Dimensional Fourier Research Articles

GENERALIZED ANALYTIC FUNCTIONS ON THE COUNTABLE PRODUCT AND DIRECT SUM OF THE COMPLEX NUMBERS

A classical result states that, in n variables, the space of the entire functionals can be identified with the space of exponential type functions via the Fourier–Borel transform. Thus, in this way the spaces of the entire and exponential type functions can be put in duality, the Martineau duality. We give a proof that the entire functionals, on the countable direct product and direct sum of the field of complex numbers, can be identified with exponential type functions in the same way. In other words, we show that the infinite dimensional Fourier–Borel transform defines Martineau dualities analogous to the finite dimensional case.

One‐Dimensional Fourier Imaging and k‐Space

AbstractMagnetic resonance imaging offers the possibility to obtain spatially resolved anatomical information. This is accomplished by taking advantage of the Larmor relationship, which dictates that the frequency of the spins depends on the local magnetic field. This unit discusses the one dimensional Fourier imaging based on this relation. The one‐to‐one mapping of the signal from a given frequency to a given spatial location is explained. The image reconstruction based on well‐known Fourier transform reconstruction method is described in detail. The Fourier transform takes the MR signal as acquired in the time domain (usually referred to as the k‐space domain) and converts it to the frequency domain where 1‐D spatially resolved information can be obtained.

Relating multipliers and transplantation for Fourier-Bessel expansions and Hankel transform

Proved are transference results that show connections between: a) multipliers for the Fourier-Bessel series and multipliers for the Hankel transform; b) maximal operators defined by Fourier-Bessel multipliers and maximal operators given by Hankel transform multipliers; c) Fourier-Bessel transplantation and Hankel transform transplantation. In some way the connections described in a) and b) can be seen as multi-dimensional extensions of the classical results of Igari, and Kenig and Tomas for the one dimensional Fourier transform. We prove our results for the non-modified Hankel transform in the power weight setting, and this allows to translate them also to the context of the modified Hankel transform. Together with Gilbert's transplantation theorem, our transference shows that harmonic analysis results for the Hankel transform of arbitrary order are consequences of corresponding results for the cosine expansions.

Estimation of Reflection Coefficients for an Open-Ended Coaxial Waveguide Radiating Into a Chiral Half-Space

A spectral-domain moment method solution is presented for estimation of the reflection coefficients of an open-ended coaxial waveguide radiating into a chiral half-space. The electromagnetic fields in the waveguide region are expanded in the normal waveguide modes, while those in the half-space are formulated in terms of plane wave spectrum using two dimensional Fourier transform. A spectral domain admittance matrix is derived through which an electric field integral equation can be obtained by matching the tangential field components on the open end interface. This integral equation is solved with the Galerkin method. The coefficients of the matrix equation are reduced to one-dimensional integrals to facilitate numerical calculation. The problem is different from the case of isotropic non-chiral medium in that chirality causes excitation of TE modes, whose effect is accounted for rigorously in the formulation. This is in contrast to plane wave reflected from a chiral media at normal incidence, where no depolarization effect can be observed. Numerical results are presented to illustrate the idea that the reflectivity of TEM mode carries information on the presence of chirality. The analysis has a potential application in developing an open-ended waveguide method for the characterization of chiral materials using only reflection measurements including depolarization effects.

Acoustic wave prediction in flowing steam–water two-phase mixture

The transient two-fluid model has been used to develop a general relation for acoustic waves with steam–water two-phase mixture in one-dimensional flowing system. The analysis is valid in principle over the whole void fraction region. Both the mechanical and thermal nonequilibrium are considered. The vapor–liquid interface heat flux is derived from the one dimensional Fourier heat conduction equation to evaluate the interphase evaporation or condensation rate. Calculations are performed for pressures from 0.07 to 16.0 MPa, void fractions from 0.0 to 1.0. Higher frequency limit of sonic velocities are only relied on the pressures and void fractions, and are insensitive to the bubble radius, tube diameter, and the velocity difference between the two phases. The predicted sonic velocities are compared with some experimental data for low pressures. Good agreement has been achieved between the predictions and the experimental data.

A Structural Analysis Method for Graphite Intercalation Compounds

ABSTRACTStudy of the microstructure of electronic materials can be enhanced by using high resolution transmission electron microscopy (TEM) combined with the technique of digitized image analysis. We show here a practical image analysis method for the microstructures of acceptor graphite intercalation compounds (GICs) with CuCl2 and FeCl3 intercalates. The two dimensional fast Fourier transform (2D-FFT) was used for the frequency analysis of the TEM pictures. It is found that the lattice images of CuCl2-GICs consist of different frequency images corresponding to specific frequencies. The detailed features of the stage-1 structure of the FeCl3-GICs is extracted quantitatively by this method from a relatively indistinct TEM picture. The stage structure of the CuCl2- and FeCl3-GICs are further investigated by analyzing the reconstruction of the TEM images by means of the two dimensional inverse FFT (2D-IFFT).

A discrete wave number transform approach for predicting pressure levels due to a line source radiating over two dimensional multi-layered materials

The problem of an acoustical line source radiating over a planar multi-layered material infinite in two of its dimensions, made up of acoustic, elastic and porous media is investigated. The fields in the different media are expanded into a superposition of plane waves using a one dimensional spatial Fourier transform along the material's interface. The problem amounts to solving for the reflection and transmission coefficients of the plane waves at each interface. The contributions of each wave are then recombined using the discrete wave number method, referred to as Bouchon's method, to get the fields in the media. The results given by the developed algorithm are successfully compared to an analytical solution considered for propagation above a locally reacting plane. ©

Precision geoid determination by spherical FFT in and around the Korean peninsula

This paper deals with the precision geoid determination by a gravimetric solution in and around the Korean peninsula. A number of data files were compiled for this work, containing now more than 69,900 point gravity data on land and ocean areas. The EGM96 global geopotential model to degree 360 was used in order to determine the long wavelength effect of the geoid surface. By applying the remove-restore technique the geoid undulations were determined by combining a geopotential model, mean free-air gravity anomalies and height in a Digital Elevation Model (DEM). Computation involves a spherical approximation to conduct the Stokes’ integration by a two dimensional spherical Fast Fourier Transform (FFT) with 100% zero-padding. A terrain correction was also computed by FFT with a spherical approximation of the Residual Terrain Model (RTM) terrain correction integration. Accuracy estimates are given for absolute geoid undulations using 78 GPS/Leveling stations. The comparative evaluation gives the bias of 0.187 meters and standard deviation of 0.28 meters, respectively. The relative accuracy achieved was of the order of 3.1 ppm for baselines between 10 and 350 kilometers.

Microwave diversity imaging using six-port reflectometer

A microwave diversity imaging system conventionally uses a vector network analyzer (VNA) to directly measure the object scattered field (amplitude and phase) over a selected frequency range and viewing angles, then reconstructs the scattering object characteristic function through two dimensional Fourier inversion. In this paper, we present a cost-effective microwave diversity imaging system using a six-port reflectometer, which measures four amplitude (or power) values to acquire the object scattered field indirectly. One can then eliminate the coherent detectors in a VNA. The calibration procedure for this microwave diversity imaging measurement is also described. Experimental results of three types of scattering objects, a metallic cylinder, four distributed line scatterers, and a 72:1 scaled B-52 aircraft model, are presented using the described six-port microwave imaging system.

Motion artifacts reduction in DWI using navigator echoes: a robust and simple correction scheme.

NMR signal phase variation caused by macroscopic motion of an object during application of the diffusion gradient is a well-known effect in diffusion-weighted imaging (DWI) using the standard pulsed gradient spin-echo sequence (PGSE). This phase error causes severe ghost artifacts in the output image when phase encoding techniques, such as two dimensional Fourier transform (2DFT) imaging, are used. One possible way to eliminate the motion effects is the navigator echo technique. The method is based on estimating the phase error from the navigator echo and using it for the correction of the image echo. The phase errors (zero and first order) for the phase correction of the image echo are usually evaluated from the navigator echo after Fourier transform (FT) in the readout direction, correcting for both translation and rotation. We present here a simple algorithm which enables evaluation and correction in the time domain of phase errors induced by motion. This approach has the advantage of improved correction of motional artifacts and minimized sensitivity to noise and inaccurate setting up of the experiment.

Quantifying SEM Resolution and Performance

The ability to reliably and quickly determine the resolution of the scanning electron microscope (SEM) is highly desirable becauseonly then can the level of performance which is being achieved be compared to that which was anticipated or specified. Further, in activities such as automated line width measurements, or in telemicroscopy, the ability to quantify the imaging performance provides the remote operator with the necessary information to monitor, document, and optimize the instrument set-up. However, compared to the equivalent task on the transmission electron microscope (TEM) an analysis of SEM performance is hampered by poor signal to noise ratios, and the absence of ideal ‘white noise’ samples.The two dimensional Fourier power spectrum of an image provides a readout of the frequency content of the information present in the micrograph. From an analysis of the power spectrum the highest spatial frequency transferred in the image can then be determined and hence the resolution of the micrograph can be specified.

Two dimensional discrete fractional Fourier transform

Fractional Fourier transform (FRFT) performs a rotation of signals in the time–frequency plane, and it has many theories and applications in time-varying signal analysis. Because of the importance of fractional Fourier transform, the implementation of discrete fractional Fourier transform will be an important issue. Recently, a discrete fractional Fourier transform (DFRFT) with discrete Hermite eigenvectors has been proposed, and it can provide similar results to match the continuous outputs. On the other hand, the two dimensional continuous fractional Fourier transform is also proposed for 2D signal analysis. This paper develops a 2D DFRFT which can preserve the rotation properties and provide similar results to continuous FRFT.

Scanning Tunnelling Microscopy Study of the Growth Mechanism for Hydrogenated Amorphous Silicon Produced by Plasma Enhanced Chemical Vapour Deposition

ABSTRACTThin films of hydrogenated amorphous silicon (a-Si:H) have been deposited by plasma-enhanced chemical vapour deposition (PECVD), and the resulting topography measured in-situ on a nanometre scale using a scanning tunnelling microscope (STM). An island structure is observed on the surface of device quality a-Si:H, which can be quantitatively analysed using a one dimensional Fourier transform of the topography. Results suggest that deposition is limited by the creation of dangling bonds on the a-Si:H surface and not by the surface transport of SiH 3 radicals at the deposition temperature (598 K). Island nucleation takes place through the abstraction of hydrogen atoms from the a-Si:H surface by plasma etching and the subsequent attachment of an SiH 3 radicals to the available sites. A thermally activated hydrogen effusion process around the edge of each island, where the step edge causes a high local hydrogen concentration, then creates further dangling bonds which allow the islands to grow. A simulation has been constructed, which confirms this two stage mechanism.

Design principles and applications of a cooled CCD camera for electron microscopy.

Cooled CCD cameras offer a number of advantages in recording electron microscope images with CCDs rather than film which include: immediate availability of the image in a digital format suitable for further computer processing, high dynamic range, excellent linearity and a high detective quantum efficiency for recording electrons. In one important respect however, film has superior properties: the spatial resolution of CCD detectors tested so far (in terms of point spread function or modulation transfer function) are inferior to film and a great deal of our effort has been spent in designing detectors with improved spatial resolution. Various instrumental contributions to spatial resolution have been analysed and in this paper we discuss the contribution of the phosphor-fibre optics system in this measurement. We have evaluated the performance of a number of detector components and parameters, e.g. different phosphors (and a scintillator), optical coupling with lens or fibre optics with various demagnification factors, to improve the detector performance. The camera described in this paper, which is based on this analysis, uses a tapered fibre optics coupling between the phosphor and the CCD and is installed on a Philips CM12 electron microscope equipped to perform cryo-microscopy. The main use of the camera so far has been in recording electron diffraction patterns from two dimensional crystals of bacteriorhodopsin--from wild type and from different trapped states during the photocycle. As one example of the type of data obtained with the CCD camera a two dimensional Fourier projection map from the trapped O-state is also included. With faster computers, it will soon be possible to undertake this type of work on an on-line basis. Also, with improvements in detector size and resolution, CCD detectors, already ideal for diffraction, will be able to compete with film in the recording of high resolution images.

SYNTHETIC APERTURE TECHNIC IN ASTRONOMY USING SLIT APERTURE TELESCOPE

The interest in a Rotating Slit-Aperture Telescope (RSAT) among other synthetic aperture telescopes is its capability of being easily coupled with a spectrograph, in order to give reconstructed images of an astronomical object as a function of the light wavelength. Each colored image is comparable with the others for fruitful astrophysical applications. The principle of image reconstruction is well known: it consists of the inversion of the set of projections (Radon transform) given by the telescope during its rotation around its optical axis. A full coverage of the two dimensional Fourier plane can be obtained by rotating the SAT. This problem has led to intense developments for medical imaging (tomography). One of the main difficulties in the reconstruction process in space may come from the jitter of the rotation axis of the RSAT. A set of projections uncorrected for this jitter produces very fuzzy reconstructed images. An elegant solution to the necessary phasing between successive projections is proposed which makes use of a small auxilliary telescope, and some numerical simulations are presented.

Adaptive segmentation of MRI data

Intensity-based classification of MR images has proven problematic, even when advanced techniques are used. Intrascan and interscan intensity inhomogeneities are a common source of difficulty. While reported methods have had some success in correcting intrascan inhomogeneities, such methods require supervision for the individual scan. This paper describes a new method called adaptive segmentation that uses knowledge of tissue intensity properties and intensity inhomogeneities to correct and segment MR images. Use of the expectation-maximization (EM) algorithm leads to a method that allows for more accurate segmentation of tissue types as well as better visualization of magnetic resonance imaging (MRI) data, that has proven to be effective in a study that includes more than 1000 brain scans. Implementation and results are described for segmenting the brain in the following types of images: axial (dual-echo spin-echo), coronal [three dimensional Fourier transform (3-DFT) gradient-echo T1-weighted] all using a conventional head coil, and a sagittal section acquired using a surface coil. The accuracy of adaptive segmentation was found to be comparable with manual segmentation, and closer to manual segmentation than supervised multivariant classification while segmenting gray and white matter.

Zooming, a Practical Strategy for Improving the Quality of Multidimensional NMR Spectra

Abstract A method has been developed, which is termed “zooming,” that provides high digital resolution in any region selected from an n -dimensional data set. The region selected can be of arbitrary size (any number of data points). In the zooming approach as implemented in the “ZOOM” computer program, the desired region of the frequency-domain spectrum is first extracted. Next, a “pseudo-FID” for the selected region is constructed by n -dimensional inverse Fourier transformation. The resulting time-domain data block is zero filled in n -dimensions and finally Fourier transformed back to the frequency domain. The basic principles behind the zooming approach are described, and its application to the analysis of 2D and 3D NMR data from proteins is demonstrated. Zooming is expected to be of even greater importance in the analysis of higher-dimensional NMR data where problems of acquisition and processing times and data storage space become more pronounced. The implementation described here is for spectra consisting of real or absolute-value data; at least one zero fill must have been applied in each dimension during the initial processing of the data.

Comparisons of spectral techniques for geoid computations over large regions

The Stokes formula is efficiently evaluated by the one-and two- dimensional (1D, 2D) fast Fourier transform (FFT) technique in the plane and on the sphere in order to obtain precise geoid determinatiover a large area such as Europe. Using a high-pass filtered spherical harmonic reference model (OSU91A truncated to different degrees), gridded gravity anomalies and geoid heights were produced and the anomalies were used as input in the FFT software. Various tests were performed with respect to the different kernel functions used, to the spherical computations in bands, as well as to windowing, edge effects and extent of the area. It is thus demonstrated that, in geoid computations over large regions, the 1D spherical FFT and the 2D multiband spherical FFT in combination with discrete spectra for the kernel functions and 100% zero-padding give better results than those obtained by the other transform techniques. Additionally, numerical tests were carried out at the same test area using the planar fast Hartley transform (FHT) instead of the FFT and the results obtained by the two attractive alternatives were compared regarding the requirements in both computer time and computer memory needed in geoid height computations.

Characterization of thin Polymeric Nanofoam films by Transmission Electron Microscopy and Small Angle Neutron Scattering

ABSTRACTThin film polymer nanofoams are produced from triblock copolymers of a fluorinated polyimide, 3F/PMDA (derived from pyromelletic dianhydride (PMDA) and 1,1-bis(4-aminophenyl)-1-phenyl-2,2,2-trifluoroethane (3F)) as the center block and polypropylene oxide (PO) as the end blocks. The nanofoam is produced using a three step process: 1.) spin casting the triblock copolymer onto a silicon substrate, 2.) thermal treatment in an Argon atmosphere to imidize the center block and 3.) thermal treatment in air to degrade the PO domains and form nanoscale voids. This process was characterized using both transmission electron microscopy (TEM) and small angle neutron scattering (SANS). For the TEM studies, Ruthenium tetroxide staining was used to enhance the contrast between the polyimide (PI) matrix and the PO microdomains or voids, which permitted a more detailed view of the microstructure of both the foamed and unfoamed materials. From the two dimensional Fourier transform of the micrographs the spatial correlation between the PO microdomains in the unfoamed material and between the voids in the foam were found. An interdomain separation distance of -37 nm was observed. SANS was performed to follow the imidization and foaming processes both in-situ and on a Si substrate. The SANS results indicated that the films are homogeneous when spun from solution and that the microphase separation of the PO domains occurs during the imidization step. The subsequent foaming step leaves the morphology generally intact with the PO domains converting to voids. A peak was found in the SANS curve with a spacing of about 26 nm, which is qualitative agreement with the TEM data.

SNR gains with superconducting coils and hyperpolarized inert gas

Magnetic resonance microscopy (MRM) has advanced from a technical challenge to a practical tool in a wide range of basic sciences (1). The nondestructive nature of the technique allows for repeated studies of the same sample, retrospective studies through any arbitrary plane, registered studies using different contrast mechanisms, and examination of valuable specimens. "Proton contrast" is provided by making use of T1, T2, T1ρ , diffusion, and magnetization contrast weightings, thus permitting direct examination of the state of water in tissues; something not possible with other microscopic techniques. The technical hurdles imposed by a limited signal-to-noise ratio (SNR) have been partially overcome by improved RF coil design and by three dimensional Fourier transform encoding for large arrays. We have introduced two additional technologies that help to further redress this SNR deficit: superconducting RF coils and inert gas hyperpolarization.It has been argued elsewhere (2,3) that the dominant source of noise in the receiving chain for small objects in high fields is the pickup coil and the preamplifier.