattering vector in ll?basis also models the Faraday rotation in the phasor form,

as shown by Cloude [2], Lee et al. [25], Bickel and Bates

[29], Freeman [30], Wang et al. [32], Bickel [50], andBebbington et al. [59] and in (39), shown at the bottom of

the page. For this reason, the RCS of a target sensed via a

circular polarization radar thus appears independent of the

Faraday rotation angle �B.

B. Modeling Circular Polarization Scattering VectorThe circular polarization scattering vector is obtained

by lexicographic ordering (22), (23) of the scattering

matrix elements represented on the left–left orthogonal

circular SU(2) ðll?Þ polarization basis in terms of the

multiplication of six SU(4) matrices and one complexnumber C ¼ SPAN expðj�Þ.

The special unitary matrices of any order n SUðnÞ are

useful for representing rotation groups and have two

interesting mathematical properties

if Rð!jÞ 2 SUðnÞ ! det Rð!jÞ

¼ 1

Rð!jÞ�1 ¼ Rð!jÞ:

�(40)

c ¼

Sll

Sll?

Sl? l

Sl? l

2666437775 ¼ RCg1ð6ÞRCg2ð8ÞRCg3ðÞRCg4ð�ÞRCg5ð�ÞRCg6ð�Þ

C

0

0

0

2666437775

¼

1 0 0 0

0 expð�j2�Þ 0 0

0 0 expðj2�Þ 0

0 0 0 1

2666437775


0 1 0 0

0 0 1 0

0 0 0 expðj2�Þ

2666437775

�


0 expðj2�Þ 0 0

0 0 expðj2�Þ 0

0 0 0 expð�j2�Þ

2666437775

1 0 0 0

0 cosð�Þ � sinð�Þ 0

0 sinð�Þ cosð�Þ 0

0 0 0 1

2666437775

�

cosð�CÞ 0 0 sinð�CÞ0 1 0 0

0 0 1 0

� sinð�CÞ 0 0 cosð�CÞ

2666437775

sinð�CÞ cosð�CÞ 0 0

cosð�CÞ � sinð�CÞ 0 0

0 0 1 0

0 0 0 �1

2666437775

C

0

0

0

2666437775

¼

sinð�CÞ cosð�CÞ exp jð�2�� 2�Þð Þcosð�CÞ cosð�Þ exp jð�2�þ 2�Þð Þcosð�CÞ sinð�Þ exp jð2�þ 2�Þð Þ

sinð�CÞ sinð�CÞ exp jð�2�þ 2�Þð Þ

2666437775SPAN expðj�Þ (39)



C. Physical Target Parameter ExtractionInversion of (39) provides the estimation of the follow-

ing characteristic parameters:

• b� ¼ ð1=4Þ argfSl?lS�ll?g measures the Faraday ro-

tation in terms of the circular cross-polarizationphase interference, and it is meaningless for non-

reciprocal target [30];

• b� ¼ ð1=4Þ argfSl?l?S�llg measures the target rota-

tion around the LOS in terms of the circular copo-

larization phase interference, and it is meaningless

for nonsymmetric target [1].

The removal of the effect of these rotations provides the

characteristic unrotated vector

c�6�8 ¼ RCg2ð�b8ÞRCg1ð�b6Þc (41)

useful for classification studies and for extracting param-

eters independent of the target orientation and of the

Faraday rotation effects useful for inversion studies.

• b� ¼ ð1=4Þ argfc��ð1Þ�c��ð2Þg is the phase

interference between copolarization and cross-polarization scattering coefficients and allows for

distinguishing anisotropic targets like dipoles and

quarter waves and is a recent contribution [20].

• b� ¼: c�C ¼ cos�1ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1=2jShh þ Svvj2Þ=kck

qÞ, for re-

ciprocal targets ð� ¼ =4Þ, �C ¼ �1�P is the main

parameter for estimating the odd–even bounce

nature of the target [14], cos2ð�Þ being the mea-

sure of the odd-bounce ratio of the instant back-

scattered power.

• c�C ¼ cos�1ðjcð2Þj= cosðb�ÞkckÞ introduced here isthe degree of balance between the cross-polarized

coefficients of the circular polarization Sinclair

matrix. � ¼ =4 for a reciprocal target but is

meaningless for �! =2. To overcome this am-

biguity, we introduce a meaningful energetic mea-

sure of nonreciprocity degree Nr.

• cNr ¼ cosðb�Þ2½cosðb�Þ2 � sinðb�Þ2� 2 ½�1 1� is the

normalized energetic difference between cross-polarized returns on the ll? basis measuring the

power of nonreciprocity. It is zero for ‘‘real’’ targets

under backscatter geometry. 1� jNrj also mea-

sures the degree of confidence in � estimation.

• c�C ¼ cos�1ðjcð1Þj= sinðc�CÞkckÞ, introduced by

Corr and Rodrigues in 2002, is the degree of ba-

lance between the copolarized coefficients of the

circular polarization Sinclair matrix. �C ¼ =4 forthe symmetric target but it is meaningless for

�! 0 [56]. To overcome this ambiguity, we in-

troduce a meaningful energetic measure of non-

symmetry degree Hel.• dHel ¼ sinð�Þ2½cosð�CÞ2 � sinð�CÞ2� 2 ½�1 1�, in-

troduced recently in [21] for characterizing the

symmetry properties of random scattering, is the

normalized energetic difference between circularcopolarized returns on the ll? basis and is useful

for assessing the left–right degree of asymmetry of

the scattering. Hel is zero for the symmetric target

providing a meaningful indicator of a target sym-

metry. 1� jHelj measures the degree of confi-

dence in � estimation.

By inverting (39), using the six estimated angles, the fol-

lowing is found:

C

0

0

0

2666437775 ¼ RCg6ð��ÞRCg5ð��CÞRCg4ð�b�ÞRCg3ð�b�Þ

� RCg2ð�b�ÞRCg1ð�b�Þc: (42)

Two parameters are extracted from the complex Cvalue:

• SPAN ¼ jCj ¼ kck2 ¼ trðSt�SÞ, introduced by

Graves [41], is a basis invariant measuring the

total amount of energy backscattered.

• b� ¼ argfCg is the target absolute phase, re-

presenting a measure of radar target range phase

shift.

D. Scattering Mechanism Interpretation Using theParticle Characterization of Radio Scattering

The wave-particle duality is an important and well-

known concept in physics introduced at the beginning of

the 20th century. Nevertheless, radar polarimetry hasbeen developed considering mainly the wave nature of the

electromagnetic radiation. In this section, the photon

circular polarization basis is used as a natural basis for

representing the radio scattering of differently shaped

targets. The analysis of the circular polarization scattering

vector characterizes the scattering process as the super-

position of the interaction of elementary particles with

matter.In the following discussion, the physical polarization of

the particles, usually represented according to the FSA, has

been transformed into the BSA conventional for radar

polarimetry, where the two conventions for circular polar-

izations on backscatter geometry describe a unique sense

of polarization in transmission and one opposite in

reception [38], [60].

• LL copolarized channel jSllj measures the rate ofreceived right spin photons (where particles are

left polarized in FSA convention) by transmitting a

wave composed by pure left spin photons. It is well

known that a left circular polarized antenna is best

matched in receiving a right-handed circularly po-

larized wave. This channel has maximum return

for dihedrals and right-handed helices, but is blind



for left-handed helices and pure odd-bouncereflectors.

• RR copolarized channel jSl? l? j measures the rate of

received left spin photons (right in FSA) by trans-

mitting a wave composed by right spin photons.

This channel has maximum return for dihedrals,

and left-handed helices, but is blind for right-

handed helices and pure odd-bounce reflectors.

For symmetric targets, the absolute value of left–left and right–right circular channels is the same

jSllj ¼ jSl?l? j [5], [50].

• LR channels jSll? j measure the rate of received left

spin photons (right in FSA) by transmitting a wave

composed of left spin photons. Due to the recipro-

city principle for backscatter geometry LR ¼ RL,

jSl? lj ¼ jSll? j. The response of this channel is maxi-

mum for symmetric trihedrals, spheres, or ele-mentary planes at orthogonal grazing.

• Other scatterers like dipoles, quarter waves, cy-

linders, and narrow diplanes have mixed behavior.

They transform an incoming pure photon left or

right state into the superposition of opposite states.

In this case, 0 G cos2ð�Þ G 1 measures the

energetic degree between copolarized and cross-

polarized returns, whereas angle 4� measures thephase interference between the outgoing LR states.

For example, the linear polarization scattered by a

dipole represents an energetic balance between all

the coefficients of the circular polarization scatter-

ing vector � ¼ =4. The energetic balance be-

tween outgoing LR states is observed in this case.

The jcj is an invariant quantity characterizing the four

main target features: the total amount of reflected powerSPAN, the odd–even bounce nature cosð�CÞ, the target

degree of reciprocity Nr, and the degree of symmetry Hel.The argfcg exploits some local effects: target phase shift �,

Faraday effect �, target orientation around LOS �, and

interference between odd–even scattering �. Differences on

� are measured between anisotropic orientation-dependent

targets like dipoles and quarter-wave devices ð� ¼ =4Þ [9],

[12], [23]. Fig. 3 gives a pictorial representation of theparticle characterization of radio scattering for target

classification studies.

By looking at Fig. 3, it can be observed that the sym-

metric targets [Fig. 3(a) and (b)], e.g., plane, diplane, di-

pole, quarter-wave devices etc., are characterized by

identical energy measured in the like polarization signa-

tures jSl? l? j ¼ jSllj. On the contrary, nonsymmetric targets

[Fig. 3(c) and (d)] (right and left helices) generate dif-ferent amplitudes on the like circular polarizations

jSl? l? j 6¼ jSllj, Hel ¼ 1.

By considering that using BSA like polarizations are

unmatched in reception, whereas cross-polarizations are

best matched, we find the following.

1) Odd-bounce reflectors (sphere, trihedral, plane)

conserve the spin of the photon on BSA conven-

tion (they are spin transformer on FSA); due to

antenna (3), we obtain jSl? lj ¼ jSll? j ¼ 1.2) Even-bounce reflector (dihedral) transforms the

photon spin on BSA convention (it is spin

conserve in FSA); this gives a maximum re-

turn in the like circular channels and is blind

in the cross-circular channels jSl? lj ¼ jSll? j ¼ 0,

jSllj ¼ jSl? l? j ¼ 1.

3) Left-handed circular polarization is transmitted,

and the left-handed helix target is modeled as aright-handed helix antenna in a reverse BSA. The

left helix receives the particle in the like right-

handed circular polarization due to reversal

geometry. By (4) V ¼ 1=2½1 j�½1 j�t ¼ 0, the signal

is unmatched and no signal is backscattered

jSllj ¼ jSll? j ¼ 0 [1], [5], [9].

4) Left-handed circular polarization is transmitted,

and the right-handed helix target is modeled asa left-handed helix antenna in a reverse BSA.

The right helix receives the particle as right-

handed circular polarization due to reversal

geometry and is matched in reception V ¼1=2½1 � j�½1 j�t ¼ 1. The target modeled being an

open circuit, it re-irradiates a left-handed polar-

ization, which in the radar receiver antenna coor-

dinates is right handed and is matched onreception: jSllj ¼ 1, jSll? j ¼ 0 [1], [5], [9]. By

transmitting right circular polarizations, the

problem is symmetric where the left helix target

is matched on reception and the right helix target

is unmatched.

It is worth noting that the particle characterization of

radio scattering, through the use of the invariant fSPAN;cos2ð�Þ;Hel;Nr;�g parameters, describes the scatteringat the macroscopic level. The quantization effect, observ-

able in very weak echo returns, will be investigated in

future works.

Fig. 3. Description of the scattering from the idealized target by

emitting left circular polarization (LC ¼ left circular polarized in red,

RC ¼ right circular polarized in green). The sense of the transmitted

polarization is according to the sense of the helix starting from

the center of the spiral.



E. Invariant Scattering Vector SubspacesBy the use of c�

b��b�, a novel in phase-reciprocal target

vector subspace cRec�b��b� 2 C4 is obtained

cRec�b��b�¼PRecc�b��b� ¼ 1 0 0 0

0 12

12

0

0 12

12

0

0 0 0 1

26643775c�

b��b� (43)

where the projection PRec operator, introduced by

Cameron, in our case, is applied after �–� removal.

More conveniently, an equivalent three-element vector

cRec�b��b�3-D 2 C3 is introduced for describing unrotated

reciprocal target subspace Rec 2 C3

cRec�b��b�3-D ¼P3-DcRec�b��b�

¼1 0 0 0

0ffiffi2p

2

ffiffi2p

20

0 0 0 1

264375cRec�b��b�: (44)

It is worth noting that, after the �-removal, like polar-ized terms ll� l?l? are also in-phase. A representation of

the modulus of the reciprocal, unrotated target vector

useful for backscatter geometry is shown in Fig. 4.

By using cRec�b��b�3-D , the symmetric target space after

�–�-removal cSym�b��b�3-D 2 C3 is a subspace of cRec�b��b�

3-D

cSym�b��b�3-D ¼PSymcRec�b��b�

3-D ¼12

0 12

0 1 012

0 12

24 35cRec�b��b�3-D : (45)

This result is in contrast to the Cameron CTD thatdecouples a reciprocal target into two symmetric (but ro-

tated) targets that do not form a subspace of the reciprocal

target vector [2], [12], [23].

Theorem 1: By removing �–� effect before decompos-

ing the target, a symmetric unrotated target that is a

member of a subspace Sym 2 C2 of the reciprocal target

space is found.Proof:

if cSym�b��b�3D1

; cSym�b��b�3D2

2 Sym

! �1cSym�b��b�3D1

þ �2cSym�b��b�3D2

2 Sym 8�1;2 2 C:

(46)

A two-element vector cSym2-D 2 C2 is also introduced

for describing the symmetric unrotated target subspace

Sym

cSym�b��b�2-D ¼P2-DcSym�b��b�

3-D

¼

ffiffi2p

20

ffiffi2p

2

0 1 0

24 35cSym�b��b�3-D : (47)

h

F. Classification Algorithm DescriptionLet

c��SPANu ¼ Rðd��ÞRðd��Þc

kck (48)

be the scattering vector obtained under test after the

removal of �–� removal and vector normalization and

c��SPANt the scattering vector of the reference cano-

nical target templates described in Table 2 [52]. The

identified class in the maximum-likelihood sense is the one

having the maximum degree of correlation

CD ¼ arg maxt c��SPANu

t�c��SPAN

t

�� n o: (49)

It is worth noting that (49) is invariant to arbitrary

shift �, relative rotations �, � shift, and vector normFig. 4. Representation of the reciprocal target using jcRec�V�C3-D j.



scaling. In our implementation, some clusters are fused:

n ¼ nþ [ n�, þ ¼ þþ [ þ�. The proposed classification

algorithm provides for symmetric targets the same six de-

cision zones of Cameron zC parameter, whereas for par-

tially symmetric targets only two helical clusters are added

(@, c), and one n for nonreciprocal targets [23]. It is worthnoting that the reference target set can be varied according

to the application of interest, by using some theoretical

scattering matrices or some templates of recorded data for

supervised classification. The possibility of supporting

both supervised and unsupervised classification algorithms

and the all-in one classification metric able to deal with

symmetric as well as nonsymmetric targets also in pre-

sence of propagation distortions � 6¼ 0, Nr 6¼ 0, are themain contributions of this paper for application of remote

sensing and surveillance.

In Table 2, the proposed �–�–� invariant decomposi-

tion has been applied to some classical scatterers, significant

for backscattering geometry, where the scattering matrix is

known in the closed form, and values of the symmetric

scatterers are ordered in terms of �C ¼ �P. Nonreciprocal

targets n are novel contributions of this paper, and theirscattering matrices are obtained by modeling Sll? 6¼ Sl?l.

G. Processing NotesIt is worth noting that the proposed decomposition has

been found suitable for developing real-time applications,

and the processing of large data sets. This is given the

reduced algorithm complexity (38), (49), compared with

Cameron identification algorithm [1], [23], [52] and a

smaller memory requirement of about twice the original

data format size.

IV. RELATIONSHIP BETWEENDETERMINISTIC TARGETDECOMPOSITION PARAMETERS

In this section, the circular polarization scattering model

is used for comparing the invariant features extracted

from the proposed CTD with the most relevant param-eters described in Section II. Circular polarization

Krogager and Cameron algorithms have been revised

and a new definition of target symmetry has been intro-

duced. Then the CTD parameters have been compared

considering three cases: reciprocal symmetric scattering,

reciprocal scattering, and nonreciprocal scattering. Some

partitions of the parameters f�;�;Nr;Helg space are

shown as varying couples of parameters and representingthe results in the plane.

A. New Definition of Target Symmetry forReciprocal Scattering ðNr ¼ 0Þ

The Kennaugh constant echo loci on the Poincare

sphere [4] have shown the equator as a plane of symmetry

for most radar targets; this property is explained through

the particle characterization of radio scattering, given the

symmetry between the scattering of left and right helicity

particles, and it introduces a necessary and sufficient con-dition for assessing the target nonsymmetry degree in the

following theorem.

Theorem 2: A reciprocal target is symmetric according

to the Kennaugh and photon transformation theories if and

only if the circular polarization copolarized returns have

identical power.

Table 2 Extracted Parameter From Classical Scatterers: Trihedral �, Cylinder i, Dipole I, Quarter-Wave Devices þ, Narrow-Diplane V, Diplane L,

Right Helix @, Left Helix ç, Nonreciprocal nþ;n�



Proof: Let �C ¼ =4, for a reciprocal targetð� ¼ =4Þ; it means that jSllj ¼ jSl? l? j ¼ k.

Shv is extracted through (38)

Shv ¼ 2j Sll þ Sl? l?ð Þ¼ 2jk exp 2jð�Þð Þ þ exp �2jð�Þð Þð Þ expðj�� 2�Þ¼ � 4k expð�j2�þ �Þ sinð�Þ: (50)

By inspection of (50), the Sinclair matrix is diagonal-

ized by forcing the target orientation angle around LOS �to zero. If � ¼ 0! Shv ¼ 0, then the target is symmetric

according to the Kennaugh–Huynen–Cameron definition

[1], [2], [4], [5], [12]. This sentence proves the sufficientcondition, but proof of the necessary condition is

needed. If a target is symmetric, then Shv ¼ 0 can be ob-

tained via a proper rotation of the Sinclair matrix around

LOS (11) by projecting SCam�1990D to the circular polari-

zation basis. By substituting Shv ¼ 0 in (38), the following

is straightforward:

Ssym�b��C ¼ 1

2

Shh � Svv jShh þ jSvv

jShh þ jSvv �Shh þ Svv

� �: (51)

Finally, considering the phasor modeling of � obtained

using the circular polarization basis, applying a backward

rotation of an angle �� to SCð�Þ, it is found that thecopolarization returns conserve the same intensity

jSllj ¼ jSl? l? j.Theorem 2 presents a faster algorithm for determining

whether the target is symmetric by comparing the power of

the circular polarization Sinclair matrix, where other algo-

rithms need the computation of a diagonalization. h

B. Equivalence Between �–� and Other SymmetryParameters for Unit Reciprocal Symmetric TargetðHel ¼ Nr ¼ 0Þ

In this section, zC is used for representing Huynen

coneigenvales; the Touzi con-eigenvalues model is writtenin terms of Huynen parameters; the � and � parameters

are used for computing zC–SSCM–TSVM parameters and

finally the components of the SDH decomposition.

1) The Relationship Between the Cameron zC Parameter andthe Kennaugh–Huynen Con-Eigenvalues: By analyzing the pa-

rametrization of the con-eigenvalues proposed by Huynen

and the Cameron’s zC parameter, the following relationshipis found, as shown in recent publications [21], [43]:

SCamD ¼

1 0

0 zC

� �

SHuy�1978D ¼m expðj2�H þ 2�HÞ

�1 0

0 tanð�HÞ2 expð�j4�HÞ

" #(52)

nevertheless the computation follows from different

methods, and the following is obtained [21], [43]:

jzCj ¼ tanð�HÞ2

argðzCÞ ¼ �4�H:

((53)

2) TVSM and Huynen Symmetric Target Parameters: By

substituting Huynen parameters, modeled according to (6)

in (17), the following is found:

tanð�sÞ expj��s ¼ expð2�HÞ � tan2ð�HÞ expð�2�HÞexpð2�HÞ þ tan2ð�HÞ expð�2�HÞ

� �:

(54)

3) The Relationships Between the Cameron zC Parameterand �–�: A unit unrotated reciprocal symmetric targetcan be modeled by substituting � ¼ =4, �C ¼ =4, � ¼� ¼ � ¼ 0, and SPAN ¼ 1, in (39), in terms of �;�parameters It can also be represented by 2-D symmetric

unrotated target vectors by applying a set of rotations and

projections, as described in (43)–(45) and (47), thus

obtaining

cSym�b��b��SPAN2-D P2-DPSymP3-DPRecc

��SPAN�¼�C¼4

¼sinð�Þ expð�j2�Þ

cosð�Þ expðj2�Þ

" #: (55)

By backprojecting cSym�b��b��SPAN2-D to the h� v basis,

the following is found:

lSym�b��b�2-D ¼ 1

�2j cosð�Þ expðj2�Þ

�1� j tanð�Þ

1þ j tanð�Þ expð�j4�Þ

" #(56)

where l stands for the ‘‘lexicographic ordering in the

horizontal, vertical basis.’’

The relationship between �–� and complex

Cameron’s zC parameter is obtained by calculating the



ratio between Huynen con-eigenvalues, where l1;2 are nor-malized con-eigenvalues extracted from (56)

zC¼min jl1;2j

max jl1;2j

; l1;2¼1 j tanð�Þ expð�j4�Þ: (57)

4) The Relationships Between �;� and SSCM, TSVM Sym-metric Target Parameters: By projecting the Touzi TSVM

model, given m ¼ 0, and kkTSVMC k ¼ 1, in [15], to the

circular polarization basis, a relationship of equivalence

between � and ��S and between � and �S is found [21]

kTSVMm¼0C ¼

ffiffiffi2p

sinð�SÞ exp jð��S � 2�Þð Þcosð�SÞ exp j

2

�

ffiffiffi2p

sinð�SÞ exp jð��S þ 2�Þð Þ

264375

!4� ¼

2� ��S

� ¼ �S:

�(58)

Similar results are obtained, when considering Touzi’s

SSCM SSCM;�B � �A parameters [54], where for

m ¼ 0, SSCM¼�S and ��S ¼ �B � �A [15] are obtained.

5) The Relationships Between �;�;�, and KrogagerDecomposition Parameters: If we consider unit, reciprocal,

and symmetric target SPAN ¼ 1, � ¼ �C ¼ =4, it follows

that the Krogager helix component is null, where the

circular polarization Sinclair matrix can be rewritten as

SC¼kD expð�j2�Þ kS exp j ’S þ

2

kS exp j ’S þ

2

kD expðj2�Þ

" #

¼sinð�Þexp�j2ð�þ�Þð Þ cosð�Þexpðj2�Þ

cosð�Þ expðj2�Þ sinð�Þexp�j2ð��Þð Þ

� �:

(59)

It follows:

kS ¼ cosð�ÞkD ¼ sinð�Þ’S ¼ 4��

2

�K ¼ �; �K ¼ �:

8>>><>>>: (60)

6) Conclusions About the Equivalence of Symmetric TargetParameters: A target is symmetric if the copolarized returns

of the circular polarization scattering matrix have the same

power. The circular polarization basis allows a fast

extraction of the ðSPAN; �;�;�Þ �–�-invariant param-

eters formally equivalent to the Huynen, Cameron,Krogager, and Touzi symmetric scattering parameters

kD

kS; ’S

� �$ð�;�Þ $ zC $ ð�H; �HÞ $ ð�S��sÞ

$ ð SSCM;�B � �AÞ: (61)

A physical justification of (61) is provided as follows: a

unit symmetric target is represented according to Camer-on, in terms of the physical target rotation of an angle �around LOS and by a two-element diagonal unit scattering

matrix (11)–(20). A unit diagonal matrix is characterized

by three DOFs where two DOFs explore the target shape

(the Cameron symmetric target space [53]) and the last

DOF is an irrelevant phase factor for classification devel-

opment. After the removal of �-effect a symmetric target

is represented uniquely by a two-component unit vector,represented in a linear h� v basis, where all the optimal

distance measures between symmetric targets are physi-

cally equivalent. Being a symmetric target dependent on

five parameters, e.g., fSPAN; �;�;�;�g, a reciprocal

target has only one free parameter more: the helicity.

C. The Relationships Between Symmetry HelicityParameters for Reciprocal Target Nr ¼ 0

In this section, the Hel indicator is written in terms of

Huynen parameters ðH; �HÞ, and H is proven not suf-

ficient for measuring the degree of nonsymmetry (heli-

city); the Cameron and Krogager helicity indicators S; kH

are modeled in terms of ð�; �;�Þ using the circular po-

larization scattering model.

1) Relationship Between Hel and Kennaugh–Huynen–Touzi Parameters ðH; �HÞ for Nonsymmetric Reciprocal Tar-get (Nr ¼ 0, Hel 6¼ 0): In Section IV, we have shown that

the �–�–� invariant Hel is well suited for characterizing

the degree of target nonsymmetry in terms of energetic

difference between ll� l?l?. In 2007, Touzi proposed the

use of Huynen H for assessing target helicity [15]. Fornonsymmetric targets, H is the ellipticity of the maximum

copolarization return. We have found that it is not possible

to trace a simple relationship between H and Hel because

H saturates for � ¼ 90�. To show this effect, the copo-

larization power spectra for � ¼ 90� and � ¼ 60� have

been represented in Fig. 5 for �C ¼ 0�, 30�, 60�, and 90�,given � ¼ � ¼ 0. The copolarization power spectrum (or

copolarization signature), introduced by Van Zyl et al.,plots the power return as a function of the polarization

antenna used [35], [61]. Through the inspection of the

maximum of the copolarization power spectrum, Huynen

ðH; HÞ parameters are found graphically.

Fig. 5(a)–(d) shows that for � ¼ 60�, there is a corre-

spondence between Hel and the Kennaugh–Huynen–

Touzi H parameters. For Hel ¼ 0:75, the co-pol max



helicity H ¼ 25�, whereas for Hel ¼ 0:37 the values of

H ¼ 12�, then there seems to exist a positive correlation

between Hel and H. This behavior is also symmetric with

respect to the positive or negative values of Hel and H.

Computing the copolarization power spectra for � ¼ 90�,varying �C [Fig. 5(e)–(h)], H ¼ =4, is constant chang-ing only in its sign. In other words, H for � ¼ =2 does

not provide the same information of Hel, but only rough

information about the left or right helicity sense, and this

parameter is also ambiguous for the dihedral � ¼ =2,

�C ¼ =4, as shown by Cameron and Leung [12]. For this

reason, H alone is not adequate for measuring the degree

of symmetry of a partially symmetric target.

In order to be more exhaustive, Hel and the maximumco-polarization helicity H have been computed for differ-

ent values of � ¼ 30�, 45�, 60�, 75�, 90�, and continuous

values of �C 2 ½0; 90��, given � ¼ � ¼ � ¼ 0 in Fig. 6.

Fig. 6(a) shows increasing values of Hel (in modulus)

for increasing values of �, where a monotonic function of

�C is shown in all ranges. Fig. 6(b) shows increasing values

of H (in modulus) for increasing values of �, but the curve

saturates for � ¼ 90�, where in this case �C informationcannot be retrieved by inverting function Hð�C; � ¼ 90�Þ.Finally, the correspondence between Hel and H has been

traced in Fig. 6(c) for � ¼ � ¼ 0� varying �, with 15� of

step size. In this case, H is not the function of Hel, but it

depends on other parameters, thus the two indicators arenot equivalent.

In 1978, Huynen proposed the power F parameter as an

indicator of the degree of target symmetry [5]. By manipu-

lating [5, eq. (6.3)–(6.4)], it is found that the circular

polarization helicity Hel is equivalent to four times the

Huynen helicity power measurement F extracted from the

power Muller matrix [5]

4F ¼ Sl? l?j j2�jSllj2 ¼m2

cos4ð�HÞcosð2�HÞ sinð2HÞ

¼Hel � SPAN: (62)

After some algebraic manipulations is found [21]

Hel ¼ cosð2�HÞcos4ð�HÞ 1þ tan4ð�HÞð Þ sinð2HÞ: (63)

The 2-D plot of Helð�H; HÞ in (63) is shown in Fig. 7.

It is interesting to note that for �H ! =4, Hel is in-

dependent of H and H becomes meaningless

lim�H!

4

Hel ¼ 0

1sinð2HÞ: (64)

Indeed, as shown in Fig. 7, Hel ¼ 0 is obtained by twosubsets of the symmetric target: H ¼ 0 symmetric target

set and �H ¼ =4 isotropic symmetric target set (trihedral,

dihedral), where H can be 6¼ 0 and, in this case, the

Kennaugh–Huynen con-eigenvalue solution degenerates

[5]. This conclusion is in agreement with Cameron’s con-

cern: ‘‘It is important to note that questions can be raised

regardless the uniqueness of the decomposition parameters

Fig. 5. Copolarization power spectra of zero oriented partially

symmetric targets (C ¼ 0�, 9 ¼ 0�). (a) � ¼ 60�, �C ¼ 0�. (b) � ¼ 60�,

�C ¼ 30�. (c) � ¼ 60�, �C ¼ 60�. (d) � ¼ 60�, �C ¼ 90�. (e) � ¼ 90�,

�C ¼ 0�. (f) � ¼ 90�, �C ¼ 30�. (g) � ¼ 90�, �C ¼ 60�. (h) � ¼ 90�,

�C ¼ 90�.

Fig. 6. Relationship, between helicity indicators. (a) Helð�;�CÞ.(b) Hð�;�CÞ. (c) Hð�;HelÞ.



derived both by Karnychev et al. [46] and R. Touzi [15],

since as indicated by Cameron and Leung [12], Huynen’s

helicity angle H is an ambiguous quantity. For example, itmay be chosen arbitrarily for a dihedral scatterer type’’ [53].

Similar comments about the use of F have been also drawn

by Huynen: ‘‘The parameter F shows a bias for RC � LC type

of circular polarization. As we shall see shortly, F is charac-

teristic of a right or left wound helix viewed on axis. We

known that F is proportional to sinð2HÞ’’ [5].

2) Modeling of the Cameron and Krogager DecompositionsUsing �; �C;� for Reciprocal Target Nr ¼ 0, a Comparison ofHelicity Indicators: Recently, both Cameron and Rais, and

Zhang et al. have proposed the circular polarization for

computing the Cameron and Krogager decompositions

[43], [62]. In Section II, we have concluded that the

Krogager decomposition performs a suboptimal represen-tation for a nonsymmetric target given the nonorthogon-

ality of its components.

Let us suppose, using our model, that the target vector

c is unit and reciprocal Nr ¼ 0, � ¼ =4. It can be mod-

eled, after deorientation and normalization in terms of

three invariant parameters ð�; �C;�Þ. By considering

SPAN ¼ 1, � ¼ 0, and � ¼ =4 in (39), the following is

found:

c�b��b��SPANj�¼4;�¼0 ¼

sinð�Þ cosð�CÞ expð�j2�Þ1ffiffi2p cosð�Þ expðj2�Þ

1ffiffi2p cosð�Þ expðj2�Þ

sinð�Þ sinð�CÞ expð�j2�Þ

266664377775:

(65)

By the analysis of �C, in (65), the sphere–diplane–helix

decomposition is straightforward, as shown in (66) at thebottom of the page.

From the analysis of (66), it follows that:

kS ¼ cosð�Þ; ’S ¼ 4�� 2

kD ¼ max

sinð�Þ sinð�CÞsinð�Þ cosð�CÞ

( )kH ¼ sinð�Þ cosð�CÞ � sinð�CÞ½ �:

8>>>><>>>>: (67)

The Cameron maximum symmetric component can be

extracted via the circular polarization scattering vector

integrating the properties of Theorem 2 and the symmetric

target subspace as described by Theorem 3.

Fig. 7. HelðH; �HÞ shows the helicity Hel as a function of the

polarizability parameter �H and maximum con-eigenvector

ellipticity H.

if � 4 ! SRec�6�8

ll?¼ 1ffiffi

2p cosð�Þ expðj2�Þ

0110

26643775þ sinð�Þ sinð�CÞ expð�j2�Þ

1001

26643775

þ sinð�Þ cosð�CÞ � sinð�CÞ½ � expð�j2�Þ

1000

26643775

if � 4 ! SRec�6�8

ll?¼ 1ffiffi

2p cosð�Þ expðj2�Þ

0110

26643775þ sinð�Þ cosð�CÞ expð�j2�Þ

1001

26643775

þ sinð�Þ sinð�CÞ � cosð�CÞ½ � expð�j2�Þ

0001

26643775

8>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>:

(66)



Theorem 3: For an unrotated reciprocal target, theCameron maximum symmetric component and the sym-

metric target subspace are equivalent.

Proof: Let c ¼ ½a b b c�t be: a, c 2 R, b 2 C a

complex unrotated reciprocal target vector. According to

Theorem 2, if a ¼ c, than c is symmetric. The Cameron

maximum symmetric component can be extracted as follows:

c ¼abbc

24 35 ¼ xbbx

24 35þ a� x00

c� x

24 35 ¼ cmaxsym þ cres (68)

minimizing the absolute value of the residual component:

@EðcresÞ=�x ¼ ð@=�xÞða2 þ c2 � 2ax� 2cxþ 2x2Þ ¼ 4x�2a� 2c ¼ 0! x ¼ aþ c=2. It follows the Cameroncircular polarization algorithm:

c ¼

abbc

26643775 ¼

aþc2

bb

aþc2

26643775þ

� c�a2

00

c�a2

26643775

¼

aþc2

bb

aþc2

26643775þ exp j

2

� � ðc�aÞ2

exp j2 4

0

0ðc�aÞ

2exp �j2

4

2664

3775¼cmax

sym þ cminsym (69)

where cmaxsym is a symmetric target formally equivalent

to cSym�b��b�3-D in (43)–(45) and cmax

sym is a symmetric target

rotated around LOS of =4 radians with respect to themaximum symmetric component as discussed by Cameron;

see [23, eq. (62)]. By substituting the model parameters of

(65) in (69), we found

c��SPAN ¼ cmaxsym þ cmin

sym

c��SPAN

¼

12sinð�Þ cosð�CÞ þ sinð�CÞ½ � expð�j2�Þ



12sinð�Þ cosð�CÞ þ sinð�CÞ½ � expð�j2�Þ

2666437775

þ

j2sinð�Þ cosð�CÞ � sinð�CÞ½ � exp j �2�þ

2

0

0j2sinð�Þ cosð�CÞ � sinð�CÞ½ � exp j �2��

2

266664

377775:(70)

h

By using the Cameron formalism, the signed Cameronnonsymmetry parameter S, recently introduced, is

derived [43]

S¼ sin�1 cminsym

� �¼sin�1 sinð�Þ cosð�CÞ � sinð�CÞ½ �ffiffiffi

2p

� �¼ sin�1 kHffiffiffi

2p� �

: (71)

Equation (71) shows that for reciprocal and nonsym-

metric targets S, kH and �; �C are strictly related and

provide an equivalent representation.

For nonsymmetric targets, the zC parameter also

depends on the �C angle. The unrotated target vector isobtained by applying a set of rotations and projections as

described in (43), (44), and (47), thus obtaining

cSym�b��b��SPAN2-D P2-DPSymP3-DPRecc

��SPAN�¼4

¼1ffiffi2p sinð�Þ cosð�CÞ þ sinð�CÞ½ � expð�j2�Þ

cosð�Þ expðj2�Þ

" #: (72)

By backprojecting cSym�b��b��SPAN2-D to the h� v basis,

we found

lSym�b��b�2-D ¼ 1

�2j cosð�Þ expðj2�Þ

�1þ jffiffi

2p tanð�Þ cosð�CÞ þ sinð�CÞ½ � expð�j4�Þ

1� jffiffi2p tanð�Þ cosð�CÞ þ sinð�CÞ½ � expð�j4�Þ

24 35(73)

where l stands for lexicographic ordering in the horizontal,

vertical basis. Using (44), the following is also found:

zC¼min jl1;2j

max jl1;2j

l1;2¼1 jffiffiffi

2p tanð�Þ cosð�CÞ þ sinð�CÞ½ � expð�j4�Þ: (74)

In conclusion, the classification algorithm developed

using the proposed decomposition is capable of gener-

alizing the results of the Cameron decomposition also for

nonsymmetric targets. The relationships between

ð�; �C;�Þ and Krogager and Cameron decomposition

parameters have been shown in (65)–(74).



D. Analysis of CTD Features for Partially ReciprocalTargets ðNr 6¼ 0Þ and in Presence of FaradayRotation ð� 6¼ 0Þ

By applying the antenna reciprocity theorem [5], [13],

it follows that the scattering matrix is symmetric

ðSaa? ¼ Sa?aÞ for backscattering geometry. The differential

analysis of the copolarized coefficients of the circular

polarization scattering matrix measures the nonreciprocal

effects in terms of the Faraday rotation phase distortion in� and amplitude distortions, measured by Nrð�; �Þ. The

nonsymmetric part of the scattering matrix accounts for a

mix of different effects, related to scattering, propagation

distortions, thermal, and speckle noises.

1) Effect of Propagation Faraday Rotation � 6¼ 0 to Recip-rocal Scattering Nr ¼ 0: By a survey of the CTD theorems,

it is found that Krogager–Huynen and Touzi CTDs havebeen developed for the six-parameter symmetric Sinclair

matrix representing reciprocal scattering [4], [5], [15]. The

generalization of the Kennaugh–Huynen CTD proposed by

Davidovitz, Boerner, Karnychev et al. and Bombrum is able

to deal with nonreciprocal scattering, but these decom-

positions extract a set of parameters dependent on the

Faraday rotation angle [44], [46], [47].

Cameron, as well as Cloude and Pottier, has used thePauli basis for extracting a reciprocal target vector, decom-

posing the scattering matrix space into two orthogonal

subspaces: reciprocal and nonreciprocal scattering [12],

[13], [23]. Unfortunately, also in this case, the parameters

extracted from Pauli-based decompositions have shown

dependence on the Faraday rotation angle [32].

More recently, Freeman has removed the Faraday ro-

tation effect via the circular polarization method proposedby Bickel and Bates [29], [30]. By applying � removal (42)

before forming reciprocal matrices, the parameters of the

fast lossless and sufficient decomposition are independent

of the Faraday rotation.

2) The Impact of the Joint Phase and Amplitude DistortionsNr 6¼ 0, � 6¼ 0: In order to assess the advantages of the

proposed decomposition in the general nonreciprocal scat-tering case, some measures of energetic loss are intro-

duced. It is worth noting that the unrotated scattering

vector c�b��b�2 C4 conserves the signal energy kc�b��b�k¼

SPAN, whereas forming reciprocal target matrices and ex-

tracting maximum symmetric scattering components are a

lossy filtering approach [23].

First, the SPAN of the reciprocal unrotated subspace is

computed

SPAN Srec��C

¼ jSllj2 þ Sl?l?j j2

þ 1

2Sll? expðj2b�Þ þ Sl?l expð�j2b�Þ�� 2

SPANðSrecÞ: (75)

Unrotated scattering vector provides in-phase Sll? andSl? l, thus obtaining a reciprocal target vector which has

higher energy compared to the Pauli basis reciprocal sub-

space projection proposed by Cameron [12].

It follows that two ratios of energy losses can be

computed

E��Loss½%� ¼ 100 �SPANðSÞ � SPAN Srec��

C

SPANðSÞ

(76)

E��Loss�Pauli½%� ¼ 100 � SPANðSÞ � SPANðSrecÞSPANðSÞ (77)

respectively, the ratio of energy loss obtained by projecting

the circular polarization scattering vector to a reciprocal

subspace (44), or using the Pauli basis reciprocal target

subspace [23].

By considering the Cameron maximum symmetric

component, the maximum symmetric SPAN is computed

SPAN Smax��sym

� �¼ SPAN cos2ð�recÞ cos2ðCÞ: (78)

The maximum symmetric SPAN allows extracting the

ratio of energy loss obtained considering the maximum

symmetric component

Emax�losssym ½%� ¼ 100 �

SPANðSÞ � SPAN Smaxsym

� �SPANðSÞ : (79)

It is worth noting that the proposed decomposition can

also be used to characterize bistatic scattering, however,

the relationships between the target shape and the charac-

teristic polarization parameters are much more compli-

cated since they inherently depend on the bistatic angle.

E. Partitions of the Circular Polarization ScatteringModel Parameter Space

In this section, the space of ð�; �; �;�Þ angles model-

ing the DOF of the unrotated target vector (41) is parti-

tioned using the proposed classification algorithm (48).

Fig. 8 presents the partition of the reciprocal targetðNr ¼ 0Þ, parameter space ð�;�; �CÞ, compared with the

partition of the Cameron complex zC disc, both obtained

via (48).

Fig. 8(a) shows the decision zones obtained with the

use of the proposed classification algorithm, modeling the

circular polarization scattering model varying �;� for

�C ¼ =4.



Fig. 8(b) shows the decision zones obtained projecting

the Cameron zC parameter in ll? basis and then applying

(48). The decision zones found numerically are equivalent

to the results found using the Cameron classification

algorithm [52].Fig. 8(c) and (d) shows the decision zones obtained for

reciprocal � ¼ =4 but partially symmetric Hel 6¼ 0 and

anisotropic � ¼ =4 targets.

Fig. 9 presents the partition of the nonsymmetric and

nonreciprocal ð�C; �Þ parameter space in the �C 6¼ =4,

� 6¼ =4 general case. Four different simulations have re-presented the boundaries of the decision zones for differ-

ent values of � ¼ 0�, 30�, 60�, 90�, given � ¼ 0�.Fig. 9(a) shows the (�C, �, � ¼ 0�, � ¼ 0�) space

divided in three decision zones: nonreciprocal+, trihedral,

and nonreciprocal-, depending only on �.

Fig. 9(b) shows the (�C, �, � ¼ 30�, � ¼ 0�) space

divided into four decision zones: nonreciprocal+, cylinder,

dipole, and nonreciprocal-, depending on ð�; �CÞ.Fig. 9(c) shows the (�C, �, � ¼ 60�, � ¼ 0�) space

divided into four decision zones: nonreciprocal +, dipole,

narrow diplane, and nonreciprocal �, depending on

ð�; �CÞ.Fig. 9(d) shows the (�C, �, � ¼ 90�, � ¼ 0�) space

divided into three decision zones: left helix, dihedral, and

right helix, depending only on �.

Fig. 9(a)–(d) also presents some symmetries in theshape of the decision zones that are related to the symme-

try of the classification problem. This property is obtained

by exploiting the symmetry properties of the circular

polarization basis scattering representation.

F. Concluding RemarksThis section has introduced the circular polarization

scattering model as the means for measuring the degree oftarget nonsymmetry simply by comparing the power of the

copolarized coefficients. The � parameter in combination

with the well-known � parameter has been proven

equivalent to represent the symmetric target scattering

space as well as the Cameron zC parameter, Huynen con-

eigenvalues, and Touzi TVSM–SSCM and Krogager SDH

parameters. For nonsymmetric scattering, the Helð�; �CÞparameter is proposed as a primary indicator better thanH providing an interpretation that is quite similar to the

new Cameron’s Sð�; �CÞ and the Krogager helix compo-

nent kHð�; �CÞ. For nonreciprocal scattering, the proposed

decomposition can provide a unique �-invariant charac-

terization capable of maximizing the energy losses and

minimizing the distortions caused by the Faraday rotation

on the polarimetric features. The partition of the ð�; �C;�;�Þ parameters space obtained via the proposed classifi-cation algorithms is shown for representing the effective-

ness of the lossless and sufficient invariant decomposition

theorem.

V. EXPERIMENTAL RESULTS:PROCESSING OF HIGH-RESOLUTIONAND FULLY POLARIMETRICSAR/ISAR DATA

A. Decomposition Assessment With AnechoicChamber UWB Data

The first data used for testing the proposed decompo-

sition theorem for the classification of point targets have

been previously described in [33]. Two photos of the

Fig. 8. (a) Partitions of the ð�;9Þ space for symmetric targets

�C ¼ =4. (b) Partition of the Cameron zC disc using the proposed

classification algorithm. (c) Partitions of the �; �C space for 9 ¼ 0

target. (d) Partitions of the �C ;9 space for anisotropic targets � ¼ =4.

Fig. 9. Partition of the ð�C ; �Þ parameter space given 9 ¼ 0, for

different values of �. (a) � ¼ 0�. (b) � ¼ 30�. (c) � ¼ 60�. (d) � ¼ 90�.



targets are shown in Fig. 10(a), where the composition ofelementary scatterers can be clearly seen.

Fig. 10(b) shows the representation of the UWB data

according to the SDH power representation, where tri-

hedrals and the flat plate are colored in blue, whereas

the dihedrals are colored in red, except for the yellow

colored dihedral (T1-W) presenting nonsymmetric (helix)

scattering.

Fig. 10(c) shows the color-composite polarimetricimages in the Pauli basis, and Fig. 10(d) shows the color-

composite polarimetric images in Sll? basis (RR ¼ jSl? l? j2,:

LL ¼ jSllj2, LR ¼ jðSll? þ Sl? lÞ=2j2Þ. B y l o o k i n g a t

Fig. 10(c), a good discrimination between the two dihed-

rals is obtained by using the red–green Pauli components,

where the 30� oriented dihedrals (T1-E) and (T2-E) have a

stronger green component, and the 0� dihedral (T1-W) is

clearly red. Fig. 10(d) gives a different information: the 0�

dihedral on the west (T1-W) has a stronger red component

due to positive helicity 20% ðRR > LLÞ than the 30�

oriented dihedral on the east (T1-E), which is greener due

to negative helicity �8% ðLL > RRÞ. The helicity informa-

tion is related to the physics of the scattering, and it should

not be confused with orientation �, which indicates the

scatterer’s rotation measured on the plane orthogonal tothe LOS.

Table 3 shows the CTD parameters, extracted from

anechoic chamber data. It should be noted that the radar

LOS is not perfectly orthogonal to the scatterers faces

ð�el ’ 17�Þ. It is worth noting that (except for odd bounces)

the � angle is estimated very well for even-bounce dihed-

rals with a precision better than 1�, due to perfect calibra-

tion and absence of noise. The � angle is between 9� and29� for odd-bounce planes and trihedrals and between 65�

and 73� for dihedrals. The differences observed when

compared to the theoretical values (respectively, 0�, 90�)are generated by imperfect orthogonal illumination, edge

diffraction, and sidelobe interference between neighbor

targets [23]. For example, the T1-N scatterer is character-

ized by � ¼ 27�; this value is significantly larger than what

is expected by a single scattering mechanism. Typically,different scattering mechanisms, e.g., double bounce, tri-

ple bounce, edge diffraction, etc., are superimposed to

obtain the LL, RR, and RL scattering components used to

calculate �. The behavior of � is meaningless for this class

of targets, whereas the energy of nonreciprocity jNrj is less

than 0.2% and � is less than 0.2� [5].

Fig. 10. Anechoic chamber data set. (a) Target pictures. (b) Sphere, diplane, helix color image. (c) Pauli basis color image R ¼ HH� VV,

G ¼ HV þ VH, B ¼ HHþ VV. (d) R ¼ RR, G ¼ LL, B ¼ RL color image.

Table 3 Decomposed Target Parameters of Two Experimental Pol-ISAR Images



In Fig. 11, the hue saturation value (HSV) color space is

used for representing the decomposed parameters in

triplets.

Fig. 11(a) shows the features for characterizing the

symmetric target ð�;�; SPANÞ, where the amplitude �parameter is confirmed as the main parameter for distin-

guishing between odd-bounce (dihedrals) and even-bounce

(planes-trihedrals) scatterers using the hue information.

Fig. 11(b) represents the features for characterizing

nonreciprocal and nonsymmetric scatterers ðHel;Nr;SPANÞ, where an average green nonsaturated image repre-

sents the symmetric and reciprocal behavior of most of the

scene.Fig. 11(c) characterizes the rotation effects �;�; SPAN,

where the different hues measure different orientation

angles where the saturation value is generally low ð�! 0Þ.

B. CTD Assessment With Airborne EMISAR FullyPolarimetric Data

In order to assess the proposed CTD for the analysis of

more realistic man made targets, airborne polarimetric

SAR data of Storebaelt are used. This data set is charac-terized by the presence of several ships. A more detailed

description can be found in [33]. The polarimetric ISAR

autofocusing technique, developed by Martorella et al., has

been applied to the largest ship on the scene in order to

compensate for the ship radial motion before applying the

CTD and evaluating the scattering vector of each image

pixel [33], [63]. The results of the application of the

proposed decomposition to the Pol-ISAR image of ship #6are shown in Fig. 12 using HSV color space. The results are

also compared with those obtained by using Krogager’s

decomposition.

Fig. 12(a) and (b) shows a great variety of colors in the

high-resolution ISAR images obtained using both the

Fig. 11. Parameter extraction from UWB data, represented according

to hue saturation value color space. (a) ð�;9;SPANÞ invariant

characterizing the symmetric scattering features. (b) ðHel;Nr;SPANÞinvariant characterizing nonreciprocity and nonsymmetry.

(c) ðV;C; SPANÞmeasuring Faraday rotation and

scattering orientation.

Fig. 12.Decomposition of POL-ISAR images of the EMISAR C-Band data

set. (a) Krogager SDH RGB color representation. (b) HSVð�;9;SPANÞinvariants characterizing the symmetric scattering features.

(c) HSVðHel;Nr;SPANÞ invariants characterizing nonreciprocity and

nonsymmetry. (c) HSVðV;C; SPANÞmeasuring Faraday rotation and

scattering orientation.



Krogager decomposition and the proposed decomposition.

This can be interpreted as a direct result of the hetero-

geneous nature of the scattering mechanisms involved in

different parts of the ship.

Fig. 12(c) represents the features for characterizing

nonreciprocal and nonsymmetric target Hel;Nr; SPAN.

An increasing contribution of nonsymmetric scattering(blue–red) can be noticed. This behavior is due to multiple

scattering mechanisms and a higher impact of speckle

noise.

Fig. 12(d) characterizes the rotation effects �;�;SPAN, where the different hues indicate different orienta-

tion angles, and the small value of saturation indicates the

small Faraday rotation angle ð�! 0Þ.

Fig. 13(a) shows the results of the application of the

proposed classification algorithm by using canonical scat-

terers, as described in Section III-F. It is worth noting that

a preselection of the scattering points has been done by

using a simple threshold on the SPAN. Fig. 13 shows, the

�;�; SPAN; Id map of the ship by overlapping the HSV

color image of the parameters �;�; SPAN, together with acharacter representing the resulting Id of the classified

pixel.

Fig. 13(b) shows the results of the application of the

Cameron classification algorithm superimposed with the

Krogager SDH, RGB color representation.

By observing Fig. 13(a), it should be noted that most of

the scatterers are characterized as symmetric targets

Fig. 13. Comparison of the proposed CTD classification with the Cameron classification applied to ship#6 [33]. (a) HSVð�;9; SPANÞcolor image with superimposed results of the proposed classification. (b) RGBðkS;kD;kHÞ color image with superimposed results of

the Cameron classification.



(298 out of 303 pixels), whereas five nonsymmetric

scatterers detected are both right helices. For this reason,

the use of fully polarimetric data is suggested to analyze the

behavior of real targets with the use of lossless and

sufficient �–�–�-invariant decomposition. By dividing

the ship in two parts through its main axis, it is clear that onthe near range side (N–E), there are more double-bounce

scatterers L and V, due to the sea–boat interaction, and

specular reflection � on the top parts. In the far range side

(S–W), the scatterers are characterized by single-bounce

�, or cylinder, dipole, and quarter-wave scatterers i;þ; I,with a few exceptions in the middle, in the stern and in the

bow, where there are also some local double-bounce scat-

tering points, possibly due to cabin, corners, or containerson the deck. Fig. 13(b) shows the application of the

Cameron decomposition, superimposed to the Krogager,

SDH color-coded ISAR image of the ship. Table 4 sum-

marizes the class distribution of the results as comparison.

Table 4 shows a similar distribution of the classification

results, obtained by using the proposed decomposition.

There is a higher rate of cylinders, diplanes, and quarter-

wave devices, whereas the Cameron classification declaresa higher rate of trihedrals, helices, and nonsymmetric

scatterers. The proposed decomposition erases the 31 non-

classification zones ‘‘nonsymmetric’’ via the all-in-one

classification metric and reduces also the number of left

and right helices declared. It is worth noting that the pro-

posed classification metric results are an optimal decision

rule for any kind of target, whereas the Cameron classi-

fication is optimal only for symmetric targets.

C. CTD Assessment With Spaceborne RADARSAT2Fully Polarimetric Data

Launched in 2007, RADARSAT-2 is a follow-on of theCanadian RADARSAT-1 spaceborne mission offering new

capabilities including multipolarization options (cross-pol,

dual-pol, or quad-pol), the ability to acquire images to the

left and right of the satellite, and improved geometric

accuracy. The satellite carries a C-band active phased array

SAR, which, operating in polarimetric mode, gives a

ground resolution ranging from 3 to 25 m and variable

swath widths from 25 to 50 km [1]. Fully polarimetric andhigh-resolution (5.4 m� 6.4 m) images of the Tangeri port

across the Gibraltar strait have been processed in order to

assess the proposed classification algorithm in the pre-

sence of severe noise disturbance and probable nonrecip-

rocal propagation distortions.

Fig. 14(a) presents Tangeri’s Geo Eye image for

comparison to color composite fully polarimetric

images obtained using the parameters of the proposeddecomposition.

Fig. 14(b) represents the Krogager SDH, RGB color

image, where the most of the sea area is characterized well

by surface scattering (blue), and the port area is more

characterized by double-bounce (red) and nonsymmetric

(green–yellow) points.

Fig. 14(c)–(e) represents the same color composite

combinations as proposed in Figs. 11 and 12. Although thesystem resolution is about 100 times coarser than of the

UWB data set and the range is quite large (686 km), good

discrimination properties are observed, in particular using

the ½� � SPAN �� parameters. In order to better detail

the effectiveness of the classification algorithm, a CFAR

detector has been applied to the SPAN image using a 100-

cell-averaging estimator.

Fig. 15(a) shows a zoom of the area of Tangeri portextracted using an optical sensor, compared with the result

of the proposed classification scheme in Fig. 15(b), and

with the Cameron classification algorithm in Fig. 15(c).

Fig. 15(b) shows the results of the proposed decom-

position theorem and the classification algorithm applied

to spaceborne fully polarimetric data, where, although the

image definition is coarser than optical, the polarimetric

sensor allows for discriminating different objects: dihe-dral scattering L and V, anisotropic target I;þ, and odd-

bounce scattering �. The application of the Cameron

algorithm shows very similar results in Fig. 15(c),

although small differences are obtained for higher values

of �.

The effect of nonreciprocal propagation and the pre-

sence of nonsymmetric scattering also seem slightly in-

creased and, for this reason, a further analysis is providedin Fig. 16, using a single range profile.

Fig. 16(a) represents the main indicators of propaga-

tion distortions and target asymmetries ½� Nr Hel�.Fig. 16(b) represents the power of the proposed un-

rotated scattering vector SPAN, compared with the power

of the three-component Pauli approximation SPANðSrecÞand the power of the maximum-symmetric component

extracted via Cameron algorithm SPANðSmaxsym Þ. Finally,

in order to better point out the energy losses obtained

using the Cameron algorithm in Fig. 16(c) and (d), the

energy losses in percent have been plotted in two

graphs.

Fig. 16(c) represents the percentages of energy lost

obtained using reciprocal target vector, comparing the

proposed reciprocal target subspace projection with the

three-element Pauli vector projection [23]. It should benoted that the new algorithm obtains smaller losses in any

case, being the cross-polarized coefficients in the phase

after � removal [23]. Fig. 16(d) represents the percentage

of energy lost obtained using the Pauli reciprocal target

vector and the Cameron maximum symmetric approxima-

tion, where the energy losses obtained under symmetri-

zation reach the 25% of the SPAN.

Table 4 Comparison of the Distribution Results of the

Two Classification Algorithms



VI. CONCLUSION

Fully polarimetric high-resolution SAR technology providesstrong information content to characterize scattering mecha-

nisms and estimate physical properties of observed targets.

In Section II, a survey of coherent target decomposi-

tions (CTD) has been detailed, specifically: 1) copolar echo

maximization (Kennaugh, Huynen, Touzi, Boerner et al.;2) additive decompositions (Krogager and Pauli); 3) cir-

cular polarization invariants (Bickel); 4) symmetric target

maximization (Cameron); and 5) polar modeling (Graves,Carrea, Souris). The features of each CTD have been

analyzed underlining their advantages and disadvantages.

In particular, an ideal decomposition would represent

the eight DOFs of the S matrix in terms of a set of features

having physical meaning, and being independent of the

effects produced by scatterers rotation around LOS ð�Þ,Faraday effect ð�Þ, and range delay ð�Þ.

This objective is obtained in Section III, where the

circular polarization basis has been found as eigenvectors

of the Faraday and orientation transformation matrices of

the scattering vector. In other words, the circular polari-

zation is the unique basis capable of representing Rð�;�Þtransformations in diagonal and phasor forms, being the

power of the circular polarization scattering vector in-

variant with respect to these transformations. A novelpoint target decomposition has been proposed, modeling

the circular polarization scattering vector in terms of the

multiplication of six SU(4) matrices and a complex

number.

The proposed scattering vector parameters have been

divided into two groups: invariant and dynamical param-

eters. The following set of features ð�;�;Nr;HelÞ are in-

variant parameters capable of representing the nature ofthe scattering mechanism observed, and they can be used

Fig. 14. Parameter extraction from RADARSAT-2 data, represented according to HSV color space. (a) Geo-Eye optical image. (b) Krogager SDH

decomposition RGB color image. (c) ð�;9; SPANÞ invariant characterizing the symmetric scattering features. (d) ðHel;Nr;SPANÞ invariant

characterizing nonreciprocity and nonsymmetry. (e) ðV;C; SPANÞmeasuring Faraday rotation and scattering orientation.



with SPAN to develop identification algorithms and inver-

sion studies.

The ð�;�;�Þ are dynamical parameters representing

the measures of 1) target orientation; 2) Faraday rotation

angle; and 3) phase rotation of the odd-bounce component.

They can be useful for measuring the ground slope, for

estimating the propagation distortions at lower frequen-cies, and for interferometric applications. The analysis of

the circular polarization coefficients has been related to

spin polarization transformation properties of elemental

packets of energy. According to the particle characteriza-

tion of radio scattering, introduced here, each scattering

vector is decomposed in terms of the degree of balance

ð�; �C; �Þ and phase interferences ð�;�;�Þ between ele-

mental scattering mechanisms, e.g., odd-bounce, even-bounce, nonsymmetric, and nonreciprocal scattering. A

novel unrotated and normalized scattering vector

c��SPAN 2 C4 has been proposed for developing

optimal classification metrics and two subvectors

cSym�b��b�2-D 2 C2; cRec�b��b�

3-D 2 C3 have been proposed forcharacterizing 2-D and 3-D complex field subspaces useful

for defining both reciprocal and symmetric targets and the

novel unrotated symmetric target subspace. The degree of

symmetry and the degree of reciprocity are directly mea-

sured using the power of the scattering matrix represented

in circular polarization.

In Section IV, the properties of the circular polariza-

tion basis and their subspaces are used for analyzing therelationships between the decomposition parameters; this

analysis has been developed in three steps.

For a symmetric target, given � ¼ �C ¼ =4, a rela-

tionship of equivalence is derived between all the most

important symmetric target parameters finding that a

symmetric target is characterized in terms of two shape

parameters ð�;�Þ formally equivalent to Cameron’s zC

and the other CTD symmetric target parameters. For re-

ciprocal and nonsymmetric target � ¼ =4, �C 6¼ =4, the

Hel parameter is presented as an invariant measure of

target nonsymmetry, where the H parameter is found

useless. The circular polarization basis is used for repre-

senting both Krogager and Cameron features by relating

them with �; �C;�. In the general case of nonreciprocal

scattering, the lossless and sufficient invariant decompo-

sition provides a representation of the target vector that is

independent of the Faraday rotation. This unique property

is useful for reducing the effect of propagation distortions

on the polarimetric features.

The proposed decomposition theorem and classifica-

tion algorithm has been tested in Section V using UWB

polarimetric radar data, airborne fully polarimetric data

(EMISAR), ISAR data of one large ship, and spaceborne

fully polarimetric (RADARSAT2) data of Gibraltar. Three

different combinations of features have been represented

in HSV color format in order to show: 1) the symmetric

target features ð�;�Þ to distinguish between trihedrals,

cylinders, dipoles, quarter-wave devices, narrow diplanes,

and diplanes; 2) the nonreciprocal and nonsymetric indi-

cators ðHel;NrÞ useful for detecting helical and nonre-

ciprocal scattering; and 3) the rotation parameters ð�;�Þmeasuring the Faraday effect and orientation. A compar-

ison between the proposed all-in-one classification algo-

rithm and the Cameron algorithm has been performed

analyzing airborne and spaceborne PolSAR data sets, thus

obtaining similar results for small values of � and a more

precise classification in the case of partially symmetric

scattering �C 6¼ =4.

Fig. 15. Geo-Eye optical image. (a) RADARSAT-2 fully polarimetric classification processing using the proposed algorithm.

(b) Classification performed using Cameron classification and SDH color coding.



The proposed decomposition results are very promising

for processing the next-generation fully polarimetric

spaceborne sensors data, as they are sufficient for model-ing all the eight DOFs of the Sinclair matrix being inva-

riant to Faraday rotation �B distortion and give a

phenomenological description of the radar scattering.

Future work will consider the generalization of the

proposed decomposition to bistatic scattering geometry

for the analysis of the random target under severe pro-

pagation distortions. The use of the proposed de-

composition in future work will provide a simplifiedmathematical model of target features and will allow

development of more efficient and faster classification

algorithms. h

Acknowledgment

The authors would like to thank Prof. J. Dall for

providing EMISAR data of Storebaelt, and the anonymous

reviewers that have helped in considerable manner to

improve the quality of the manuscript. R. Paladini

would like to thank Professor Emeritus W. M. Boerner

for helpful comments about the proposed technique;

Prof. K. Mahdjoubi of the Institute of Electronics andTelecommunications of Rennes (IETR), University of

Rennes, Rennes, France, for a helpful introduction on

helical antennas; Dr. C. Lardeux for endless discussions on

polarimetry; and Prof. D. Bini of the Department of

Mathematics, Pisa University, Pisa, Italy, for helpful advice

on the singular value decomposition.

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Fig. 16. Range profile processing. (a) Nonsymmetry characteristic parameters ðV;Nr;HelÞ. (b) Comparison of different power measurements

SPAN;SPANðSrecÞ;SPANðSmaxsym Þ. (c) Energetic losses E��loss�V�C½%�;E��loss�Pauli½%�. (d) Energetic losses E��loss�Pauli½%�;Emax�loss

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ABOUT T HE AUTHO RS

Riccardo Paladini (Member, IEEE) was born in

Viareggio, Italy, in October 1979. He received the

Laurea (M.S.) degree in telecommunication engi-

neering from the University of Pisa, Pisa, Italy, in

2007, where he also worked toward the Ph.D.

degree in remote sensing between 2008 and 2010,

in collaboration with the University of Rennes-1,

Rennes, France. He received the Ph.D. degree

Europeaus with a dissertation titled ‘‘Polarimetric

radar target decomposition and classification’’

from the Department of Information Engineering, University of Pisa.

He was a Visiting Scientist at the University of Rennes-1 from February

to October 2010. His main areas of research are development of radar

target classification and identification systems based on polarimetric

high-resolution images, polarimetric target decomposition theorems,

statistical radar scattering characterization, and high-frequency over-

the-horizon clutter modeling.

Laurent Ferro Famil (Member, IEEE) received the

Laurea degree in electronics systems and com-

puter engineering, the M.S. degree in electronics,

and the Ph.D. degree from the University of

Nantes, Nantes, France, in 1996, 1996, and 2000,

respectively.

He is currently with the University of Rennes 1,

Rennes, France, where he was an Associate

Professor in 2001, has been a Full Professor since

2011, and is currently the Head of the Radar

Polarimetry Remote Sensing Group, Institute of Electronics and Tele-

communications of Rennes (IETR). His current activities in education are

concerned with analog electronics, digital communications, microwave

theory, signal processing, and polarimetric synthetic aperture radar

(SAR) remote sensing. He is particularly interested in polarimetric SAR

signal processing, radar polarimetry theory, and natural media remote

sensing using multibaseline polarimetric and interferometric SAR data,

with application to classification, electromagnetic scattering modeling

and physical parameter retrieval, time–frequency analysis, and 3-D

reconstruction of environments.

Eric Pottier (Fellow, IEEE) received the M.Sc. and

Ph.D. degrees in signal processing and telecom-

munication from the University of Rennes 1,

Rennes, France, in 1987 and 1990, respectively,

and the Habilitation from the University of

Nantes, Nantes, France, in 1998.

Since 1999, he has been a Full Professor at the

University of Rennes 1, where he is currently the

Deputy Director of the Institute of Electronics and

Telecommunications of Rennes (IETR) and also

Head of the Image and Remote Sensing GroupVSAPHIR Team. His

current activities of research and education are centered in the topics

of analog electronics, microwave theory, and radar imaging with

emphasis in radar polarimetry. His research covers a wide spectrum

of areas from radar image processing (SAR, ISAR), polarimetric

scattering modeling, supervised/unsupervised polarimetric segmenta-

tion, and classification to fundamentals and basic theory of polarimetry.

Since 1989, he has supervised more than 45 research students to

graduation (M.Sc. and Ph.D.) in radar polarimetry covering areas from

theory to remote sensing applications. He has chaired and organized

more than 50 sessions in international conferences and was member of

the technical and scientific committees of more than 35 international

symposia or conferences. He has been invited to present 48 presenta-

tions in international conferences. He has published nine book

chapters, more than 50 papers in refereed journals, and presented

more than 340 papers during international conferences, symposia, and

workshops. He has published a book coauthored with Dr. J.-S. Lee:

Polarimetric Radar Imaging: From basics to applications (Boca Raton,

FL: CRC Press, 397 pp., 2009, ISBN: 978-1-4200-5497-2). He has pre-

sented advanced courses and seminars on radar polarimetry to a wide

range of organizations (DLR, NASDA, JRC, RESTEC, IECAS, INPE, ASF)

and events (ISAP2000, EUSAR04-06-10, NATO-04-06, PolInSAR05-11,

JAXA06, IGARSS03-05-07-08-09-10-11).

Dr. Pottier was presented the Best Paper Award at the Third

European Conference on Synthetic Aperture Radar (EUSAR2000) and

received the 2007 IEEE GRS-S Letters Prize Paper Award. He is a

recipient of the 2007 IEEE GRS-S Education Award ‘‘In recognition of

his significant educational contributions to Geoscience and Remote

Sensing.’’ He has been elevated to IEEE Fellow (January 2011) with the

accompanying citation: ‘‘for contributions to polarimetric synthetic

aperture radar.’’

Marco Martorella (Senior Member, IEEE) was

born in Portoferraio, Italy, in June 1973. He

received the Telecommunication Engineering

Laurea (cum laude) and Ph.D. degrees from the

University of Pisa, Pisa, Italy, in 1999 and 2003,

respectively.

He became a Postdoctoral Researcher in 2003,

a Researcher/Lecturer in 2005 and a permanent

Senior Researcher/Lecturer in 2008 at the De-

partment of Information Engineering, University

of Pisa. He has coauthored about 25 journal papers and 50 conference

papers. He has given lectures and seminars in several research

institutions in the United States, Australia, South Africa, Asia, South

America, and Europe and presented tutorials on ISAR at IEE/IEEE Radar

Conferences. His research interests are mainly in the field of radar

imaging.

Dr. Martorella received the Australia–Italy award for young re-

searchers in 2008.

Fabrizio Berizzi (Senior Member, IEEE) received

the Electronic Enginering Laurea and Ph.D. de-

grees from the University of Pisa, Pisa, Italy, in

1990 and 1994, respectively.

He has been a full Professor at the University of

Pisa since November 2009. He teaches remote

sensing systems, signal theory, and digital com-

munications at the University of Pisa and electric

communications at the Italian Navy Academic. He

has been working on ISAR, SAR, and radar systems

since 1990. He published more than 100 papers and the book Radar

Remote Sensing Systems (Milan, Italy: Apogeo, in Italian).

Prof. Berizzi was a Guest Editor of the Special Issue on ISAR of the

EURASIP Journal on Advances in Signal Processing in June 2006 and

Chairman of the special session ‘‘Radar target imaging,’’ ‘‘ISAR imaging,’’

and ‘‘ISAR’’ at the 2003 and 2008 IEEE International Radar Conference

(Adelaide, Australia) and the 2008 IEEE Radar Conference (Rome, Italy).

Since 1992, he has been involved in several scientific projects as a

Principal Investigator, funded by the University Ministry, the Defence

Ministry, Italian and European Space Agencies, Industries, Tuscany

region, the European Space Agency (ESA), and the European Defence

Agency.



Enzo Dalle Mese (Life Fellow, IEEE) graduated in

electronic engineering from the University of Pisa,

Pisa, Italy, in 1968.

Currently, he is a Full Professor of Radar

Technique at the Faculty of Engineering, Univer-

sity of Pisa. He teaches also at the Naval Academy

of the Italian Navy at Leghorn. He spent time as a

visiting professor at universities and research

centers in different countries (Australia, China,

United Kingdom) and worked as consultant for a

number of national industries in the field of radar and telecommunica-

tions. He has coauthored more than 200 scientific papers. During 1993–

1995, he was the Director of the Department of Information Engineering,

University of Pisa. During 1994–2006, he taught information theory and

coding and radar theory and technique at the University of Siena, Siena,

Italy. Since 2001, he has cooperated with the Italian Inter-University

Consortium for Telecommunications (CNIT) by managing a number of

projects of military interest in the field of radar design and radar signal

processing. He is the Chairman of the Ph.D. postgraduate course on

remote sensing of the University of Pisa and the Chairman of the Laurea

course in telecommunication at the University of Pisa; since January 2011,

he has been the Director of the Radar and Surveillance Systems (RaSS)

National Laboratory of CNIT. He manages a number of projects supported

by the Italian Ministry of Defense, the European Union, the Italian Space

Agency, Industries, and others. His major fields of interest are: radar

systems, remote sensing, and radar signal processing.



Documents

Point Target Classification via Fast Lossless and Sufficient $\Omega$–$\Psi$–$\Phi$ Invariant Decomposition of High-Resolution and Fully Polarimetric SAR/ISAR Data