On Properness Of Quaternion Valued Random Variables

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Introduction. Properness of complex valued random variables and vectors is a well-known subject in signal processing and information theory (Neeser, F.D. ...
On Properness Of Quaternion Valued Random Variables By Pierre-Olivier Amblard and Nicolas Le Bihan Laboratoire des Images et des Signaux, CNRS UMR 5083, 961 Rue de la Houille Blanche, Domaine Universitaire, BP 46, 38402 Saint Martin d’H`eres Cedex, France e-mail: [email protected], [email protected]

Abstract In this paper, we present the concept of properness for quaternion random variables and emphasize some second order geometrical consequences on the four dimensional probability distribution for such variables. Properness is stated in terms of the invariance of the variable distribution under Clifford translations.

1. Introduction Properness of complex valued random variables and vectors is a well-known subject in signal processing and information theory (Neeser, F.D. & Massey, J.L. (1993)). In the last decade, geometric algebras, and among them quaternions, have found applications in signal and image processing. Examples are the modelling and the analysis of color images (Sangwine S.J. (1996)), the definition of quaternion valued Fourier transforms for greyscale images (B¨ ulow & Sommer (2001)), or the quaternion representation of 3D- or 4D-sensor measurements for polarization modelling (Le Bihan, N. & Mars, J. (2004)). In all these works, variables, signals or images were considered as deterministic quantities. However, many applications need a stochastic modelling of the observed phenomena (e.g. polarized magnetic disturbances, electromagnetic waves carrying random codes, noise in color image processing. . . ). In this paper, we examine the extension of properness to the case of quaternion random variables, and link it with some geometrical invariance properties of the distributions of such variables.

2. Quaternions and quaternion random variables After a recall on the definition of quaternions and their properties, we introduce the concept of quaternion random variable and pay attention to its possible vector representations. 2.1. Definition and properties Quaternions algebra is a four dimensional hypercomplex numbers system discovered by Sir R.W. Hamilton in 1843 (Hamilton W.R. (1843)). A quaternion q has a real part and a three dimensional imaginary part such as: q = a + ib + jc + kd

(2.1)

and with the following relations between the imaginary units: ij = k = −ji and i = j = k = ijk = −1. Quaternions form a noncommutative division algebra, noted √H, so that for q1 , q2 ∈ H, q1 q2 #= q2 q1 generally. The conjugate of q ∈ H is q¯ = a−ib−jc−kd, its norm is |q| = a2 + b2 + c2 + d2 and its inverse is q −1 = q¯/|q|2 . Note that, as C ⊂ H, conjugation of a complex number z will be noted z¯. A quaternion is called pure when its real part is null and unit if its norm equals one. If q is a pure unit quaternion, then: |q|2 = 1 and q 2 = −1. Euler formula extends to H, so that any quaternion q can be written: q = |q|eξθ , where ξ is a pure unit π quaternion usually called the axis; θ is the angle. Any pure unit quaternions µ can thus be written: µ = eµ 2 . Conjugation is an anti-involution over H (q1 q2 = q¯2 q¯1 ), but there exist three important involutions, noted qi , qj and qk , defined as: 2

2

2

qi = −iqi , qj = −jqj , qk = −kqk

(2.2)

q → eµ1 θ qe−µ1 θ

(2.3)

These involutions are isometries and are special cases of 4D rotations (see Coxeter H.S.M. (1946)). In fact, for any quaternion q the mapping:

where µ1 is a pure unit quaternion and θ ∈ [0, 2π), leaves invariant the plane spanned by {1, µ1 } while it performs a clockwise rotation of angle 2θ in the plane spanned by {µ2 , µ3 }, assuming {1, µ1 , µ2 , µ3 } is an orthonormal basis of H. Note that for involutions in (2.2), the mapping axis are respectively i, j and k with angle θ = π/2. Particular isometries of interest in this work are the so called Clifford translations. There are two types of such translations (see Coxeter H.S.M. (1946)). A left Clifford translation is the mapping: q → eµ1 θ q

(2.4)

ˆ = [a b c d]T , q ˜ = [z1 z¯1 z2 z¯2 ]T , q ˘ = [q qi qj qk ]T q

(2.5)

while the mapping q → qeµ1 θ is a right Clifford translation. A left Clifford translation performs a clockwise rotation of angle θ in the plane spanned by {1, µ1 } as well as in in the plane spanned by {µ2 , µ3 }. However, a right Clifford translation performs a clockwise rotation in {1, µ1 } and an counterclockwise rotation in {µ2 , µ3 } (both of angle θ). It possible to consider quaternions as complexified complex numbers, such that: q = z1 +z2 j where z1 , z2 ∈ Ci . Due to noncommutativity of quaternions product, the order in z2 j is important. In this notation, known as the Cayley-Dickson notation, z1 = a + ib and z2 = c + id. A quaternion can be seen as a four dimensional real vector, i.e. an element of R4 . Also, it is possible to obtain the four real elements (a, b, c, d) from combinations of z1 and z2 and their conjugates, or from combinations of q and its three involutions. This allows to introduce for a quaternion random variable q, the three following vector representations: These representations are linked the following way: ˜ = A[RC] q ˆ and q ˘ = A[CH] q ˜ q

(2.6)

with:

    1 i 0 0 1 0 0 j 1 −i 0 0  [CH] 1 0 0 −j   = A[RC] =  (2.7) 0 0 1 i  ; A 0 1 j 0  0 0 1 −i 0 1 −j 0 Vector representations are of interest in the study of a quaternion random variable as they allow easier geometrical interpretations in 4D space. 2.2. Quaternion random variables A quaternion valued random variable is defined unambiguously as a real valued four dimensional random vector. As such, a quaternion random variable q is fully described by the joint probability density function (pdf) of the four components a, b, c, d of its vector representation qˆ , or equivalently by the characteristic function. However, special features of the pdf, such as symmetries under some transformations, may not be easily revealed by this representation. Precisely, using the complex representation q˜ or the quaternion representation q˘ may reveal ˆ, q ˜ and q ˘ are linked by relations (2.6), using the complex or the quaternion more easily these features. Since q representations amounts to define the pdf of the quaternion variable on C4 and on H4 . This can be done if one consider that z1 , z¯1 , z2 , z¯2 are algebraically independent variables, and q, qi , qj , qk are algebraically independent variables also. This can be rigorously formalized–as was done for complex variables in (Amblard, P.O. et. al. (1996)). However, we restrict in the following to first and second order statistics only. When considering a quaternion random variable q, the mathematical expectation of q is given as follows: E[q] = E[a] + E[b]i + E[c]j + E[d]k

(2.8)

where the expectation of real valued random variables (a, b, c, d) is taken in the classical sense. Without loss of generality, the considered quaternion random variables are supposed centered (i.e. E[q] = 0) in the sequel. ˆ , its covariance matrix is: Using the real vector representation of q, noted q ( ' (2.9) Λqˆ = E qˆqˆ T

that contains second order statistical relationships between a, b, c and d. Using the two other representations, it is possible to define a complex and a quaternion representation of the covariance matrix given by: ) ˜† ] Λq˜ = E[˜ qq (2.10) ˘† ] qq Λq˘ = E[˘

Λq˜ contains the second order cross-moments between z1 , z¯1 , z2 and z¯2 , and Λq˘ the second order cross-moments between q, qi , qj and qk . Operator † stands for conjugation-transposition.

3. Properness of quaternion random variables In (Vakhania, N.N. (1998)), it was shown that there exists two levels of properness for a quaternion random variable, namely C- and H-properness. Vakhania proposed a definition of properness based on the fact that the real representation of the covariance matrix commutes with either the real matrix representation of i or with both the real representations of i and j. This properness can be interpreted as the invariance of the pdf under some specific rotations of angle π/2. Here, we extend the definition to an arbitrary axis and angle and examine the consequences of so defined properness on the second order statistical relationships between components of the quaternion random variable. 3.1. C-properness Definition 1. A quaternion valued random variable q is called Cη -proper if: d

q = eηϕ q, ∀ϕ

(3.1)

for one and only one imaginary unit η = i, j or k.

Clearly, a Cη -proper quaternion random variable has a distribution that is invariant by left Clifford translation of axis η and of any angle ϕ (i.e. simultaneous rotations in of angle ϕ in two orthogonal planes of 4D space). As an example, we now study the case of a Ci -proper quaternion random variable. Property 1. The real, complex and quaternion representation covariance matrices of a Ci -proper quaternion random variable q have the following structures:  2   2    σ1 Σ ∆ 0 0 Σ1 0 0 τβ τγ 0 Ω 2 0    σ12 τγ −τβ  0   and Λ ˜ =  0 Σ1 Ω2 0  and Λ ˘ = ∆i Σ 0 (3.2) Λqˆ =  2 q q  τβ 0 0 τγ σ2 0  Ω Σ2 0  0 Σ ∆ 2 2 τγ −τβ 0 Ω 0 0 Σ2 σ2 0 0 ∆i Σ

where σ12 = E[a2 ] = E[b2 ], σ22 = E[c2 ] = E[d2 ], τβ = E[ac] = −E[bd] and τγ = E[ad] = E[bc] are real coefficients corresponding to cross-covariances between pairs of variables (a, b) and (c, d). We also have Σ21 = E[|z1 |2 ] = 2σ12 , Σ22 = E[|z2 |2 ] = 2σ22 and Ω = E[z1 z2 ] = 2(τβ + τγ ). Finally, we have Σ = E[|q|2 ] = E[|qi,j,k |2 ] = 2σ12 + 2σ22 and ∆ = E[q q¯i ] = E[qj q¯k ] = 2σ12 − 2σ22 + 2(τβ + iτγ )j.

Proof. Ci -properness of q involves that E[q q¯] = eiϕ E[q q¯]e−iϕ . Using vector representations of q given in (2.5) and transition matrices in (2.7), the structures of Λqˆ , Λq˜ and Λq˘ come out by straightforward calculation. Thus, a Ci -proper quaternion random variable q is correlated with the variable qi while it is decorrelated with variables qj and qk . Ci -properness involves an invariance of the distribution under left Clifford translation. It means that the distribution is left invariant by simultaneous rotations of angle π/2 in the planes spanned by {1, i} and {j, k}. Looking at the complex representation of q, Ci -properness is equivalent to the second order circularity of both z1 and z2 . Recall that a complex random variable z is circular if its pdf is invariant under any rotations; second order circularity of z is achieved if the real and imaginary parts of z have same variance and are uncorrelated. Furthermore, Ci -properness does not require uncorrelation between z1 and z2 . As an example, consider a Gaussian Ci -proper random variable with σ12 = 1, σ22 = 1.5, τβ = 0.7 and τγ = 0.2. In figure 1, we plot 104 samples of this variable. Looking at planes {1, i} and {j, k}, we can see that z1 and z2 are both circular: the distributions are invariant under rotations. However, the correlation between z1 and z2 is revealed by looking at the distributions in the planes {1, j} and {i, k}: the ellipsis parameter are governed by τβ , whereas looking at the distributions in the planes {1, k} and {i, j} reveals parameter τγ . 3.2. H-properness Definition 2. A quaternion random variable q is said to be H−proper if: d

and for any pure unit quaternion η.

q = eηϕ q, ∀ϕ

(3.3)

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Figure 1. C-proper Gaussian random variable. Three possible representations: 1- planes {1, i} and {j, k} (upper-left), 2- planes {1, j} and {i, k} (upper-right), 3- planes {1, k} and {i, j} (lower-center).

In this case, the real, complex and quaternion representations of the covariance matrix have the following structure: (3.4) Λqˆ = σ 2 I4 ; Λq˜ = 2σ 2 I4 ; Λqˆ = 4σ 2 I4 where I4 is the 4 × 4 identity matrix and σ 2 = E[a2 ] = E[b2 ] = E[c2 ] = E[d2 ]. Clearly, the distribution of a H-proper quaternion random variable is invariant under any four dimensional isometric transformation. In the Gaussian case, the distribution of a H-proper variable is contained in a 4D hypersphere.

4. Discussion We have extended a definition of properness for quaternion valued random variables based on the invariance of the pdf under the action of left Clifford translations. This allows to go deeper in the study of the symmetries of the pdf. We have however restricted the implications of properness to the second order statistics. Of course, for Gaussian variables the analysis is full since Gaussian are entirely described by the second order statistics. Further work will consist in the use of higher order statistics to characterize some 4D geometrical properties of quaternion valued random variable distributions. For example, we are working on the notion of n-th order properness, for which the rotational invariance are no longer continuous but rather discrete 2π/n. This notion will be of importance in the study of 4D constellations used for communication purposes, see for example (Zetterberg, L. H. & Br¨ andstr¨om (1977)). REFERENCES Amblard, P.O. & Lacoume J.L. & Gaeta M. 1996 Statistics for Complex Random Variables and Signals : Part 1 and 2 Signal Processing Vol. 53, pp. 1–25. Neeser, F.D. & Massey, J.L. 1993 Proper Complex Random Process with applications to information theory IEEE Trans. on Information Theory Vol. 39, No. 4, pp. 1293–1302. Hamilton W.R. 1843 On quaternions Proceeding of the Royal Irish Academy Coxeter H.S.M. 1946 Quaternions and reflections The american mathematical monthly Vol. 53, pp. 136–146. Sangwine, S. J. 1996 Fourier transforms of colour images using quaternions, or hypercomplex, numbers Electronics letters Vol. 32, No. 21, pp. 1979–1980. ¨ low, T. & Sommer, G. 2001 Hypercomplex Signals– A Novel Extension of the Analytic Signal to the MultidimenBu sional Case IEEE Trans. on Signal Processing vol. 49, No. 11, pp. 2844–2852. Le Bihan, N. & Mars, J. 2004 Singular Value Decomposition of matrices of quaternions: a new tool for vector-sensor signal processing Signal Processing Vol. 2004. Vakhania, N.N. 1998 Random vectors with values in quaternions Hilbert spaces Th. Probab. Appl. Vol. 43, No. 1, pp. 99–115 ¨ ndstro ¨ m, H. 1977 Codes for combined phase and amplitude modulated signals in a fourZetterberg, L. H. & Bra dimensional space IEEE Trans. on Communications Vol. 25, No. 9, pp. 943–950.

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