Showing papers on "Fourier transform published in 2005"

Seismic Ray Theory

Abstract: Preface 1. Introduction 2. The elastodynamics and its simple solutions 3. Seismic rays and travel times 4. Dynamic ray tracing. Paraxial ray methods 5. Ray amplitudes 6. Ray synthetic seismograms Appendix. Fourier transform, Hilbert transform and analytical signals References Index.

Tuned Lamb-Wave Excitation and Detection with Piezoelectric Wafer Active Sensors for Structural Health Monitoring

Abstract: The capability of embedded piezoelectric wafer active sensors (PWAS) to excite and detect tuned Lamb waves for structural health monitoring is explored. First, a brief review of Lamb waves theory is presented. Second, the PWAS operating principles and their structural coupling through a thin adhesive layer are analyzed. Then, a model of the Lamb waves tuning mechanism with PWAS transducers is described. The model uses the space domain Fourier transform. The analysis is performed in the wavenumber space. The inverse Fourier transform is used to return into the physical space. The integrals are evaluated with the residues theorem. A general solution is obtained for a generic expression of the interface shear stress distribution. The general solution is reduced to a closed-form expression for the case of ideal bonding which admits a closed-form Fourier transform of the interfacial shear stress. It is shown that the strain wave response varies like sin a, whereas the displacement response varies like sinc a. ...

Hilbert-Huang transform and its applications

Abstract: The Hilbert-Huang Transform (HHT) represents a desperate attempt to break the suffocating hold on the field of data analysis by the twin assumptions of linearity and stationarity. Unlike spectrograms, wavelet analysis, or the Wigner-Ville Distribution, HHT is truly a time-frequency analysis, but it does not require an a priori functional basis and, therefore, the convolution computation of frequency. The method provides a magnifying glass to examine the data, and also offers a different view of data from nonlinear processes, with the results no longer shackled by spurious harmonics — the artifacts of imposing a linearity property on a nonlinear system or of limiting by the uncertainty principle, and a consequence of Fourier transform pairs in data analysis. This is the first HHT book containing papers covering a wide variety of interests. The chapters are divided into mathematical aspects and applications, with the applications further grouped into geophysics, structural safety and visualization.

Understanding NMR Spectroscopy

Abstract: 1. What this book is about and who should read it. 2. Setting the scene. 3. Energy levels and NMR spectra. 4. The vector model. 5. Fourier transformation and data processing. 6. The quantum mechanics of one spin. 7. Product operators. 8. Two-dimensional NMR. 9. Relaxation and the NOE. 10. Advanced topics in tow-dimensional NMR. 11. Coherence selection: phase cycling and field gradient pulses. 12. How the spectrometer works. Appendix: Some mathematical topics. Index.

Fourier slice photography

Abstract: This paper contributes to the theory of photograph formation from light fields. The main result is a theorem that, in the Fourier domain, a photograph formed by a full lens aperture is a 2D slice in the 4D light field. Photographs focused at different depths correspond to slices at different trajectories in the 4D space. The paper demonstrates the utility of this theorem in two different ways. First, the theorem is used to analyze the performance of digital refocusing, where one computes photographs focused at different depths from a single light field. The analysis shows in closed form that the sharpness of refocused photographs increases linearly with directional resolution. Second, the theorem yields a Fourier-domain algorithm for digital refocusing, where we extract the appropriate 2D slice of the light field's Fourier transform, and perform an inverse 2D Fourier transform. This method is faster than previous approaches.

Spectral decomposition of seismic data with continuous-wavelet transform

Abstract: This paper presents a new methodology for computing a time-frequency map for nonstationary signals using the continuous-wavelet transform (CWT). The conventional method of producing a time-frequency map using the short time Fourier transform (STFT) limits time-frequency resolution by a predefined window length. In contrast, the CWT method does not require preselecting a window length and does not have a fixed time-frequency resolution over the timefrequency space. CWT uses dilation and translation of a wavelet to produce a time-scale map. A single scale encompasses a frequency band and is inversely proportional to the time support of the dilated wavelet. Previous workers have converted a time-scale map into a time-frequency map by taking the center frequencies of each scale. We transform the time-scale map by taking the Fourier transform of the inverse CWT to produce a time-frequency map. Thus, a time-scale map is converted into a time-frequency map in which the amplitudes of individual frequencies rather than frequency bands are represented. We refer to such a map as the time-frequency CWT (TFCWT). We validate our approach with a nonstationary synthetic example and compare the results with the STFT and a typical CWT spectrum. Two field examples illustrate that the TFCWT potentially can be used to detect frequency shadows caused by hydrocarbons and to identify subtle stratigraphic features for reservoir characterization.

Wavelet analysis of extended x-ray absorption fine structure data

Abstract: Fourier transform (FT) is a fundamental step for the data reduction and interpretation of extended x-ray absorption fine structure (EXAFS) spectra. The FT separates backscattering atoms by their radial distance from the absorbing atom (so-called shells). We suggest to routinely complement the FT by a wavelet transform (WT), which provides not only radial distance resolution, but resolves in $k$ space. This information eases the discrimination of atoms by their elemental nature, especially if these atoms are at the same distance. We present an in-depth analysis of the Morlet wavelet, which has specific advantages for EXAFS analysis, including the possibility to estimate Morlet parameter values optimized either for elemental or for spatial resolution. Using an experimental spectrum of $\mathrm{Zn}\text{\ensuremath{-}}\mathrm{Al}$ layered double hydroxide, we demonstrate the discrimination of $\mathrm{Al}$ and $\mathrm{Zn}$ at a similar crystallographic position, in spite of destructive interference substantially reducing signal information. Finally, the extension to multiple scattering paths leads to a deeper understanding of the resolution properties of the WT.

Gaussian split Ewald: A fast Ewald mesh method for molecular simulation.

Abstract: Gaussian split Ewald (GSE) is a versatile Ewald mesh method that is fast and accurate when used with both real-space and k-space Poisson solvers. While real-space methods are known to be asymptotically superior to k-space methods in terms of both computational cost and parallelization efficiency, k-space methods such as smooth particle-mesh Ewald (SPME) have thus far remained dominant because they have been more efficient than existing real-space methods for simulations of typical systems in the size range of current practical interest. Real-space GSE, however, is approximately a factor of 2 faster than previously described real-space Ewald methods for the level of force accuracy typically required in biomolecular simulations, and is competitive with leading k-space methods even for systems of moderate size. Alternatively, GSE may be combined with a k-space Poisson solver, providing a conveniently tunable k-space method that performs comparably to SPME. The GSE method follows naturally from a uniform framework that we introduce to concisely describe the differences between existing Ewald mesh methods.

A geometrical interpretation of large amplitude oscillatory shear response

Abstract: Although the stress of oscillatory shear flow can be decomposed into elastic and viscous parts in the linear regime, it is not yet known how to decompose the stress of a large amplitude oscillatory shear (LAOS) flow. We developed a method of analyzing LAOS data, which decomposes the stress into elastic and viscous components on the basis of a sound mathematical and physical foundation. This method is based on the symmetry of the stress and is a generalization of linear viscoelasticity from the viewpoint of geometry. The proposed method is more powerful than previous methods such as Fourier transform analysis and the Lissajous plot, in that it is more sensitive to the presence of nonlinearities and it is easier to determine nonlinear parameters.

Fourier Analysis in Convex Geometry

Abstract: Introduction Basic concepts Volume and the Fourier transform Intersection bodies The Busemann-Petty problem Intersection bodies and $L_p$-spaces Extremal sections of $\ell_q$-balls Projections and the Fourier transform Bibliography Index.

Biosensing using porous silicon double-layer interferometers: reflective interferometric Fourier transform spectroscopy.

Abstract: A simple, chip-based implementation of a double-beam interferometer that can separate biomolecules based on size and that can compensate for changes in matrix composition is introduced. The interferometric biosensor uses a double-layer of porous Si comprised of a top layer with large pores and a bottom layer with smaller pores. The structure is shown to provide an on-chip reference channel analogous to a double-beam spectrometer, but where the reference and sample compartments are stacked one on top of the other. The reflectivity spectrum of this structure displays a complicated interference pattern whose individual components can be resolved by fitting of the reflectivity data to a simple interference model or by fast Fourier transform (FFT). Shifts of the FFT peaks indicate biomolecule penetration into the different layers. The small molecule, sucrose, penetrates into both porous Si layers, whereas the large protein, bovine serum albumin (BSA), only enters the large pores. BSA can be detected even in a large (100-fold by mass) excess of sucrose from the FFT spectrum. Detection can be accomplished either by computing the weighted difference in the frequencies of two peaks or by computing the ratio of the intensities of two peaks in the FFT spectrum.

Instantaneous complex conjugate resolved spectral domain and swept-source OCT using 3x3 fiber couplers

Abstract: We report that the complex conjugate artifact in Fourier domain optical coherence tomography approaches (including spectral domain and swept source OCT) may be resolved by the use of novel interferometer designs based on 3x3 and higher order fiber couplers. Interferometers built from NxN (N>2) truly fused fiber couplers provide simultaneous access to non-complementary phase components of the complex interferometric signal. These phase components may be converted to quadrature components by trigonometric manipulation, then inverse Fourier transformed to obtain A-scans and images with resolved complex conjugate artifact. We demonstrate instantaneous complex conjugate resolved Fourier domain OCT using 3x3 couplers in both spectral domain and swept source implementations. Complex conjugate artifact suppression by factors of ~20dB and ~25dB are demonstrated for spectral domain and swept source implementations, respectively.

WARP: accurate retrieval of shapes using phase of Fourier descriptors and time warping distance

Abstract: Effective and efficient retrieval of similar shapes from large image databases is still a challenging problem in spite of the high relevance that shape information can have in describing image contents. We propose a novel Fourier-based approach, called WARP, for matching and retrieving similar shapes. The unique characteristics of WARP are the exploitation of the phase of Fourier coefficients and the use of the dynamic time warping (DTW) distance to compare shape descriptors. While phase information provides a more accurate description of object boundaries than using only the amplitude of Fourier coefficients, the DTW distance permits us to accurately match images even in the presence of (limited) phase shillings. In terms of classical precision/recall measures, we experimentally demonstrate that WARP can gain, say, up to 35 percent in precision at a 20 percent recall level with respect to Fourier-based techniques that use neither phase nor DTW distance.

Reconstruction of solid models from oriented point sets

Abstract: In this paper we present a novel approach to the surface reconstruction problem that takes as its input an oriented point set and returns a solid, water-tight model. The idea of our approach is to use Stokes' Theorem to compute the characteristic function of the solid model (the function that is equal to one inside the model and zero outside of it). Specifically, we provide an efficient method for computing the Fourier coefficients of the characteristic function using only the surface samples and normals, we compute the inverse Fourier transform to get back the characteristic function, and we use iso-surfacing techniques to extract the boundary of the solid model.The advantage of our approach is that it provides an automatic, simple, and efficient method for computing the solid model represented by a point set without requiring the establishment of adjacency relations between samples or iteratively solving large systems of linear equations. Furthermore, our approach can be directly applied to models with holes and cracks, providing a method for hole-filling and zippering of disconnected polygonal models.

Improved time bounds for near-optimal sparse Fourier representations

Abstract: •We study the problem of finding a Fourier representation R of m terms for a given discrete signal A of length N. The Fast Fourier Transform (FFT) can find the optimal N-term representation in time O(N log N) time, but our goal is to get sublinear time algorithms when m << N. Suppose ||A||2 ≤M||A-Ropt||2, where Ropt is the optimal output. The previously best known algorithms output R such that ||A-R||22≤(1+e))||A-Ropt||22 with probability at least 1-δ in time* poly(m,log(1/δ),log N,log M,1/e). Although this is sublinear in the input size, the dominating expression is the polynomial factor in m which, for published algorithms, is greater than or equal to the bottleneck at m2 that we identify below. Our experience with these algorithms shows that this is serious limitation in theory and in practice. Our algorithm beats this m2 bottleneck. Our main result is a significantly improved algorithm for this problem and the d-dimensional analog. Our algorithm outputs an R with the same approximation guarantees but it runs in time m•poly(log(1/δ),log N,log M,1/e). A version of the algorithm holds for all N, though the details differ slightly according to the factorization of N. For the d-dimensional problem of size N1 × N2 × •• × Nd, the linear-in-m algorithm extends efficiently to higher dimensions for certain factorizations of the Ni's; we give a quadratic-in-m algorithm that works for any values of Ni's. This article replaces several earlier, unpublished drafts.

Application of fast Fourier transform cross-correlation for the alignment of large chromatographic and spectral datasets.

Abstract: Preprocessing of chromatographic and spectral data is an important aspect of analytical sciences. In particular, recent advances in proteomics have resulted in the generation of large data sets that require analysis. To assist accurate comparison of chemical signals, we propose two methods for the alignment of multiple spectral data sets. Based on methods previously described, each chromatograph or spectrum to be aligned is divided and aligned as individual segments to a reference. However, our methods make use of fast Fourier transform for the rapid computation of a cross-correlation function that enables alignments between samples to be optimized. The proposed methods are demonstrated in comparison with an existing method on a chromatographic and a mass spectral data set. It is shown that our methods provide an advantage of speed and a reduction of the number of input parameters required. The software implementations for the proposed alignment methods are available under the downloads section at http://ptcl.chem.ox.ac.uk/~jwong/specalign.

High-accuracy, midinfrared (450cm−1⩽ω⩽4000cm−1) refractive index values of silicon

Abstract: The real and imaginary parts of the refractive index, n(ω) and k(ω), of silicon were measured as a function of photon frequency ω using Fourier transform infrared (FTIR) transmission spectral data. An accurate mechanical measurement of the wafer’s thickness, t, was required, and two FTIR spectra were used: one of high resolution (Δω=0.1to0.5cm−1) yielding a typical channel spectrum (Fabry–Perot fringes) dependent mainly on t and n(ω), and one of low resolution (Δω=4.0cm−1) yielding an absorption spectrum dependent mainly on t and k(ω). A procedure was developed to first get initial estimates for n(ω) for the high-resolution spectrum and then calculate k(ω) from the faster low-resolution spectrum with minimal measurement drift. Then both initial n and final k values were used together as starting point data for a fit to the high-resolution spectrum. A previously derived transmission formula for a convergent incident beam was used for the fit. The accuracy of n(ω) determined using this procedure is mostly d...

Fourier Transform Raman and Fourier Transform Infrared Spectroscopy Studies of Silk Fibroin

Abstract: The chemical and conformational structures of Bombyx mori silk were studied with the complementary techniques of Fourier transform Raman spectroscopy and Fourier transform infrared spectroscopy. The Fourier trans- form Raman spectrum of silk showed strong bands for the photosensitive aromatic amino acids tyrosine, tryptophan, and phenylalanine. Intensive UV/ozone irradiation reduced the Raman intensities of the amide III band and the tyrosine signals due to peptide chain scission of the silk polymer and the photochemical changes in the tyrosine residues on the silk molecules. However, the Raman spectroscopy changes for tryptophan were less clearly defined because of peak overlapping with other amino acid signals and the low concentration of tryptophan. The Fourier transform infrared (attenuated total reflectance) technique, coupled with sec- ond-derivative spectroscopy analysis, demonstrated a de- crease in the crystallinity degree and tyrosine content of UV/ozone-irradiated silk, as indicated by changes in the Fourier transform infrared bands of amide III doublet and tyrosine signals. © 2005 Wiley Periodicals, Inc. J Appl Polym Sci 96: 1999 -2004, 2005

A smooth particle-mesh Ewald algorithm for Stokes suspension simulations: The sedimentation of fibers

Abstract: Large-scale simulations of non-Brownian rigid fibers sedimenting under gravity at zero Reynolds number have been performed using a fast algorithm. The mathematical formulation follows the previous simulations by Butler and Shaqfeh [“Dynamic simulations of the inhomogeneous sedimentation of rigid fibres,” J. Fluid Mech. 468, 205 (2002)]. The motion of the fibers is described using slender-body theory, and the line distribution of point forces along their lengths is approximated by a Legendre polynomial in which only the total force, torque, and particle stresslet are retained. Periodic boundary conditions are used to simulate an infinite suspension, and both far-field hydrodynamic interactions and short-range lubrication forces are considered in all simulations. The calculation of the hydrodynamic interactions, which is typically the bottleneck for large systems with periodic boundary conditions, is accelerated using a smooth particle-mesh Ewald (SPME) algorithm previously used in molecular dynamics simulations. In SPME the slowly decaying Green’s function is split into two fast-converging sums: the first involves the distribution of point forces and accounts for the singular short-range part of the interactions, while the second is expressed in terms of the Fourier transform of the force distribution and accounts for the smooth and long-range part. Because of its smoothness, the second sum can be computed efficiently on an underlying grid using the fast Fourier transform algorithm, resulting in a significant speed-up of the calculations. Systems of up to 512 fibers were simulated on a single-processor workstation, providing a different insight into the formation, structure, and dynamics of the inhomogeneities that occur in sedimenting fiber suspensions.

Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms

Abstract: By use of matrix-based techniques it is shown how the space–bandwidth product (SBP) of a signal, as indicated by the location of the signal energy in the Wigner distribution function, can be tracked through any quadratic-phase optical system whose operation is described by the linear canonical transform. Then, applying the regular uniform sampling criteria imposed by the SBP and linking the criteria explicitly to a decomposition of the optical matrix of the system, it is shown how numerical algorithms (employing interpolation and decimation), which exhibit both invertibility and additivity, can be implemented. Algorithms appearing in the literature for a variety of transforms (Fresnel, fractional Fourier) are shown to be special cases of our general approach. The method is shown to allow the existing algorithms to be optimized and is also shown to permit the invention of many new algorithms.

Clifford Fourier transform on vector fields

Abstract: Image processing and computer vision have robust methods for feature extraction and the computation of derivatives of scalar fields. Furthermore, interpolation and the effects of applying a filter can be analyzed in detail and can be advantages when applying these methods to vector fields to obtain a solid theoretical basis for feature extraction. We recently introduced the Clifford convolution, which is an extension of the classical convolution on scalar fields and provides a unified notation for the convolution of scalar and vector fields. It has attractive geometric properties that allow pattern matching on vector fields. In image processing, the convolution and the Fourier transform operators are closely related by the convolution theorem and, in this paper, we extend the Fourier transform to include general elements of Clifford Algebra, called multivectors, including scalars and vectors. The resulting convolution and derivative theorems are extensions of those for convolution and the Fourier transform on scalar fields. The Clifford Fourier transform allows a frequency analysis of vector fields and the behavior of vector-valued filters. In frequency space, vectors are transformed into general multivectors of the Clifford Algebra. Many basic vector-valued patterns, such as source, sink, saddle points, and potential vortices, can be described by a few multivectors in frequency space.

A fast IE-FFT algorithm for solving PEC scattering problems

Abstract: This paper presents a novel fast integral equation method, termed IE-FFT, for solving large electromagnetic scattering problems Similar to other fast integral equation methods, the IE-FFT algorithm starts by partitioning the basis functions into multilevel clustering groups Subsequently, the entire impedance matrix is decomposed into two parts: one for the self and/or near field couplings, and one for well-separated group couplings The IE-FFT algorithm employs two discretizations one is for the unknown current on an unstructured triangular mesh, and the other is a uniform Cartesian grid for interpolating the Green's function By interpolating the Green's function on a regular Cartesian grid, the couplings between two well-separated groups can be computed using the fast Fourier transform (FFT) Consequently, the IE-FFT algorithm does not require the knowledge of addition theorem It simply utilizes the Toeplitz property of the coefficient matrix and is therefore applicable to both static and wave propagation problems

Fourier's law for a microscopic model of heat conduction

Abstract: We consider a chain of N harmonic oscillators perturbed by a conservative stochastic dynamics and coupled at the boundaries to two gaussian thermostats at different temperatures. The stochastic perturbation is given by a diffusion process that exchange momentum between nearest neighbor oscillators conserving the total kinetic energy. The resulting total dynamics is a degenerate hypoelliptic diffusion with a smooth stationary state. We prove that the stationary state, in the limit as N→ ∞, satisfies Fourier’s law and the linear profile for the energy average