singular value decomposition ofquaternion matrices: a new...

Signal Processing 84 (2004) 1177–1199www.elsevier.com/locate/sigpro

Singular value decomposition of quaternion matrices: a new toolfor vector-sensor signal processing

Nicolas Le Bihan∗, J*erome MarsLaboratoire des Images et des Signaux, ENSIEG, CNRS UMR 5083, 961 Rue de la Houille Blanche, Domaine Universitaire,

B.P. 46, 38402 Saint Martin d’H-eres Cedex, France

Received 31 May 2002; received in revised form 30 October 2003

Abstract

We present a new approach for vector-sensor signal modelling and processing, based on the use of quaternion algebra.We introduce the concept of quaternionic signal and give some primary tools to characterize it. We then study the problemof vector-sensor array signals, and introduce a subspace method for wave separation on such arrays. For this purpose, weexpose the extension of the Singular Value Decomposition (SVD) algorithm to the 9eld of quaternions. We discuss in moredetails Singular Value Decomposition for matrices of Quaternions (SVDQ) and linear algebra over quaternions 9eld. TheSVDQ allows to calculate the best rank-� approximation of a quaternion matrix and can be used in subspace method forwave separation over vector-sensor array. As the SVDQ takes into account the relationship between components, we showthat SVDQ is more e>cient than classical SVD in polarized wave separation problem.? 2004 Elsevier B.V. All rights reserved.

Keywords: Vector-sensor array; Quaternion model for vector-sensor signal; Quaternion matrix algebra; Singular Value Decomposition ofQuaternion matrix (SVDQ); Subspace method over quaternion 9eld; Polarized wave separation

1. Introduction

Vector-sensor signal processing concerns manyareas such as acoustic [25], seismic [1,18],communications [2,16] and electromagnetics [24,25].A vector-sensor is generally composed of threecomponents which record three-dimensional (3D)vibrations propagating through the medium. Thecomponents of a vector-sensor are, in fact, dipolesin the case of electromagnetic waves and directional

∗ Corresponding author. Tel.: +33-4-76-82-64-86;fax: +33-4-76-82-63-84.

E-mail address: [email protected] (N. Le Bihan).

geophones in the case of elastic waves. Sensors are po-sitioned close together and record vibrations in threeorthogonal directions. A schematic representation ofpropagation and recording of polarized waves usingvector-sensor array is presented in Fig. 1. The use ofmulticomponent sensors allows to recover polariza-tion of the received waves. This physical informationis helpful in path recovering and waves identi9cationor classi9cation. Classical approaches in vector-sensorsignal processing propose to concatenate the compo-nents into a long vector before processing [1,24,25]. Inthis paper, we propose a diHerent approach to processvector-sensor signals, based on a quaternionic model.

A polarized signal, obtained by recording a polar-ized wave using a vector-sensor, is a signal which

0165-1684/$ - see front matter ? 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.sigpro.2004.04.001

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1178 N. Le Bihan, J. Mars / Signal Processing 84 (2004) 1177–1199

y

z

x

y

z

x

y

z

x

Source

Vector sensors

Wave fronts

Fig. 1. Schematic representation of a vector-sensor array during apolarized seismic wave acquisition.

samples live in 3D space (in three componentsvector-sensor case). We de9ne the quaternionicsignal to be a time series with samples beingquaternions. The three components of the signalare placed into the imaginary parts of the quater-nion. In this case the possibly non-linear rela-tions between them are kept unchanged. This isnot the case when using long-vector approachas this model assume intrinsically that only lin-ear relations exist between components. Thismotivates the introduction of quaternions to modelvector-sensor signals. The quaternionic signal modelallows then to develop new tools for vector-sensorsignals, based on quaternion algebra.

We concentrate in this paper on the presentationof the extension of a classical and very useful toolin numerical signal processing: the Singular ValueDecomposition (SVD). We present the extension ofSVD to the quaternion 9eld and we propose a newsimple subspace method for vector-sensor signal pro-cessing. This new approach (that takes into accountrelationship between components) is applied on somesimulated data.

The paper is organized as follows. In Section 2, wegive a brief recall on quaternions and some of theirproperties. In Section 3, a review of some quater-nion linear matrix algebra theorems is presented, aswell as a presentation of the isomorphism betweenquaternion and complex 9elds. Using this isomor-phism, quaternion matrix operations can be computedby means of classical complex matrix computation al-gorithms. Particular attention is paid to the extensionof the SVD to matrices with quaternions entries, calledSVDQ. In Section 4, using de9nition of right Hilbertspace from Section 1, we can study the quaternion

one-dimensional (1D) signal by means of quaternionvectors, and the bidimensional (2D) signals by meansof quaternion matrices. We propose a new model forvector-sensor array signals. This model allows us tomanipulate polarized signals and to include polariza-tion property directly in the processing. We also intro-duce the mixture model for the polarized wave sepa-ration technique presented next. Section 5 is devotedto introduction of a new subspace method for waveseparation on vector-sensor arrays based on SVDQ.Its e>ciency is then evaluated on synthetic polar-ized dataset in Section 6. Conclusions are given inSection 7.

2. Quaternions

Quaternions are an extension of complex numbersfrom the 2D plane to the 3D space (as well as 4Dspace) and form one of the four existing division alge-bra (real R, complex C, quaternions H and octonionsO). Firstly discovered by Sir Hamilton [12] in 1843,they can be regarded as a special case of hypercom-plex numbers systems [17]. Quaternion algebra is alsothe even CliHord algebra Cl+3 [21].

Quaternions are mostly used in computer vision be-cause of their ability to represent rotations in 3D and4D spaces. In fact, a quaternion can represent any ro-tation of the rotation group SO(3). Also, two quater-nions can represent any rotation from the group ofrotations SO(4) [21]. Real and complex numbers arespecial cases of quaternions (i.e. R ⊂ C ⊂ H).

We now give the de9nition of quaternions andintroduce diHerent notations for them. Then webrieMy introduce some basic properties. Interestedreader may 9nd some more complete material in[5,10,12,13,17,21].

2.1. De<nition

A quaternion q is an element of the 4D normedalgebra over the real number with basis {1; i; j; k}.So, q has a real part and three imaginary parts:

q = a + bi + cj + dk; (1)

where a; b; c; d∈R, and i; j; k are imaginary units.a is called the real part and q − a the vector part.

N. Le Bihan, J. Mars / Signal Processing 84 (2004) 1177–1199 1179

The multiplication rules between the three imaginarynumbers are:

i2 = j2 = k2 = ijk = −1;

ij = −ji = k;

ki = −ik = j;

jk = −kj = i: (2)

Note that the multiplication of quaternions is not com-mutative, and this is due to the relations betweenthe three imaginary parts. 1 Several notations existfor quaternions [5,10,12,13,28]. A quaternion q canuniquely be expressed as

q = � + j; (3)

where � = a + bi and = c + di ∈C. This is cur-rently known as the Cayley–Dickson notation. � canbe seen as the projection of q on the complex plane[15]. We will later use the Cayley–Dickson notationfor quaternion matrix processing.

It is well known that the Euler formula also general-izes to quaternion [13]. This allows to interpret quater-nions in terms of modulus and phase. Any quaternionq = a + bi + cj + dk can thus be written as

q = e�� with

=√

a2 + b2 + c2 + d2;

� =bi + cj + dk√b2 + c2 + d2

;

� = arctan

(√b2 + c2 + d2

a

):

(4)

In this polar formulation, is the modulus of q, � is apure unitary quaternion (see Section 2.2 for de9nition)sometimes called eigenaxis [28] and � is the angle(or eigenangle [28]) between the real part and the 3Dimaginary (or vector) part. This can be rewritten in atrigonometric version as

e�� = (cos � + � sin �)

= (cos � + sin � cos sin i

+ sin � sin sin j + sin �k); (5)

1 Some commutative versions of quaternions, named bicomplexnumbers, have been introduced (see [5,10] for details), whichhave found some applications in digital signal processing [31] andsource separation [36].

where and are respectively the azimuth and eleva-tion angles that de9ne the position of � in the 3D space.The polar notation will be used in the quaternionmodel for vector signals in Section 4.

2.2. Properties

We now give some properties of quaternions. In-terested reader will 9nd an extensive list of quater-nions properties in [12,13,17,37]. We focus here onthe properties that we need in our work.

The conjugate of a quaternion, Oq, is de9ned as

Oq = a − bi − cj − dk: (6)

A pure quaternion is a quaternion with a null realpart (a = 0). Quaternions with non-null real parts aresometimes called full quaternions.

The norm of a quaternion is

|q| =√

q Oq =√

Oqq =√

a2 + b2 + c2 + d2: (7)

q is called a unit quaternion if its norm is 1. Theinverse of a quaternion is

q−1 =Oq

|q|2 : (8)

Multiplication of quaternions is not commutative.Given two quaternions q1 and q2, we have

q1q2 �= q2q1: (9)

This property involves the possible de9nition of twokinds of vector spaces overH. In order to study signalswith values in H, we now give some details on thede9nition of Hilbert spaces over the quaternion 9eld.This de9nition is necessary for validation of the modelthat we propose for vector signals in Section 4.

2.3. Quaternion Hilbert space

Here we will consider right vector spaces over H.As said before,H is a division algebra, andHN can beseen as a vector space of dimension N overH. In sucha vector space, matrices operate on the opposite side ofHN compared to scalars. In order to recover classicalmatrix calculus rules, we choose the convention thatmatrices operate on the left side, and so scalars operateon the right side. This is why we will use right vectorspaces here.


De�nition 1. A vector space of dimension N , namelyHN , over the 9eld of quaternions H is a right vectorspace if:

∀v∈HN and ∀�; �∈H: (v�)� = v(��): (10)

AsH is not commutative, Eq. (10) is not equivalentto the property that holds in the case of left vectorspaces overH: �(�v)=(��)v. In our case (right vectorspace), the order of factors in matrix multiplicationmust be carefully kept.

We now introduce some basic de9nitions neededfor quaternionic numerical signal processing over aquaternion Hilbert space.

2.3.1. Quaternion vectorClassically, in numerical signal processing, signals

are represented in a Hilbert space. Similarly, for thecase of quaternionic signals, we can work on a quater-nion Hilbert space. This quaternion Hilbert space is aright vector space. In this Hilbert space, a quaternionicdiscretized signal of N samples is a vector whose ele-ments are quaternions: x= [x1 x2 : : : xN ]T ∈HN , andxi ∈H.

Over this vector space, a scalar product can bede9ned as

〈x; y〉 = x/y =N∑

�=1

Ox�y�; (11)

where / denotes the quaternion transposition-conjugateoperator. Due to the fact that for quaternions we haveq1q2 = q2q1 (because q1q2 = q2 q1 and Oq = q), thescalar product has the following property:

〈x; y〉 = 〈y; x〉: (12)

Two quaternion vectors x∈HN and y∈HN are saidorthogonal if 〈x; y〉 = 0.

The de9nition of the norm for quaternion vectors is

‖x‖ =√

〈x; x〉: (13)

A metric can also be de9ned in order to measure thedistance between two quaternion vectors x and y:

d(x; y) = ‖x − y‖ = [(x − y)/(x − y)]1=2: (14)

These de9nitions are nothing more than the exten-sion to the quaternion case of the classical de9nitionsfrequently used in signal processing. We also give

here the de9nition of the outer product of two quater-nion vectors.

Given x∈HN and y∈HM , their outer product isgiven by

A = [a�]�; = x ◦ y = [x�y]�;; (15)

where A∈HN×M is a rank-1 quaternion matrix. Theouter product of x and x (i.e. x ◦ x or the alternativenotation xx/) is a hermitian rank-1 matrix. Rank-1matrices will be encountered when considering theSVD of quaternion matrices.

2.3.2. Quaternion matrixA given set of N signals from HM (noted xp with

16p6N ) can be arranged in a matrix X such as

X =

xT1

xT2

...

xTN

: (16)

This matrix is quaternion valued and de9nes a vectorspace X∈HN×M . The study of quaternion matriceswill be made in Section 3. Such a matrix will be usedto represent a set of discretized signals recorded on avector-sensor array (see Section 4).

2.4. Linear subspaces over quaternion vector spaces

We give here the de9nitions of basis and subspacesover a quaternion vector space. Recall that we considerhere a right vector space as de9ned in Section 2.3.

De�nition 2. Consider a set of vectors, not neces-sarily linearly independent, (e1; e2; : : : ; ep), e� ∈HN .The right span Sr of these vectors is the set of allvectors e∈HN that may be generated from linearcombination of the set:

Sr(e1; e2; : : : ; ep) =

{e: e =

p∑�=1

e�a�

}; (17)

where a� ∈H.

The span Sr is itself a subspace of HN . We nowintroduce the de9nition of right linearly independencefor a set of quaternion vectors.


De�nition 3. A set of quaternion valued vectors(e1; e2; : : : ; ep) is said to be right linearly independentif: e1c1 + e2c2 + · · · + epcp = 0 holds only whenc1 = c2 = c3 = · · · = cp = 0, with c� ∈H.

As in complex and real cases, the concept of a basisover a quaternion vector space can be stated:

De�nition 4. If, in the set (e1; e2; : : : ; ep), right lin-early dependent vectors are removed and all of theright linearly independent ones are retained, the newset (e(1); e(2); : : : ; e(p)) is a right basis.

We can now introduce the direct subspaces concept:

De�nition 5. The right linearly independent set ofvectors (e(1); e(2); : : : ; e(p)) can be divided in m disjointsets of vectors. Each disjoint set spans its respectivesubspaceSr

q . Therefore,Sr can be written as a directsum of disjoint right linearly independent subspaces:

Sr =Sr1 ⊕Sr

2 ⊕ · · · ⊕Srm: (18)

In the case where the basis is orthogonalized (via aQR process for example), the setSr can be expressedas a direct sum of mutually orthogonal subspaces.

De�nition 6. The set of vectors can be orthogonalizedand the orthogonal set of vectors is noted (u(1); u(2); : : : ;u(p)) with 〈u(�); u()〉=0 for all � �= . ThenSr formsa linearly independent orthogonal basis. So the setSr

can be written as a direct sum of m linear independentand orthogonal subspaces:

Sr =Ur1 ⊕Ur

2 ⊕ · · · ⊕Urm; (19)

where

Ur! =

{e: e =

∑i!∈I!

ui!bi!

}(20)

with I! the!th disjoint subset of {1; 2; : : :} and bi! ∈Hthe coe>cients of the vectors in the !th subset. 2

Existence of a set of orthogonal quaternion vectorswas also postulated in Lemma 5.2 in [37].

2 In the general case, coe>cients bi! take values in H. However,as R ⊂ C ⊂ H, linear combinations of quaternionic vectors withreal or complex coe>cients can be encountered and are just specialcases.

The decomposition in direct orthogonal subspaceswill be exploited in the polarized wave separationtechnique presented in Section 5.

Now, in order to develop some quaternionic sig-nal processing tools, we introduce some elements ofquaternion matrix algebra which are the basics neededfor vector signal processing.

3. Quaternion matrix algebra

The study of quaternion matrices has been an active9eld of research for several years, with starting pointin 1936 [35], and is still in development. Important re-search works on this subject were carried out by Zhang[37], Huang and So [15], Thompson [32], among oth-ers. In this section, we give some elements of quater-nion matrix algebra. After introducing the complex no-tation for vectors and quaternion matrices, we presentthe extension of classical matrix algebra concepts tothe quaternion 9eld. An extensive summary of the ba-sic properties of quaternion matrices can be found in[3,20,37]. We then give the de9nition of the SVD of aQuaternion matrix (SVDQ) and propose an algorithmfor its computation. We also demonstrate that, as inreal and complex case, the SVDQ allows to de9ne thebest rank-� truncation of a quaternion matrix.

3.1. Complex representation of a quaternion vector

De�nition 7. Given a quaternion vector x∈HN ex-pressed using the Cayley–Dickson notation such as:x=x1+x2j, with x1 and x2 ∈CN . We can then de9nea bijection f : HN → C2N such as

f(x) =

[x1

−x∗2

]; (21)

where ∗ is the complex conjugate operator.

Linear properties are invariant under bijection f.We will see the usefulness of this notation in comput-ing the left and right singular vectors of a quaternionmatrix (Section 3.4.1.).

3.2. Complex representation of a quaternion matrix

De�nition 8. Given a quaternion matrix A∈HN×M ,its expression using the Cayley–Dickson notation is


A = A1 + A2j, where A1 and A2 are complex matri-ces (i.e. ∈CN×M ). One can then de9ne the complexadjoint matrix [37], denoted by A ∈C2N×2M , corre-sponding to the quaternion matrix A, as follows:

A =

(A1 A2

−A∗2 A∗

1

): (22)

Some properties of A are given in [20,35,37]. Wegive here a summary of these properties for givenmatrices A and B∈HN×M and their complex adjoint A and B ∈C2N×2M :

• A is normal, hermitian or unitary if and only if Ais normal, hermitian or unitary,

• AB = A B,• A+B = A + B,• A−1 = ( A)−1 if A−1 exists• A/ = ( A)†,

where † denotes the complex conjugate-transpositionoperator. This complex notation for quaternion matri-ces can be used to compute quaternion matrix decom-position using complex decomposition algorithm.

We now focus on some classical matrix propertiesin the quaternion case.

3.3. Properties of quaternion matrices

We do not intend to list the whole set of propertiesof matrices of quaternions, but will give those whichare useful for our discussion.

3.3.1. Basic operationsSome well-known equalities for complex and real

matrices stand for matrices of quaternions, whereassome others are no more valid. Here is a short list ofrelations that stand for quaternion matrices A∈HN×M

and B∈HM×P:

• (AB)/ = B/A/,• AB �= AB in general,• (A/)−1 = (A−1)/,• (A)−1 �= A−1 in general.

See [37] for a more complete list. The most commonway to study quaternion matrices is to use their com-plex adjoint (complex representation of a quaternion

matrix) introduced in [20]. We now give some generalde9nitions for matrices over H.

3.3.2. Rank of a quaternion matrix

De�nition 9. As in complex and real matrix case, therank of a matrix of quaternions A is the minimumnumber of quaternion matrices of rank-1 that yield Aby a linear combination.

Property 1. The rank of a quaternion matrix A is r ifand only if the rank of its complex adjoint A is 2r.

This property was 9rst stated by Wolf [35]. In ad-dition to this de9nition, we indicate that the rank of Ais r if A has r non-zero singular values.

3.4. Singular value decomposition of a quaternionmatrix (SVDQ)

We study the de9nition and computation of the SVDfor matrices of Quaternions (SVDQ). Regarding thewide use of SVD and its extensions in signal process-ing [8,9,23,30,34], we can envisage huge potentialityfor SVDQ with applications in domains connected tosignal and image processing, especially for vector sig-nals and vector images cases. Here we propose to useSVDQ for vector-sensor array processing. The 9rststatement of this decomposition can be found, to ourknowledge, in [37].

Proposition 10. For any matrix A∈HN×M , of rankr, there exist two quaternion unitary matrices 3 Uand V such that

A = U

((r 0

0 0

)V/; (23)

where (r is a real diagonal matrix and has r non-nullentries on its diagonal (i.e. singular values of A).U∈HN×N and V∈HM×M contain respectively theleft and right quaternion singular vectors of S.

Proof. Demonstration of existence of the SVDQ isgiven in [37]. It is based on the proof of existence ofthe polar decomposition for any quaternion matrix. As

3 A quaternion unitary matrix W∈HN×N has the followingproperty: WW/=W/W=IN , with IN ∈RN×N the identity matrix.


polar decomposition and SVD are equivalent (even forquaternion matrices), existence of the polar decom-position induces existence of the SVDQ (see [37] fordetails).

The SVDQ can also be written as follows:

A =r∑

n=1

unv/n%n; (24)

where un are the left singular vectors (columns of U)and vn the right singular vectors (columns of V). %r

are the real singular values. This expression showsthat the SVDQ, as does the SVD in real and complexcase, decomposes the matrix into a sum of r rank-1quaternion matrices. The SVDQ is hence said to berank revealing: the number of non-null singular valuesequals the rank of the matrix.

We now propose a method for the computation ofthe SVDQ.

3.4.1. Computation of the SVDQWe propose to compute the SVDQ using the iso-

morphism between HN×M and C2N×2M . Notice thatquaternion algorithms have, up to now, only beenproposed for QR decomposition [6], but no quater-nionic algorithm has been proposed for computing theSVDQ.

The singular elements of a quaternion matrix A canbe obtained from the SVD of its complex adjoint ma-trix A. Firstly, recall that A admits a (complex) SVD:

A = U A

((2r 0

0 0

)(V A)† =

2r∑n′=1

%n′u An′ v A†

n′ ; (25)

where the columns of U A and V A are elements ofC2N and C2M respectively and given by

u An′ =

[u An′

−( Su An′ )∗

]and v An′ =

[v An′

−( Sv An′ )∗

](26)

with u An′ ∈C2N (with u A

n′ ∈CN and Su An′ ∈CN ) and

v An′ ∈C2M (with v An′ ∈CM and Sv An′ ∈CM ). The specialstructure of A (see Eq. (22)) induces the structuresin u A

n′ and v An′ which represent quaternion vectors intheir complex notations (see Eq. (21)). This structure

in singular vectors of A is always found when con-sidering complex adjoint matrices. 4

Considering the diagonal matrix (2r , it has thespecial structure:

(2r = diag(%1; %1; %2; %2; : : : ; %r; %r): (27)

This structure in the diagonal entries comes from thefact that the singular values of A are the square rootsof the eigenvalues of A( A)† and ( A)† A. 5

Now, for recovering the singular values and vectorsof A∈HN×M (i.e. %n; un and vn) from the singularelements of A ∈C2N×2M , one must proceed that way:

(1) Compute the SVD of A.(2) Associate the nth right (resp. left) singular vec-

tor of A with the n′ =(2n−1)th right (resp. left)singular vector of A, using the following equiv-alences:

un = u An′ + Su A

n′ j;

vn = v An′ + Sv An′ j; (28)

where u An′ , Su A

n′ and v An′ , Sv An′ are given in Eq. (26).(3) AHect the nth element of the diagonal matrix (r

(Eq. (23)), with the n′ = (2n − 1)th diagonalelement of the diagonal matrix (2r (Eq. (25)).

The computation of the SVDQ for a N×M quaternionmatrix is equivalent to the computation of the SVDof a 2N × 2M complex matrix (its complex adjoint)which can be performed using classical algorithms ofSVD over C2N×2M [11].

3.5. Best rank-� approximation of a quaternionmatrix

The Best rank-� approximation investigates theproblem of the approximation of a given matrix, in

4 This is due to the fact that left and right singular vectors of A are the eigenvectors of A( A)† and ( A)† A. Now, A( A)† = AA/ and ( A)† A = A/A, which are complex adjoint matricesthemselves. It was shown in [22,37] that eigenvectors of anycomplex adjoint matrix have the special structure given in Eq.(21).

5 Eigenvalues of complex adjoint matrices appear by conjugatepairs along the diagonal, and as A( A)† is hermitian, its eigen-values are real and appear by pairs along the diagonal (see [37]for details). So does the singular values of A and this inducesthe special structure of (2r .


an optimal least-squares sense, by a matrix of 9xedrank-�. In real and complex cases, the solution is ob-tained by the truncation of the SVD (see [29] for ex-ample). We show here that in the quaternion case,the truncation of the SVDQ also gives the best rank-�approximation of a matrix.

Proposition 11. Consider a matrix A∈HN×M , withits SVDQ given by Eq. (24). De<ne A�, composed ofthe � <rst singular elements (notedAn) of the SVDQof A. A� is the rank-� truncation of A and can bewritten as

A� =�∑

n=1

An; (29)

where a singular element is given by An = unv/n%n.Matrices An have ranks equal to 1, and so A� is arank-� matrix. Thus de<ned, A� is the Best rank-�approximation of A.

Proof. The square error between the original matrixand the rank-� truncation is

&2 = tr[(A − A�)/(A − A�)]; (30)

where tr is the trace operator. Using the complex ad-joint notation for quaternion matrices, we obtain theequivalent equation:

&2C = tr[( A − A2�)/( A − A2�)]: (31)

This error is called the Frobenius norm of A − A2� .Assuming that the truncation of rank 2� of A is

A2� = (U A)(2�(V A)†; (32)

where (2� = diag[%1; %1; %2; %2; : : : ; %�; %�; 0; : : : ; 0],then it can be demonstrated [29] that

&2C =2r∑

n=2�+1

%2n: (33)

Remember that the singular values of A are the sameas those of A. A has twice more singular values dueto its construction formula and size. Therefore, if (30)is used, the square error expression will be similar to(33), with twice less values for the index n and so&2 = &2C.

A property of the error is that it is orthogonal to theapproximated matrix A�. This is true for A as well asfor A:

(A − A�)A/� = 0: (34)

This makes the rank-� truncation of the SVDQ of aquaternion matrix the Best rank-� approximation ofthe matrix, in the mean square sense [29].

4. Quaternionic model for vector-sensor signals

Signals recorded on a vector-sensor array are of-ten arranged in a long-vector form [1,24,25] beforeprocessing, or sometimes, processed component-wise.The use of long-vectors may be restrictive if all possi-ble ways to build this vector are not examined [7,19].A consequence of processing components separatelyis that the relations between the signal components arenot taken into account. The approach proposed hereavoid component-wise and long-vector techniques andaim to process vector signals in a concise way usingquaternionic signals.

4.1. Quaternionic signals

In order to process simultaneously the whole dataset and to preserve the polarization relations, we pro-pose to encode vector-sensor signals as quaternion val-ued signals. Such a model allows to extend the con-cept of modulus and phase to vector-signals havingsamples evoluting in 3D space. The module is linkedwith signal magnitude while the 3D phase is directlyrepresentative of the 3D Lissajous plot (polarizationcurve).

4.1.1. Single polarized signalIn the case of a vector-sensor made of three or-

thogonal components (x; y and z), each one recordinga signal of M time samples, we can write the set ofthree discretized signals (x(m); y(m) and z(m), withm= 1; : : : ; M) in a quaternionic single one, s(m) suchas

s(m) = x(m)i + y(m)j + z(m)k: (35)

This model is close to the one given by Sangwine [27]for representing colour images. 6

6 In Sangwine’s work, the three imaginary parts of a purequaternion are used to represent the red, blue and green componentsof a pixel. Notice that in the colour image case, no polarizationrelations exist between colour channels, but the pixel value livesalso in 3D space.


−1−0.5

00.5

1

−1

−0.50

0.5

1 −1

−0.5

0

0.5

1

Fig. 2. Three-dimensional plot of the eigenaxis of an unpolarizedwhite Gaussian noise. Each point on the S2 sphere correspondsto a position of the eigenaxis at a given time.

Here, we consider a vector-sensor array composedof N (n = 1; : : : ; N ) three components vector-sensors.The quaternionic signal recorded on the sensor atposition n, noted sn(m), can be written using polarnotation (Eq. (5)):

sn(m) = n(m)e�n(m))=2 = n(m)e�n(m) (36)

with �n(m))=2 = �n(m). The angle between real andimaginary part of the quaternionic signal equals )=2here as the real part is null (cf. Eq. (5)). Consideringthe quaternionic signal described in (36), it is possibleto understand the signal into its envelope part and itsphase term (which is a 3D phase). So, the 3D curvedescribed by �n(m) between t =1 and t =M is linkedwith the polarization of the recorded wave. It is infact a normalized version of its Lissajous plot. Asexamples, we present in Figs. 2–4, behaviours of �(m)for three diHerent cases: the linearly polarized, theelliptically polarized and the non-polarized case.

In Fig. 2, signals x(m); y(m) and z(m) are threeindependent Gaussian noises, so no relations exist be-tween them and �(m) is distributed over the wholesurface of the sphere (i.e.S2). In Fig. 3, signals y(m)and z(m) are of the form �x(m), with �∈R. Thereexists a linear relation (polarization is said to be lin-ear) between the three components. So, �(m) takestwo values on the sphere that can be connected with aline (second value is not visible on the 9gure as it ismasked by the sphere, but as the line passes throughthe origin, one can easily guess where this point is by

−1−0.5

00.5

1

−1−0.5

00.5

1−1

−0.5

0

0.5

1

Fig. 3. Three-dimensional plot of the eigenaxis of a linearly po-larized Gaussian noise. Each point on the S2 sphere correspondsto a position of the eigenaxis at a given time.

−1−0.5

00.5

1

−1−0.5

00.5

1 −1

−0.5

0

0.5

1

Fig. 4. Three-dimensional plot of the eigenaxis of an ellipticallypolarized Gaussian noise. Each point on theS2 sphere correspondsto a position of the eigenaxis at a given time.

point symmetry). In Fig. 4, y(m) and z(m) are of theform �. +[x(m)] where �∈R and operator +[:] con-sists in phase shifting the considered signal (explicitexpression of +[:] is given in Section 4.1.2). Phaseshifts and amplitude ratios between components of thevector signal induces an elliptical behaviour of theeigenangle �(m).

This three examples illustrate how �(m) render thepolarization relations between the components of vec-tor signals. We now investigate the eHect of a phaseshift on an elliptically polarized vector signal.


4.1.2. Attenuation and phase shift of a polarizedsignal

During propagation of polarized waves, the mediumcan aHect the waveform by attenuating or phase shift-ing it (for example when the medium is anisotropic).Attenuation consists in multiplication of the polarizedwaveform by a real scalar, while phase shift consistsin a rotation of its eigenaxis in the 3D space (a proofof this correspondence between phase shift and 3D ro-tation is given in Appendix A). Thus, an attenuated(by a factor �) and shifted (of a quantity !) polarizedsignal s?(m) can be represented as

s?(m) = �(m)e!�(m)!−1; (37)

where �∈R and !∈H, with |!|=1. The !�(m)!−1

term consists in a rotation of the original eigenaxis(see [17] for rotations using quaternions). Notice thatthe phase shift is no more an angle here (like it is for1D signals) but a quaternion that represent a rotationin 3D space.

It must be noticed that the most general case ofphase shifting for three components sensors would bethat propagation introduces three diHerent phase termson the three components. A more special case is whenthe three components are phase shifted of the samequantity. This also correspond to a rotation in 3D spaceof the Lissajous plot. This is what happens in the caseof dispersive waves (Section 4.1.4.).

4.1.3. Multisensors polarized signalsNow considering a set of N vector-sensors, the col-

lected vector data set S can be written as a matrixwhose rows are the signals recorded on vector-sensorsthat constitute the array

S =

s1(m)

s2(m)

...

sN (m)

: (38)

In this way, the set of signals recorded on thevector-sensor array, S, is a matrix of size N×M whichelements are pure quaternions (i.e., S∈HN×M ).

4.1.4. Dispersive wavesDispersive waves are guided waves having dif-

ferent phase and group velocities. Group velocity

0

2

4

6

8

10

120 20 40 60 80 100 120

Sen

sors

Time samples

Fig. 5. Dispersive wave collected on single-component sensors.Group velocity (wave packet velocity) and phase velocity (ripplesvelocity) are diHerent.

corresponds to the velocity of the wave packet whilephase velocity corresponds to the velocity of waveripples. These concepts are well known in Physics,see for example [4] for details.

Consider now a set of N vector-sensors recordinga dispersive wave. As group velocity is the velocityof the wave packet, it will correspond to the veloc-ity of the envelope of the quaternionic vector-signalrecorded (i.e. of |s?(m)|=(m)). The wave propaga-tion velocity induces a constant time delay - betweensignals recorded on two adjacent sensors. So that thedelay between the 9rst sensor and the last one of thearray is (N − 1)- (for an array with N sensors). Asan example, one component of a dispersive wave ispresented in Fig. 5.

The slope of the wave packet on the “sensors vs.time samples” representation can be related to thepropagation velocity of the wave. Waves impingingthe whole set of sensors at the same time are said tohave in<nite velocity (see Fig. 6). Such con9gurationhappens after velocity correction (see details in Sec-tion 4.3).

The diHerence between phase and group velocitycorresponds to a constant phase shift from one sen-sor to an other. Such eHect also exists when consid-ering sets of scalar sensors [33]. We consider herethat for such dispersive waves, the signal recorded onthe (n + 1)th sensor is a phase shifted (of a constant


0

2

4

6

8

10

120 20 40 60 80 100 120

Time samples

Sen

sors

Fig. 6. Dispersive wave collected on single-component sensors,after group velocity correction. The wave packets impinge everysensor at the same time. Phase velocity is not corrected. This waveis said to be with <nite (group) velocity.

quantity !) version of signal recorded at the nth sen-sor. Assuming, that the signal recorded on the 9rstsensor has an arbitrary phase �(m), the signal recordedon the nth sensor can be written as

s?n (m) = (m)eq(n−1)�(m)q−(n−1)

: (39)

So phase shift only aHects the eigenaxis of the quater-nionic signal. We will show in Section 5 that this phaseshift does not aHect the dimension of the subspacenecessary for characterisation of the dispersive wave.

4.2. Quaternionic noise

We brieMy introduce the concept of white quater-nionic noise that will be encountered in the mixturemodel in Section 4.3.

De�nition 12. Consider a three-components centerednoise b(m) recorded on a single vector-sensor. Sucha noise is said quaternionic unpolarized white andGaussian if

b(m) = bx(m)i + by(m)j + bz(m)k; (40)

where bx(m); by(m) and bz(m) are Gaussian whitenoise (with variances .x; .y and .z, and with E[bbT]=diag(.x; .y; .z), where b = [bx(m) by(m) bz(m)]T).

There is no relations between components of asuch noise (so it is said unpolarized), and consider-ing its polar notation (Eq. (5)), its eigenangle has abehaviour similar to the one given in Fig. 2.

4.3. Mixture model

We now present the mixture model that will beused in the polarized wave separation technique(Section 5).

4.3.1. General modelThe received signal on the nth sensor results from

the superposition of P (possibly dispersive) polarizedwaves gp(m). These waves have propagated througha medium (supposed isotropic) that only aHect wavesby attenuating, time delaying or phase shifting them.Possibly, additive zero-mean Gaussian white quater-nionic noise bn(m) is recorded:

sn(m) =P∑

p=1

hnpgp(m − (n − 1)-p) + bn(m) (41)

with hnp ∈R the magnitude of wave p at sensorn; gp(m) ∈HN the normalized quaternionic (polar-ized) waveform of wave p. -p is the time delay ofwaveform p between two consecutive sensors. Thisdelay is linked (via the distance between sensorsWm) with the waveform velocity cp : -p = Wm=cp.bn(m) is supposed spatially white, unpolarized andindependent from the source waves. Spatial whitenessinduces that two quaternionic noises collected on twodiHerent sensors are decorrelated:

〈b�(m); b(m)〉 = .21�; (42)

where .2 is the variance of the centered noise (i.e.squared norm of the quaternion vector correspondingto b(m)) and 1� is the Kronecker delta. 7

Clearly, the mixture considered here (Eq. (41)) isan instantaneous mixture.

4.3.2. Pre-processingThe pre-processing step consists in applying a

group velocity correction to a given wave (herep=1), in order that its group velocity becomes in<nite

7 1� =

{1 if � = ;

0 if � �= :


(i.e. -1 = 0, waveforms arriving simultaneously onthe whole set of vector-sensors). This pre-processingstep can be visualized by the action that leads fromFigs. 5 to 6. If pre-processing is operated on g1, themixture given in (41) becomes

sn(m)=hn1g1(m)+P∑

p=2

hnpgp(m−(n−1)-′p)+b′

n(m);

(43)

where -′p and b′

n(m) are the modi9ed time delays ofother waves and the modi9ed noise (due to delaycorrection to have g1(m) with in<nite velocity).

The quaternion subspace method presented in nextsection exploits the low rank dimension of a the sub-space necessary to represent an in9nite velocity waveg1(m) and thus separate it from the other waves andnoise (under orthogonality assumption).

5. Quaternionic subspace method for polarizedwave separation

A direct and simple method to process wave sep-aration on a real-valued 2D dataset is to decomposethe original vector space de9ned by the data intotwo orthogonal subspaces, one called the “signal sub-space” and the other one the “noise subspace”. Thiswell-known decomposition technique [29] is basedon the SVD of the original data set and can be ex-tended to a vector-sensor 2D dataset, using the SVDQpresented above.

5.1. SVDQ and orthogonal subspaces

As stated in Section 2.4, it is possible to decom-pose a quaternionic vector space into two orthogonalsubspaces. Now, considering that a given quaternionmatrix S∈HN×M de9nes such a vector space, its de-composition in two orthogonal subspaces can be di-rectly obtained from its SVDQ. The 9rst subspace isspanned by the singular vectors corresponding to high-est singular values and the second one is spanned bythe last singular vectors. This can written as

S = [U1 |U2]

21 0

0 22

0

0 0

[V/

1

V/2

]= S1 + S2;

(44)

where S1 is the subspace generated by the rank-� trun-cation of the SVDQ of S (� highest singular values).Assuming S has rank r, then rank(S) = rank(S1) +rank(S2) (i.e. rank(S1)=� and rank(S2)=r−�). Theuse of SVDQ to build the subspaces ensures orthogo-nality between them in the sense of the inner productde9ned in (11):

〈u(1)p ; u(2)

q 〉 = 0 and 〈v(1)p ; v(2)

q 〉 = 0 (45)

for any p = 1; : : : ; � and q = � + 1; : : : ; r; where u(1)

and v(1) (resp. u(2) and v(2)) are columns of U1 andV1 (resp. U2 and V2).

We will exploit the orthogonality between sub-spaces to develop a wave separation technique forvector-sensor 2D array.

5.2. Polarized wave separation

Consider a set of N signals, sn(m), collected on avector-sensor array, as introduced in Section 4.1. Thus,following the quaternionic signal model, the data setcan be arranged in a quaternion matrix S∈HN×M .This quaternionic data set is a mixture of polarizedwaves and noise. Assuming that a pre-processing stephas been performed in order to have one polarizedwave that has in9nite velocity, S can be written usingthe mixture model presented in Eq. (43).

The aim of the technique presented here is to esti-mate the in<nite velocity wave and to remove it fromthe data set. In order to do so, the quaternion matrix Sis decomposed into orthogonal subspaces called sig-nal subspace and noise subspace like:

S = Ssignal + Snoise: (46)

According to the SVDQ de9nition (Eq. (24)), it ispossible to rewrite the decomposition into subspacesas

S =�∑

n=1

unv/n%n +r∑

n=�+1

unv/n%n; (47)

where r is the rank of S. As SVDQ is a canonicaldecomposition of the original quaternion data set,it expresses S as a sum of r rank-1 matrices. Thesignal subspace is then constructed with the rank-�truncation of S and the noise subspace with the re-maining part of S (sum of the r − � other rank-1matrices).


Determination of � is done by looking at the set ofsingular values (see for example Fig. 9) and 9nding agap in the set. The � highest singular values are thenkept to build the signal subspace. As independence isstated in the mixture, the orthogonal bases estimatedwith the SVDQ do not lead to a separation into inde-pendent subspaces. As a consequence, the signal sub-space will contain some noise (in fact the noise can beseen as a quaternionic full rank matrix as it is spatiallywhite, and so some parts of the noise will remain inthe rank-� truncation of the original data set).

Practically, the pre-processing ensures that the in-<nite velocity wave s1(m) is represented by a rank-1matrix (Ssignal, with � = 1). Waves with 9nite veloc-ity are full rank matrices, and thus are considered asnoise in the processing. Links between subspaces rankand polarized waves are presented in Appendix B. Tosummarize, the proposed subspace method lead to anestimate of the in<nite velocity using a rank-1 trunca-tion of the original matrix S that is corrupted by thepart of the additive noise which is spatially and tem-porally correlated with the wave. 8 The technique isstill valid for small rank of signal subspace (�¿ 1)if there remains a signi9cant gap in the set of singu-lar values. The larger is the rank of the signal sub-space, the more one get some unexpected correlatednoise in it.

In the following section, we present somesimulation results showing the behaviour of the sub-space technique based on SVDQ, as well as compar-ison with results obtained using a component-wiseapproach.

6. Simulation results

In the following simulations, the number of sensorsis N =10 and the number of time samples is M =128.For each example, quaternionic and component-wiseapproaches results are presented. The component-wiseapproach consists in performing (real) subspacesdecompositions separately on the three realmatrices that contain array signals of the threecomponents.

8 Recall that correlation is taken in the sense of the quaternionicscalar product.

6.1. Example 1: One elliptically polarized waveadded with noise

The three components of the original simulated po-larized wave with in<nite velocity are presented inFig. 7. In Fig. 8, the mixture consists in the wave andsome additive noise. The three components are pre-sented, as three pure imaginary parts of a quaternionmatrix (Fig. 8) using the quaternion model.

Singular values of the quaternionic mixture matrixare presented in Fig. 9. Due to the in<nite velocitypolarized wave, one singular value has greater mag-nitude than the others (see Fig. 9). So the signal sub-space will be constructed using a rank-1 truncation ofthe SVDQ. Result obtained for the signal subspace ispresented in Fig. 11.

In order to compare the SVDQ with the component-wise approach, SVD is computed on the three compo-nents independently. The three sets of singular valuesare presented in Fig. 10. It can be seen that one ofthe singular values is greater in each of the three sets.However, the ratio between the 9rst and the secondsingular values in the three sets is always lower thanthe ratio in the SVDQ case. For example, on the jcomponent singular values set, the 9rst singular valueis not so diHerent from the others. This is because thewave is buried in noise on this component, and be-cause the magnitude of noise is spread on all singu-lar values with a relatively high level. This does notallow to distinct a singular value among others (thatcould be attributed to the polarized wave). For thisreason, there is no possibility of distinction betweensignal and noise part when computing this componentindependently.

The component-wise signal subspace has been con-structed using a rank-1 truncation of the SVD for eachcomponent. It is presented in Fig. 12.

The correspondence between the original polarizedwave (Fig. 7) and the signal subspace obtained bySVDQ (Fig. 11) shows that the technique gives satis-fying results. Although noise corrupts the wave front,it is possible to distinguish the coherent wave from theambient noise (Fig. 11). In particular, on componentj of the mixture (Fig. 8), where the wave is buriedin the noise, the SVDQ based technique allows to re-cover the wave. On the opposite, the component-wiseapproach does not lead to good estimation of the orig-inal wave. The result is presented in Fig. 12. One can


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Sen

sors

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Sen

sors

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k imaginary part

Sen

sors

Time samples

Fig. 7. Original signal with in9nite envelope velocity recorded on three components array.

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i imaginary part

Sen

sors

Time samples

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Sen

sors

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k imaginary part

Sen

sors

Time samples

Fig. 8. Mixture 1: one wave and noise.

see that the signal subspace obtained in Fig. 12 is cor-rupted by noise, and the wave is not fully recovered.

By computing each component separately, thecomponent-wise approach does not take advantage ofthe relations between the components (polarizationproperty) and the wave front is not correctly esti-mated. This result shows that the quaternion modelapproach gives better results by taking advantage ofthe relationship between signal components (Fig. 11).

Also, this example shows that the wave is recov-ered on the three components, and even on the j

component, where its component was not viewable inthe mixture (Fig. 8). This indicates that the use of theall components simultaneously in the processing, al-lows to extract signal components that were lost innoise. Phase shifts and magnitude ratio are recoveredfor the signal part, which means that an identi9ca-tion of the estimated wave is possible after the pro-cessing, using the estimated polarization. This is notpossible using the component-wise technique and, insuch cases, the SVDQ is required to enhance signal tonoise ratio and estimate the polarization of the wave.


1 2 3 4 5 6 7 8 9 108

9

10

11

12

13

14

15

16

17

Singular value number

Mag

nitu

de

Fig. 9. Mixture 1: singular values obtained by SVDQ.

1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

16

Mag

nitu

de


i componentj componentk component

Fig. 10. Mixture 1: singular values obtained by SVD independently on the three components.

6.2. Example 2: Several polarized waves corruptedby noise

In this example, it is shown how a dispersivewave with strong magnitude and elliptical polariza-tion can be extracted from the original data. Afterestimating this wave, the other waves with lowestamplitudes can be recovered. The considered mix-ture is presented in Fig. 13. It is made of four wavescorrupted by noise. One wave has greater ampli-tude on the three components, and has an in9nite

velocity (pre-processing step has been performed).This 9rst wave is also a dispersive wave (see Section4.1.4.). Other ones have lower magnitude on diHerentcomponents and are also dispersive waves. All thewaves that are in the mixture have diHerent ellipticalpolarizations.

As in the 9rst example, the SVDQ and the SVDtechniques are compared on this data set. In Figs. 14and 15, we present respectively the set of singularvalues obtained by SVDQ and the three sets of singularvalues obtained by component-wise SVD.


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Sen

sors

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120 50 100

j imaginary part

Sen

sors

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k imaginary part

Sen

sors

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Fig. 11. Mixture 1: signal subspace using SVDQ.

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sors

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Time samples

Sen

sors

Fig. 12. Mixture 1: signal subspace using classical SVD component-wise.

The sets of singular values obtained by the threecomponent-wise SVDs (Fig. 14) have the same be-haviour, showing a high amplitude singular value andthe others with lower magnitude. This leads to theconstruction of the signal subspace using three rank-1truncations of SVDs. The corresponding signal sub-space is then presented in Fig. 16.

In the SVDQ case (Fig. 15), the set of singu-lar values has globally the same behaviour as inthe component-wise SVD case, except that the ratio

between the 9rst and the second singular values isgreater using SVDQ (Fig. 15) than in the SVD case(Fig. 14). The signal subspace is also obtained byrank-1 truncation of the SVDQ (Fig. 17).

In signal subspaces obtained via SVDQ and SVDtechniques, the 9rst wave seems to be well estimated.However, the result is better in the SVDQ case.This can be seen in the noise subspaces presented inFig. 18 for the SVD component-wise and in Fig. 19for the SVDQ.


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Fig. 13. Mixture 2: four waves and noise.

1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

Mag

nitu

de


i componentj componentk component

Fig. 14. Mixture 2: singular values obtained by SVD independently on the three components.

In the noise subspace obtained by SVDQ (Fig. 19),the 9rst wave is no more visible, due to the goodestimation of its three components using the quater-nion approach. On the other hand, in the noise sub-space obtained by component-wise SVD (Fig. 18),some parts of the 9rst wave can be seen (especially onthe last sensors of the three components). This is dueto a worse estimation of the in<nite velocity wave inthe signal subspaces when processing each componentseparately. The dispersion of the 9rst wave is the originof the problem encountered by the component-wiseapproach. It is known that a dispersive wave needs

more than a rank-1 truncation of the SVD to be wellestimated [18].

Using SVDQ, the 9rst wave has been well estimatedand so waves of lower magnitude can be recoveredon the three components (Fig. 19). The polarizationand dispersion of the 9rst wave are also estimated cor-rectly in the signal subspace. The quaternion subspacemethod is not sensitive to phase shifts (dispersion orelliptical polarization) in the way that such physicalquantities do not increase the rank of the signal sub-space. This is not possible using component-wise ap-proach (or subspace method over the real 9eld R).


1 2 3 4 5 6 7 8 9 100

5

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20

25


Mag

nitu

de

Fig. 15. Mixture 2: singular values obtained by SVDQ.

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sors

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Sen

sors

Time samples

Fig. 16. Mixture 2: signal subspace obtained by classical SVD component-wise.

The sensitivity of signal subspace estimation to sen-sor rotation is very small due to the fact that it cor-responds to a rotation in 3D. This can be seen as aphase shift of the quaternionic signal and so it doesnot aHect estimation of the signal subspace. So theSVDQ based technique gives the same performancewhen sensor rotation occurs.

7. Conclusion

We have proposed a new model for vector-sensorarray signals based on quaternion algebra. It provides

a simple notation for vector signal manipulation andprocessing. The quaternion approach proposed is validfor sensors having four components (encountered inGeophysics when a pressure sensor is added to a tripletof velocity measurement sensor) and three compo-nents. It also includes the case of two componentssensors, reducing then to complex case (as C ⊂ H).

Quaternion model allows to include wave polariza-tion in algorithms and to take advantage of the signalredundancy on its components. The existence of phaseshifts (elliptical polarization or dispersion of waves)between the components does not increase the rank of


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Fig. 17. Mixture 2: signal subspace using SVDQ.

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Fig. 18. Mixture 2: noise subspace obtained by classical SVD component-wise.

signal subspace in SVDQ approach, whereas it doesin the SVD component-wise approach.

We have proposed a new quaternion subspacemethod that can be used for polarized wave separa-tion on vector-sensor array. Its superiority upon thecomponent-wise approach (over R) has been shown.The SVDQ technique may also be interesting to re-cover the component of a polarized signal that islost in noise. This con9guration could be found in

communication problem, when one signal is lostin noise in a diversity of polarization transmissiontechnique.

As said above, SVDQ leads to orthogonal bases thatfail to separate independent waves and noise. A nat-ural extension of the proposed method would includehigher order statistics (ICA-like technique) in orderto enhance the separation results and release the con-straint of uncorrelation to higher order independence.


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Fig. 19. Mixture 2: noise subspace using SVDQ.

The proposed method is in fact linked with PCA andcould be seen as the standardisation step in a blindpolarized source separation algorithm.

As perspectives to this work, the study of sensorswith an arbitrary number l of components could beinvestigated. The use of hypercomplex number of di-mension l to encode could there be used, as well as theuse of CliHord Algebras [14,21]. These algebras havealready been used to develop pattern recognition algo-rithm on hyperspectral images [26]. Results obtainedherein and in other work using geometric algebra tohandle vector signals tend to show that such algebraare a powerful tool for developing algorithm speciallydedicated to vector-sensors signal processing.

Appendix A. Phase shift of polarized signals

Using the quaternionic signal representation, anypolarized signal is expressed as

sn(m) = n(m)e�n(m))=2 = n(m)e�n(m): (A.1)

This is to say that its magnitude is, at time sample m,equal to (m), and the distribution of magnitude on thethree components is carried by the eigenangle �(m).As |�(m)|m=m0 = 1; ∀m0, this 3D phase correspondsto a series of point on the 3D unit sphere S2. So, aconstant phase shift, i.e. independent of m, consists in

a rotation of all time samples in 3D space. This is to saythat a phase shifted version, s?n (m), of a quaternionicsignal sn(m) can be written as

s?n (m) = qsn(m)q−1; (A.2)

where q∈H is a unit quaternion (see [17] for rota-tions using quaternions). In fact, the rotation opera-tor q[:]q−1 only aHects the phase of the quaternionicsignal, so that we can write

s?n (m) = ?n (m)e�

?n (m) = n(m)eq�n(m)q−1

: (A.3)

Proof. As s?n (m) = qsn(m)q−1, the square of themodulus of s?n (m) is given by

?2n (m) = (qsn(m)q−1)(qsn(m)q−1)∗

= qsn(m)q−1(q−1)∗sn(m)∗q∗ (A.4)

=1

|q|2 (qsn(m)q−1qsn(m)∗q∗)

=1

|q|2 qsn(m)sn(m)∗q∗ (A.5)

=1

|q|2 q|sn(m)|2q∗ = |sn(m)|2: (A.6)


So

?n (m) = n(m) (A.7)

which proves that s?n (m) and sn(m) have the samemodulus. Then, as any quaternion q = a + bi+ cj+dk = Re(q) + Vec(q), with Re(q) = a and Vec(q) =bi + cj + dk, we can write

s?n (m) = qsn(m)q−1

= qRe(sn(m))q−1 + qVec(sn(m))q−1 (A.8)

then

|Vec(s?n (m))| = Vec(s?n (m))Vec(s?n (m))∗ (A.9)

and so, using the same argument given in (A.4), (A.5)and (A.6), we get

|Vec(s?n (m))| = |Vec(sn(m))|: (A.10)

Then, it is straightforward, using de9nitions in Eq. (4),that the eigenaxis of s?n (m) and sn(m) are linked as

�?(n) = q�(n)q−1: (A.11)

As polarization information is encoded in the eige-naxis of the quaternion signal, it can be useful to vi-sualize the diHerent positions of �(m) with time (i.e.for the increasing values of m).

The eHect of a phase shift on a polarized signal is arotation of the polarization curve (Lissajous plot) onthe 3D unit sphere.

Appendix B. Subspaces rank and polarized waves

The subspace method presented in Section 5 is apowerful tool for wave separation in the case wherethe signal subspace is a low rank truncation of thedata representing the original data set. In the followingderivations, we consider that signals and noises col-lected on vector-sensor arrays are matrices of HN×M

B.1. In<nite velocity polarized waves are rank-1quaternion matrices

When considering the SVDQ as a sum of rank-1 ma-trices, the 9rst so-called rank-1 matrix Sr=1 ∈HN×M

is constructed using the 9rst left and the 9rst rightsingular vectors, and can be expressed as their outer

product, multiplied by the corresponding singularvalue:

S(r=1) = %uv/ = %

u1

u2

...

uN

[ Ov1 Ov2 : : : OvM ] (B.1)

with un; Ovm ∈H and %∈R. Then the 9rst row of S(r=1)

is the row vector s(r=1)1 = %[u1 Ov1 u1 Ov2 : : : u1 OvM ] =

[%s1 %s2 : : : %sM ]. The expression of other rowsfollows immediately.

Now considering the case where the recorded polar-ized wave has in<nite velocity (and is not dispersive),the signals recorded from the second to the N th sen-sors are just copies of the signal recorded on the 9rstsensor sT, up to a (real) scalar amplitude term. This isto say that for in9nite velocity waves, the quaternionmatrix, say Sinf , that represents its behaviour on thearray has the form:

Sinf

=

s1 s2 : : : sM�1s1 �1s2 : : : �1sM�2s1 �2s2 : : : �2sM

......

�N−1s1 �N−1s2 : : : �N−1sM

=

sT

�1sT

�2sT

...

�N−1sT

;

(B.2)

where �i ∈R and si ∈H. The rank of Sinf is 1 as it canbe written as

Sinf = %′

u′1

u′2

...

u′N

[ Ov1 Ov2 : : : OvM ]

= %′

1%′ u1

�1

%′ u1

...�(N−1)

%′ u1

[ Ov1 Ov2 : : : OvM ]; (B.3)


where %′ is such that vector u is normalized, and u1v/=sT. Consequently, when only an in<nite velocity is inthe data set, it is a rank-1 matrix.

B.2. In<nite velocity polarized and dispersive wavesare rank-1 quaternion matrices

As explained in Section 4.1.4 and Appendix A, adispersive wave recorded on a vector-sensor array canbe written as

Sinf =disp =

sT

�1qsTq−1

�2q2sTq−2

...

�N−1qN−1sTq−(N−1)

; (B.4)

where q∈H and |q|=1. This matrix can also be writtenas the product of two quaternion vectors weighted bya real constant:

Sinf =disp = %′′

u′′1

u′′2

...

u′′N

[ Ov′′1 Ov′′

2 : : : Ov′′M ]

= %′′

1%′′ u1

�1

%′′ qu1

...�(N−1)

%′′ q(N−1)u1

×[ Ov1 Ov2q : : : OvMq−(N−1)]: (B.5)

This demonstrates that a dispersive wave is a rank-1quaternionic matrix.

B.3. Non-in<nite velocity polarized waves are fullrank quaternion matrices

When no pre-processing step has been performedon the data set, a recorded polarized wave impingesthe set of sensors with diHerent arrival times. Thus,

each sensor records a delayed version of the signalrecorded on the 9rst one (one of the three componentsof such a signal is presented in Fig. 5). As we considerinstantaneous mixture, a delayed version of a wave isanother wave, so that the array of N sensors will seeN waves. Denoting the diHerent waves as sT1 ; : : : ; s

TN ,

the data matrix Snon-inf is

Snon-inf =

sT1

sT2

sT3...

sTN

: (B.6)

There is no reason for the lines of Snon-inf to be cor-related, and so any of the quaternionic signals sTi withi = 2; : : : ; N cannot be written as sT1 multiplied by a(real) scalar. Then, each of the N signals sT1 ; s

T2 ; : : : ; s

TN

is represented by a rank-1 matrix (see Sections B.1and B.2). As a consequence, a non-in9nite velocitypolarized (possibly dispersive) wave is represented bya sum of N rank-1 quaternionic matrices:

Snon-inf =N∑i=1

uiv/i %i (B.7)

This makes Snon-inf a rank-N matrix.

B.4. Non-polarized noise is full rank quaternionmatrix

Consider that each of the N vector-sensors of thearray has recorded a white Gaussian noise. Noiserecorded on one vector-sensor is independent of noisesrecorded on other vector-sensors. This is to say that

〈b�(m); b(m)〉 = .21�; (B.8)

where and � are the sensor position in the ar-ray, and the scalar product 〈:; :〉 is de9ned inEq. (11). 1� is the Kronecker symbol de9ned andused in Section 5.1. Thus, the set of N Gaussian noisesrecorded on the N vector-sensors form an orthogonalbasis (which is also an independent basis). Conse-quently, the rank of the quaternion matrix that repre-sents the noise, say B∈HN×M , is a full rank matrix(rank(B) = min(N;M)).


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