Singular Value Decomposition of Quaternion Matrices

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

Singular value decomposition ofquaternion matrices: a new tool for vector-sensor signal processing Nicolas Le Bihan∗,JÃerˆome Mars

Laboratoire des Images et des Signaux, ENSIEG, CNRS UMR 5083, 961Rue de la Houille Blanche, Domaine Universitaire, B.P. 46, 38402 Saint Martin d’HeresÂ 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 ofquaternionic signal and give some primary tools to characterize it. We then study the problem ofvector-sensor array signals, and introduce a subspace method forwave separation on such arrays. For this purpose, we expose the extension ofthe Singular Value Decomposition (SVD) algorithm to the ÿeld ofquaternions. We discuss in more details Singular Value Decomposition for matrices of Quaternions (SVDQ) and linear algebra over quaternions ÿeld. The SVDQ allows to calculate the best rank- approximation ofa quaternion matrix and can be used in subspace method for wave separation over vector-sensor array. As the SVDQ takes into account the relationship between components, we show that 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 of Quaternion matrix (SVDQ); Subspace method over quaternion ÿeld; Polarized wave separation

1. Introduction geophones in the case ofelastic waves. Sensors are po- sitioned close together and record vibrations in three Vector-sensor signal processing concerns many orthogonal directions. A schematic representation of areas such as acoustic [25], seismic [1,18], propagation and recording ofpolarized waves using communications [2,16] and electromagnetics [24,25]. vector-sensor array is presented in Fig. 1. The use of A vector-sensor is generally composed ofthree multicomponent sensors allows to recover polariza- components which record three-dimensional (3D) tion ofthe received waves. This physical information vibrations propagating through the medium. The is helpful in path recovering and waves identiÿcation components ofa vector-sensor are, in fact,dipoles or classiÿcation. Classical approaches in vector-sensor in the case ofelectromagnetic waves and directional signal processing propose to concatenate the components into a long vector before processing [1,24,25]. In this paper, we propose a diHerent approach to process vector-sensor signals, based on a quaternionic model. ∗ Corresponding author. Tel.: +33-4-76-82-64-86; fax: +33-4-76-82-63-84. A polarized signal, obtained by recording a polar- E-mail address: [email protected] (N. Le Bihan). ized wave using a vector-sensor, is a signal which

Vector sensors one-dimensional (1D) signal by means ofquaternion z z z vectors, and the bidimensional (2D) signals by means y y y Source ofquaternion matrices. We propose a new model for x x x Wave fronts vector-sensor array signals. This model allows us to manipulate polarized signals and to include polarization property directly in the processing. We also introduce the mixture model for the polarized wave separation technique presented next. Section 5 is devoted to introduction ofa new subspace method forwave Fig. 1. Schematic representation ofa vector-sensor array during a separation on vector-sensor arrays based on SVDQ. polarized seismic wave acquisition. Its e>ciency is then evaluated on synthetic polarized dataset in Section 6. Conclusions are given in Section 7. samples live in 3D space (in three components vector-sensor case). We deÿne the quaternionic signal to be a time series with samples being 2. Quaternions quaternions. The three components ofthe signal are placed into the imaginary parts ofthe quater- Quaternions are an extension ofcomplex numbers nion. In this case the possibly non-linear rela- from the 2D plane to the 3D space (as well as 4D tions between them are kept unchanged. This is space) and form one of the four existing division alge- not the case when using long-vector approach bra (real R, complex C, quaternions H and octonions as this model assume intrinsically that only lin- O). Firstly discovered by Sir Hamilton [12] in 1843, ear relations exist between components. This they can be regarded as a special case of hypercom- motivates the introduction ofquaternions to model plex numbers systems [17]. Quaternion algebra is also vector-sensor signals. The quaternionic signal model + the even CliHord algebra Cl3 [21]. allows then to develop new tools for vector-sensor Quaternions are mostly used in computer vision be- signals, based on quaternion algebra. cause oftheir ability to represent rotations in 3D and We concentrate in this paper on the presentation 4D spaces. In fact, a quaternion can represent any ro- ofthe extension ofa classical and very usefultool tation ofthe rotation group SO(3). Also, two quater- in numerical signal processing: the Singular Value nions can represent any rotation from the group of Decomposition (SVD). We present the extension of rotations SO(4) [21]. Real and complex numbers are SVD to the quaternion ÿeld and we propose a new special cases ofquaternions (i.e. R ⊂ C ⊂ H). simple subspace method for vector-sensor signal pro- We now give the deÿnition ofquaternions and cessing. This new approach (that takes into account introduce diHerent notations for them. Then we relationship between components) is applied on some brieMy introduce some basic properties. Interested simulated data. reader may ÿnd some more complete material in The paper is organized as follows. In Section 2,we [5,10,12,13,17,21]. give a briefrecall on quaternions and some oftheir properties. In Section 3, a review ofsome quaternion linear matrix algebra theorems is presented, as 2.1. Deÿnition well as a presentation ofthe isomorphism between quaternion and complex ÿelds. Using this isomor- A quaternion q is an element ofthe 4D normed phism, quaternion matrix operations can be computed algebra over the real number with basis {1; i; j; k}. by means ofclassical complex matrix computation al- So, q has a real part and three imaginary parts: gorithms. Particular attention is paid to the extension q = a + bi + cj + dk; (1) ofthe SVD to matrices with quaternions entries, called SVDQ. In Section 4, using deÿnition ofright Hilbert where a; b; c; d ∈ R, and i; j; k are imaginary units. space from Section 1, we can study the quaternion 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 imaginary where and are respectively the azimuth and eleva- numbers are: tion angles that deÿne the position of in the 3D space. The polar notation will be used in the quaternion i2 = j2 = k2 = ijk = −1; model for vector signals in Section 4. ij = −ji = k; 2.2. Properties ki = −ik = j;

jk = −kj = i: (2) We now give some properties ofquaternions. In- terested reader will ÿnd an extensive list ofquater- Note that the multiplication ofquaternions is not com- nions properties in [12,13,17,37]. We focus here on mutative, and this is due to the relations between the properties that we need in our work. 1 the three imaginary parts. Several notations exist The conjugate ofa quaternion, Oq, is deÿned as for quaternions [5,10,12,13,28]. A quaternion q can uniquely be expressed as qO = a − bi − cj − dk: (6) q = + ÿj; (3) A pure quaternion is a quaternion with a null real where = a + bi and ÿ = c + di ∈ C. This is cur- part (a = 0). Quaternions with non-null real parts are rently known as the Cayley–Dickson notation. can sometimes called full quaternions. be seen as the projection of q on the complex plane The norm ofa quaternion is [15]. We will later use the Cayley–Dickson notation |q| = qqO = qqO = a2 + b2 + c2 + d2: (7) for quaternion matrix processing. It is well known that the Euler formula also general- q is called a unit quaternion ifits norm is 1. The izes to quaternion [13]. This allows to interpret quater- inverse ofa quaternion is nions in terms ofmodulus and phase. Any quaternion qO q−1 = : (8) q = a + bi + cj + dk can thus be written as 2  |q|  2 2 2 2  = a + b + c + d ; Multiplication ofquaternions is not commutative.   bi + cj + dk Given two quaternions q and q , we have  = √ ; 1 2 Â 2 2 2 q = e with b + c + d (4) q q = q q : (9)  √ 1 2 2 1  2 2 2  b + c + d  Â = arctan : This property involves the possible deÿnition oftwo a kinds ofvector spaces over H. In order to study signals In this polar formulation, is the modulus of q, is a with values in H, we now give some details on the pure unitary quaternion (see Section 2.2 for deÿnition) deÿnition ofHilbert spaces over the quaternion ÿeld. sometimes called eigenaxis [28] and Â is the angle This deÿnition is necessary for validation of the model (or eigenangle [28]) between the real part and the 3D that we propose for vector signals in Section 4. imaginary (or vector) part. This can be rewritten in a trigonometric version as 2.3. Quaternion Hilbert space e Â = (cos Â + sin Â) Here we will consider right vector spaces over H. = (cos Â + sin Â cos sin i As said before, H is a division algebra, and HN can be seen as a vector space ofdimension N over H. In such + sin Â sin sin j + sin Âk); (5) a vector space, matrices operate on the opposite side of HN compared to scalars. In order to recover classical matrix calculus rules, we choose the convention that 1 Some commutative versions ofquaternions, named bicomplex numbers, have been introduced (see [5,10] for details), which matrices operate on the left side, and so scalars operate have found some applications in digital signal processing [31] and on the right side. This is why we will use right vector source separation [36]. spaces here. 1180 N. Le Bihan, J. Mars / Signal Processing 84 (2004) 1177–1199

Deÿnition 1. A vector space ofdimension N, namely here the deÿnition ofthe outer product oftwo quater- HN , over the ÿeld ofquaternions H is a right vector nion vectors. space if: Given x ∈ HN and y ∈ HM , their outer product is given by ∀v ∈ HN and ∀; ∈ H:(v) = v(): (10)

A =[aÿ];ÿ = x ◦ y =[xyÿ];ÿ; (15) As H is not commutative, Eq. (10) is not equivalent N×M to the property that holds in the case ofleftvector where A ∈ H is a rank-1 quaternion matrix. The spaces over H: (v)=()v. In our case (right vector outer product of x and x (i.e. x ◦ x or the alternative / space), the order offactorsin matrix multiplication notation xx ) is a hermitian rank-1 matrix. Rank-1 must be carefully kept. matrices will be encountered when considering the We now introduce some basic deÿnitions needed SVD ofquaternion matrices. for quaternionic numerical signal processing over a quaternion Hilbert space. 2.3.2. Quaternion matrix M A given set of N signals from H (noted xp with 2.3.1. Quaternion vector 1 6 p 6 N) can be arranged in a matrix X such as  T  Classically, in numerical signal processing, signals x1 are represented in a Hilbert space. Similarly, for the    xT  case ofquaternionic signals, we can work on a quater-  2  X =   : (16) nion Hilbert space. This quaternion Hilbert space is a  .   .  right vector space. In this Hilbert space, a quaternionic   T discretized signal of N samples is a vector whose ele- xN T N ments are quaternions: x =[x1 x2 ::: xN ] ∈ H , and x ∈ H. This matrix is quaternion valued and deÿnes a vector i N×M Over this vector space, a scalar product can be space X ∈ H . The study ofquaternion matrices deÿned as will be made in Section 3. Such a matrix will be used to represent a set ofdiscretized signals recorded on a N / vector-sensor array (see Section 4). x; y = x y = xOy; (11) =1 2.4. Linear subspaces over quaternion vector spaces where / denotes the quaternion transposition-conjugate operator. Due to the fact that for quaternions we have We give here the deÿnitions ofbasis and subspaces q1q2 = q2q1 (because q1q2 = q2 q1 and qO = q), the over a quaternion vector space. Recall that we consider scalar product has the following property: here a right vector space as deÿned in Section 2.3. x; y = y; x : (12) Deÿnition 2. Consider a set ofvectors, not neces- N N N Two quaternion vectors x ∈ H and y ∈ H are said sarily linearly independent, (e1; e2;:::;ep), e ∈ H . orthogonal if x; y =0. The right span Sr ofthese vectors is the set ofall The deÿnition ofthe norm for quaternion vectors is vectors e ∈ HN that may be generated from linear combination ofthe set: x = x; x : (13) p r A metric can also be deÿned in order to measure the S (e1; e2;:::;ep)= e: e = ea ; (17) distance between two quaternion vectors x and y: =1 d(x; y)= x − y =[(x − y)/(x − y)]1=2: (14) where a ∈ H. These deÿnitions are nothing more than the exten- The span Sr is itselfa subspace of HN .Wenow sion to the quaternion case ofthe classical deÿnitions introduce the deÿnition of right linearly independence frequently used in signal processing. We also give for a set of quaternion vectors. N. Le Bihan, J. Mars / Signal Processing 84 (2004) 1177–1199 1181

Deÿnition 3. A set ofquaternion valued vectors The decomposition in direct orthogonal subspaces (e1; e2;:::;ep) is said to be right linearly independent will be exploited in the polarized wave separation if: e1c1 + e2c2 + ··· + epcp = 0 holds only when technique presented in Section 5. c1 = c2 = c3 = ···= cp = 0, with c ∈ H. Now, in order to develop some quaternionic signal processing tools, we introduce some elements of As in complex and real cases, the concept ofa basis quaternion matrix algebra which are the basics needed over a quaternion vector space can be stated: for vector signal processing.

Deÿnition 4. If, in the set (e1; e2;:::;ep), right linearly dependent vectors are removed and all ofthe 3. Quaternion matrix algebra right linearly independent ones are retained, the new set (e(1); e(2);:::;e(p))isaright basis. The study ofquaternion matrices has been an active ÿeld ofresearch forseveral years, with starting point We can now introduce the direct subspaces concept: in 1936 [35], and is still in development. Important re- search works on this subject were carried out by Zhang Deÿnition 5. The right linearly independent set of [37], Huang and So [15], Thompson [32], among oth- vectors (e(1); e(2);:::;e(p)) can be divided in m disjoint ers. In this section, we give some elements ofquater- sets ofvectors. Each disjoint set spans its respective nion matrix algebra. After introducing the complex no- r r tation for vectors and quaternion matrices, we present subspace Sq . Therefore, S can be written as a direct sum ofdisjoint right linearly independent subspaces: the extension ofclassical matrix algebra concepts to the quaternion ÿeld. An extensive summary ofthe ba- Sr = Sr ⊕ Sr ⊕···⊕Sr : (18) 1 2 m sic properties ofquaternion matrices can be foundin [3,20,37]. We then give the deÿnition ofthe SVD ofa In the case where the basis is orthogonalized (via a Quaternion matrix (SVDQ) and propose an algorithm QR process for example), the set Sr can be expressed for its computation. We also demonstrate that, as in as a direct sum ofmutually orthogonal subspaces. real and complex case, the SVDQ allows to deÿne the best rank- truncation ofa quaternion matrix. Deÿnition 6. The set ofvectors can be orthogonalized and the orthogonal set ofvectors is noted ( u(1); u(2);:::; r 3.1. Complex representation of a quaternion vector u(p)) with u(); u(ÿ) =0 for all = ÿ. Then S forms a linearly independent orthogonal basis. So the set Sr Deÿnition 7. Given a quaternion vector x ∈ HN ex- can be written as a direct sum of m linear independent pressed using the Cayley–Dickson notation such as: and orthogonal subspaces: N x=x1 +x2j, with x1 and x2 ∈ C . We can then deÿne Sr = Ur ⊕ Ur ⊕···⊕Ur ; (19) a bijection f : HN → C2N such as 1 2 m where x1 f(x)= ; (21) −x∗ Ur (20) 2 ! = e: e = ui! bi!

i!∈I! where ∗ is the complex conjugate operator. with I the !th disjoint subset of {1; 2;:::} and b ∈ H ! i! the coe>cients ofthe vectors in the !th subset. 2 Linear properties are invariant under bijection f. We will see the usefulness of this notation in comput- Existence ofa set oforthogonal quaternion vectors ing the left and right singular vectors of a quaternion was also postulated in Lemma 5.2 in [37]. matrix (Section 3.4.1.).

3.2. Complex representation of a quaternion matrix 2 In the general case, coe>cients bi! take values in H. However, as R ⊂ C ⊂ H, linear combinations ofquaternionic vectors with N×M real or complex coe>cients can be encountered and are just special Deÿnition 8. Given a quaternion matrix A ∈ H , cases. its expression using the Cayley–Dickson notation is 1182 N. Le Bihan, J. Mars / Signal Processing 84 (2004) 1177–1199

A = A1 + A2j, where A1 and A2 are complex matri- matrix) introduced in [20]. We now give some general ces (i.e. ∈ CN×M ). One can then deÿne the complex deÿnitions for matrices over H. 2N×2M adjoint matrix [37], denoted by A ∈ C , corresponding to the quaternion matrix A, as follows: 3.3.2. Rank of a quaternion matrix A1 A2 = : (22) Deÿnition 9. As in complex and real matrix case, the A ∗ ∗ −A2 A1 rank ofa matrix ofquaternions A is the minimum number ofquaternion matrices ofrank-1 that yield A Some properties of A are given in [20,35,37]. We by a linear combination. give here a summary ofthese properties forgiven matrices A and B ∈ HN×M and their complex adjoint Property 1. The rank ofa quaternion matrix A is r if 2N×2M A and B ∈ C : and only ifthe rank ofits complex adjoint A is 2r.

• A is normal, hermitian or unitary ifand only if A This property was ÿrst stated by Wolf[ 35]. In ad- is normal, hermitian or unitary, dition to this deÿnition, we indicate that the rank of A • AB = A B, is r if A has r non-zero singular values. • A+B = A + B, −1 −1 • A−1 =( A) if A exists 3.4. Singular value decomposition of a quaternion † • A/ =( A) , matrix (SVDQ) where † denotes the complex conjugate-transposition We study the deÿnition and computation ofthe SVD operator. This complex notation for quaternion matri- for matrices of Quaternions (SVDQ). Regarding the ces can be used to compute quaternion matrix decom- wide use ofSVD and its extensions in signal process- position using complex decomposition algorithm. ing [8,9,23,30,34], we can envisage huge potentiality We now focus on some classical matrix properties for SVDQ with applications in domains connected to in the quaternion case. signal and image processing, especially for vector signals and vector images cases. Here we propose to use 3.3. Properties of quaternion matrices SVDQ for vector-sensor array processing. The ÿrst statement ofthis decomposition can be found,to our We do not intend to list the whole set ofproperties knowledge, in [37]. ofmatrices ofquaternions, but will give those which are useful for our discussion. Proposition 10. For any matrix A ∈ HN×M , of rank r, there exist two quaternion unitary matrices 3 U and V such that 3.3.1. Basic operations Some well-known equalities for complex and real (r 0 A = U V/; (23) matrices stand for matrices of quaternions, whereas 00 some others are no more valid. Here is a short list of relations that stand for quaternion matrices A ∈ HN×M where (r is a real diagonal matrix and has r non-null and B ∈ HM×P: entries on its diagonal (i.e. singular values of A). U ∈ HN×N and V ∈ HM×M contain respectively the • (AB)/ = B/A/, left and right quaternion singular vectors of S. • AB = A B in general, • (A/)−1 =(A−1)/, Proof. Demonstration ofexistence ofthe SVDQ is • (A)−1 = A−1 in general. given in [37]. It is based on the proofofexistence of the polar decomposition for any quaternion matrix. As See [37] for a more complete list. The most common way to study quaternion matrices is to use their com- 3 A quaternion unitary matrix W ∈ HN×N has the following / / N×N plex adjoint (complex representation ofa quaternion property: WW =W W=IN , with IN ∈ R the identity matrix. N. Le Bihan, J. Mars / Signal Processing 84 (2004) 1177–1199 1183 polar decomposition and SVD are equivalent (even for in singular vectors of A is always found when con- quaternion matrices), existence ofthe polar decom- sidering complex adjoint matrices. 4 position induces existence ofthe SVDQ (see [ 37]for Considering the diagonal matrix (2r, it has the details). special structure: ( = diag(% ; % ; % ; % ; :::; % ; % ): (27) The SVDQ can also be written as follows: 2r 1 1 2 2 r r This structure in the diagonal entries comes from the r fact that the singular values of are the square roots / A A = unv %n; (24) † † 5 n ofthe eigenvalues of A( A) and ( A) A. n=1 Now, for recovering the singular values and vectors N×M of A ∈ H (i.e. %n; un and vn) from the singular where un are the left singular vectors (columns of U) 2N×2M elements of A ∈ C , one must proceed that way: and vn the right singular vectors (columns of V). %r are the real singular values. This expression shows (1) Compute the SVD of A. that the SVDQ, as does the SVD in real and complex (2) Associate the nth right (resp. left) singular vec- case, decomposes the matrix into a sum of r rank-1 tor of A with the n =(2n−1)th right (resp. left) quaternion matrices. The SVDQ is hence said to be singular vector of A, using the following equiv- rank revealing: the number ofnon-null singular values alences: equals the rank ofthe matrix. A A We now propose a method for the computation of un = u˙n + uSn j; the SVDQ. A A vn = v˙n + vSn j; (28)

A A A A where u˙n , uSn and v˙n , vSn are given in Eq. (26). 3.4.1. Computation of the SVDQ (3) AHect the nth element ofthe diagonal matrix (r We propose to compute the SVDQ using the iso- (Eq. (23)), with the n =(2n − 1)th diagonal morphism between HN×M and C2N×2M . Notice that element ofthe diagonal matrix (2r (Eq. (25)). quaternion algorithms have, up to now, only been proposed for QR decomposition [6], but no quater- The computation ofthe SVDQ fora N ×M quaternion nionic algorithm has been proposed for computing the matrix is equivalent to the computation ofthe SVD SVDQ. ofa 2 N × 2M complex matrix (its complex adjoint) The singular elements ofa quaternion matrix A can which can be performed using classical algorithms of be obtained from the SVD of its complex adjoint ma- SVD over C2N×2M [11]. trix A. Firstly, recall that A admits a (complex) SVD: 3.5. Best rank- approximation of a quaternion 2r (2r 0 matrix A A † A A† A = U (V ) = %n u v ; (25) 00 n n n=1 The Best rank- approximation investigates the problem ofthe approximation ofa given matrix, in where the columns of U A and V A are elements of C2N and C2M respectively and given by 4 This is due to the fact that left and right singular vectors of A A u˙ v˙ are the eigenvectors of ( )† and ( )† . Now, ( )† = A n A n A A A A A A A u = and v = (26) † n A ∗ n A ∗ AA/ and ( A) A = A/A, which are complex adjoint matrices −(uSn ) −(vSn ) themselves. It was shown in [22,37] that eigenvectors ofany complex adjoint matrix have the special structure given in Eq. A 2N A N A N (21). with u ∈ C (with u˙ ∈ C and uS ∈ C ) and n n n 5 v A ∈ C2M (with v˙ A ∈ CM and vS A ∈ CM ). The special Eigenvalues of complex adjoint matrices appear by conjugate n n n † structure of pairs along the diagonal, and as A( A) is hermitian, its eigen- A (see Eq. (22)) induces the structures values are real and appear by pairs along the diagonal (see [37] in u A and v A which represent quaternion vectors in n n for details). So does the singular values of A and this induces their complex notations (see Eq. (21)). This structure the special structure of (2r. 1184 N. Le Bihan, J. Mars / Signal Processing 84 (2004) 1177–1199 an optimal least-squares sense, by a matrix ofÿxed This makes the rank- truncation ofthe SVDQ ofa rank-. In real and complex cases, the solution is ob- quaternion matrix the Best rank- approximation of tained by the truncation ofthe SVD (see [ 29] for ex- the matrix, in the mean square sense [29]. ample). We show here that in the quaternion case, the truncation ofthe SVDQ also gives the best rank- approximation ofa matrix. 4. Quaternionic model for vector-sensor signals

Proposition 11. Consider a matrix A ∈ HN×M , with Signals recorded on a vector-sensor array are ofits SVDQ given by Eq. (24). Deÿne A, composed of ten arranged in a long-vector form [1,24,25] before the ÿrst singular elements (noted An) of the SVDQ processing, or sometimes, processed component-wise. of A. A is the rank- truncation of A and can be The use oflong-vectors may be restrictive ifall possi- written as ble ways to build this vector are not examined [7,19]. A consequence ofprocessing components separately A = An; (29) is that the relations between the signal components are n=1 not taken into account. The approach proposed here / where a singular element is given by An = unvn%n. avoid component-wise and long-vector techniques and Matrices An have ranks equal to 1, and so A is a aim to process vector signals in a concise way using rank- matrix. Thus deÿned, A is the Best rank- quaternionic signals. approximation of A. 4.1. Quaternionic signals Proof. The square error between the original matrix and the rank- truncation is In order to process simultaneously the whole data 2 / & =tr[(A − A) (A − A)]; (30) set and to preserve the polarization relations, we pro- where tr is the trace operator. Using the complex ad- pose to encode vector-sensor signals as quaternion val- joint notation for quaternion matrices, we obtain the ued signals. Such a model allows to extend the con- equivalent equation: cept ofmodulus and phase to vector-signals having samples evoluting in 3D space. The module is linked &2 =tr[( − )/( − )]: (31) C A A2 A A2 with signal magnitude while the 3D phase is directly

This error is called the Frobenius norm of A − A2 . representative ofthe 3D Lissajous plot (polarization Assuming that the truncation ofrank 2 of A is curve). =(U A )( (V A )†; (32) A2 2 4.1.1. Single polarized signal where ( = diag[% ; % ; % ; % ; :::; % ; % ; 0;:::;0], 2 1 1 2 2 In the case ofa vector-sensor made ofthree or- then it can be demonstrated [29] that thogonal components (x; y and z), each one recording 2r &2 = %2: (33) a signal of M time samples, we can write the set of C n three discretized signals (x(m);y(m) and z(m), with n=2+1 m =1;:::;M) in a quaternionic single one, s(m) such Remember that the singular values of are the same A as as those of A. A has twice more singular values due to its construction formula and size. Therefore, if (30) s(m)=x(m)i + y(m)j + z(m)k: (35) is used, the square error expression will be similar to (33), with twice less values for the index n and so This model is close to the one given by Sangwine [27] 6 2 2 for representing colour images. & = &C.

A property ofthe error is that it is orthogonal to the 6 In Sangwine’s work, the three imaginary parts ofa pure quaternion are used to represent the red, blue and green components approximated matrix A. This is true for A as well as for A: ofa pixel. Notice that in the colour image case, no polarization relations exist between colour channels, but the pixel value lives / (A − A)A =0: (34) also in 3D space. N. Le Bihan, J. Mars / Signal Processing 84 (2004) 1177–1199 1185

1 1

0.5 0.5

0 0

−0.5 −0.5

−1 −1 1 1 0.5 1 0.5 1 0 0.5 0 0.5 0 0 − − 0.5 −0.5 0.5 −0.5 −1 −1 −1 −1

Fig. 2. Three-dimensional plot ofthe eigenaxis ofan unpolarized Fig. 3. Three-dimensional plot ofthe eigenaxis ofa linearly po- white Gaussian noise. Each point on the S2 sphere corresponds larized Gaussian noise. Each point on the S2 sphere corresponds to a position ofthe eigenaxis at a given time. to a position ofthe eigenaxis at a given time.

Here, we consider a vector-sensor array composed of N (n =1;:::;N) three components vector-sensors. 1 The quaternionic signal recorded on the sensor at 0.5 position n, noted sn(m), can be written using polar notation (Eq. (5)): 0 n(m))=2 n(m) sn(m)= n(m)e = n(m)e (36) −0.5 with n(m))=2=n(m). The angle between real and imaginary part ofthe quaternionic signal equals )=2 −1 here as the real part is null (cf. Eq. (5)). Considering 1 0.5 1 the quaternionic signal described in (36), it is possible 0 0.5 to understand the signal into its envelope part and its − 0 0.5 −0.5 phase term (which is a 3D phase). So, the 3D curve −1 −1 described by n(m) between t = 1 and t = M is linked with the polarization ofthe recorded wave. It is in Fig. 4. Three-dimensional plot ofthe eigenaxis ofan elliptically polarized Gaussian noise. Each point on the S2 sphere corresponds fact a normalized version of its Lissajous plot. As to a position ofthe eigenaxis at a given time. examples, we present in Figs. 2–4, behaviours of (m) for three diHerent cases: the linearly polarized, the elliptically polarized and the non-polarized case. In Fig. 2, signals x(m);y(m) and z(m) are three point symmetry). In Fig. 4, y(m) and z(m) are ofthe independent Gaussian noises, so no relations exist be- form . +[x(m)] where ∈ R and operator +[:] con- tween them and (m) is distributed over the whole sists in phase shifting the considered signal (explicit surface of the sphere (i.e. S2). In Fig. 3, signals y(m) expression of +[:] is given in Section 4.1.2). Phase and z(m) are ofthe form x(m), with ∈ R. There shifts and amplitude ratios between components of the exists a linear relation (polarization is said to be lin- vector signal induces an elliptical behaviour ofthe ear) between the three components. So, (m) takes eigenangle (m). two values on the sphere that can be connected with a This three examples illustrate how (m) render the line (second value is not visible on the ÿgure as it is polarization relations between the components ofvec- masked by the sphere, but as the line passes through tor signals. We now investigate the eHect ofa phase the origin, one can easily guess where this point is by shift on an elliptically polarized vector signal. 1186 N. Le Bihan, J. Mars / Signal Processing 84 (2004) 1177–1199

4.1.2. Attenuation and phase shift of a polarized signal 0 During propagation ofpolarized waves, the medium can aHect the waveform by attenuating or phase shift- 2 ing it (for example when the medium is anisotropic). Attenuation consists in multiplication ofthe polarized 4 waveform by a real scalar, while phase shift consists 6

in a rotation ofits eigenaxis in the 3D space (a proof Sensors ofthis correspondence between phase shiftand 3D rotation is given in Appendix A). Thus, an attenuated 8 (by a factor ) and shifted (of a quantity !) polarized signal s?(m) can be represented as 10

? !(m)!−1 s (m)= (m)e ; (37) 12 0 20 40 60 80 100 120 where ∈ R and ! ∈ H, with |!| = 1. The !(m)!−1 Time samples term consists in a rotation ofthe original eigenaxis Fig. 5. Dispersive wave collected on single-component sensors. (see [17] for rotations using quaternions). Notice that Group velocity (wave packet velocity) and phase velocity (ripples the phase shift is no more an angle here (like it is for velocity) are diHerent. 1D signals) but a quaternion that represent a rotation in 3D space. It must be noticed that the most general case of corresponds to the velocity ofthe wave packet while phase shifting for three components sensors would be phase velocity corresponds to the velocity ofwave that propagation introduces three diHerent phase terms ripples. These concepts are well known in Physics, on the three components. A more special case is when see for example [4] for details. the three components are phase shifted of the same Consider now a set of N vector-sensors recording quantity. This also correspond to a rotation in 3D space a dispersive wave. As group velocity is the velocity ofthe Lissajous plot. This is what happens in the case ofthe wave packet, it will correspond to the veloc- ofdispersive waves (Section 4.1.4.). ity ofthe envelope ofthe quaternionic vector-signal recorded (i.e. of |s?(m)| = (m)). The wave propaga- 4.1.3. Multisensors polarized signals tion velocity induces a constant time delay - between Now considering a set of N vector-sensors, the col- signals recorded on two adjacent sensors. So that the lected vector data set S can be written as a matrix delay between the ÿrst sensor and the last one ofthe whose rows are the signals recorded on vector-sensors array is (N − 1)- (for an array with N sensors). As that constitute the array an example, one component ofa dispersive wave is   presented in Fig. 5. s1(m)   The slope ofthe wave packet on the “sensors vs.    s2(m)  time samples” representation can be related to the   S =  .  : (38) propagation velocity ofthe wave. Waves impinging  .   .  the whole set ofsensors at the same time are said to have inÿnite velocity (see Fig. 6). Such conÿguration s (m) N happens after velocity correction (see details in Sec- In this way, the set ofsignals recorded on the tion 4.3). vector-sensor array, S, is a matrix ofsize N ×M which The diHerence between phase and group velocity elements are pure quaternions (i.e., S ∈ HN×M ). corresponds to a constant phase shift from one sensor to an other. Such eHect also exists when consid- 4.1.4. Dispersive waves ering sets ofscalar sensors [ 33]. We consider here Dispersive waves are guided waves having dif- that for such dispersive waves, the signal recorded on ferent phase and group velocities. Group velocity the (n + 1)th sensor is a phase shifted (of a constant N. Le Bihan, J. Mars / Signal Processing 84 (2004) 1177–1199 1187

There is no relations between components ofa 0 such noise (so it is said unpolarized), and considering its polar notation (Eq. (5)), its eigenangle has a 2 behaviour similar to the one given in Fig. 2.

4 4.3. Mixture model

6 We now present the mixture model that will be Sensors used in the polarized wave separation technique 8 (Section 5).

10 4.3.1. General model The received signal on the nth sensor results from 12 0 20 40 60 80 100 120 the superposition of P (possibly dispersive) polarized Time samples waves gp(m). These waves have propagated through a medium (supposed isotropic) that only aHect waves Fig. 6. Dispersive wave collected on single-component sensors, after group velocity correction. The wave packets impinge every by attenuating, time delaying or phase shifting them. sensor at the same time. Phase velocity is not corrected. This wave Possibly, additive zero-mean Gaussian white quater- is said to be with ÿnite (group) velocity. nionic noise bn(m) is recorded: P sn(m)= hnpgp(m − (n − 1)-p)+bn(m) (41) quantity !) version ofsignal recorded at the nth sen- p=1 sor. Assuming, that the signal recorded on the ÿrst with hnp ∈ R the magnitude ofwave p at sensor sensor has an arbitrary phase (m), the signal recorded n; g (m) ∈ HN the normalized quaternionic (polar- on the nth sensor can be written as p ized) waveform of wave p. -p is the time delay of s?(m)= (m)eq(n−1)(m)q−(n−1) : (39) waveform p between two consecutive sensors. This n delay is linked (via the distance between sensors So phase shift only aHects the eigenaxis ofthe quater- Wm) with the waveform velocity cp : -p =Wm=cp. nionic signal. We will show in Section 5 that this phase bn(m) is supposed spatially white, unpolarized and shift does not aHect the dimension of the subspace independent from the source waves. Spatial whiteness necessary for characterisation of the dispersive wave. induces that two quaternionic noises collected on two diHerent sensors are decorrelated: 2 4.2. Quaternionic noise b(m);bÿ(m) = . 1ÿ; (42) where .2 is the variance ofthe centered noise (i.e. We brieMy introduce the concept of white quater- squared norm ofthe quaternion vector corresponding nionic noise that will be encountered in the mixture to b(m)) and 1 is the Kronecker delta. 7 model in Section 4.3. ÿ Clearly, the mixture considered here (Eq. (41)) is an instantaneous mixture. Deÿnition 12. Consider a three-components centered noise b(m) recorded on a single vector-sensor. Such 4.3.2. Pre-processing a noise is said quaternionic unpolarized white and The pre-processing step consists in applying a Gaussian if group velocity correction to a given wave (here p=1), in order that its group velocity becomes inÿnite b(m)=bx(m)i + by(m)j + bz(m)k; (40) where bx(m);by(m) and bz(m) are Gaussian white 1if = ÿ; T 7 noise (with variances .x;.y and .z, and with E[bb ]= 1ÿ = T 0if = ÿ: diag(.x;.y;.z), where b =[bx(m) by(m) bz(m)] ). 1188 N. Le Bihan, J. Mars / Signal Processing 84 (2004) 1177–1199

(i.e. -1 = 0, waveforms arriving simultaneously on where S1 is the subspace generated by the rank- trun- the whole set ofvector-sensors). This pre-processing cation ofthe SVDQ of S ( highest singular values). step can be visualized by the action that leads from Assuming S has rank r, then rank(S) = rank(S1)+ Figs. 5 to 6. Ifpre-processing is operated on g1, the rank(S2) (i.e. rank(S1)= and rank(S2)=r −). The mixture given in (41) becomes use ofSVDQ to build the subspaces ensures orthogo- P nality between them in the sense ofthe inner product sn(m)=hn1g1(m)+ hnpgp(m−(n−1)-p)+bn(m); deÿned in (11): p=2 u(1); u(2) = 0 and v(1); v(2) =0 (45) (43) p q p q where - and b (m) are the modiÿed time delays of for any p =1;:::; and q = +1;:::;r; where u(1) p n (1) (2) (2) other waves and the modiÿed noise (due to delay and v (resp. u and v ) are columns of U1 and V (resp. U and V ). correction to have g1(m) with inÿnite velocity). 1 2 2 The quaternion subspace method presented in next We will exploit the orthogonality between sub- section exploits the low rank dimension ofa the subspaces to develop a wave separation technique for space necessary to represent an inÿnite velocity wave vector-sensor 2D array. g1(m) and thus separate it from the other waves and noise (under orthogonality assumption). 5.2. Polarized wave separation

5. Quaternionic subspace method for polarized Consider a set of N signals, sn(m), collected on a wave separation vector-sensor array, as introduced in Section 4.1. Thus, following the quaternionic signal model, the data set A direct and simple method to process wave sep- can be arranged in a quaternion matrix S ∈ HN×M . aration on a real-valued 2D dataset is to decompose This quaternionic data set is a mixture ofpolarized the original vector space deÿned by the data into waves and noise. Assuming that a pre-processing step two orthogonal subspaces, one called the “signal sub- has been performed in order to have one polarized space” and the other one the “noise subspace”. This wave that has inÿnite velocity, S can be written using well-known decomposition technique [29] is based the mixture model presented in Eq. (43). on the SVD ofthe original data set and can be ex- The aim ofthe technique presented here is to esti- tended to a vector-sensor 2D dataset, using the SVDQ mate the inÿnite velocity wave and to remove it from presented above. the data set. In order to do so, the quaternion matrix S is decomposed into orthogonal subspaces called sig- 5.1. SVDQ and orthogonal subspaces nal subspace and noise subspace like: S = S + S : (46) As stated in Section 2.4, it is possible to decom- signal noise pose a quaternionic vector space into two orthogonal According to the SVDQ deÿnition (Eq. (24)), it is subspaces. Now, considering that a given quaternion possible to rewrite the decomposition into subspaces matrix S ∈ HN×M deÿnes such a vector space, its de- as composition in two orthogonal subspaces can be di- r rectly obtained from its SVDQ. The ÿrst subspace is / / S = unvn%n + unvn%n; (47) spanned by the singular vectors corresponding to high- n=1 n=+1 est singular values and the second one is spanned by where r is the rank of S. As SVDQ is a canonical the last singular vectors. This can written as   decomposition ofthe original quaternion data set, 21 0 it expresses S as a sum of r rank-1 matrices. The  0  V/   1 signal subspace is then constructed with the rank- S =[U1 | U2] 0 22 = S1 + S2;   / truncation of S and the noise subspace with the re- V2 0 0 maining part of S (sum ofthe r − other rank-1 (44) matrices). N. Le Bihan, J. Mars / Signal Processing 84 (2004) 1177–1199 1189

Determination of is done by looking at the set of 6.1. Example 1: One elliptically polarized wave singular values (see for example Fig. 9) and ÿnding a added with noise gap in the set. The highest singular values are then kept to build the signal subspace. As independence is The three components ofthe original simulated po- stated in the mixture, the orthogonal bases estimated larized wave with inÿnite velocity are presented in with the SVDQ do not lead to a separation into inde- Fig. 7. In Fig. 8, the mixture consists in the wave and pendent subspaces. As a consequence, the signal sub- some additive noise. The three components are pre- space will contain some noise (in fact the noise can be sented, as three pure imaginary parts ofa quaternion seen as a quaternionic full rank matrix as it is spatially matrix (Fig. 8) using the quaternion model. white, and so some parts ofthe noise will remain in Singular values ofthe quaternionic mixture matrix the rank- truncation ofthe original data set). are presented in Fig. 9. Due to the inÿnite velocity Practically, the pre-processing ensures that the in- polarized wave, one singular value has greater mag- ÿnite velocity wave s1(m) is represented by a rank-1 nitude than the others (see Fig. 9). So the signal sub- matrix (Ssignal, with = 1). Waves with ÿnite veloc- space will be constructed using a rank-1 truncation of ity are full rank matrices, and thus are considered as the SVDQ. Result obtained for the signal subspace is noise in the processing. Links between subspaces rank presented in Fig. 11. and polarized waves are presented in Appendix B.To In order to compare the SVDQ with the component- summarize, the proposed subspace method lead to an wise approach, SVD is computed on the three compo- estimate ofthe inÿnite velocity using a rank-1 trunca- nents independently. The three sets ofsingular values tion ofthe original matrix S that is corrupted by the are presented in Fig. 10. It can be seen that one of part ofthe additive noise which is spatially and tem- the singular values is greater in each ofthe three sets. porally correlated with the wave. 8 The technique is However, the ratio between the ÿrst and the second still valid for small rank of signal subspace (¿1) singular values in the three sets is always lower than ifthere remains a signiÿcant gap in the set ofsingu- the ratio in the SVDQ case. For example, on the j lar values. The larger is the rank ofthe signal sub- component singular values set, the ÿrst singular value space, the more one get some unexpected correlated is not so diHerent from the others. This is because the noise in it. wave is buried in noise on this component, and be- In the following section, we present some cause the magnitude ofnoise is spread on all singu- simulation results showing the behaviour ofthe sub- lar values with a relatively high level. This does not space technique based on SVDQ, as well as compar- allow to distinct a singular value among others (that ison with results obtained using a component-wise could be attributed to the polarized wave). For this approach. reason, there is no possibility ofdistinction between signal and noise part when computing this component independently. 6. Simulation results The component-wise signal subspace has been constructed using a rank-1 truncation ofthe SVD foreach In the following simulations, the number of sensors component. It is presented in Fig. 12. is N =10 and the number oftime samples is M =128. The correspondence between the original polarized For each example, quaternionic and component-wise wave (Fig. 7) and the signal subspace obtained by approaches results are presented. The component-wise SVDQ (Fig. 11) shows that the technique gives satis- approach consists in performing (real) subspaces fying results. Although noise corrupts the wave front, decompositions separately on the three real it is possible to distinguish the coherent wave from the matrices that contain array signals ofthe three ambient noise (Fig. 11). In particular, on component components. j ofthe mixture (Fig. 8), where the wave is buried in the noise, the SVDQ based technique allows to recover the wave. On the opposite, the component-wise 8 Recall that correlation is taken in the sense ofthe quaternionic approach does not lead to good estimation ofthe orig- scalar product. inal wave. The result is presented in Fig. 12. One can 1190 N. Le Bihan, J. Mars / Signal Processing 84 (2004) 1177–1199

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

see that the signal subspace obtained in Fig. 12 is cor- component, where its component was not viewable in rupted by noise, and the wave is not fully recovered. the mixture (Fig. 8). This indicates that the use ofthe By computing each component separately, the all components simultaneously in the processing, al- component-wise approach does not take advantage of lows to extract signal components that were lost in the relations between the components (polarization noise. Phase shifts and magnitude ratio are recovered property) and the wave front is not correctly esti- for the signal part, which means that an identiÿca- mated. This result shows that the quaternion model tion ofthe estimated wave is possible afterthe pro- approach gives better results by taking advantage of cessing, using the estimated polarization. This is not the relationship between signal components (Fig. 11). possible using the component-wise technique and, in Also, this example shows that the wave is recov- such cases, the SVDQ is required to enhance signal to ered on the three components, and even on the j noise ratio and estimate the polarization ofthe wave. N. Le Bihan, J. Mars / Signal Processing 84 (2004) 1177–1199 1191

12 Magnitude 11

8 1 2 3 4 5 6 7 8 9 10 Singular value number

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

16 i component j component k component 14

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

6.2. Example 2: Several polarized waves corrupted velocity (pre-processing step has been performed). by noise This ÿrst wave is also a dispersive wave (see Section 4.1.4.). Other ones have lower magnitude on diHerent In this example, it is shown how a dispersive components and are also dispersive waves. All the wave with strong magnitude and elliptical polariza- waves that are in the mixture have diHerent elliptical tion can be extracted from the original data. After polarizations. estimating this wave, the other waves with lowest As in the ÿrst example, the SVDQ and the SVD amplitudes can be recovered. The considered mix- techniques are compared on this data set. In Figs. 14 ture is presented in Fig. 13. It is made offourwaves and 15, we present respectively the set ofsingular corrupted by noise. One wave has greater ampli- values obtained by SVDQ and the three sets ofsingular tude on the three components, and has an inÿnite values obtained by component-wise SVD. 1192 N. Le Bihan, J. Mars / Signal Processing 84 (2004) 1177–1199

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

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

The sets ofsingular values obtained by the three between the ÿrst and the second singular values is component-wise SVDs (Fig. 14) have the same be- greater using SVDQ (Fig. 15) than in the SVD case haviour, showing a high amplitude singular value and (Fig. 14). The signal subspace is also obtained by the others with lower magnitude. This leads to the rank-1 truncation ofthe SVDQ (Fig. 17). construction ofthe signal subspace using three rank-1 In signal subspaces obtained via SVDQ and SVD truncations ofSVDs. The corresponding signal sub- techniques, the ÿrst wave seems to be well estimated. space is then presented in Fig. 16. However, the result is better in the SVDQ case. In the SVDQ case (Fig. 15), the set ofsingu- This can be seen in the noise subspaces presented in lar values has globally the same behaviour as in Fig. 18 for the SVD component-wise and in Fig. 19 the component-wise SVD case, except that the ratio for the SVDQ. N. Le Bihan, J. Mars / Signal Processing 84 (2004) 1177–1199 1193

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

25 i component j component k component 20

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

In the noise subspace obtained by SVDQ (Fig. 19), more than a rank-1 truncation ofthe SVD to be well the ÿrst wave is no more visible, due to the good estimated [18]. estimation ofits three components using the quater- Using SVDQ, the ÿrst wave has been well estimated nion approach. On the other hand, in the noise sub- and so waves oflower magnitude can be recovered space obtained by component-wise SVD (Fig. 18), on the three components (Fig. 19). The polarization some parts ofthe ÿrst wave can be seen (especially on and dispersion ofthe ÿrst wave are also estimated cor- the last sensors ofthe three components). This is due rectly in the signal subspace. The quaternion subspace to a worse estimation ofthe inÿnite velocity wave in method is not sensitive to phase shifts (dispersion or the signal subspaces when processing each component elliptical polarization) in the way that such physical separately. The dispersion ofthe ÿrst wave is the origin quantities do not increase the rank ofthe signal sub- ofthe problem encountered by the component-wise space. This is not possible using component-wise ap- approach. It is known that a dispersive wave needs proach (or subspace method over the real ÿeld R). 1194 N. Le Bihan, J. Mars / Signal Processing 84 (2004) 1177–1199

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

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

The sensitivity of signal subspace estimation to sen- a simple notation for vector signal manipulation and sor rotation is very small due to the fact that it cor- processing. The quaternion approach proposed is valid responds to a rotation in 3D. This can be seen as a for sensors having four components (encountered in phase shift of the quaternionic signal and so it does Geophysics when a pressure sensor is added to a triplet not aHect estimation ofthe signal subspace.Sothe ofvelocity measurement sensor) and three compo- SVDQ based technique gives the same performance nents. It also includes the case oftwo components when sensor rotation occurs. sensors, reducing then to complex case (as C ⊂ H). Quaternion model allows to include wave polariza- 7. Conclusion tion in algorithms and to take advantage ofthe signal redundancy on its components. The existence ofphase We have proposed a new model for vector-sensor shifts (elliptical polarization or dispersion of waves) array signals based on quaternion algebra. It provides between the components does not increase the rank of N. Le Bihan, J. Mars / Signal Processing 84 (2004) 1177–1199 1195

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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 does communication problem, when one signal is lost in the SVD component-wise approach. in noise in a diversity ofpolarization transmission We have proposed a new quaternion subspace technique. method that can be used for polarized wave separa- As said above, SVDQ leads to orthogonal bases that tion on vector-sensor array. Its superiority upon the fail to separate independent waves and noise. A nat- component-wise approach (over R) has been shown. ural extension ofthe proposed method would include The SVDQ technique may also be interesting to re- higher order statistics (ICA-like technique) in order cover the component ofa polarized signal that is to enhance the separation results and release the con- lost in noise. This conÿguration could be found in straint ofuncorrelation to higher order independence. 1196 N. Le Bihan, J. Mars / Signal Processing 84 (2004) 1177–1199

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

The proposed method is in fact linked with PCA and a rotation ofall time samples in 3D space. This is to say ? could be seen as the standardisation step in a blind that a phase shifted version, sn (m), ofa quaternionic polarized source separation algorithm. signal sn(m) can be written as As perspectives to this work, the study ofsensors ? −1 with an arbitrary number l ofcomponents could be sn (m)=qsn(m)q ; (A.2) investigated. The use ofhypercomplex number ofdimension l to encode could there be used, as well as the where q ∈ H is a unit quaternion (see [17] for rota- use ofCliHord Algebras [ 14,21]. These algebras have tions using quaternions). In fact, the rotation opera- already been used to develop pattern recognition algo- tor q[:]q−1 only aHects the phase ofthe quaternionic rithm on hyperspectral images [26]. Results obtained signal, so that we can write herein and in other work using geometric algebra to ? −1 handle vector signals tend to show that such algebra ? ? n (m) qn(m)q sn (m)= n (m)e = n(m)e : (A.3) are a powerful tool for developing algorithm specially dedicated to vector-sensors signal processing. ? −1 Proof. As sn (m)=qsn(m)q , the square ofthe ? modulus of sn (m) is given by

?2 −1 −1 ∗ Appendix A. Phase shift of polarized signals n (m)=(qsn(m)q )(qsn(m)q ) = qs (m)q−1(q−1)∗s (m)∗q∗ (A.4) Using the quaternionic signal representation, any n n polarized signal is expressed as 1 s (m)= (m)e n(m))=2 = (m)en(m): (A.1) = (qs (m)q−1qs (m)∗q∗) n n n |q|2 n n This is to say that its magnitude is, at time sample m, 1 equal to (m), and the distribution ofmagnitude on the = qs (m)s (m)∗q∗ (A.5) |q|2 n n three components is carried by the eigenangle (m). As |(m)| m=m0 =1; ∀m0, this 3D phase corresponds to a series ofpoint on the 3D unit sphere S2. So, a 1 2 ∗ 2 = q|sn(m)| q = |sn(m)| : (A.6) constant phase shift, i.e. independent of m, consists in |q|2 N. Le Bihan, J. Mars / Signal Processing 84 (2004) 1177–1199 1197

So product, multiplied by the corresponding singular

? value: n (m)= n(m) (A.7)   u1 which proves that s?(m) and s (m) have the same   n n   modulus. Then, as any quaternion q = a + bi + cj +  u2  (r=1) /   dk = Re(q) + Vec(q), with Re(q)=a and Vec(q)= S = %uv = %  .  [Ov1 vO2 ::: vOM ] (B.1)  .  bi + cj + dk, we can write  . 

? −1 uN sn (m)=qsn(m)q (r=1) = qRe(s (m))q−1 + qVec(s (m))q−1 (A.8) with un; vOm ∈ H and % ∈ R. Then the ÿrst row of S n n (r=1) is the row vector s1 = %[u1vO1 u1vO2 ::: u1vOM ]= then [%s1 %s2 ::: %sM ]. The expression ofother rows |Vec(s?(m))| = Vec(s?(m))Vec(s?(m))∗ (A.9) follows immediately. n n n Now considering the case where the recorded polar- and so, using the same argument given in (A.4), (A.5) ized wave has inÿnite velocity (and is not dispersive), and (A.6), we get the signals recorded from the second to the Nth sensors are just copies ofthe signal recorded on the ÿrst |Vec(s?(m))| = |Vec(s (m))|: (A.10) n n sensor sT, up to a (real) scalar amplitude term. This is Then, it is straightforward, using deÿnitions in Eq. (4), to say that for inÿnite velocity waves, the quaternion ? matrix, say Sinf , that represents its behaviour on the that the eigenaxis of sn (m) and sn(m) are linked as array has the form: ?(n)=q(n)q−1: (A.11) Sinf   As polarization information is encoded in the eige-   T s1 s2 ::: sM s naxis ofthe quaternion signal, it can be usefulto vi-      sT  sualize the diHerent positions of (m) with time (i.e.  1s1 1s2 ::: 1sM   1      for the increasing values of m).  s s ::: s   sT  =  2 1 2 2 2 M  =  2  ; The eHect ofa phase shifton a polarized signal is a      . .   .  rotation ofthe polarization curve (Lissajous plot) on  . .   .  the 3D unit sphere. T N−1s1 N−1s2 ::: N−1sM N−1s (B.2)

inf Appendix B. Subspaces rank and polarized waves where i ∈ R and si ∈ H. The rank of S is1asitcan be written as The subspace method presented in Section 5 is a   u powerful tool for wave separation in the case where 1   the signal subspace is a low rank truncation ofthe  u2  inf   data representing the original data set. In the following S = %   [Ov1 vO2 ::: vOM ]  .  derivations, we consider that signals and noises col-  .  lected on vector-sensor arrays are matrices of HN×M uN  1  B.1. Inÿnite velocity polarized waves are rank-1 u  % 1  quaternion matrices    1   u1   %  When considering the SVDQ as a sum ofrank-1 ma- = %   [Ov1 vO2 ::: vOM ]; (B.3)  .  trices, the ÿrst so-called rank-1 matrix Sr=1 ∈ HN×M  .    is constructed using the ÿrst left and the ÿrst right (N−1) u singular vectors, and can be expressed as their outer % 1 1198 N. Le Bihan, J. Mars / Signal Processing 84 (2004) 1177–1199

/ where % is such that vector u is normalized, and u1v = each sensor records a delayed version ofthe signal sT. Consequently, when only an inÿnite velocity is in recorded on the ÿrst one (one ofthe three components the data set, it is a rank-1 matrix. ofsuch a signal is presented in Fig. 5). As we consider instantaneous mixture, a delayed version ofa wave is B.2. Inÿnite velocity polarized and dispersive waves another wave, so that the array of N sensors will see T T are rank-1quaternion matrices N waves. Denoting the diHerent waves as s1 ;:::;sN , the data matrix Snon-inf is As explained in Section 4.1.4 and Appendix A,a  T  s1 dispersive wave recorded on a vector-sensor array can    T  be written as  s2     T  s non-inf  T  S =  s3  : (B.6)      T −1   .   1qs q   .     .  inf =disp  2 T −2  S =  2q s q  ; (B.4) T   sN    .   .  There is no reason for the lines of Snon-inf to be cor- T qN−1sTq−(N−1) related, and so any ofthe quaternionic signals si with N−1 T i =2;:::;N cannot be written as s1 multiplied by a where q ∈ H and |q|=1. This matrix can also be written T T T (real) scalar. Then, each ofthe N signals s1 ; s2 ;:::;sN as the product oftwo quaternion vectors weighted by is represented by a rank-1 matrix (see Sections B.1 a real constant: and B.2). As a consequence, a non-inÿnite velocity   polarized (possibly dispersive) wave is represented by u1   a sum of N rank-1 quaternionic matrices:    u2  Sinf =disp = %   [Ov vO ::: vO ] N  .  1 2 M Snon-inf = u v/% (B.7)  .  i i i   i=1 u N This makes Snon-inf a rank-N matrix.  1  u  % 1  B.4. Non-polarized noise is full rank quaternion     matrix  1 qu   % 1  = %     Consider that each ofthe N vector-sensors ofthe  .   .  array has recorded a white Gaussian noise. Noise   recorded on one vector-sensor is independent ofnoises (N−1) q(N−1)u % 1 recorded on other vector-sensors. This is to say that −(N−1) 2 ×[Ov1 vO2q ::: vOM q ]: (B.5) b(m);bÿ(m) = . 1ÿ; (B.8) This demonstrates that a dispersive wave is a rank-1 where ÿ and are the sensor position in the ar- quaternionic matrix. ray, and the scalar product :; : is deÿned in Eq. (11). 1ÿ is the Kronecker symbol deÿned and B.3. Non-inÿnite velocity polarized waves are full used in Section 5.1. Thus, the set of N Gaussian noises rank quaternion matrices recorded on the N vector-sensors form an orthogonal basis (which is also an independent basis). Conse- When no pre-processing step has been performed quently, the rank ofthe quaternion matrix that repre- on the data set, a recorded polarized wave impinges sents the noise, say B ∈ HN×M , is a full rank matrix the set ofsensors with diHerent arrival times. Thus, (rank(B) = min(N; M)). N. Le Bihan, J. Mars / Signal Processing 84 (2004) 1177–1199 1199

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