Quaternion involutions and anti-involutions

Ell, Todd A.; Sangwine, Stephen J.

doi:10.1016/j.camwa.2006.10.029

Cited by 104 publications

(66 citation statements)

References 5 publications

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“…In particular, if µ in (2) is a pure unit quaternion, then the quaternion rotation (2) becomes the quaternion involution given in [31]. Some important properties of the quaternion rotation (see [14,26]) are:…”

Section: Quaternion Algebramentioning

confidence: 99%

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Optimization in Quaternion Dynamic Systems: Gradient, Hessian, and Learning Algorithms

Xia

Mandic

2016

IEEE Trans. Neural Netw. Learning Syst.

112

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The optimization of real scalar functions of quaternion variables, such as the mean square error or array output power, underpins many practical applications. Solutions often require the calculation of the gradient and Hessian, however, real functions of quaternion variables are essentially non-analytic. To address this issue, we propose new definitions of quaternion gradient and Hessian, based on the novel generalized HR (GHR) calculus, thus making possible efficient derivation of optimization algorithms directly in the quaternion field, rather than transforming the problem to the real domain, as is current practice. In addition, unlike the existing quaternion gradients, the GHR calculus allows for the product and chain rule, and for a one-to-one correspondence of the proposed quaternion gradient and Hessian with their real counterparts. Properties of the quaternion gradient and Hessian relevant to numerical applications are elaborated, and the results illuminate the usefulness of the GHR calculus in greatly simplifying the derivation of the quaternion least mean squares, and in quaternion least square and Newton algorithm. The proposed gradient and Hessian are also shown to enable the same generic forms as the corresponding real-and complex-valued algorithms, further illustrating the advantages in algorithm design and evaluation.

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Section: Quaternion Algebramentioning

confidence: 99%

“…The HR calculus can be considered as a generalization of the complex CR calculus [12,21,32] to the quaternion field, as the basis for the HR calculus is the use of involutions (generalized conjugate) [31]. However, the traditional product rule does not apply within the HR calculus because of the non-commutativity of quaternion algebra.…”

Section: Introductionmentioning

confidence: 99%

Optimization in Quaternion Dynamic Systems: Gradient, Hessian, and Learning Algorithms

Xia

Mandic

2016

IEEE Trans. Neural Netw. Learning Syst.

112

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“…Before proving these theorems, we mention that according to our knowledge there are only a relatively few results ( [2], [3], [4], [5], [7], [8]) about functional equations in connection with quaternions.…”

Section: Quaternions and Vector Productsmentioning

confidence: 99%

Functional equations for vector products and quaternions

Nyul

2012

Aequat. Math.

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Abstract. In our paper we find all functions f : R × R 3 → H and g : R 3 → H satisfying f (r, v)f (s, w) = − v, w + f (rs, svThese functional equations were motivated by the well-known identities for vector products and quaternions, which can be obtained from the solutions f (r, (v1, v2, v3) Mathematics Subject Classification (2010). 39B52, 16K20.Keywords. functional equation, vector product, quaternion. Quaternions and vector productsLet H = {r+v 1 i+v 2 j +v 3 k | r, v 1 , v 2 , v 3 ∈ R} be the skew field of quaternions with the basic relations i 2 = j 2 = k 2 = ijk = −1. We call r the real part and v 1 i + v 2 j + v 3 k the imaginary part of the quaternion r + v 1 i + v 2 j + v 3 k ∈ H. A quaternion is purely imaginary if its real part equals 0. The absolute value of the above quaternion is defined by r 2 + v 2 1 + v 2 2 + v 2 3 . It is well-known (see e.g. [1], [6], [9], [10]) that identifying the purely imaginary quaternion v 1 i+v 2 j +v 3 k with the vector (v 1 , v 2 , v 3 ) ∈ R 3 we have (r + v)(s + w) = (rs − v, w ) + (sv + rw + v × w) (r, s ∈ R, v, w ∈ R 3 ), and for purely imaginary quaternionswhere on the left-hand sides quaternionic products stand, v, w and v × w denote the standard inner product (dot product) and the cross product of v, w ∈ R 3 , respectively.

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“…f (q), as well as its components f a (q), f b (q), f c (q), and f d (q), can be viewed as functions of q a , q b , q c and q d , which can be expressed in terms of q and its involutions [9]:…”

Section: A Differentiation With Respect To a Quaternion-valued Vectormentioning

confidence: 99%

A general quaternion-valued gradient operator and its applications to computational fluid dynamics and adaptive beamforming

Jiang

Liu

2014

2014 19th International Conference on Digital Signal Processing

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Abstract-Quaternion-valued signal processing has received increasing attention recently. One key operation involved in derivation of all kinds of adaptive algorithms is the gradient operator. Although there have been some derivations of this operator in literature with different level of details, it is still not fully clear how this operator can be derived in the most general case and how it can be applied to various signal processing problems. In this work, we will give a general derivation of the quaternion-valued gradient operator and then apply it to two different areas. One is to combine with the classic computational fluid dynamics (CFD) approach in wind profile prediction and the other one is to apply the result to the adaptive beamforming problem for vector sensor arrays.

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Quaternion involutions and anti-involutions

Cited by 104 publications

References 5 publications

Optimization in Quaternion Dynamic Systems: Gradient, Hessian, and Learning Algorithms

Optimization in Quaternion Dynamic Systems: Gradient, Hessian, and Learning Algorithms

Functional equations for vector products and quaternions

A general quaternion-valued gradient operator and its applications to computational fluid dynamics and adaptive beamforming

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