Fourier Domain Representation Research Articles

Efficient calculation of the three-dimensional sound pressure field around a slab track

The wavenumber domain Boundary Element (2.5D BE) method is well suited to calculate the acoustic sound field around structures with a constant cross-section along one dimension, such as noise barriers or railway track. By expressing the sound field along this dimension in wavenumber domain, the numerical model is reduced from a 3D model to 2D model at each wavenumber. A consequence of the required discrete Fourier domain representation is that the sound field is represented by periodically repeating sections, of which only one section is physically meaningful. The resolution and the number of required wavenumbers increases with the desired length and spatial discretisation of this section. Describing the sound field adequately to auralise the sound without disturbing artefacts requires a large number of wavenumbers (and thus 2D BE computations), which is not feasible for large geometries. Here, a method is introduced that allows the calculation of the 3D sound field by solving a single 2D BE problem for a dense frequency spectrum and interpolating at higher wavenumbers. The calculation efficiency is further increased by precalculating the acoustic transfer functions between each BE surface element and receiver positions. Combining these two methods allows for efficient calculation of the 3D sound field around acoustically rigid structures such as slab tracks. The numerical approach is validated by comparison with a standard 2.5D BEM calculation and an analytical solution. Precalculated transfer functions to calculate the sound radiation from railway track, which are made available online, are illustrated. An example application is presented.

Ring and Radius Sampling Based Phasor Field Diffraction Algorithm for Non-Line-of-Sight Reconstruction.

Non-Line-of-Sight (NLOS) imaging reconstructs occluded scenes based on indirect diffuse reflections. The computational complexity and memory consumption of existing NLOS reconstruction algorithms make them challenging to be implemented in real-time. This paper presents a fast and memory-efficient phasor field-diffraction-based NLOS reconstruction algorithm. In the proposed algorithm, the radial property of the Rayleigh Sommerfeld diffraction (RSD) kernels along with the linear property of Fourier transform are utilized to reconstruct the Fourier domain representations of RSD kernels using a set of kernel bases. Moreover, memory consumption is further reduced by sampling the kernel bases in a radius direction and constructing them during the run-time. According to the analysis, the memory efficiency can be improved by as much as 220×. Experimental results show that compared with the original RSD algorithm, the reconstruction time of the proposed algorithm is significantly reduced with little impact on the final imaging quality.

Retrospective correction of intensity inhomogeneity with sparsity constraints in transform-domain: Application to brain MRI

An effective retrospective correction method is introduced in this paper for intensity inhomogeneity which is an inherent artifact in MR images. Intensity inhomogeneity problem is formulated as the decomposition of acquired image into true image and bias field which are expected to have sparse approximation in suitable transform domains based on their known properties. Piecewise constant nature of the true image lends itself to have a sparse approximation in framelet domain. While spatially smooth property of the bias field supports a sparse representation in Fourier domain. The algorithm attains optimal results by seeking the sparsest solutions for the unknown variables in the search space through L1 norm minimization. The objective function associated with defined problem is convex and is efficiently solved by the linearized alternating direction method. Thus, the method estimates the optimal true image and bias field simultaneously in an L1 norm minimization framework by promoting sparsity of the solutions in suitable transform domains. Furthermore, the methodology doesn't require any preprocessing, any predefined specifications or parametric models that are critically controlled by user-defined parameters. The qualitative and quantitative validation of the proposed methodology in simulated and real human brain MR images demonstrates the efficacy and superiority in performance compared to some of the distinguished algorithms for intensity inhomogeneity correction.

A Fourier Domain Training Framework for Convolutional Neural Networks Based on the Fourier Domain Pyramid Pooling Method and Fourier Domain Exponential Linear Unit

convolutional neural networks (CNNs) in the frequency domain is of great significance for extending the deep learning principle to the frequency domain. However, the frequency domain representation of the convnet architecture is highly demanding due to their complicated Fourier domain training features. Therefore, high accuracy and unambiguous representation strategies are needed for training convolutional neural networks entirely in the Fourier domain. Being founded on the bin decomposition mechanism and the non-saturated activation theory, this paper proposes an accurate, stable and efficient Fourier domain training framework for convolutional neural networks. The framework contains two important Fourier domain representations: one is the Fourier domain exponential linear unit, and the other is the pyramid pooling layer. The former alleviates the vanishing phenomenon and makes CNNs easier to converge in the Fourier domain; the latter avoids the original cropping or warping steps and improves the classification accuracy. With the framework, the Fourier domain training accuracy is improved without sacrificing the throughput of the graphic processing unit (GPU). With the Re-50 as the backbone, the top-1 and top-5 classification errors are reduced from 28.85 and 9.55 to 18.63 and 4.05, respectively, while the speedup ratios of the framework can reach up to 4.9877 and 1.8997, respectively, at a batch size of 128 on an NVIDIA GEFORCE RTX 2080 GPU (8.92 TFLOPS). The average difference between the classification value and the ground truth value is only 0.21 on the MetaGram-1 set, which indicates great goodness-of-fit and robustness of the framework. This investigation illustrates that the proposed Fourier domain CNN framework using the sophisticated Fourier domain representation strategy is highly efficient and accurate. Therefore, it may serve as a baseline framework to establish the training pipelines for Fourier domain CNNs, which can improve the deep learning accuracy of CNNs and extend the Fourier domain representation strategy to other deep learning networks.

Sparse fast Fourier transform for exactly sparse signals and signals with additive Gaussian noise

In recent years, the Fourier domain representation of sparse signals has been very attractive. Sparse fast Fourier transform (or sparse FFT) is a new technique which computes the Fourier transform in a compressed way, using only a subset of the input data. Sparse FFT computes the desired transform in sublinear time, which means in an amount of time that is smaller than the size of data. In big data problems and medical imaging to reduce the time that patient spends in MRI machine, FFT algorithm is not ‘fast’ enough anymore; therefore, the concept of sparse FFT is very important. Similar to compressed sensing, sparse FFT algorithm computes just the important components in the frequency domain in sublinear time. In this work, sparse FFT algorithm is studied and implemented on MATLAB and its performance is compared with Ann Arbor FFT. A filter is used to hash the frequencies in the n dimensional frequency-sparse signal into B bins, where $$B=n/16$$ . The filter is used for analyzing an important fraction of the whole signal, and therefore, instead of computing n point FFT, B point FFT is computed, and this results in a faster Fourier transform. The probability of success of the implemented algorithm is investigated for noiseless and noisy signals. It is deduced that as the sparsity increases, the probability of perfect transform also increases. If the performances of the algorithm in both cases are compared, it is clearly seen that the performances degrade when there is noise. Therefore, it can be concluded that the algorithm should be improved especially for noisy considerations. The solvability boundary for a constant probability of error is deducted and added to give insight for future studies.

A phase-space formulation and Gaussian approximation of the filtering equations for nonlinear quantum stochastic systems

This paper is concerned with a filtering problem for a class of nonlinear quantum stochastic systems with multichannel nondemolition measurements. The system-observation dynamics are governed by a Markovian Hudson-Parthasarathy quantum stochastic differential equation driven by quantum Wiener processes of bosonic fields in vacuum state. The Hamiltonian and system-field coupling operators, as functions of the system variables, are assumed to be represented in a Weyl quantization form. Using the Wigner-Moyal phase-space framework, we obtain a stochastic integro-differential equation for the posterior quasi-characteristic function (QCF) of the system conditioned on the measurements. This equation is a spatial Fourier domain representation of the Belavkin-Kushner-Stratonovich stochastic master equation driven by the innovation process associated with the measurements. We discuss a specific form of the posterior QCF dynamics in the case of linear system-field coupling and outline a Gaussian approximation of the posterior quantum state.

A tutorial survey on waves propagating in periodic media: Electronic, photonic and phononic crystals. Perception of the Bloch theorem in both real and Fourier domains

This paper presents a review of the most widely-used methods in order to determine the structure of eigenmodes propagating in periodic materials. Both real and Fourier domain methods are outlined. The basic concepts such as eigensolutions and their k-labeling, reciprocal lattice, Brillouin zones, etc., are gradually introduced and explained. Special attention is devoted on the physical aspect and all non usual nomenclatures are defined. In a similar way, all indispensable mathematics are described and their physical content expounded. For completeness, all nonessential notions in a first reading are maintained but deferred in Appendix. Going on with the tutorial, we show how Brillouin exploited the correspondence between the real and Fourier domain representations in order to explain the band structure and especially its periodicity in the Fourier domain. Then, following the way paved by Brillouin, we show how the Bloch theorem may be deduced from general considerations concerning Fourier analysis. To this aim, using nowadays available mathematical tools inspired from the fields of discrete signal analysis we have built a formalism which allows a comprehensive vision of the two domain correspondence. This formalism, developed on mathematical tools well fitted to describe periodic media, introduces appreciable shortcuts and appears to be versatile and easily transposable to different physical domains. Formalism application examples are given in the case of solid-state, photonic and phononic crystals.

Measurement Errors in Multivariate Chemical Data

Modern analytical measurements are commonly presented in the form of vectors (e.g., spectra) or higher order data structures such as matrices, and these are often subjected to multivariate data analysis strategies to extract information. One aspect of these measurements that is often poorly understood is the underlying nature of the measurement errors and how these affect the ability to obtain chemical information. This Account outlines some of the methods that can be used to characterize multivariate measurement errors and how this information can be used to improve the results of data analysis. Characterization includes general classifications of error, Fourier domain representations, and the error covariance matrix. The calculation and interpretation of error covariance and correlation matrices are illustrated using experimental measurements, and data analysis methods that make use of this error information are briefly reviewed. A simple example is presented to show how information about measurement errors allows for more effective extraction of meaningful chemical variance in the data.

Compressed sensing SAR imaging based on sparse representation in fractional Fourier domain

Compressed sensing (CS) is a new technique of utilizing a priori knowledge on sparsity of data in a certain domain for minimizing necessary number of measurements. Based on this idea, this paper proposes a novel synthetic aperture radar (SAR) imaging approach by exploiting sparseness of echo data in the fractional Fourier domain. The effectiveness and robustness of the approach are assessed by some numerical experiments under various noisy conditions and different measurement matrices. Experimental results have shown that, the obtained images by using the CS technique depend on measurement matrix and have higher output signal to noise ratio than traditional pulse compression technique. Finally simulated and real data are also processed and the achieved results show that the proposed approach is capable of reconstructing the image of targets and effectively suppressing noise.

Splines of Complex Order: Fourier, Filters and Fractional Derivatives

For the Schoenberg (polynomial) B-splines, interesting relations between their functional representation, Dirichlet averages, and difference operators are known. We use these relations to extend B-splines to an arbitrary (infinite) sequence of knots and to higher dimensions. The Fourier domain representation of multidimensional complex B-splines, their two-scale relation and an explicit time domain representation are given for the case of equidistantly distributed knots on rays. In addition, we show that complex B-splines are solutions of the sampling/interpolation problem $${D^z}f = \sum\limits_{k = 0}^\infty {{a_k}\delta ( \bullet - k)}$$ of complex order z, Re z > 1.

Packed domain Rayleigh-Sommerfeld wavefield propagation for large targets

For applications in the domain of digital holographic microscopy, we present a fast algorithm to propagate scalar wave fields from a small source area to an extended, parallel target area of coarser sampling pitch, using the first Rayleigh-Sommerfeld diffraction formula. Our algorithm can take full advantage of the fast Fourier transform by decomposing the convolution kernel of the propagation into several convolution kernel patches. Using partial overlapping of the patches together with a soft blending function, the Fourier spectrum of these patches can be reduced to a low number of significant components, which can be stored in a compact sparse array structure. This allows for rapid evaluation of the partial convolution results by skipping over negligible components through the Fourier domain pointwise multiplication and direct mapping of the remaining multiplication results into a Fourier domain representation of the coarsly sampled target patch. The algorithm has been verified experimentally at a numerical aperture of 0.62, not showing any significant resolution limitations.

Comparison of post-Newtonian templates for compact binary inspiral signals in gravitational-wave detectors

The two-body dynamics in general relativity has been solved perturbatively using the post-Newtonian (PN) approximation. The evolution of the orbital phase and the emitted gravitational radiation are now known to a rather high order up to $\mathcal{O}({v}^{8})$, $v$ being the characteristic velocity of the binary. The orbital evolution, however, cannot be specified uniquely due to the inherent freedom in the choice of parameter used in the PN expansion, as well as the method pursued in solving the relevant differential equations. The goal of this paper is to determine the (dis)agreement between different PN waveform families in the context of initial and advanced gravitational-wave detectors. The waveforms employed in our analysis are those that are currently used by Initial LIGO/Virgo, that is, the time-domain PN models TaylorT1, TaylorT2, TaylorT3, the Fourier-domain representation TaylorF2 (or stationary phase approximant), and the effective-one-body model, and two more recent models, TaylorT4 and TaylorEt. For these models we examine their overlaps with one another for a number of different binaries at 2PN, 3PN, and 3.5PN orders to quantify their differences. We then study the overlaps of these families with the prototype effective-one-body family, currently used by Initial LIGO, calibrated to numerical-relativity simulations to help us decide whether there exist preferred families, in terms of detectability and computational cost, that are the most appropriate as search templates. We conclude that as long as the total mass remains less than a certain upper limit ${M}_{\mathrm{crit}}$, all template families at 3.5PN order (except TaylorT3 and TaylorEt) are equally good for the purpose of detection. The value of ${M}_{\mathrm{crit}}$ is found to be $\ensuremath{\sim}12{M}_{\ensuremath{\bigodot}}$ for Initial, Enhanced, and Advanced LIGO. From a purely computational point of view, we recommend that 3.5PN TaylorF2 be used below ${M}_{\mathrm{crit}}$ and that the effective-one-body model calibrated to numerical-relativity simulations be used for total binary mass $M>{M}_{\mathrm{crit}}$.

A modified k-means algorithm for circular invariant clustering

Several important pattern recognition applications are based on feature vector extraction and vector clustering. Directional patterns are commonly represented by rotation-variant vectors Fd formed from features uniformly extracted in M directions. It is often desirable that pattern recognition algorithms are invariant under pattern rotation. This paper introduces a distance measure and a K-means-based algorithm, namely, Circular K-means (CK-means) to cluster vectors containing directional information, such as Fd, in a circular-shift invariant manner. A circular shift of Fd corresponds to pattern rotation, thus, the algorithm is rotation invariant. An efficient Fourier domain representation of the proposed measure is presented to reduce computational complexity. A split and merge approach (SMCK-means), suited to the proposed CK-means technique, is proposed to reduce the possibility of converging at local minima and to estimate the correct number of clusters. Experiments performed for textural images illustrate the superior performance of the proposed algorithm for clustering directional vectors Fd, compared to the alternative approach that uses the original K-means and rotation-invariant feature vectors transformed from Fd.

Erratum: Parameter estimation of inspiralling compact binaries using 3.5 post-Newtonian gravitational wave phasing: The nonspinning case [Phys. Rev. D71, 084008 (2005)

(Abridged) We revisit the problem of parameter estimation of gravitational-wave chirp signals from inspiralling non-spinning compact binaries in the light of the recent extension of the post-Newtonian (PN) phasing formula to order $(v/c)^7$ beyond the leading Newtonian order. We study in detail the implications of higher post-Newtonian orders from 1PN up to 3.5PN in steps of 0.5PN ($\sim v/c$), and examine their convergence. In both initial and advanced detectors the estimation of the chirp mass (${\cal M}$) and symmetric mass ratio ($\eta$) improve at higher PN orders but oscillate with every half-a-PN order. We compare parameter estimation in different detectors and assess their relative performance in two different ways: at a {\it fixed SNR,} with the aim of understanding how the bandwidth improves parameter estimation, and for a {\it fixed source}, to gauge the importance of sensitivity. Errors in parameter estimation at a fixed SNR are smaller for VIRGO than for both initial and advanced LIGO. However, for sources at a fixed distance it is advanced LIGO that achieves the lowest errors owing to its greater sensitivity. Finally, we compute the amplitude corrections due to the `frequency-sweep' in the Fourier domain representation of the waveform within the stationary phase approximation and discuss its implication on parameter estimation.

Parameter estimation of inspiralling compact binaries using 3.5 post-Newtonian gravitational wave phasing: The nonspinning case

We revisit the problem of parameter estimation of gravitational-wave chirp signals from inspiralling nonspinning compact binaries in the light of the recent extension of the post-Newtonian (PN) phasing formula to order $(v/c{)}^{7}$ beyond the leading Newtonian order. We study in detail the implications of higher post-Newtonian orders from 1PN up to 3.5PN in steps of 0.5PN ($\ensuremath{\sim}v/c$), and examine their convergence. In both initial and advanced detectors the estimation of the chirp mass ($\mathcal{M}$) and symmetric mass ratio ($\ensuremath{\eta}$) improve at higher PN orders but oscillate with every half-a-PN order. In initial LIGO, for a $10{M}_{\ensuremath{\bigodot}}--10{M}_{\ensuremath{\bigodot}}$ binary at a signal-to-noise ratio (SNR) of 10, the improvement in the estimation of $\mathcal{M}$ ($\ensuremath{\eta}$) at 3.5PN relative to 2PN is $\ensuremath{\sim}19%$ (52%). We compare parameter estimation in different detectors and assess their relative performance in two different ways: at a fixed SNR, with the aim of understanding how the bandwidth improves parameter estimation, and for a fixed source, to gauge the importance of sensitivity. Errors in parameter estimation at a fixed SNR are smaller for VIRGO than for both initial and advanced LIGO. This is because of the larger bandwidth over which it observes the signals. However, for sources at a fixed distance it is advanced LIGO that achieves the lowest errors owing to its greater sensitivity. Finally, we compute the amplitude corrections due to the ``frequency-sweep'' in the Fourier domain representation of the waveform within the stationary phase approximation and discuss its implication on parameter estimation. We find that the amplitude corrections change the errors in $\mathcal{M}$ and $\ensuremath{\eta}$ by less than 10% for initial LIGO at a signal-to-noise ratio of 10. Our analysis makes explicit the significance of higher PN order modeling of the inspiralling compact binary on parameter estimation.

Fourier domain representation of planar curves for recognition in multiple views

Recognition of planar shapes is an important problem in computer vision and pattern recognition. The same planar object contour imaged from different cameras or from different viewpoints looks different and their recognition is non-trivial. Traditional shape recognition deals with views of the shapes that differ only by simple rotations, translations, and scaling. However, shapes suffer more serious deformation between two general views and hence recognition approaches designed to handle translations, rotations, and/or scaling would prove to be insufficient. Many algebraic relations between matching primitives in multiple views have been identified recently. In this paper, we explore how shape properties and multiview relations can be combined to recognize planar shapes across multiple views. We propose novel recognition constraints that a planar shape boundary must satisfy in multiple views. The constraints are on the rank of a Fourier-domain measurement matrix computed from the points on the shape boundary. Our method can additionally compute the correspondence between the curve points after a match is established. We demonstrate the applications of these constraints experimentally on a number of synthetic and real images.

Frequency-domainP-approximant filters for time-truncated inspiral gravitational wave signals from compact binaries

Frequency-domain filters for time-windowed gravitational waves from inspiralling compact binaries are constructed which combine the excellent performance of our previously developed time-domain P-approximants with the analytic convenience of the stationary phase approximation without a serious loss in event rate. These Fourier-domain representations incorporate the ``edge oscillations'' due to the (assumed) abrupt shut-off of the time-domain signal caused by the relativistic plunge at the last stable orbit. These new analytic approximations, the SPP-approximants, are not only `effectual' for detection and `faithful' for parameter estimation, but are also computationally inexpensive to generate (and are `faster' by factors up to 10, as compared to the corresponding time-domain templates). The SPP approximants should provide data analysts the Fourier-domain templates for massive black hole binaries of total mass m less than about 40 solar mases, the most likely sources for LIGO and VIRGO.

Speckle-motion artifact under tissue shearing

Research has shown that, for a rotating phantom, the speckle pattern may not replicate the phantom motion, rather it may show a large lateral translation component in addition to rotation. This translation effect was labeled speckle-motion artifact. An image formation model has been shown to explain the phenomenon, pointing to the curvature of the imaging system point spread function (PSF) at the origin of this effect. The present paper extends this analysis and proposes a model, which predicts that a lateral motion artifact also would occur with shear motion. In the model, the artifact is found to be proportional to the shear angle and dependent of shear orientation, being maximal for shear that runs parallel to the axial direction; as for rotation, the artifact increases with frequency and beamwidth. This would mean that, when viewing a parabolic flow in the far field or with a highly curved PSF, an apparent contraction/expansion pattern in the direction of the vessel wall would be superimposed to the real velocity profile. In elastography, when viewing an inclusion subjected to an axial strain, four motion artifact regions are expected near the inclusion. The model is developed using the Fourier domain representation of the speckles for tissue-motion compensated signals, also called Lagrangian speckle. It can explain the artifact in terms of a simple spectral translation of a parabolic phase profile; given this, it is shown the artifact would be proportional to the lateral derivative of the axial displacement field. The spectral representation of Lagrangian speckle, for shear, also provides a simple geometrical interpretation for speckle decorrelation in terms of the shear strength and orientation, and in terms of the beam characteristics, i.e., the axial and lateral bandwidth.

Quantification of Overlapping VIS-Absorption Bands by Fourier-Domain Fitting of a Generalized Band Shape Function

Parameter estimation of overlapping (near-)symmetrical absorption bands with reference to quantitative determination may advantageously be performed by a new procedure that involves fitting a generalized four-parameter band shape function to the Fourier-domain representation of the second-derivative spectrum. Important features of the method are: insensitivity to linear and weakly curved baselines, inherent noise reduction, fast calculation of objective function, and band simulation capability much better than that with the Gauss or Lorentz shapes.

A new isostatic residual gravity map of the conterminous United States with a discussion on the significance of isostatic residual anomalies

To display more clearly the gravity anomalies caused by geologic bodies in the upper parts of the crust, a new colored isostatic residual gravity map of the conterminous United States has been prepared using the gravity data set compiled for the Gravity Anomaly Map of the United States (Society of Exploration Geophysicists, 1982). The new isostatic residual gravity map is based on an Airy‐Heiskanen model of local compensation, in which the surface load requiring compensation is defined by 5‐min topographic and bathymetric data sets. A colored first‐vertical‐derivative map of isostatic residual gravity further enhances the short‐wavelength anomalies produced by bodies at or near the surface and emphasizes the regional fabrics and trends in the gravity field. For the purpose of displaying gravity anomalies caused by shallow bodies of geologic significance, the nature of the isostatic model and the values of its parameters are of lesser importance than the application of an isostatic correction of some sort. Most isostatic models result in residual gravity maps that appear nearly identical in their main patterns and features. Anomalies on isostatic residual gravity maps should not be casually interpreted in terms of “undercompensation” or “overcompensation” because large‐amplitude anomalies can be produced by crustal bodies in complete local isostatic equilibrium. Many isostatic residual gravity anomalies less than several hundred kilometers wide can be related to known geologic bodies. We present here a classification scheme that attempts to categorize such anomalies on the basis of tectonic environment. In general, highs correlate with intruded or accreted mafic material or with upthrusted crustal sections, whereas lows occur over low‐density sedimentary or volcanic sections, felsic intrusive bodies, or down‐warped crustal sections. Although some longer‐wavelength anomalies, such as the broad gravity high centered over Montana, could be manifestations of density contrasts deep in the mantle, many such anomalies can also be modeled by geologically reasonable density contrasts that are isostatically compensated and confined to depths of less than several hundred kilometers, so that their source bodies need not be deep. The fact that certain of these broader anomalies have well‐defined boundaries which correlate with near‐surface geologic features increases the likelihood that their sources lie entirely within the lithosphere. If so, then the density contrasts required to explain the gravity data imply fundamental anomalies in chemical composition or thermal state for the crustal and upper mantle columns under these regions. We have investigated spectral analysis as a method to quantitatively characterize regional anomaly patterns. Contoured plots of normalized amplitude spectra were prepared for various areas of the isostatic residual gravity field of the United States. These Fourier domain representations show characteristic patterns that can be interpreted in terms of the trends and wavelengths of anomalies and may help to more objectively distinguish geologic basements with different origins or tectonic histories.