Showing papers on "Fourier transform published in 2015"

A new Definition of Fractional Derivative without Singular Kernel

Abstract: In the paper, we present a new definition of fractional deriva tive with a smooth kernel which takes on two different representations for the temporal and spatial variable. The first works on the time variables; thus it is suitable to use th e Laplace transform. The second definition is related to the spatial va riables, by a non-local fractional derivative, for which it is more convenient to work with the Fourier transform. The interest for this new approach with a regular kernel was born from the prospect that there is a class of non-local systems, which have the ability to descri be the material heterogeneities and the fluctuations of diff erent scales, which cannot be well described by classical local theories or by fractional models with singular kernel.

Phase Retrieval via Matrix Completion

Abstract: This paper develops a novel framework for phase retrieval, a problem which arises in X-ray crystallography, diffraction imaging, astronomical imaging, and many other applications. Our approach, cal...

Second-Order Synchrosqueezing Transform or Invertible Reassignment? Towards Ideal Time-Frequency Representations

Abstract: This paper considers the analysis of multicomponent signals, defined as superpositions of real or complex modulated waves. It introduces two new post-transformations for the short-time Fourier transform that achieve a compact time-frequency representation while allowing for the separation and the reconstruction of the modes. These two new transformations thus benefit from both the synchrosqueezing transform (which allows for reconstruction) and the reassignment method (which achieves a compact time-frequency representation). Numerical experiments on real and synthetic signals demonstrate the efficiency of these new transformations, and illustrate their differences.

Perfect vortex beam: Fourier transformation of a Bessel beam.

Abstract: We derive a mathematical description of a perfect vortex beam as the Fourier transformation of a Bessel beam. Building on this development, we experimentally generate Bessel–Gauss beams of different orders and Fourier transform them to form perfect vortex beams. By controlling the radial wave vector of a Bessel–Gauss beam, we can control the ring radius of the generated beam. Our theoretical predictions match with the experimental results and also provide an explanation for previous published works. We find the perfect vortex resembles that of an orbital angular momentum (OAM) mode supported in annular profiled waveguides. Our prefect vortex beam generation method can be used to excite OAM modes in an annular core fiber.

Local Fractional Integral Transforms and Their Applications

Abstract: Local Fractional Integral Transforms and Their Applications provides information on how local fractional calculus has been successfully applied to describe the numerous widespread real-world phenomena in the fields of physical sciences and engineering sciences that involve non-differentiable behaviors. The methods of integral transforms via local fractional calculus have been used to solve various local fractional ordinary and local fractional partial differential equations and also to figure out the presence of the fractal phenomenon. The book presents the basics of the local fractional derivative operators and investigates some new results in the area of local integral transforms.Provides applications of local fractional Fourier SeriesDiscusses definitions for local fractional Laplace transformsExplains local fractional Laplace transforms coupled with analytical methods

Experimental robustness of Fourier ptychography phase retrieval algorithms

Abstract: Fourier ptychography is a new computational microscopy technique that provides gigapixel-scale intensity and phase images with both wide field-of-view and high resolution. By capturing a stack of low-resolution images under different illumination angles, an inverse algorithm can be used to computationally reconstruct the high-resolution complex field. Here, we compare and classify multiple proposed inverse algorithms in terms of experimental robustness. We find that the main sources of error are noise, aberrations and mis-calibration (i.e. model mis-match). Using simulations and experiments, we demonstrate that the choice of cost function plays a critical role, with amplitude-based cost functions performing better than intensity-based ones. The reason for this is that Fourier ptychography datasets consist of images from both brightfield and darkfield illumination, representing a large range of measured intensities. Both noise (e.g. Poisson noise) and model mis-match errors are shown to scale with intensity. Hence, algorithms that use an appropriate cost function will be more tolerant to both noise and model mis-match. Given these insights, we propose a global Newton’s method algorithm which is robust and accurate. Finally, we discuss the impact of procedures for algorithmic correction of aberrations and mis-calibration.

Multichannel compressive sensing MRI using noiselet encoding.

Abstract: The incoherence between measurement and sparsifying transform matrices and the restricted isometry property (RIP) of measurement matrix are two of the key factors in determining the performance of compressive sensing (CS). In CS-MRI, the randomly under-sampled Fourier matrix is used as the measurement matrix and the wavelet transform is usually used as sparsifying transform matrix. However, the incoherence between the randomly under-sampled Fourier matrix and the wavelet matrix is not optimal, which can deteriorate the performance of CS-MRI. Using the mathematical result that noiselets are maximally incoherent with wavelets, this paper introduces the noiselet unitary bases as the measurement matrix to improve the incoherence and RIP in CS-MRI. Based on an empirical RIP analysis that compares the multichannel noiselet and multichannel Fourier measurement matrices in CS-MRI, we propose a multichannel compressive sensing (MCS) framework to take the advantage of multichannel data acquisition used in MRI scanners. Simulations are presented in the MCS framework to compare the performance of noiselet encoding reconstructions and Fourier encoding reconstructions at different acceleration factors. The comparisons indicate that multichannel noiselet measurement matrix has better RIP than that of its Fourier counterpart, and that noiselet encoded MCS-MRI outperforms Fourier encoded MCS-MRI in preserving image resolution and can achieve higher acceleration factors. To demonstrate the feasibility of the proposed noiselet encoding scheme, a pulse sequences with tailored spatially selective RF excitation pulses was designed and implemented on a 3T scanner to acquire the data in the noiselet domain from a phantom and a human brain. The results indicate that noislet encoding preserves image resolution better than Fouirer encoding.

Multiresolution Graph Fourier Transform for Compression of Piecewise Smooth Images

Abstract: Piecewise smooth (PWS) images (e.g., depth maps or animation images) contain unique signal characteristics such as sharp object boundaries and slowly varying interior surfaces. Leveraging on recent advances in graph signal processing, in this paper, we propose to compress the PWS images using suitable graph Fourier transforms (GFTs) to minimize the total signal representation cost of each pixel block, considering both the sparsity of the signal’s transform coefficients and the compactness of transform description. Unlike fixed transforms, such as the discrete cosine transform, we can adapt GFT to a particular class of pixel blocks. In particular, we select one among a defined search space of GFTs to minimize total representation cost via our proposed algorithms, leveraging on graph optimization techniques, such as spectral clustering and minimum graph cuts. Furthermore, for practical implementation of GFT, we introduce two techniques to reduce computation complexity. First, at the encoder, we low-pass filter and downsample a high-resolution (HR) pixel block to obtain a low-resolution (LR) one, so that a LR-GFT can be employed. At the decoder, upsampling and interpolation are performed adaptively along HR boundaries coded using arithmetic edge coding, so that sharp object boundaries can be well preserved. Second, instead of computing GFT from a graph in real-time via eigen-decomposition, the most popular LR-GFTs are pre-computed and stored in a table for lookup during encoding and decoding. Using depth maps and computer-graphics images as examples of the PWS images, experimental results show that our proposed multiresolution-GFT scheme outperforms H.264 intra by 6.8 dB on average in peak signal-to-noise ratio at the same bit rate.

Fourier magnetic imaging with nanoscale resolution and compressed sensing speed-up using electronic spins in diamond

Abstract: Optically detected magnetic resonance using nitrogen-vacancy (NV) colour centres in diamond is a leading modality for nanoscale magnetic field imaging, as it provides single electron spin sensitivity, three-dimensional resolution better than 1 nm (ref. 5) and applicability to a wide range of physical and biological samples under ambient conditions. To date, however, NV-diamond magnetic imaging has been performed using 'real-space' techniques, which are either limited by optical diffraction to ∼250 nm resolution or require slow, point-by-point scanning for nanoscale resolution, for example, using an atomic force microscope, magnetic tip, or super-resolution optical imaging. Here, we introduce an alternative technique of Fourier magnetic imaging using NV-diamond. In analogy with conventional magnetic resonance imaging (MRI), we employ pulsed magnetic field gradients to phase-encode spatial information on NV electronic spins in wavenumber or 'k-space' followed by a fast Fourier transform to yield real-space images with nanoscale resolution, wide field of view and compressed sensing speed-up.

New Gravitational Memories

Abstract: The conventional gravitational memory effect is a relative displacement in the position of two detectors induced by radiative energy flux. We find a new type of gravitational `spin memory' in which beams on clockwise and counterclockwise orbits acquire a relative delay induced by radiative angular momentum flux. It has recently been shown that the displacement memory formula is a Fourier transform in time of Weinberg's soft graviton theorem. Here we see that the spin memory formula is a Fourier transform in time of the recently-discovered subleading soft graviton theorem.

Power System Harmonics

Abstract: The French mathematician J. B. J. Fourier showed that arbitrary periodic functions could be represented by an infinite series of sinusoids of harmonically related frequencies. This chapter first defines periodic functions and orthogonal functions. A periodic function can be expanded in a Fourier series. The Fourier series of a periodic function is the sum of sinusoidal components of different frequencies. The chapter then illustrates the functions of odd or skew symmetry, even symmetry and half-wave symmetry. The odd and even symmetry has been obtained with the triangular function by shifting the origin. Fourier analysis of a continuous periodic signal in the time domain gives a series of discrete frequency components in the frequency domain. The chapter describes Dirichlet conditions and notion of power spectrum. Finally, it explains the function of convolution, which is generally carried out in the frequency domain.

Optimization of sampling pattern and the design of Fourier ptychographic illuminator.

Abstract: Fourier ptychography (FP) is a recently developed imaging approach that facilitates high-resolution imaging beyond the cutoff frequency of the employed optics. In the original FP approach, a periodic LED array is used for sample illumination, and therefore, the scanning pattern is a uniform grid in the Fourier space. Such a uniform sampling scheme leads to 3 major problems for FP, namely: 1) it requires a large number of raw images, 2) it introduces the raster grid artefacts in the reconstruction process, and 3) it requires a high-dynamic-range detector. Here, we investigate scanning sequences and sampling patterns to optimize the FP approach. For most biological samples, signal energy is concentrated at low-frequency region, and as such, we can perform non-uniform Fourier sampling in FP by considering the signal structure. In contrast, conventional ptychography perform uniform sampling over the entire real space. To implement the non-uniform Fourier sampling scheme in FP, we have designed and built an illuminator using LEDs mounted on a 3D-printed plastic case. The advantages of this illuminator are threefold in that: 1) it reduces the number of image acquisitions by at least 50% (68 raw images versus 137 in the original FP setup), 2) it departs from the translational symmetry of sampling to solve the raster grid artifact problem, and 3) it reduces the dynamic range of the captured images 6 fold. The results reported in this paper significantly shortened acquisition time and improved quality of FP reconstructions. It may provide new insights for developing Fourier ptychographic imaging platforms and find important applications in digital pathology.

A Fast Maneuvering Target Motion Parameters Estimation Algorithm Based on ACCF

Abstract: This letter considers the motion parameters estimation problem for a maneuvering target with arbitrary parameterized motion. The slant range of the target is modeled as a polynomial function in terms of its multiple motion parameters and a fast estimation method based on adjacent cross correlation function (ACCF) is proposed, where the iterative adjacent cross correlation operation is employed to remove the range migration and reduce the order of Doppler frequency migration. Then the motion parameters are estimated via Fourier transform. Compared with the generalized Radon Fourier transform (GRFT), the proposed method can estimate the parameters without searching procedure and acquire close estimation performance at high signal-to-noise ratio (SNR) with a much lower computational cost. Finally, simulations are provided to demonstrate the effectiveness.

Comparative analysis of imaging configurations and objectives for Fourier microscopy.

Abstract: Fourier microscopy is becoming an increasingly important tool for the analysis of optical nanostructures and quantum emitters. However, achieving quantitative Fourier space measurements requires a thorough understanding of the impact of aberrations introduced by optical microscopes that have been optimized for conventional real-space imaging. Here we present a detailed framework for analyzing the performance of microscope objectives for several common Fourier imaging configurations. To this end, we model objectives from Nikon, Olympus, and Zeiss using parameters that were inferred from patent literature and confirmed, where possible, by physical disassembly. We then examine the aberrations most relevant to Fourier microscopy, including the alignment tolerances of apodization factors for different objective classes, the effect of magnification on the modulation transfer function, and vignetting-induced reductions of the effective numerical aperture for wide-field measurements. Based on this analysis, we identify an optimal objective class and imaging configuration for Fourier microscopy. In addition, the Zemax files for the objectives and setups used in this analysis have been made publicly available as a resource for future studies.

Arc Fault and Flash Signal Analysis in DC Distribution Systems Using Wavelet Transformation

Abstract: Arc faults have always been a concern for electrical systems, as they can cause fires, personnel shock hazard, and system failure. Existing commercialized techniques that rely on pattern recognition in the time domain or frequency domain analysis using a Fourier transform do not work well, because the signal-to-noise ratio is low and the arc signal is not periodic. Instead, wavelet transform (WT) provides a time and frequency approach to analyze target signals with multiple resolutions. In this paper, a new approach using WT for arc fault analysis in dc systems is proposed. The process of detecting an arc fault involves signal analysis and then feature identification. The focus of this paper is on the former. Simulation models are synthesized to study the theoretical results of the proposed methodology and traditional fast Fourier transform analysis on arcing faults. Experimental data from the dc system of a photovoltaic array is also shown to validate the approach.

Compressive Sensing of a Taylor-Fourier Multifrequency Model for Synchrophasor Estimation

Abstract: Synchrophasor measurements, performed by phasor measurement units (PMUs), are becoming increasingly important for power system network monitoring. Synchrophasor standards define test signals for verification of PMU compliance, and set acceptance limits in each test condition for two performance classes ( $P$ and $M$ ). Several PMU algorithms have been proposed to deal with steady-state and dynamic operating conditions identified by the standard. Research and discussion arising from design, implementation, testing and characterization of PMUs evidenced that some disturbances, such as interharmonic interfering signals, can seriously degrade synchrophasor measurement accuracy. In this paper, a new compressive sensing (CS) approach is introduced and applied to synchrophasor measurements using a CS Taylor–Fourier (TF) multifrequency (CSTFM) model. The aim is to exploit, in a joint method, the properties of CS and the TF transform to identify the most relevant components of the signal, even under dynamic conditions, and to model them in the estimation procedure, thus limiting the impact of harmonic and interhamonic interferences. The CSTFM approach is verified using composite tests derived from the test conditions of the synchrophasor standard and simulation results are presented to show its potentialities.

Fast Numerical Nonlinear Fourier Transforms

Abstract: The nonlinear Fourier transform, which is also known as the forward scattering transform, decomposes a periodic signal into nonlinearly interacting waves. In contrast to the common Fourier transform, these waves no longer have to be sinusoidal. Physically relevant waveforms are often available for the analysis instead. The details of the transform depend on the waveforms underlying the analysis, which in turn are specified through the implicit assumption that the signal is governed by a certain evolution equation. For example, water waves generated by the Korteweg–de Vries equation can be expressed in terms of cnoidal waves. Light waves in optical fiber governed by the nonlinear Schrodinger equation (NSE) are another example. Nonlinear analogs of classic problems such as spectral analysis and filtering arise in many applications, with information transmission in optical fiber, as proposed by Yousefi and Kschischang, being a very recent one. The nonlinear Fourier transform is eminently suited to address them—at least from a theoretical point of view. Although numerical algorithms are available for computing the transform, a fast nonlinear Fourier transform that is similarly effective as the fast Fourier transform is for computing the common Fourier transform has not been available so far. The goal of this paper is to address this problem. Two fast numerical methods for computing the nonlinear Fourier transform with respect to the NSE are presented. The first method achieves a runtime of $O(D^{2})$ floating point operations, where $D$ is the number of sample points. The second method applies only to the case where the NSE is defocusing, but it achieves an $O(D\log ^{2}D)$ runtime. Extensions of the results to other evolution equations are discussed as well.

Elliptic Fourier transformation-based histograms of oriented gradients for rotationally invariant object detection in remote-sensing images

Abstract: High-resolution remote-sensing images are widely used for object detection but are affected by various factors. During the detection process, the orientation sensitivity of the image features is crucial to the detection performance. This study presents a novel rotationally invariant object detection descriptor that can address the difficulties with object detection that are caused by different object orientations. We use orientation normalization, feature space mapping, and an elliptic Fourier transform to achieve rotational invariance of the histogram of oriented gradients. Validation experiments indicate that the proposed descriptor is robust to rotation, noise, and compression. We use this novel image descriptor to detect aircraft and cars in remote-sensing images. The results show that the proposed method offers robust rotational invariance in object detection.

Asymmetric optical cryptosystem based on coherent superposition and equal modulus decomposition.

Abstract: A novel asymmetric cryptosystem based on coherent superposition, which is free from silhouette problem, is proposed. Being different from the phase-truncated Fourier transform-based cryptosystem, the encryption process uses equal modulus decomposition (EMD) to create an effective trapdoor one-way function. As a result, the proposed method achieves high robustness against the special attack based on iterative Fourier transform. Simulation results are presented to prove the validity of the proposed system.

Non-uniform sampling: post-Fourier era of NMR data collection and processing.

Abstract: The invention of multidimensional techniques in the 1970s revolutionized NMR, making it the general tool of structural analysis of molecules and materials. In the most straightforward approach, the signal sampling in the indirect dimensions of a multidimensional experiment is performed in the same manner as in the direct dimension, i.e. with a grid of equally spaced points. This results in lengthy experiments with a resolution often far from optimum. To circumvent this problem, numerous sparse-sampling techniques have been developed in the last three decades, including two traditionally distinct approaches: the radial sampling and non-uniform sampling. This mini review discusses the sparse signal sampling and reconstruction techniques from the point of view of an underdetermined linear algebra problem that arises when a full, equally spaced set of sampled points is replaced with sparse sampling. Additional assumptions that are introduced to solve the problem, as well as the shape of the undersampled Fourier transform operator (visualized as so-called point spread function), are shown to be the main differences between various sparse-sampling methods.

Size-dependent generalized thermoelasticity using Eringen's nonlocal model

Abstract: Ultrafast lasers, even the novel laser burst technology, have been widely using in numerous applications, especially in micro-machining. The mechanism of ultrafast laser and matter interaction, such as: heat transfer, deformation of nanostructure, has caught numerous theoretical and experimental research interests. However, in such cases the classical models of thermo-elasticity may be challenged to give accurate responses: firstly, Fourier's law of heat conduction law may break down under the high heat flux and low temperature conditions; secondly, classical elasticity may fail as the external characteristic length (or time) approaches to the internal characteristic length (or time). In this work, to simulate transient thermo-elastic responses of the nanostructure subjected to a sudden thermal loading, classical thermo-elastic models are extended in two aspects: in mechanical sense, Eringen's nonlocal elasticity (differential constitutive relations) is employed to depict the size-dependence; while the memory dependence of heat conduction is considered using Caputo fractional derivative and m emory d ependent d erivative (MDD). In a separated section, the concept of Nonlocal operator and Memory dependent operator are proposed by revisiting Eringen's integral-type nonlocal theory, and comparing fractional calculus, MDD, and a newly reported “Most nature fractional derivative and integral”. In the numerical part, a thermo-elastic medium subjected to a sudden heating at one end is considered, and an analytical technique based on Laplace transform is adopted. While the inverse Laplace transform is numerically implemented by using an efficient and pragmatic algorithm ‘NILT’. Numerical results, i.e., temperature vs. position, displacement vs. position and stress vs. position, are shown graphically, and the influences of nonlocal scale parameter on them are also evaluated. It is concluded that nonlocal scale parameter's effect on the deformation and stress are significant, which is excessively important is determining material's failure when subjected to ultrafast laser like heating, although its effect on the temperature is negligible.

Nanoscale-resolved chemical identification of thin organic films using infrared near-field spectroscopy and standard Fourier transform infrared references

Abstract: We establish a solid basis for the interpretation of infrared near-field spectra of thin organic films on highly reflective substrates and provide guidelines for their straightforward comparison to standard far-field Fourier transform infrared (FTIR) spectra. Particularly, we study the spectral behavior of near-field absorption and near-field phase, both quantities signifying the presence of a molecular resonance. We demonstrate that the near-field phase spectra only weakly depend on the film thickness and can be used for an approximate comparison with grazing incidence FTIR (GI-FTIR) spectra. In contrast, the near-field absorption spectra can be compared more precisely with far-field spectra: for ultrathin films they match well GI-FTIR spectra, while for thick films a good agreement with standard transmission FTIR spectra is found. Our results are based on experimental data obtained by nanoscale FTIR (nano-FTIR) spectroscopy and supported by a comprehensive theoretical analysis.

Uniform energy bound and asymptotics for the Maxwell field on a slowly rotating Kerr black hole exterior

Abstract: We consider the Maxwell equation in the exterior of a very slowly rotating Kerr black hole. For this system, we prove the boundedness of a positive definite energy on each hypersurface of constant t. We also prove the convergence of each solution to a stationary Coulomb solution. We separate a general solution into the charged, Coulomb part and the uncharged part. Convergence to the Coulomb solutions follows from the fact that the uncharged part satisfies a Morawetz estimate, i.e. that a spatially localized energy density is integrable in time. For the unchanged part, we study both the full Maxwell equation and the Fackerell–Ipser equation for one component. To treat the Fackerell–Ipser equation, we use a Fourier transform in t. For the Fackerell–Ipser equation, we prove a refined Morawetz estimate that controls 3/2 derivatives with no loss near the orbiting null geodesics.