Showing papers on "Fourier transform published in 2014"

Integral transforms and their applications

Abstract: INTEGRAL TRANSFORMS Brief Historical Introduction Basic Concepts and Definitions FOURIER TRANSFORMS AND THEIR APPLICATIONS Introduction The Fourier Integral Formulas Definition of the Fourier Transform and Examples Fourier Transforms of Generalized Functions Basic Properties of Fourier Transforms Poisson's Summation Formula The Shannon Sampling Theorem Gibbs' Phenomenon Heisenberg's Uncertainty Principle Applications of Fourier Transforms to Ordinary Differential Eqn Solutions of Integral Equations Solutions of Partial Differential Equations Fourier Cosine and Sine Transforms with Examples Properties of Fourier Cosine and Sine Transforms Applications of Fourier Cosine and Sine Transforms to Partial DE Evaluation of Definite Integrals Applications of Fourier Transforms in Mathematical Statistics Multiple Fourier Transforms and Their Applications Exercises LAPLACE TRANSFORMS AND THEIR BASIC PROPERTIES Introduction Definition of the Laplace Transform and Examples Existence Conditions for the Laplace Transform Basic Properties of Laplace Transforms The Convolution Theorem and Properties of Convolution Differentiation and Integration of Laplace Transforms The Inverse Laplace Transform and Examples Tauberian Theorems and Watson's Lemma Exercises APPLICATIONS OF LAPLACE TRANSFORMS Introduction Solutions of Ordinary Differential Equations Partial Differential Equations, Initial and Boundary Value Problems Solutions of Integral Equations Solutions of Boundary Value Problems Evaluation of Definite Integrals Solutions of Difference and Differential-Difference Equations Applications of the Joint Laplace and Fourier Transform Summation of Infinite Series Transfer Function and Impulse Response Function Exercises FRACTIONAL CALCULUS AND ITS APPLICATIONS Introduction Historical Comments Fractional Derivatives and Integrals Applications of Fractional Calculus Exercises APPLICATIONS OF INTEGRAL TRANSFORMS TO FRACTIONAL DIFFERENTIAL EQUATIONS Introduction Laplace Transforms of Fractional Integrals Fractional Ordinary Differential Equations Fractional Integral Equations Initial Value Problems for Fractional Differential Equations Green's Functions of Fractional Differential Equations Fractional Partial Differential Equations Exercises HANKEL TRANSFORMS AND THEIR APPLICATIONS Introduction The Hankel Transform and Examples Operational Properties of the Hankel Transform Applications of Hankel Transforms to Partial Differential Equations Exercises MELLIN TRANSFORMS AND THEIR APPLICATIONS Introduction Definition of the Mellin Transform and Examples Basic Operational Properties Applications of Mellin Transforms Mellin Transforms of the Weyl Fractional Integral and Derivative Application of Mellin Transforms to Summation of Series Generalized Mellin Transforms Exercises HILBERT AND STIELTJES TRANSFORMS Introduction Definition of the Hilbert Transform and Examples Basic Properties of Hilbert Transforms Hilbert Transforms in the Complex Plane Applications of Hilbert Transforms Asymptotic Expansions of One-Sided Hilbert Transforms Definition of the Stieltjes Transform and Examples Basic Operational Properties of Stieltjes Transforms Inversion Theorems for Stieltjes Transforms Applications of Stieltjes Transforms The Generalized Stieltjes Transform Basic Properties of the Generalized Stieltjes Transform Exercises FINITE FOURIER SINE AND COSINE TRANSFORMS Introduction Definitions of the Finite Fourier Sine and Cosine Transforms and Examples Basic Properties of Finite Fourier Sine and Cosine Transforms Applications of Finite Fourier Sine and Cosine Transforms Multiple Finite Fourier Transforms and Their Applications Exercises FINITE LAPLACE TRANSFORMS Introduction Definition of the Finite Laplace Transform and Examples Basic Operational Properties of the Finite Laplace Transform Applications of Finite Laplace Transforms Tauberian Theorems Exercises Z TRANSFORMS Introduction Dynamic Linear Systems and Impulse Response Definition of the Z Transform and Examples Basic Operational Properties The Inverse Z Transform and Examples Applications of Z Transforms to Finite Difference Equations Summation of Infinite Series Exercises FINITE HANKEL TRANSFORMS Introduction Definition of the Finite Hankel Transform and Examples Basic Operational Properties Applications of Finite Hankel Transforms Exercises LEGENDRE TRANSFORMS Introduction Definition of the Legendre Transform and examples Basic Operational Properties of Legendre Transforms Applications of Legendre Transforms to Boundary Value Problems Exercises JACOBI AND GEGENBAUER TRANSFORMS Introduction Definition of the Jacobi Transform and Examples Basic Operational Properties Applications of Jacobi Transforms to the Generalized Heat Conduction Problem The Gegenbauer Transform and its Basic Operational Properties Application of the Gegenbauer Transform LAGUERRE TRANSFORMS Introduction Definition of the Laguerre Transform and Examples Basic Operational Properties Applications of Laguerre Transforms Exercises HERMITE TRANSFORMS Introduction Definition of the Hermite Transform and Examples Basic Operational Properties Exercises THE RADON TRANSFORM AND ITS APPLICATION Introduction Radon Transform Properties of Radon Transform Radon Transform of Derivatives Derivatives of Radon Transform Convolution Theorem for Radon Transform Inverse of Radon Transform Exercises WAVELETS AND WAVELET TRANSFORMS Brief Historical Remarks Continuous Wavelet Transforms The Discrete Wavelet Transform Examples of Orthonormal Wavelets Exercises Appendix A Some Special Functions and Their Properties A-1 Gamma, Beta, and Error Functions A-2 Bessel and Airy Functions A-3 Legendre and Associated Legendre Functions A-4 Jacobi and Gegenbauer Polynomials A-5 Laguerre and Associated Laguerre Functions A-6 Hermite and Weber-Hermite Functions A-7 Hurwitz and Riemann zeta Functions Appendix B Tables of Integral Transforms B-1 Fourier Transforms B-2 Fourier Cosine Transforms B-3 Fourier Sine Transforms B-4 Laplace Transforms B-5 Hankel Transforms B-6 Mellin Transforms B-7 Hilbert Transforms B-8 Stieltjes Transforms B-9 Finite Fourier Cosine Transforms B-10 Finite Fourier Sine Transforms B-11 Finite Laplace Transforms B-12 Z Transforms B-13 Finite Hankel Transforms Answers and Hints to Selected Exercises Bibliography Index

Embedded pupil function recovery for Fourier ptychographic microscopy

Abstract: We develop and test a pupil function determination algorithm, termed embedded pupil function recovery (EPRY), which can be incorporated into the Fourier ptychographic microscopy (FPM) algorithm and recover both the Fourier spectrum of sample and the pupil function of imaging system simultaneously. This EPRY-FPM algorithm eliminates the requirement of the previous FPM algorithm for a priori knowledge of the aberration in the imaging system to reconstruct a high quality image. We experimentally demonstrate the effectiveness of this algorithm by reconstructing high resolution, large field-of-view images of biological samples. We also illustrate that the pupil function we retrieve can be used to study the spatially varying aberration of a large field-of-view imaging system. We believe that this algorithm adds more flexibility to FPM and can be a powerful tool for the characterization of an imaging system’s aberration.

On the computational complexity of the empirical mode decomposition algorithm

Abstract: It has been claimed that the empirical mode decomposition (EMD) and its improved version the ensemble EMD (EEMD) are computation intensive. In this study we will prove that the time complexity of the EMD/EEMD, which has never been analyzed before, is actually equivalent to that of the Fourier Transform. Numerical examples are presented to verify that EMD/EEMD is, in fact, a computationally efficient method.

Maneuvering Target Detection via Radon-Fractional Fourier Transform-Based Long-Time Coherent Integration

Abstract: Long-time coherent integration technique is one of the most important methods for the improvement of radar detection ability of a weak maneuvering target, whereas the integration performance may be greatly influenced by the across range unit (ARU) and Doppler frequency migration (DFM) effects. In this paper, a novel representation known as Radon-fractional Fourier transform (RFRFT) is proposed and investigated to solve the above problems simultaneously. It can not only eliminate the effect of DFM by selecting a proper rotation angle but also achieve long-time coherent integration without ARU effect. The RFRFT can be regarded as a special Doppler filter bank composed of filters with different rotation angles, which indicates a generalization of the traditional moving target detection (MTD) and FRFT methods. Some useful properties and the likelihood ratio test detector of RFRFT are derived for maneuvering target detection. Finally, numerical experiments of aerial target and marine target detection are carried out using simulated and real radar datasets. The results demonstrate that for integration gain and detection ability, the proposed method is superior to MTD, FRFT, and Radon-Fourier transform under low signal-to-clutter/noise ratio (SCR/SNR) environments. Moreover, the trajectory of target can be easily obtained via RFRFT as well.

Fourier spectral methods for fractional-in-space reaction-diffusion equations

Abstract: Fractional differential equations are becoming increasingly used as a powerful modelling approach for understanding the many aspects of nonlocality and spatial heterogeneity. However, the numerical approximation of these models is demanding and imposes a number of computational constraints. In this paper, we introduce Fourier spectral methods as an attractive and easy-to-code alternative for the integration of fractional-in-space reaction-diffusion equations described by the fractional Laplacian in bounded rectangular domains of \(\mathbb {R}^n\). The main advantages of the proposed schemes is that they yield a fully diagonal representation of the fractional operator, with increased accuracy and efficiency when compared to low-order counterparts, and a completely straightforward extension to two and three spatial dimensions. Our approach is illustrated by solving several problems of practical interest, including the fractional Allen–Cahn, FitzHugh–Nagumo and Gray–Scott models, together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator.

Optimal point spread function design for 3D imaging.

Abstract: Optical imaging of single nanoscale objects such as a quantum dot, metallic nanoparticle, or a single molecule provides a powerful window into a variety of biological or material systems, and the physical problem of extracting maximum information from single emitters is an important goal. One application is single-particle tracking (SPT [1]), which relies upon extracting the spatial trajectory of a single moving molecular label, quantum dot, or metallic nanoparticle from a series of images. For example, a single mRNA particle can be localized and followed in a living cell in real-time [2]. Another application of single-molecule localization is “super-resolution” (SR) microscopy, [3–5] which works by ensuring that only a sparse subset of labels on an extended object (e.g. a cellular structure) are emitting in each imaging frame. One localizes the single emitters just as in SPT; the multitude of localizations are then reconstructed into a single, high-resolution image. This enables the spatial resolving power of SR microscopy to surpass the classical diffraction resolution limit by 5- to 10-fold. Historically, single-particle localization was used for 2D imaging, namely, inferring the x,y coordinates of each emitter, e.g. by centroid-fitting of by fitting to a 2D Gaussian [6]. However, the third spatial dimension, z, or the depth of an emitter, can also be inferred from its measured 2D image. This can be done by considering how the shape of the microscope’s point spread function (PSF) varies with emitter position. The PSF of a microscope is the image that is detected when observing a point source. For a standard microscope, to a good approximation, the PSF in focus (i.e. z=0) resembles a circular Airy pattern, and its shape is invariant to lateral shifts (x,y) of the emitter – however it will change upon defocus (z). Unfortunately, the standard PSF spreads out (defocuses) quickly with z which limits the range over which z can be determined. Importantly, to obtain much more useful 3D position information, the PSF of the microscope can be altered – for example by pupil (Fourier) plane processing [7,8]. Phase modulating the electromagnetic field in the Fourier plane is a low-loss method to encode z-information in the shape of the image on the camera. Examples of this include astigmatic PSFs [9,10], double-helix (DH-PSF) microscopy [11–13] or segmented phase ramps [14]. The precision to which a single emitter can be localized depends on several factors. These include the emitter’s brightness (detected photon flux), background fluorescence, detector pixel size, and detection noise [15,16]. Another key factor is the shape of the PSF itself. For example, in astigmatism-based 3D imaging, the PSF is altered to have an elliptical shape, and the z position of the emitter can be determined by the relative widths of the PSF along the two principal axes [9,10]. The double-helix PSF [12,13] is composed of two spots, with the angle between a line connecting them and the camera axis encoding the z position of the emitter. Among existing PSFs for 3D imaging, the double-helix PSF has been shown to allow a larger depth of field than astigmatism (~2–3 μm vs ~0.5–0.7 μm) [17], and a recently suggested PSF based on accelerating beams [18] demonstrates high, uniform precision over a 3 μm range. The purpose of this paper is to fundamentally improve upon these previous schemes. Here, we address the problem of finding a feasible and optimally informative PSF. Namely, we ask the question – given an imaging scenario with certain characteristics (e.g. magnification, noise level, pixel size, emitter signal) – what is the pupil plane pattern that would yield maximal physical information about the 3D position of an emitter, and what is the resulting optimal PSF? In other words – since localization precision depends on the PSF of the system – can we design the system to have a PSF that would yield the best possible precision in determining x, y, and z, compared to any other PSF? We regard such a PSF as optimally informative. A powerful measure of the effectiveness of a PSF for encoding an emitter’s position is based on Fisher information [17,19,20], a concept from statistical information theory. Fisher information is a mathematical measure of the sensitivity of an observable quantity (the PSF) to changes in its underlying parameters (emitter position). Using the Fisher information function, one may compute the Cramer-Rao lower bound (CRLB), which is the theoretical best-case x,y,z precision that can be attained (with any unbiased estimator) given a PSF and a noise model. With the right estimator, the best-case localization precision represented by the CRLB can be approached in practice [21–23]. Traditionally, the CRLB has been used as an analysis tool, i.e. to evaluate the performance of an existing PSF design, which is often conceived using physical intuition and reasonable requirements (e.g. a significant change of the PSF over the z-range of interest, and concentration of emitted light into small spots). The CRLB has also been used to fine-tune an existing PSF [24]. To find the optimal pupil plane pattern, and thereby the optimal PSF, we propose a new approach to PSF design –we treat the PSF as a free design parameter of the imaging system, and generate PSFs with optimal photon-efficient 3D position encoding, with no prior constraints on the shape of the PSF. This is achieved by CRLB optimization – that is, we directly solve the mathematical optimization problem of minimizing the CRLB (and hence improving the precision bound) of the system, and use the resulting PSF. Such a PSF will provide optimal precision by definition. Physically reasonable requirements are accounted for by using realistic imaging and noise models, including pixelation, photon shot-noise Poisson statistics, and background fluorescence. This enables us to demonstrate, for typical experimental conditions and without scanning, the highest theoretical precision to date over a 3 μm axial range, as well as <50 nm experimental precision for an unprecedented ~5 μm range.

Light Field Reconstruction Using Sparsity in the Continuous Fourier Domain

Abstract: Sparsity in the Fourier domain is an important property that enables the dense reconstruction of signals, such as 4D light fields, from a small set of samples. The sparsity of natural spectra is often derived from continuous arguments, but reconstruction algorithms typically work in the discrete Fourier domain. These algorithms usually assume that sparsity derived from continuous principles will hold under discrete sampling. This article makes the critical observation that sparsity is much greater in the continuous Fourier spectrum than in the discrete spectrum. This difference is caused by a windowing effect. When we sample a signal over a finite window, we convolve its spectrum by an infinite sinc, which destroys much of the sparsity that was in the continuous domain. Based on this observation, we propose an approach to reconstruction that optimizes for sparsity in the continuous Fourier spectrum. We describe the theory behind our approach and discuss how it can be used to reduce sampling requirements and improve reconstruction quality. Finally, we demonstrate the power of our approach by showing how it can be applied to the task of recovering non-Lambertian light fields from a small number of 1D viewpoint trajectories.

Anisotropic failure of Fourier theory in time-domain thermoreflectance experiments

Abstract: The applicability of Fourier's law to heat transfer problems relies on the assumption that heat carriers have mean free paths smaller than important length scales of the temperature profile. This assumption is not generally valid in nanoscale thermal transport problems where spacing between boundaries is small (<1 μm), and temperature gradients vary rapidly in space. Here we study the limits to Fourier theory for analysing three-dimensional heat transfer problems in systems with an interface. We characterize the relationship between the failure of Fourier theory, phonon mean free paths, important length scales of the temperature profile and interfacial-phonon scattering by time-domain thermoreflectance experiments on Si, Si0.99Ge0.01, boron-doped Si and MgO crystals. The failure of Fourier theory causes anisotropic thermal transport. In situations where Fourier theory fails, a simple radiative boundary condition on the heat diffusion equation cannot adequately describe interfacial thermal transport.

Information Transmission Using the Nonlinear Fourier Transform, Part I: Mathematical Tools

Abstract: The nonlinear Fourier transform (NFT), a powerful tool in soliton theory and exactly solvable models, is a method for solving integrable partial differential equations governing wave propagation in certain nonlinear media. The NFT decorrelates signal degrees-of-freedom in such models, in much the same way that the Fourier transform does for linear systems. In this three-part series of papers, this observation is exploited for data transmission over integrable channels, such as optical fibers, where pulse propagation is governed by the nonlinear Schrodinger equation. In this transmission scheme, which can be viewed as a nonlinear analogue of orthogonal frequency-division multiplexing commonly used in linear channels, information is encoded in the nonlinear frequencies and their spectral amplitudes. Unlike most other fiber-optic transmission schemes, this technique deals with both dispersion and nonlinearity directly and unconditionally without the need for dispersion or nonlinearity compensation methods. This paper explains the mathematical tools that underlie the method.

The fourier-based synchrosqueezing transform

Abstract: The short-time Fourier transform (STFT) and the continuous wavelet transform (CWT) are extensively used to analyze and process multicomponent signals, i.e. superpositions of modulated waves. The synchrosqueezing is a post-processing method which circumvents the uncertainty relation inherent to these linear transforms, by reassigning the coefficients in scale or frequency. Originally introduced in the setting of the CWT, it provides a sharp, concentrated representation, while remaining invertible. This technique received a renewed interest with the recent publication of an approximation result related to the application of the synchrosqueezing to multi-component signals. In the current paper, we adapt the formulation of the synchrosqueezing to the STFT and state a similar theoretical result to that obtained in the CWT framework. The emphasis is put on the differences with the CWT-based synchrosqueezing with numerical experiments illustrating our statements.

Information Transmission Using the Nonlinear Fourier Transform, Part II: Numerical Methods

Abstract: In this paper, numerical methods are suggested to compute the discrete and the continuous spectrum of a signal with respect to the Zakharov-Shabat system, a Lax operator underlying numerous integrable communication channels including the nonlinear Schrodinger channel, modeling pulse propagation in optical fibers. These methods are subsequently tested and their ability to estimate the spectrum are compared against each other. These methods are used to compute the spectrum of various signals commonly used in the optical fiber communications. It is found that the layer peeling and the spectral methods are suitable schemes to estimate the nonlinear spectra with good accuracy. To illustrate the structure of the spectrum, the locus of the eigenvalues is determined under amplitude and phase modulation in a number of examples. It is observed that in some cases, as signal parameters vary, eigenvalues collide and change their course of motion. The real axis is typically the place from which new eigenvalues originate or, are absorbed into after traveling a trajectory in the complex plane.

Spectral estimation—What is new? What is next?

Abstract: Spectral estimation, and corresponding time-frequency representation for nonstationary signals, is a cornerstone in geophysical signal processing and interpretation. The last 10-15 years have seen the development of many new high-resolution decompositions that are often fundamentally different from Fourier and wavelet transforms. These conventional techniques, like the short-time Fourier transform and the continuous wavelet transform, show some limitations in terms of resolution (localization) due to the trade-off between time and frequency localizations and smearing due to the finite size of the time series of their template. Well-known techniques, like autoregressive methods and basis pursuit, and recently developed techniques, such as empirical mode decomposition and the synchrosqueezing transform, can achieve higher time-frequency localization due to reduced spectral smearing and leakage. We first review the theory of various established and novel techniques, pointing out their assumptions, adaptability, and expected time-frequency localization. We illustrate their performances on a provided collection of benchmark signals, including a laughing voice, a volcano tremor, a microseismic event, and a global earthquake, with the intention to provide a fair comparison of the pros and cons of each method. Finally, their outcomes are discussed and possible avenues for improvements are proposed.

Stable and Robust Sampling Strategies for Compressive Imaging

Abstract: In many signal processing applications, one wishes to acquire images that are sparse in transform domains such as spatial finite differences or wavelets using frequency domain samples. For such applications, overwhelming empirical evidence suggests that superior image reconstruction can be obtained through variable density sampling strategies that concentrate on lower frequencies. The wavelet and Fourier transform domains are not incoherent because low-order wavelets and low-order frequencies are correlated, so compressive sensing theory does not immediately imply sampling strategies and reconstruction guarantees. In this paper, we turn to a more refined notion of coherence-the so-called local coherence-measuring for each sensing vector separately how correlated it is to the sparsity basis. For Fourier measurements and Haar wavelet sparsity, the local coherence can be controlled and bounded explicitly, so for matrices comprised of frequencies sampled from a suitable inverse square power-law density, we can prove the restricted isometry property with near-optimal embedding dimensions. Consequently, the variable-density sampling strategy we provide allows for image reconstructions that are stable to sparsity defects and robust to measurement noise. Our results cover both reconstruction by l1-minimization and total variation minimization. The local coherence framework developed in this paper should be of independent interest, as it implies that for optimal sparse recovery results, it suffices to have bounded average coherence from sensing basis to sparsity basis-as opposed to bounded maximal coherence-as long as the sampling strategy is adapted accordingly.

High-resolution fluorescence imaging via pattern-illuminated Fourier ptychography

Abstract: Fluorescence microscopy plays a vital role in modern biological research and clinical diagnosis. Here, we report an imaging approach, termed pattern-illuminated Fourier ptychography (FP), for fluorescence imaging beyond the diffraction limit of the employed optics. This approach iteratively recovers a high-resolution fluorescence image from many pattern-illuminated low-resolution intensity measurements. The recovery process starts with one low-resolution measurement as the initial guess. This initial guess is then sequentially updated by other measurements, both in the spatial and Fourier domains. In the spatial domain, we use the pattern-illuminated low-resolution images as intensity constraints for the sample estimate. In the Fourier domain, we use the incoherent optical-transfer-function of the objective lens as the object support constraint for the solution. The sequential updating process is then repeated until the sample estimate converges, typically for 5-20 times. Different from the conventional structured illumination microscopy, any unknown pattern can be used for sample illumination in the reported framework. In particular, we are able to recover both the high-resolution sample image and the unknown illumination pattern at the same time. As a demonstration, we improved the resolution of a conventional fluorescence microscope beyond the diffraction limit of the employed optics. The reported approach may provide an alternative solution for structure illumination microscopy and find applications in wide-field, high-resolution fluorescence imaging.

Encoding complex fields by using a phase-only optical element

Abstract: We show that the amplitude and phase information from a two-dimensional complex field can be synthesized from a phase-only optical element with micrometric resolution. The principle of the method is based on the combination of two spatially sampled phase elements by using a low-pass filter at the Fourier plane of a 4-f optical system. The proposed encoding technique was theoretically demonstrated, as well as experimentally validated with the help of a phase-only spatial light modulator for phase encoding, a conventional CMOS camera to measure the amplitude of the complex field, and a Shack-Hartmann wavefront sensor to determine its phase.

Detection and Extraction of Target With Micromotion in Spiky Sea Clutter Via Short-Time Fractional Fourier Transform

Abstract: In order to effectively detect moving targets in heavy sea clutter, the micro-Doppler (m-D) effect is studied and an effective algorithm based on short-time fractional Fourier transform (STFRFT) is proposed for target detection and m-D signal extraction. Firstly, the mathematical model of target with micromotion at sea, including translation and rotation movement, is established, which can be approximated as the sum of linear-frequency-modulated signals within a short time. Then, due to the high-power, time-varying, and target-like properties of sea spikes, which may result in poor detection performance, sea spikes are identified and eliminated before target detection to improve signal-to-clutter ratio (SCR). By taking the absolute amplitude of signals in the best STFRFT domain (STFRFD) as the test statistic, and comparing it with the threshold determined by a constant false alarm rate detector, micromotion target can be declared or not. STFRFT with Gaussian window is employed to provide time-frequency distribution of m-D signals, and the instantaneous frequency of each component can be extracted and estimated precisely by STFRFD filtering. In the end, datasets from the intelligent pixel processing radar with HH and VV polarizations are used to verify the validity of this proposed algorithm. Two shore-based experiments are also conducted using an X-band sea search radar and an S-band sea surveillance radar, respectively. The results demonstrate that the proposed method not only achieves high detection probability in a low-SCR environment but also outperforms the short-time Fourier transform-based method.

Extending Single-Molecule Microscopy Using Optical Fourier Processing

Abstract: This article surveys the recent application of optical Fourier processing to the long-established but still expanding field of single-molecule imaging and microscopy. A variety of single-molecule studies can benefit from the additional image information that can be obtained by modulating the Fourier, or pupil, plane of a widefield microscope. After briefly reviewing several current applications, we present a comprehensive and computationally efficient theoretical model for simulating single-molecule fluorescence as it propagates through an imaging system. Furthermore, we describe how phase/amplitude-modulating optics inserted in the imaging pathway may be modeled, especially at the Fourier plane. Finally, we discuss selected recent applications of Fourier processing methods to measure the orientation, depth, and rotational mobility of single fluorescent molecules.

An FFT-based Galerkin method for homogenization of periodic media

Abstract: In 1994, Moulinec and Suquet introduced an efficient technique for the numerical resolution of the cell problem arising in homogenization of periodic media. The scheme is based on a fixed-point iterative solution to an integral equation of the Lippmann-Schwinger type, with action of its kernel efficiently evaluated by the Fast Fourier Transform techniques. The aim of this work is to demonstrate that the Moulinec-Suquet setting is actually equivalent to a Galerkin discretization of the cell problem, based on approximation spaces spanned by trigonometric polynomials and a suitable numerical integration scheme. For the latter framework and scalar elliptic problems, we prove convergence of the approximate solution to the weak solution, including a-priori estimates for the rate of convergence for sufficiently regular data and the effects of numerical integration. Moreover, we also show that the variational structure implies that the resulting non-symmetric system of linear equations can be solved by the conjugate gradient method. Apart from providing a theoretical support to Fast Fourier Transform-based methods for numerical homogenization, these findings significantly improve on the performance of the original solver and pave the way to similar developments for its many generalizations proposed in the literature.

Nonlinear inverse synthesis for high spectral efficiency transmission in optical fibers

Abstract: In linear communication channels, spectral components (modes) defined by the Fourier transform of the signal propagate without interactions with each other. In certain nonlinear channels, such as the one modelled by the classical nonlinear Schr\"odinger equation, there are nonlinear modes (nonlinear signal spectrum) that also propagate without interacting with each other and without corresponding nonlinear cross talk; effectively, in a linear manner. Here, we describe in a constructive way how to introduce such nonlinear modes for a given input signal. We investigate the performance of the nonlinear inverse synthesis (NIS) method, in which the information is encoded directly onto the continuous part of the nonlinear signal spectrum. This transmission technique, combined with the appropriate distributed Raman amplification, can provide an effective eigenvalue division multiplexing with high spectral efficiency, thanks to highly suppressed channel cross talk. The proposed NIS approach can be integrated with any modulation formats. Here, we demonstrate numerically the feasibility of merging the NIS technique in a burst mode with high spectral efficiency methods, such as orthogonal frequency division multiplexing and Nyquist pulse shaping with advanced modulation formats (e.g., QPSK, 16QAM, and 64QAM), showing a performance improvement up to 4.5 dB, which is comparable to results achievable with multi-step per span digital back propagation.

Time-Frequency Analysis of Seismic Data Using Synchrosqueezing Transform

Abstract: Time-frequency analysis can provide useful information in seismic data processing and interpretation. An accurate time-frequency representation is important in highlighting subtle geologic structures and in detecting anomalies associated with hydrocarbon reservoirs. The popular methods, like short-time Fourier transform and wavelet analysis, have limitations in dealing with fast varying instantaneous frequencies, which is often the characteristic of seismic data. The synchrosqueezing transform (SST) is a promising tool to provide a detailed time-frequency representation. We apply the SST to seismic data and show its potential to seismic signal processing applications.

Fourier transform and the global Gan–Gross–Prasad conjecture for unitary groups

Abstract: By the relative trace formula approach of Jacquet�Rallis, we prove the global Gan�Gross�Prasad conjecture for unitary groups under some local restrictions for the automorphic representations

Discussion and a new attack of the optical asymmetric cryptosystem based on phase-truncated Fourier transform

Abstract: A discussion and a cryptanalysis of the optical phase-truncated Fourier-transform-based cryptosystem are presented in this paper. The concept of an optical asymmetric cryptosystem, which was introduced into the optical image encryption scheme based on phase-truncated Fourier transforms in 2010, is suggested to be retained in optical encryption. A new method of attack is also proposed to simultaneously obtain the main information of the original image, the two decryption keys from its cyphertext, and the public keys based on the modified amplitude-phase retrieval algorithm. The numerical results illustrate that the computing efficiency of the algorithm is improved and the number of iterations is much less than that by the specific attack, which has two iteration loops.

Recent Developments in the Sparse Fourier Transform: A compressed Fourier transform for big data

Abstract: The discrete Fourier transform (DFT) is a fundamental component of numerous computational techniques in signal processing and scientific computing. The most popular means of computing the DFT is the fast Fourier transform (FFT). However, with the emergence of big data problems, in which the size of the processed data sets can easily exceed terabytes, the "fast" in FFT is often no longer fast enough. In addition, in many big data applications it is hard to acquire a sufficient amount of data to compute the desired Fourier transform in the first place. The sparse Fourier transform (SFT) addresses the big data setting by computing a compressed Fourier transform using only a subset of the input data, in time smaller than the data set size. The goal of this article is to survey these recent developments, explain the basic techniques with examples and applications in big data, demonstrate tradeoffs in empirical performance of the algorithms, and discuss the connection between the SFT and other techniques for massive data analysis such as streaming algorithms and compressive sensing.

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 $O(D\log^2D)$ runtime. Extensions of the results to other evolution equations are discussed as well.

ISAR Imaging of a Ship Target Based on Parameter Estimation of Multicomponent Quadratic Frequency-Modulated Signals

Abstract: High-resolution inverse synthetic aperture radar (ISAR) imaging of a ship target is a challenging task because of fluctuation with the ocean waves. The images obtained with a standard range-Doppler algorithm are usually blurred. Consequently, the range-instantaneous-Doppler (RID) technique should be used to improve the image quality. In this paper, the received signal in a range cell is modeled as a multicomponent quadratic frequency-modulated (QFM) signal after range compression and motion compensation, and then a new RID ISAR imaging algorithm is proposed that introduces a new method for estimating parameters of the QFM signal. By defining a new function and using the scaled Fourier transform (SCFT) with respect to the time axis, the coherent integration of auto-terms can be realized via the subsequent Fourier transformation with respect to the lag-time axis, and a peak can be obtained in the 2-D frequency plane, which is appropriate for parameter estimation of the QFM signal to reconstruct RID images. The proposed algorithm is accurate and fast since the defined function has moderate order nonlinearity and the SCFT can be performed via chirp z-transform. Experiments demonstrate the performance of the new algorithm. Comparisons with existing algorithms are also given, which show that the proposed algorithm can efficiently produce a focused image with less fake scatterers.

Communication: The description of strong correlation within self-consistent Green's function second-order perturbation theory

Abstract: We report an implementation of self-consistent Green's function many-body theory within a second-order approximation (GF2) for application with molecular systems. This is done by iterative solution of the Dyson equation expressed in matrix form in an atomic orbital basis, where the Green's function and self-energy are built on the imaginary frequency and imaginary time domain, respectively, and fast Fourier transform is used to efficiently transform these quantities as needed. We apply this method to several archetypical examples of strong correlation, such as a H32 finite lattice that displays a highly multireference electronic ground state even at equilibrium lattice spacing. In all cases, GF2 gives a physically meaningful description of the metal to insulator transition in these systems, without resorting to spin-symmetry breaking. Our results show that self-consistent Green's function many-body theory offers a viable route to describing strong correlations while remaining within a computationally tractable single-particle formalism.

Theory and modelling of constant-Q P- and S-waves using fractional spatial derivatives

Abstract: I have developed and solved the constant-Q model for the attenuation of P- and S-waves in the time domain using a new modeling algorithm based on fractional derivatives. The model requires time derivatives of order m 2 applied to the strain components, where m 0,1,... and 1/tan 1 1/Q, with Q the P-wave or S-wave quality factor. The derivatives are computed with the Grunwald-Letnikov and central-difference fractional approximations, which are extensions of the standard finite-difference operators for derivatives of integer order. The modeling uses the Fourier method to compute the spatial derivatives, and therefore can handle complex geometries and general materialproperty variability. I verified the results by comparison with the 2D analytical solution obtained for wave propagation in homogeneous Pierre Shale. Moreover, the modeling algorithm was used to compute synthetic seismograms in heterogeneous media corresponding to a crosswell seismic experi ment.

Content adaptive illumination for Fourier ptychography.

Abstract: Fourier ptychography (FP) is a recently reported technique, for large field-of-view and high-resolution imaging. Specifically, FP captures a set of low-resolution images, under angularly varying illuminations, and stitches them together in the Fourier domain. One of FP’s main disadvantages is its long capturing process, due to the requisite large number of incident illumination angles. In this Letter, utilizing the sparsity of natural images in the Fourier domain, we propose a highly efficient method, termed adaptive Fourier ptychography (AFP), which applies content adaptive illumination for FP, to capture the most informative parts of the scene’s spatial spectrum. We validate the effectiveness and efficiency of the reported framework, with both simulated and real experiments. Results show that the proposed AFP could shorten the acquisition time of conventional FP, by around 30%–60%.

On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions

Abstract: We prove that the set of exceptional $${\lambda\in (1/2,1)}$$ such that the associated Bernoulli convolution is singular has zero Hausdorff dimension, and likewise for biased Bernoulli convolutions, with the exceptional set independent of the bias. This improves previous results by Erdos, Kahane, Solomyak, Peres and Schlag, and Hochman. A theorem of this kind is also obtained for convolutions of homogeneous self-similar measures. The proofs are very short, and rely on old and new results on the dimensions of self-similar measures and their convolutions, and the decay of their Fourier transform.