Hamiltonian Quaternions Research Articles

An active contour model for medical image segmentation using a quaternion framework

This paper presents a new method for segmenting medical images is based on Hamiltonian quaternions and the associative algebra, method of the active contour model and LPA-ICI (local polynomial approximation - the intersection of confidence intervals) anisotropic gradient. Since for segmentation tasks, the image is usually converted to grayscale, this leads to the loss of important information about color, saturation, and other important information associated color. To solve this problem, we use the quaternion framework to represent a color image to consider all three channels simultaneously when segmenting the RGB image. As a method of noise reduction, adaptive filtering based on local polynomial estimates using the ICI rule is used. The presented new approach allows obtaining clearer and more detailed boundaries of objects of interest. The experiments performed on real medical images (Z-line detection) show that our segmentation method of more efficient compared with the current state-of-art methods.

Nonclassical quaternions and pentanions in problems of inertial orientation

The article considers the nonclassical quaternions and pentanions of helf-rotations of solidbody and their application in problems of control and orientation of moving objects. In contrast to classicalrationed Hamiltonian quaternions of complete rotations the nonclassical quaternions of helf-rotations maybe null, they have variable rates, depending on the angle of Euler finite rotation

Equations and algorithms for determining the inertial attitude and apparent velocity of a moving object in quaternion and biquaternion 4D orthogonal operators

We consider equations and algorithms describing the operation of strapdown inertial navigation systems (SINS) intended for determining the inertial attitude parameters (the Rodrigues–Hamilton (Euler) parameters) and the apparent velocity of a moving object. The construction of these equations and algorithms is based on the Kotelnikov–Study transference principle, Hamiltonian quaternions and Clifford biquaternions, and differential equations in four-dimensional (quaternion and biquaternion) orthogonal operators.

Fast-Decodable Asymmetric Space-Time Codes From Division Algebras

Multiple-input double-output (MIDO) codes are important in the near-future wireless communications, where the portable end-user device is physically small and will typically contain at most two receive antennas. Especially tempting is the 4 x 2 channel due to its immediate applicability in the digital video broadcasting (DVB). Such channels optimally employ rate-two space-time (ST) codes consisting of (4 x 4) matrices. Unfortunately, such codes are in general very complex to decode, hence setting forth a call for constructions with reduced complexity. Recently, some reduced complexity constructions have been proposed, but they have mainly been based on different ad hoc methods and have resulted in isolated examples rather than in a more general class of codes. In this paper, it will be shown that a family of division algebra based MIDO codes will always result in at least 37.5% worst-case complexity reduction, while maintaining full diversity and, for the first time, the non-vanishing determinant (NVD) property. The reduction follows from the fact that, similarly to the Alamouti code, the codes will be subsets of matrix rings of the Hamiltonian quaternions, hence allowing simplified decoding. At the moment, such reductions are among the best known for rate-two MIDO codes. Several explicit constructions are presented and shown to have excellent performance through computer simulations.

Maximal Orders in the Design of Dense Space-Time Lattice Codes

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In this paper, we construct explicit rate-one, full-diversity, geometrically dense matrix lattices with large, nonvanishing determinants (NVDs) for four transmit antenna multiple-input–single-output (MISO) space-time (ST) applications. The constructions are based on the theory of rings of algebraic integers and related subrings of the Hamiltonian quaternions and can be extended to a larger number of Tx antennas. The usage of ideals guarantees an NVD larger than one and an easy way to present the exact proofs for the minimum determinants. The idea of finding denser sublattices within a given division algebra is then generalized to a multiple-input–multiple-output (MIMO) case with an arbitrary number of Tx antennas by using the theory of cyclic division algebras (CDAs) and maximal orders. It is also shown that the explicit constructions in this paper all have a simple decoding method based on sphere decoding. Related to the decoding complexity, the notion of sensitivity is introduced, and experimental evidence indicating a connection between sensitivity, decoding complexity, and performance is provided. Simulations in a quasi-static Rayleigh fading channel show that our dense quaternionic constructions outperform both the earlier rectangular lattices and the rotated quasi-orthogonal ABBA lattice as well as the diagonal algebraic space-time (DAST) lattice. We also show that our quaternionic lattice is better than the DAST lattice in terms of the diversity-multiplexing gain tradeoff (DMT). </para>

On the Multiplicative Complexity of the Inversion and Division of Hamiltonian Quaternions

The multiplicative complexity of a finite set of rational functions is the number of essential multiplications and divisions that are necessary and sufficient to compute these rational functions. We prove that the multiplicative complexity of inversion in the division algebra \H of Hamiltonian quaternions over the reals, that is, the multiplicative complexity of the coordinates of the inverse of a generic element from \H , is exactly eight. Furthermore, we show that the multiplicative complexity of the left and right division of Hamiltonian quaternions is at least eleven.

Traces in Arithmetic Fuchsian Groups

Let A be a quaternion algebra over an algebraic number fieldFof degreenover Q such that A⊗OR≌M2(R)×Hn−1, where H are the Hamiltonian quaternions. If O is an order in A, then O1={x∈O:N(x)=1} can be embedded inSL2(R) as a Fuchsian group. Let TO={t:t=Tr(γ),γ∈O1} be the set of traces of the elements in O1and TO(r)={t∈TO:0⩽t⩽2r}. The first purpose of this paper is to show thatlimr→∞|TO(r)|r=22n−1DF∏p|d(O)mp(O),whereDFis the discriminant ofF,d(O) is the (reduced) discriminant of O, andmp(O) only depends on the completion Opfr;of O for nonzero prime ideals p dividingd(O). We also calculate the invariantsmp(O) for important families of orders.

Generators of Large Subgroups of Units of Integral Group Rings of Nilpotent Groups

Let G be a finite nilpotent group so that all simple components ( D) n × n , n ≥ 2 of Q G satisfy the congruence subgroup theorem. Suppose that for all odd primes p dividing | G| the Hamiltonian quaternions H split over the pth cyclotomic field Q (ζ p ). Then new units B 3 are introduced so that 〈 B 1, B 2, B ′ 2, B 3〉 is of finite index in U( Z G).

The use of quaternions to generalize the Kolosov-Muskhelishvili method to three-dimensional problems of the theory of elasticity

A boundary equation is obtained for the first fundamental problem of the theory of elasticity using the Kolosov-Muskhelishvili method, by replacing the field of complex numbers by the field of Hamiltonian quaternions. A specific solution is given for the space with an elliptical cavity, subjected to a uniform tensile force at infinity.

Eisenstein-series on real, complex, and quaternionic half-spaces

The real, complex, and quaternionic half-spaces are introduced in certain analogy with the Siegel half-space. The modified symplec- tic group acts on the attached half-space in the usual way. At first properties of these half-spaces considered as symmetric spaces are derived. Then fundamental domain with respect to the modified modular group, which consists of integral modified symplectic matri- ces, is constructed. The behavior of convergence of the corresponding Eisenstein-series is determined carefully. The Fourier-coefficients of the Eisenstein-series are calculated explicitly, whenever the degree is sufficiently small. Introduction. The present paper deals with half-spaces, which are built in analogy with the Siegel half-space, and the corresponding non- analytic Eisenstein-series. The roots can be traced back to C. L. SiegePs paper Die Modulgruppe in einer einfachen involutorischen Algebra (30). A special case of these investigations is considered and contin- ued by the examination of the Riemannian geometry as well as the attached Eisenstein-series. To be more precise, throughout this paper let F stand for R, C or H, where H is the skew-field of real Hamiltonian quaternions. Just as in (16) let r = r(F) = dim RF and denote the standard basis of F over R by 1 = β\,..., er. Given = £); =1 Ujβj e F, aj e R, put Re(ά) := a and let *-+ = 2 Re(#) - denote the canonical conjugation in F. Then A^n resp. A e Mat(w F), means that A is an n x n matrix with entries in F and A! denotes the transpose of A. The letter / is reserved for the identity matrix and 0 for the zero matrix of appropriate size. GL(fl F) stands for the group of units in the ring Mat(rc F). The half-space &(n; F) consists of all Z e Mat(π; F) such that Z+z' becomes positive definite Hermitian matrix. Thus i^(n C) equals the Hermitian half-space, which was investigated by H. Braun (3), But the remaining cases are related, because X(n H) can always be embedded into the Hermitian half-space of degree In. The attached modified symplectic group MSρ(w F) consists of the automorphs of the symmetric matrix Q = (^Q), / = /W, having the

Arithmetic of second-order matrices

One presents information from the arithmetic of second-order matrices used at the application of the discrete ergodic method to indefinite ternary quadratic forms (see U. M. Pachev's paper in this issue, pp. 51–84). In particular, one has developed the theory of the rotations of vector-matrices, similar to the well-known theory of B. A. Venkov for the arithmetic of the Hamiltonian quaternions.

Space, time and geometry

Space, time, and their pseudo-euclidian geometry have an algebraic origin from the Hamiltonian quaternions and their product systems. They have unique mathematical properties yielding also the large dimensionless fundamental constants and particle masses of physics. All natural laws must be Lorentz-invariant, also the gravitational field equations. Einstein's application of Riemannian geometry is only a calculation device valid only for very small dimensionless gravitational potentials. In this case his field equations transform asymptotically into linear field equations for tensor potentials very similar to the electromagnetic ones for vector potentials. All numerical observations known up to date only this case. Kazuo Kondo et al. used this calculation device for many physical problems related with tensors.