full mean square performance bounds on quaternion estimators for improper data novel physical insights are provided into the mean square performance bounds of quaternion valued widely linear wl semi widely linear swl and strictly linear estimators for the generality of quaternion valued gaussian data this is achieved by first defining three kinds of complementary mean square errors cmses of these estimators and by further exploiting the corresponding degrees of mathbb h improperness second order noncircularity next the investigation of the bounds of the attainable cmses by these classes of estimators shows that only a joint consideration of the proposed cmse analysis and the standard mse analysis provides enough degrees of freedom for a detailed account of the mse performance the so established framework for the analysis of estimators for mathbb h improper data is shown to be capable of measuring error power distribution for each data channel an important finding which is not possible to obtain through the standard mse analysis only simulations in the system identification setting support the analysis

vector sparse representation of color image using quaternion matrix analysis traditional sparse image models treat color image pixel as a scalar which represents color channels separately or concatenate color channels as a monochrome image in this paper we propose a vector sparse representation model for color images using quaternion matrix analysis as a new tool for color image representation its potential applications in several image processing tasks are presented including color image reconstruction denoising inpainting and super resolution the proposed model represents the color image as a quaternion matrix where a quaternion based dictionary learning algorithm is presented using the k quaternion singular value decomposition qsvd generalized k means clustering for qsvd method it conducts the sparse basis selection in quaternion space which uniformly transforms the channel images to an orthogonal color space in this new color space it is significant that the inherent color structures can be completely preserved during vector reconstruction moreover the proposed sparse model is more efficient comparing with the current sparse models for image restoration tasks due to lower redundancy between the atoms of different color channels the experimental results demonstrate that the proposed sparse image model avoids the hue bias issue successfully and shows its potential as a general and powerful tool in color image analysis and processing domain

performance bounds of quaternion estimators the quaternion widely linear wl estimator has been recently introduced for optimal second order modeling of the generality of quaternion data both second order circular proper and second order noncircular improper experimental evidence exists of its performance advantage over the conventional strictly linear sl as well as the semi wl swl estimators for improper data however rigorous theoretical and practical performance bounds are still missing in the literature yet this is crucial for the development of quaternion valued learning systems for 3 d and 4 d data to this end based on the orthogonality principle we introduce a rigorous closed form solution to quantify the degree of performance benefits in terms of the mean square error obtained when using the wl models the cases when the optimal wl estimation can simplify into the swl or the sl estimation are also discussed

quaternion involutions and anti involutions an involution or anti involution is a self inverse linear mapping in this paper we study quaternion involutions and anti involutions we review formal axioms for such involutions and anti involutions we present two mappings one a quaternion involution and one an anti involution and a geometric interpretation of each as reflections we present results on the composition of these mappings and show that the quaternion conjugate may be expressed using three mutually perpendicular anti involutions finally we show that projection of a vector or quaternion can be expressed concisely using three mutually perpendicular anti involutions

diffusive stability of oscillations in reaction diffusion systems we study nonlinear stability of spatially homogeneous oscillations in reaction diffusion systems assuming absence of unstable linear modes and linear diffusive behavior for the neutral phase we prove that spatially localized perturbations decay algebraically with the diffusive rate t n 2 in space dimension n we also compute the leading order term in the asymptotic expansion of the solution and show that it corresponds to a spatially localized modulation of the phase our approach is based on a normal form transformation in the kinetics ode which partially decouples the phase equation at the expense of making the whole system quasilinear stability is then obtained by a global fixed point argument in temporally weighted sobolev spaces corresponding author arnd scheel

generic instability of spatial unfoldings of almost homoclinic periodic orbits we consider spatially homogeneous time periodic solutions of general partial differential equations we prove that when such a solution is close enough to a homoclinic orbit or a homoclinic bifurcation for the differential equation governing the spatially homogeneous solutions of the pde then it is generically unstable with respect to large wavelength perturbations moreover the instability is of one of the two following types either the well known kuramoto phase instability corresponding to a floquet multiplier becoming larger than 1 or a fundamentally different kind of instability occurring with a period doubling at an intrinsic finite wavelength and corresponding to a floquet multiplier becoming smaller than 1

the dirichlet problem for nonlinear second order elliptic equations i monge ampegre equation on considere le probleme de dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture d un domaine borne dans r n avec une frontiere c

global exact controllability of a class of quasilinear hyperbolic systems in this paper we consider a distributed parameter boundary control system described by a reducible quasilinear hyperbolic system formula posed on the domain 0 x 1 with the nonlinear boundary conditions formula in the case that the system islinear degenerate we show that it isgloballyexactly boundary controllable

optimization in quaternion dynamic systems gradient hessian and learning algorithms the optimization of real scalar functions of quaternion variables such as the mean square error or array output power underpins many practical applications solutions typically require the calculation of the gradient and hessian however real functions of quaternion variables are essentially nonanalytic which are prohibitive to the development of quaternion valued learning systems to address this issue we propose new definitions of quaternion gradient and hessian based on the novel generalized hamilton real ghr calculus thus making a possible efficient derivation of general optimization algorithms directly in the quaternion field rather than using the isomorphism with 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 novel quaternion gradient and hessian with their real counterparts properties of the quaternion gradient and hessian relevant to numerical applications are also introduced opening a new avenue of research in quaternion optimization and greatly simplified the derivations of learning algorithms the proposed ghr calculus is shown to yield the same generic algorithm forms as the corresponding real and complex valued algorithms advantages of the proposed framework are illuminated over illustrative simulations in quaternion signal processing and neural networks

a quaternion widely linear adaptive filter a quaternion widely linear qwl model for quaternion valued mean square error mse estimation is proposed the augmented statistics are first introduced into the field of quaternions and it is demonstrated that this allows for capturing the complete second order statistics available the qwl model is next incorporated into the quaternion least mean square qlms algorithm to yield the widely linear qlms wl qlms this allows for a unified approach to adaptive filtering of both q proper and q improper signals leading to improved accuracies compared to the qlms class of algorithms simulations on both benchmark and real world data support the analysis

fast independent component analysis algorithm for quaternion valued signals an extension of the fast independent component analysis algorithm is proposed for the blind separation of both bbq proper and bbq improper quaternion valued signals this is achieved by maximizing a negentropy based cost function and is derived rigorously using the recently developed mbi bbh bbr calculus in order to implement newton optimization in the augmented quaternion statistics framework it is shown that the use of augmented statistics and the associated widely linear modeling provides theoretical and practical advantages when dealing with general quaternion signals with noncircular rotation dependent distributions simulations using both benchmark and real world quaternion valued signals support the approach

inverse observability estimates for second order hyperbolic equations with variable coefficients abstract we consider a general second order hyperbolic equation defined on an open bounded domain r n with variable coefficients in both the elliptic principal part and in the first order terms as well at first no boundary conditions b c are imposed our main result theorem 3 5 is a reconstruction or inverse estimate for solutions w under checkable conditions on the coefficients of the principal part the h1 l2 energy at time t t or at time t 0 is dominated by the l2 norms of the boundary traces w a and wt modulo an interior lower order term once homogeneous b c are imposed our results yield under a uniqueness theorem needed to absorb the lower order term continuous observability estimates for both the dirichlet and neumann case with an explicit sharp observability time hence by duality exact controllability results moreover no artificial geometrical conditions are imposed on the controlled part of the boundary in the neumann case in contrast with existing literature the first step of our method employs a riemann geometry approach to reduce the original variable coefficient principal part problem in r n to a problem on an appropriate riemann manifold determined by the coefficients of the principal part where the principal part is the laplacian in our second step we employ explicit carleman estimates at the differential level to take care of the variable first order energy level terms in our third step we employ micro local analysis yielding a sharp trace estimate to remove artificial geometrical conditions on the controlled part of the boundary in the neumann case

quaternion reproducing kernel hilbert spaces existence and uniqueness conditions the existence and uniqueness conditions of quaternion reproducing kernel hilbert spaces qrkhs are established in order to provide a mathematical foundation for the development of quaternion valued kernel learning algorithms this is achieved through a rigorous account of left quaternion hilbert spaces which makes it possible to generalise standard rkhs to quaternion rkhs quaternion versions of the riesz representation and moore aronszajn theorems are next introduced thus underpinning kernel estimation algorithms operating on quaternion valued feature spaces the difference between the proposed quaternion kernel concept and the existing real and vector approaches is also established in terms of both theoretical advantages and computational complexity the enhanced estimation ability of the so introduced quaternion valued kernels over their real and vector valued counterparts is validated through kernel ridge regression applications simulations on real world 3d inertial body sensor data and nonlinear channel equalisation using novel quaternion cubic and gaussian kernels support the approach

a class of quaternion kalman filters the existing kalman filters for quaternion valued signals do not operate fully in the quaternion domain and are combined with the real kalman filter to enable the tracking in 3 d spaces using the recently introduced hr calculus we develop the fully quaternion valued kalman filter qkf and quaternion extended kalman filter qekf allowing for the tracking of 3 d and 4 d signals directly in the quaternion domain to consider the second order noncircularity of signals we employ the recently developed augmented quaternion statistics to derive the widely linear qkf wl qkf and widely linear qekf wl qekf to reduce computational requirements of the widely linear algorithms their efficient implementation are proposed and it is shown that the quaternion widely linear model can be simplified when processing 3 d data further reducing the computational requirements simulations using both synthetic and real world circular and noncircular signals illustrate the advantages offered by widely linear over strictly linear quaternion kalman filters

exponential decay rates for the solutions of euler bernoulli equations with boundary dissipation occurring in the moments only abstract we shall derive uniform decay rates for the total energy of the solutions to the euler bernoulli equations wherein energy dissipation occurs on the boundary the main feature which distinguishes this paper from other related works is the fact that the boundary feedback is acting through the moments only the key technical element responsible for the proof of the main result is a new regularity estimate for the solutions to the nonhomogeneous schrodinger equation

a note on the boundary stabilization of the wave equation we study the energy decay rates of the wave equation in a domain where boundary damping is present we generalize the geometrical conditions obtained earlier in j math purer appl 58 1979 pp 249 273 by using some more general multipliers of strauss comm pure appl math 28 1975 pp 265 278 the interaction between distributed damping and boundary damping is discussed a regulator problem is also formally discussed by the synthesis method

riccati equations for hyperbolic partial differential equations with l 2 0 t l 2 γ dirichlet boundary terms this paper studies the quadratic optimal control problem for second order linear hyperbolic partial differential equations defined on a bounded domain omega subset r n with boundary gamma both the finite interval case 0 t t infty and the infinite interval case t infty regulator problem are considered the distinguishing feature of the paper which differentiates it from previous scarce literature on the subject is that the controls are only l 2 0 t l 2 gamma functions which act in the dirichlet b c and that the corresponding solutions are penalized in the l 2 0 t l 2 omega norm smoother controls particularly in space were taken in the few previous works on this subject the well posedness of this formulation stems from recent results by the authors about regularity of second order hyperbolic mixed problems l t 1 l t 3 under minimal assumptions the optimal control is synthesized in a pointwise feedback form through an operator which is s

consimilarity of quaternions and coneigenvalues of quaternion matrices first of all by characterizing solutions of the quaternion equation a x x b this paper studies consimilarity of quaternions and some related consequences for the important role of coneigenvalues in consimilarity tranformations of quaternion matrices this paper further derives the relations between principle right coneigenvalues of a quaternion matrix and eigenvalues of the corresponding real representation matrix then based on the real representation matrix an effective algorithm is presented to calculate all coneigenvalues and the associated coneigenvectors of a quaternion matrix finally two numerical examples are given to verify the effectiveness of the proposed algorithm

decay of solutions of wave equations in a bounded region with boundary dissipation abstract an energy decay rate is obtained for solutions of wave type equations in a bounded region in rn whose boundary consists partly of a nontrapping reflecting surface and partly of an energy absorbing surface

hamilton and jacobi come full circle jacobi algorithms for structured hamiltonian eigenproblems we develop jacobi algorithms for solving the complete eigenproblem for hamiltonian and skew hamiltonian matrices that are also symmetric or skew symmetric based on the direct solution of 4 4 and in one case 8 8 subproblems these structure preserving algorithms produce symplectic orthogonal bases for the invariant subspaces associated with a matrix in any one of the four classes under consideration the key step in the construction of the algorithms is a quaternion characterization of the 4 4 symplectic orthogonal group and the subspaces of 4 4 hamiltonian skew hamiltonian symmetric and skew symmetric matrices in addition to preserving structure these algorithms are inherently parallelizable numerically stable and show asymptotic quadratic convergence

the theory of quaternion matrix derivatives a systematic framework for the calculation of the derivatives of quaternion matrix functions with respect to quaternion matrix variables is introduced the proposed approach is equipped with the matrix product and chain rules and applies to both analytic and nonanalytic functions of quaternion variables this rectifies a mathematical shortcut in the existing methods which incorrectly use the traditional product rule we also show that within the proposed framework the derivatives of quaternion matrix functions can be calculated directly without using quaternion differentials or resorting to the isomorphism with real vectors illustrative examples show how the proposed quaternion matrix derivatives can be used as an important tool for solving optimization problems in signal processing applications

matrix lsqr algorithm for structured solutions to quaternionic least squares problem abstract in this paper we employ matrix lsqr algorithm to deal with quaternionic least squares problem in order to find the minimum norm solutions with kinds of special structures and propose a strategy to accelerate convergence rate of the algorithm via right left preconditioning of the coefficient matrices we mainly focus on analyzing the minimum norm hermitian solution and the minimum norm bihermitian solution to the quaternionic least squares problem i j k other structured solutions also can be obtained using the proposed technique a number of numerical experiments are performed to show the efficiency of the preconditioned matrix lsqr algorithm

hypercomplex signals a novel extension of the analytic signal to the multidimensional case the construction of gabor s 1946 complex signal which is also known as the analytic signal provides direct access to a real one dimensional 1 d signal s local amplitude and phase the complex signal is built from a real signal by adding its hilbert transform which is a phase shifted version of the signal as an imaginary part to the signal since its introduction the complex signal has become an important tool in signal processing with applications for example in narrowband communication different approaches to an n d analytic or complex signal have been proposed in the past we review these approaches and propose the hypercomplex signal as a novel extension of the complex signal to n d this extension leads to a new definition of local phase which reveals information on the intrinsic dimensionality of the signal the different approaches are unified by expressing all of them as combinations of the signal and its partial and total hilbert transforms examples that clarify how the approaches differ in their definitions of local phase and amplitude are shown an example is provided for the two dimensional 2 d hypercomplex signal which shows how the novel phase concept can be used in texture segmentation

uniform exponential energy decay of wave equations in a bounded region with l2 0 l2 γ feedback control in the dirichlet boundary conditions etant donne un ouvert borne r n n 2 on considere l equation d onde w tt w dans o avec des conditions initiales w o x w o x l 2 w 1 o x w 1 x h 1 et des conditions aux limites non homogenes de type dirichlet w t u t sur o etant la frontiere de

performance analysis of deficient length quaternion least mean square adaptive filters quaternion adaptive filters have been widely used for processing 3d and 4d phenomena deficient length quaternion adaptive filters are explicitly or implicitly used in many practical applications where the length of system impulse response is large or unknown however their statistical behaviors are yet to be fully understood as theoretical results on the class of full length quaternion least mean square qlms algorithms do not necessarily apply to their deficient length versions this article fills this void and analyses the mean and mean square convergence of the deficient length qlms algorithms both for the strictly linear and widely linear cases transient and steady state performance is characterised by exploiting the augmented statistics of noncircular quaternion random vectors a novel decorrelation technique in the quaternion domain is shown to allow for the development of intuitive closed form solutions for correlated quaternion gaussian inputs thus unveiling the relationship between the algorithm behaviour and the noncircularity of quaternion input data the analysis also provides a general framework whereby the strictly linear and semi widely linear qlms algorithms can be seen as deficient length versions of the widely linear qlms numerical simulations validate the accuracy of the theoretical results and support the behaviour of the considered algorithms

criteria for the stability of spatial extensions of fixed points and periodic orbits of differential equations in dimension 2 abstract we consider partial differential equations that can be viewed as spatial extensions of two dimensional differential equations with respect to a coupling matrix we provide analytical and geometrical criteria governing the stability of spatially homogeneous stationary solutions and the phase stability of spatially homogeneous time periodic solutions we distinguish cases where the differential equation has a conservative or a dissipative behavior the geometrical criteria roughly speaking relate to the sense of rotation of on one hand the coupling matrix and on the other the flow of the differential equation around the spatially homogeneous solution

dirichlet boundary stabilization of the wave equation with damping feedback of finite range abstract a closed loop system consisting of the wave equation with a feedback acting in the dirichlet boundary condition in the form of a nonlocal one dimensional range operator defined on the velocity vector damping is considered beside the well posedness question generation of a feedback c 0 semigroup the boundary feedback stabilization problem is studied while asymptotic decay to zero in the uniform operator norm can never occur for the considered class of closed loop systems however checkable sufficient conditions on the vectors in the boundary conditions are provided that guarantee asymptotic decay to zero in the strong norm of appropriate sobolev spaces these conditions include but are not limited to the standard case of dissipativity of the feedback system the starting point of this approach is a functional analytic model for the study of nonsmooth boundary input hyperbolic equations recently developed i lasiecka and r triggiani appl math optim 7 1 1981 35 93