Radial Fourier Research Articles

Asymmetric Interaction of a Rigid Disc Embedded in a Transversely Isotropic Half-Space

An analytical treatment has been presented for the lateral response of a rigid circular disc embedded in a transversely isotropic half-space. By means of a complete representation of using displacement potentials, it is shown that the governing equations of equilibrium for this class of problems can be uncoupled into two fourth-order partial differential equations. Using cylindrical co-ordinate system and applying Hankel integral transform in the radial direction and Fourier series, the problem may be changed to a system of four separate integral equations, which, in turn, are reduced to a pair of Fredholm equation of the second kind, whose solutions are then computed. In addition to furnishing a unified view of existing solutions for zero and infinite embedments, the present treatment reveals a severe boundary-layer phenomenon, which is apt to be of interest to this class of problems in general. The present solutions are analytically and numerically in exact agreement with the existing solutions for a half-space with isotropic material properties. To confirm the accuracy of the numerical evaluation of the integrals involved, numerical results are included for cases of different degrees of the material anisotropy. The contact pressure, desired force–displacement relationship, stress and displacement field are explicitly found.

Wave based analysis of the Green’s function for a layered cylindrical shell

Cylindrical shells composed of concentric layers may be designed to affect the way that elastic waves are generated and propagated, particularly when some layers are anisotropic. To aid the design process, the present work develops a wave based analysis of the Green's function for a layered cylindrical shell in which the response is given as a sum of waves propagating in the axial coordinate. The analysis assumes linear Hookean materials for each layer. It uses finite element discretizations in the radial coordinate and Fourier series expansions in the circumferential coordinate, leading to linear equations in the axial wavenumber domain that relate shell displacements and forces. Inversion to the axial domain is accomplished via a state-space formulation that is evaluated using residue integration. The resulting expression for the Green's function for each circumferential harmonic is a summation over the natural waves of the shell. The finite element discretization in the radial direction allows the approach to be used for arbitrarily thick shells. The approach is benchmarked to results from an isotropic shell and numerical examples are given for a shell composed of a fiber-reinforced material. The numerical examples illustrate the effect of fiber orientation on the Green's function.

Improved bounds for Stein's square functions

We prove a weighted norm inequality for the maximal Bochner--Riesz operator and the associated square-function. This yields new $L^p(R^d)$ bounds on classes of radial Fourier multipliers for $p\ge 2+4/d$ with $d\ge 2$, as well as space-time regularity results for the wave and Schr\"odinger equations.

Automated and adaptable quantification of cellular alignment from microscopic images for tissue engineering applications.

Cellular alignment plays a critical role in functional, physical, and biological characteristics of many tissue types, such as muscle, tendon, nerve, and cornea. Current efforts toward regeneration of these tissues include replicating the cellular microenvironment by developing biomaterials that facilitate cellular alignment. To assess the functional effectiveness of the engineered microenvironments, one essential criterion is quantification of cellular alignment. Therefore, there is a need for rapid, accurate, and adaptable methodologies to quantify cellular alignment for tissue engineering applications. To address this need, we developed an automated method, binarization-based extraction of alignment score (BEAS), to determine cell orientation distribution in a wide variety of microscopic images. This method combines a sequenced application of median and band-pass filters, locally adaptive thresholding approaches and image processing techniques. Cellular alignment score is obtained by applying a robust scoring algorithm to the orientation distribution. We validated the BEAS method by comparing the results with the existing approaches reported in literature (i.e., manual, radial fast Fourier transform-radial sum, and gradient based approaches). Validation results indicated that the BEAS method resulted in statistically comparable alignment scores with the manual method (coefficient of determination R(2)=0.92). Therefore, the BEAS method introduced in this study could enable accurate, convenient, and adaptable evaluation of engineered tissue constructs and biomaterials in terms of cellular alignment and organization.

Floquet analysis of two-dimensional perturbed Keplerian flows in cataclysmic variables

<i>Aims. <i/>We apply Floquet analysis to study the dynamical stability of two-dimensional axisymmetric Keplerian flows distorted by an <i>m<i/> = 2 epicyclic periodic forcing term, representing a tidal perturbation in a thin accretion disk (AD) of a cataclysmic variable (CV). This tidal mode is responsible for the disk truncation mechanism.<i>Methods. <i/>Floquet analysis reveals, with a positive Floquet exponent, exponentially growing modes of the local instabilities that can cause the disruption of the AD. The equations of the inviscid fluid used to model the AD describe an oversimplified two-dimensional tidally-distorted accretion disk. Viscosity and the other properties expected in a CV AD are taken from the Shakura-Sunyaev <i>α<i/>-disk solution and introduced in the model through the thermodynamical quantities present in the equations. In particular, we address this investigation in search of parametric resonances of the radial Fourier components of the velocity field in the multi-dimensional parameter space of the solutions describing a fluid stream in a CV AD: the mass of the primary <i>M<i/><sub>1<sub/>, the orbital period <i>p<i/> of the system, the orbital fiducial radius <i>r<i/> of the fluid stream and the normalized Fourier mode of the radial motion, <i>k<i/>.<i>Results. <i/>We find that, in region of the <i>r<i/>3:2 orbital resonance radius, where superhumps (SH) form, the <i>m<i/> = 2 tidal epicyclic modes alone would not introduce any perturbation in this region of the AD, and therefore the Floquet exponent of the <i>m<i/> = 2 tidal force is found to be either negative or zero. However, only by introducing the 3:1 tidal perturbation, will the dynamics of the fluid streams at the SH radius become unstable indicating the onset of SH. Finally, Floquet analysis of the Fourier decomposition of the derivative along the radial <i>x<i/>-direction shows the important role of the low Fourier modes of the radial velocity perturbations in the disk truncation mechanism.

On Radial and Conical Fourier Multipliers

We investigate connections between radial Fourier multipliers on ℝ d and certain conical Fourier multipliers on ℝd+1. As an application we obtain a new weak type endpoint bound for the Bochner–Riesz multipliers associated with the light cone in ℝd+1, where d≥4, and results on characterizations of L p →L p,ν inequalities for convolutions with radial kernels.

A comparative image analysis of discrete radial Fourier transforms

Fourier transform is not often applied to pattern and object recognition, because of its inefficiency in image description and calculation. In order to solve this problem, a bi-discrete radial cosine transform is proposed. This new transform has two superior properties. Firstly, by using Mukundan’s square-to-circular transformation, it projects images in square to images in discrete circle girds. Secondly, different from the classical radial orthogonal plane, the new orthogonal plane is composed of discrete cosine functions orthogonal in both radial and circumferential directions. Experiment results indicate that the proposed bi-discrete radial cosine transform has an excellent performance in terms of efficient image reconstruction and computation.

SEMINUMERICAL MODEL TO STUDY TEMPERATURE DISTRIBUTION IN PERIPHERAL LAYERS OF ELLIPTICAL AND TAPERED SHAPED HUMAN LIMBS

In this article, a seminumerical approach has been developed to study temperature distribution in peripheral layers of tapered shaped human limbs, which are elliptical in shape. The model is three-dimensional which incorporates the important biophysical parameters such as blood mass flow rate, thermal conductivity and rate of metabolic heat generation. Appropriate boundary conditions have been framed using biophysical conditions. The finite element method has been employed along radial and angular directions and Fourier series method along axial direction to obtain temperature profiles in the region. The results have been used to study relationships among various physical and physiological parameters. MATLAB 7.0 has been used to simulate the model and obtain numerical results.

Reference free structure determination through eigenvectors of center of mass operators

Recovering the three-dimensional structure of molecules is important for understanding their functionality. We describe a spectral graph algorithm for reconstructing the three-dimensional structure of molecules from their cryo-electron microscopy images taken at random unknown orientations. We first identify a one-to-one correspondence between radial lines in three-dimensional Fourier space of the molecule and points on the unit sphere. The problem is then reduced to determining the coordinates of points on the sphere given a subset of their pairwise geodesic distances. To recover those coordinates, we exploit the special geometry of the problem, as rendered by the Fourier projection–slice theorem, to construct a weighted graph whose vertices are the radial Fourier lines and whose edges are linked using the common line property. The graph organizes the radial lines on the sphere in a global manner that reveals the acquisition direction of each image. This organization is derived from a global computation of a few eigenvectors of the graph's sparse adjacency matrix. Once the directions are obtained, the molecule can be reconstructed using classical tomography methods. The presented algorithm is direct (as opposed to iterative refinement schemes), does not require any prior model for the reconstructed object, and is shown to have favorable computational and numerical properties. Moreover, the algorithm does not impose any assumption on the distribution of the projection orientations. Physically, this means that the algorithm is applicable to molecules that have unknown spatial preference.

Adaptive backstepping dynamic surface control for systems with periodic disturbances using neural networks

This paper addresses the adaptive neural network tracking control problem for a class of strict-feedback systems with unknown non-linearly parameterised and time-varying disturbed function of known periods. Radial basis function neural network and Fourier series expansion are combined into a new function approximator to model each suitable disturbed function in systems. Dynamic surface control approach is used to solve the problem of ‘explosion of complexity’ in backstepping design procedure. The uniform boundedness of all closed-loop signals is guaranteed. The tracking error is proved to converge to a small residual set around the origin. A simulation example is provided to illustrate the effectiveness of the control scheme designed.

An application of one-sided Jacobi polynomials for spectral modeling of vector fields in polar coordinates

A spectral tau-method is proposed for solving vector field equations defined in polar coordinates. The method employs one-sided Jacobi polynomials as radial expansion functions and Fourier exponentials as azimuthal expansion functions. All the regularity requirements of the vector field at the origin and the physical boundary conditions at a circumferential boundary are exactly satisfied by adjusting the additional tau-coefficients of the radial expansion polynomials of the highest order. The proposed method is applied to linear and nonlinear-dispersive time evolution equations of hyperbolic-type describing internal Kelvin and Poincaré waves in a shallow, stratified lake on a rotating plane.

Inherent length‐scales of periodic solar wind number density structures

We present an analysis of the radial length‐scales of periodic solar wind number density structures. We converted 11 years (1995–2005) of solar wind number density data into radial length series segments and Fourier analyzed them to identify all spectral peaks with radial wavelengths between 72 (116) and 900 (900) Mm for slow (fast) wind intervals. Our window length for the spectral analysis was 9072 Mm, approximately equivalent to 7 (4) h of data for the slow (fast) solar wind. We required that spectral peaks pass both an amplitude test and a harmonic F‐test at the 95% confidence level simultaneously. From the occurrence distributions of these spectral peaks for slow and fast wind, we find that periodic number density structures occur more often at certain radial length‐scales than at others, and are consistently observed within each speed range over most of the 11‐year interval. For the slow wind, those length‐scales are L ∼ 73, 120, 136, and 180 Mm. For the fast wind, those length‐scales are L ∼ 187, 270 and 400 Mm. The results argue for the existence of inherent radial length‐scales in the solar wind number density.

Characterizations of Hankel multipliers

We give characterizations of radial Fourier multipliers as acting on radial L p functions, 1 < p < 2d/(d + 1), in terms of Lebesgue space norms for Fourier localized pieces of the convolution kernel. This is a special case of corresponding results for general Hankel multipliers. Besides L p − L q bounds we also characte- rize weak type inequalities and intermediate inequalities involving Lorentz spaces. Applications include results on interpolation of multiplier spaces.

An Evaluation of Image Sampling and Compression for Human Iris Recognition

The resilience of identity verification systems to subsampling and compression of human iris images is investigated for three high-performance iris-matching algorithms. For evaluation, 2156 images from 308 irises from the extended Chinese Academy of Sciences Institute of Automation database were mapped into a rectangular format with 512 pixels circumferentially and 80 radially. For identity verification, the 48 rows that were closest to the pupil were taken and images were resized by subsampling their Fourier coefficients. Negligible degradation in verification is observed if at least 171 circumferential and 16 radial Fourier coefficients are preserved, which would correspond to sampling the polar image at 342 times 32 pixels. With JPEG2000 compression, improved matching performance is observed down to 0.3 b/pixel (bpp), attributed to noise reduction without a significant loss of texture. To ensure that the iris-matching algorithms studied are not degraded by image compression, it is recommended that normalized iris images should be exchanged at 512 times 80 pixel resolution, compressed by JPEG 2000 to 0.5 bpp. This achieves a smaller file size than the ANSI/INCITS 379-2004 iris image interchange format.

Direct numerical simulation of supercritical annular electroconvection

We use direct numerical simulation to study electrically driven convection in an annular thin film. The simulation models a laboratory experiment that consists of a weakly conducting, submicron thick liquid crystal film suspended between two concentric electrodes. The film is driven to convect by imposing a sufficiently large voltage across it. The flow is driven by a surface charge density inversion which is unstable to the imposed electrical force. This mechanism is closely analogous to the mass density inversion which is unstable to the buoyancy force in conventional, thermally driven Rayleigh-Bénard convection. The simulation uses a pseudospectral method with Chebyshev polynomials in the radial direction and Fourier modes in the azimuthal direction. The numerical results, which are in good agreement with previous experimental data and theoretical predictions, reveal several insights. The mode competition near a codimension-two point exhibits hysteresis. The primary bifurcation is supercritical for a broad range of fluid and geometrical parameters.

Generic orthogonal moments: Jacobi–Fourier moments for invariant image description

A multi-distorted invariant orthogonal moments, Jacobi–Fourier Moments (JFM), were proposed. The integral kernel of the moments was composed of radial Jacobi polynomial and angular Fourier complex componential factor. The variation of two parameters in Jacobi polynomial, α and β , can form various types of orthogonal moments: Legendre–Fourier Moments ( α = 1 , β = 1 ) ; Chebyshev–Fourier Moments ( α = 2 , β = 3 2 ) ; Orthogonal Fourier–Mellin Moments ( α = 2 , β = 2 ) ; Zernike Moments and pseudo-Zernike Moments, and so on. Therefore, Jacobi–Fourier Moments are generic expressions of orthogonal moments formed by a radial orthogonal polynomial and angular Fourier complex component factor, providing a common mathematical tool for performance analysis of the orthogonal moments. In the paper, Jacobi–Fourier Moments were calculated for a deterministic image, and the original image was reconstructed with the moments. The relationship between Jacobi–Fourier Moments and other orthogonal moments was studied. Theoretical analysis and experimental investigation were conducted in terms of the description performance and noise sensibility of the JFM.

Fourier Dissection of Early-Type Galaxy Bars

This paper reports on a near-infrared survey of early-type galaxies designed to provide information on bar strengths, bulges, disks, and bar parameters in a statistically well-defined sample of S0-Sa galaxies. Early-type galaxies have the advantage that their bars are relatively free of the effects of dust, star formation, and spiral structure that complicate bar studies in later type galaxies. We describe the survey and present results on a detailed analysis of the relative Fourier intensity amplitudes of bars in 26 early-type galaxies. We also evaluate the symmetry assumption of these amplitudes with radius, used recently for bar-spiral separation in later type galaxies. The results show a wide variety of radial Fourier profiles of bars, ranging from simple symmetric profiles that can be represented in terms of a single Gaussian component to both symmetric and asymmetric profiles that can be represented by two overlapping Gaussian components. More complicated profiles than these are also found, often due to multiple barlike features including extended ovals or lenses. Based on the gravitational bar torque indicator Qb, double-Gaussian bars are stronger on average than single-Gaussian bars, at least for our small sample. We show that published numerical simulations in which the bar transfers a large amount of angular momentum to the halo can account for many of the observed profiles. The range of possibilities encountered in models seems well-represented in the observed systems.

AEGIS: An adaptive ideal-magnetohydrodynamics shooting code for axisymmetric plasma stability

A new linear ideal-magnetohydrodynamics stability code for axisymmetric plasmas, AEGIS, is described. The AEGIS code employs adaptive shooting in the radial direction and Fourier decomposition in the poloidal direction. The general solution is a linear combination of the independent solutions of the Euler–Lagrange equations solved by the adaptive shooting. A multiple-region matching technique is used to overcome the numerical difficulty associated with the stiff nature of the independent solutions. Benchmarks with other MHD codes show good agreement. Because it is adaptive, the AEGIS code has very good resolution near the singular surfaces of MHD modes. AEGIS has the additional advantage of allowing the investigation of modes with not only low mode numbers, but also intermediate to high mode numbers.

Harmonic Finite Element Analysis for Anisotropic Viscoelasticity

A numerical methodology for predicting the linear anisotropic viscoelastic behavior of bodies that can be uniquely described in cylindrical coordinates is devised. The methodology is an extension of the Fourier-finite element method and is based on a combination of finite element discretization in the radial and longitudinal directions and Fourier decomposition in the angular direction. The proposed method is capable of handling inhomogeneous polar-orthotropic viscoelastic material properties. In general, the method provides dimensional reduction in the finite element formulation, which leads to a reduction in the complexity of the numerical model including the meshing process. Overall, the method is shown to be competent mainly because of the efficient exploitation of the angular description of the geometry and the properties distribution. A numerical example is given for the analysis of an orthotropic viscoelastic ring under asymmetric shear loading.

Fourier Integral Operators on Noncompact Symmetric Spaces of Real Rank One

Let X=G/K be a noncompact symmetric space of real rank one. The purpose of this paper is to investigate Lp boundedness properties of a certain class of radial Fourier integral operators on the space X. We will prove that if uτ is the solution at some fixed time τ of the natural wave equation on X with initial data f and g and 1<p<∞, then ‖uτ‖Lp(X)⩽Cp(τ)(‖f‖Lpbp(X)+(1+τ)‖g‖Lpbp−1(X)). We will obtain both the precise behavior of the norm Cp(τ) and the sharp regularity assumptions on the functions f and g (i.e., the exponent bp) that make this inequality possible. In the second part of the paper we deal with the analog of E. M. Stein's maximal spherical averages and prove exponential decay estimates (of a highly non-euclidean nature) on the Lp norm of supT⩽τ⩽T+1|f*dστ(z)|, where dστ is a normalized spherical measure.