Showing papers on "Fourier transform published in 2018"

Experimental pulse NMR : a nuts and bolts approach

Abstract: * Basics of Pulse and Fourier Transform NMR * Details of Pulse and Fourier Transform NMR * Relaxation * NMR of Solids * NMR Hardware * Practical Techniques * Postscript

Real-time full-field characterization of transient dissipative soliton dynamics in a mode-locked laser

Abstract: Dissipative solitons are remarkably localized states of a physical system that arise from the dynamical balance between nonlinearity, dispersion and environmental energy exchange. They are the most universal form of soliton that can exist, and are seen in far-from-equilibrium systems in many fields, including chemistry, biology and physics. There has been particular interest in studying their properties in mode-locked lasers, but experiments have been limited by the inability to track the dynamical soliton evolution in real time. Here, we use simultaneous dispersive Fourier transform and time-lens measurements to completely characterize the spectral and temporal evolution of ultrashort dissipative solitons as their dynamics pass through a transient unstable regime with complex break-up and collisions before stabilization. Further insight is obtained from reconstruction of the soliton amplitude and phase and calculation of the corresponding complex-valued eigenvalue spectrum. These findings show how real-time measurements provide new insights into ultrafast transient dynamics in optics. The simultaneous use of dispersive Fourier transform and time-lens measurements allows complete characterization of the unstable spectral and temporal evolution of ultrashort dissipative solitons, providing further insight into ultrafast transient dynamics in optics.

ATOMO: Communication-efficient Learning via Atomic Sparsification

Abstract: Distributed model training suffers from communication overheads due to frequent gradient updates transmitted between compute nodes. To mitigate these overheads, several studies propose the use of sparsified stochastic gradients. We argue that these are facets of a general sparsification method that can operate on any possible atomic decomposition. Notable examples include element-wise, singular value, and Fourier decompositions. We present ATOMO, a general framework for atomic sparsification of stochastic gradients. Given a gradient, an atomic decomposition, and a sparsity budget, ATOMO gives a random unbiased sparsification of the atoms minimizing variance. We show that recent methods such as QSGD and TernGrad are special cases of ATOMO, and that sparsifiying the singular value decomposition of neural networks gradients, rather than their coordinates, can lead to significantly faster distributed training.

Efficient High Dimensional Bayesian Optimization with Additivity and Quadrature Fourier Features

Abstract: We develop an efficient and provably no-regret Bayesian optimization (BO) algorithm for optimization of black-box functions in high dimensions We assume a generalized additive model with possibly overlapping variable groups When the groups do not overlap, we are able to provide the first provably no-regret \emph{polynomial time} (in the number of evaluations of the acquisition function) algorithm for solving high dimensional BO To make the optimization efficient and feasible, we introduce a novel deterministic Fourier Features approximation based on numerical integration with detailed analysis for the squared exponential kernel The error of this approximation decreases \emph{exponentially} with the number of features, and allows for a precise approximation of both posterior mean and variance In addition, the kernel matrix inversion improves in its complexity from cubic to essentially linear in the number of data points measured in basic arithmetic operations

Single-Image Depth Estimation Based on Fourier Domain Analysis

Abstract: We propose a deep learning algorithm for single-image depth estimation based on the Fourier frequency domain analysis. First, we develop a convolutional neural network structure and propose a new loss function, called depth-balanced Euclidean loss, to train the network reliably for a wide range of depths. Then, we generate multiple depth map candidates by cropping input images with various cropping ratios. In general, a cropped image with a small ratio yields depth details more faithfully, while that with a large ratio provides the overall depth distribution more reliably. To take advantage of these complementary properties, we combine the multiple candidates in the frequency domain. Experimental results demonstrate that proposed algorithm provides the state-of-art performance. Furthermore, through the frequency domain analysis, we validate the efficacy of the proposed algorithm in most frequency bands.

Simultaneous spatial, spectral, and 3D compressive imaging via efficient Fourier single-pixel measurements

Abstract: Single-pixel imaging can capture images using a detector without spatial resolution, which enables imaging in various situations that are challenging or impossible with conventional pixelated detectors. Here we report a compressive single-pixel imaging approach that can simultaneously encode and recover spatial, spectral, and 3D information of the object. In this approach, we modulate and condense the object information in the Fourier space and detect the light signals using a single-pixel detector. The data-compressing operation is similar to conventional compression algorithms that selectively store the largest coefficients of a transform domain. In our implementation, we selectively sample the largest Fourier coefficients, and no iterative optimization process is needed in the recovery process. We demonstrate an 88% compression ratio for producing a high-quality full-color 3D image. The reported approach provides a solution for information multiplexing in single-pixel imaging settings. It may also generate new insights for developing multi-modality computational imaging systems.

Metasurface Enabled Wide-Angle Fourier Lens.

Abstract: Fourier optics, the principle of using Fourier transformation to understand the functionalities of optical elements, lies at the heart of modern optics, and it has been widely applied to optical information processing, imaging, holography, etc. While a simple thin lens is capable of resolving Fourier components of an arbitrary optical wavefront, its operation is limited to near normal light incidence, i.e., the paraxial approximation, which puts a severe constraint on the resolvable Fourier domain. As a result, high-order Fourier components are lost, resulting in extinction of high-resolution information of an image. Other high numerical aperture Fourier lenses usually suffer from the bulky size and costly designs. Here, a dielectric metasurface consisting of high-aspect-ratio silicon waveguide array is demonstrated experimentally, which is capable of performing 1D Fourier transform for a large incident angle range and a broad operating bandwidth. Thus, the device significantly expands the operational Fourier space, benefitting from the large numerical aperture and negligible angular dispersion at large incident angles. The Fourier metasurface will not only facilitate efficient manipulation of spatial spectrum of free-space optical wavefront, but also be readily integrated into micro-optical platforms due to its compact size.

Non-Convex Phase Retrieval From STFT Measurements

Abstract: The problem of recovering a one-dimensional signal from its Fourier transform magnitude, called Fourier phase retrieval, is ill-posed in most cases. We consider the closely-related problem of recovering a signal from its phaseless short-time Fourier transform (STFT) measurements. This problem arises naturally in several applications, such as ultra-short laser pulse characterization and ptychography. The redundancy offered by the STFT enables unique recovery under mild conditions. We show that in some cases the unique solution can be obtained by the principal eigenvector of a matrix, constructed as the solution of a simple least-squares problem. When these conditions are not met, we suggest using the principal eigenvector of this matrix to initialize non-convex local optimization algorithms and propose two such methods. The first is based on minimizing the empirical risk loss function, while the second maximizes a quadratic function on the manifold of phases. We prove that under appropriate conditions, the proposed initialization is close to the underlying signal. We then analyze the geometry of the empirical risk loss function and show numerically that both gradient algorithms converge to the underlying signal even with small redundancy in the measurements. In addition, the algorithms are robust to noise.

Wavelet analysis of extended X-ray absorption fine structure data: Theory, application

Abstract: Fourier transform (FT) plays an indispensable role in the quantitative analysis of extended X-ray absorption fine structure (EXAFS). The fitting of FT-EXAFS has already solved many scientific issues. However, FT is not well suited for signals which involve transient processes. More and more complex and obscure systems require to be studied with the development of modern science and technology, especially the complex system showing overlapped single-/multi-scattering pathways in EXAFS spectrum, the unknown system involving atoms with similar atom numbers and some other unusual systems that cannot be solved only by the conventional FT and fitting method. Wavelet transform (WT) of EXAFS spectrums discerns the contribution of each pathway not only in R-space but also in k-space at the same time. The maximums of k-R contour map of the WT coefficients' modulus represent the contributions of specific pathways. Together with a priori knowledge or analysis of the system, WT k-R map helps us better understand the local structure and improve the fitting model. The most critical issues of WT analysis are how to improve the resolution with the least loss of information, and how to identify the contributions of different pathways quickly and accurately. To meet wider applications in the future, the WT method for EXAFS analysis still need to be improved.

Observation of Fourier transform limited lines in hexagonal boron nitride

Abstract: Single defect centers in layered hexagonal boron nitride are promising candidates as single-photon sources for quantum optics and nanophotonics applications. However, spectral instability hinders many applications. Here, we perform resonant excitation measurements and observe Fourier transform limited linewidths down to $\ensuremath{\approx}50$ MHz. We investigated the optical properties of more than 600 single-photon emitters (SPEs) in hBN. The SPEs exhibit narrow zero-phonon lines distributed over a spectral range from 580 to 800 nm and with dipolelike emission with a high polarization contrast. Finally, the emitters withstand transfer to a foreign photonic platform, namely, a silver mirror, which makes them compatible with photonic devices such as optical resonators and paves the way to quantum photonics applications.

Clebsch–Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network

Abstract: Recent work by Cohen et al. has achieved state-of-the-art results for learning spherical images in a rotation invariant way by using ideas from group representation theory and noncommutative harmonic analysis. In this paper we propose a generalization of this work that generally exhibits improved performace, but from an implementation point of view is actually simpler. An unusual feature of the proposed architecture is that it uses the Clebsch--Gordan transform as its only source of nonlinearity, thus avoiding repeated forward and backward Fourier transforms. The underlying ideas of the paper generalize to constructing neural networks that are invariant to the action of other compact groups.

Modulation Over Nonlinear Fourier Spectrum: Continuous and Discrete Spectrum

Abstract: Nonlinear frequency division multiplexed (NFDM) systems are considered when data are modulated in both parts of the nonlinear Fourier spectrum: Continuous spectrum and discrete spectrum An efficient algorithm is introduced to generate a time-domain signal from a given nonlinear spectrum The transmission of such NFDM symbols is experimentally demonstrated over 1460 km standard single-mode fiber with EDFA-only amplification In each NFDM symbol, the continuous spectrum is modulated by 64 $\times$ 05 Gbaud orthogonal frequency-division multiplexing (OFDM) symbols with 32-QAM format whereas the discrete spectrum contains four eigenvalues with the same imaginary part, each one is modulated by 8-PSK format, resulting a line rate of 553 Gb/s The cross-talk between different nonlinear modes is quantified in terms of cross-correlation and the performance loss is computed in terms of mutual information when each nonlinear modes is detected individually

Solving Fourier ptychographic imaging problems via neural network modeling and TensorFlow.

Abstract: Fourier ptychography is a recently developed imaging approach for large field-of-view and high-resolution microscopy. Here we model the Fourier ptychographic forward imaging process using a convolutional neural network (CNN) and recover the complex object information in a network training process. In this approach, the input of the network is the point spread function in the spatial domain or the coherent transfer function in the Fourier domain. The object is treated as 2D learnable weights of a convolutional or a multiplication layer. The output of the network is modeled as the loss function we aim to minimize. The batch size of the network corresponds to the number of captured low-resolution images in one forward/backward pass. We use a popular open-source machine learning library, TensorFlow, for setting up the network and conducting the optimization process. We analyze the performance of different learning rates, different solvers, and different batch sizes. It is shown that a large batch size with the Adam optimizer achieves the best performance in general. To accelerate the phase retrieval process, we also discuss a strategy to implement Fourier-magnitude projection using a multiplication neural network model. Since convolution and multiplication are the two most-common operations in imaging modeling, the reported approach may provide a new perspective to examine many coherent and incoherent systems. As a demonstration, we discuss the extensions of the reported networks for modeling single-pixel imaging and structured illumination microscopy (SIM). 4-frame resolution doubling is demonstrated using a neural network for SIM. The link between imaging systems and neural network modeling may enable the use of machine-learning hardware such as neural engine and tensor processing unit for accelerating the image reconstruction process. We have made our implementation code open-source for researchers.

A fast and efficient approach to color-image encryption based on compressive sensing and fractional Fourier transform

Abstract: This paper introduces a novel method of fast and efficient measurement matrices and random phase masks for color image encryption, in which Kronecker product (KP) is combined with chaotic map. The encryption scheme is based on two-dimension (2D) compressive sensing (CS) and fraction Fourier transform (FrFT). In this algorithm, the KP is employed to extend low dimension seed matrices to obtain high dimension measurement matrices and random phase masks. The low dimension seed matrices are generated by controlling chaotic map. The original image is simultaneously encrypted and compressed by the 2D CS, then re-encrypted with FrFT. The proposed encryption scheme fulfills high speed, low complexity and high security. Numerical simulation results demonstrate the excellent performance and security of the proposed scheme.

Total Variation--Based Phase Retrieval for Poisson Noise Removal

Abstract: Phase retrieval plays an important role in vast industrial and scientific applications. We consider a noisy phase retrieval problem in which the magnitudes of the Fourier transform (or a general linear transform) of an underling object are corrupted by Poisson noise, since any optical sensors detect photons, and the number of detected photons follows the Poisson distribution. We propose a variational model for phase retrieval based on a total variation regularization as an image prior and maximum a posteriori estimation of a Poisson noise model, which is referred to as “TV-PoiPR”. We also propose an efficient numerical algorithm based on an alternating direction method of multipliers and establish its convergence. Extensive experiments for coded diffraction, holographic, and ptychographic patterns are conducted using both real- and complex-valued images to demonstrate the effectiveness of our proposed methods.

A parallel non-uniform fast Fourier transform library based on an "exponential of semicircle" kernel

Abstract: The nonuniform fast Fourier transform (NUFFT) generalizes the FFT to off-grid data. Its many applications include image reconstruction, data analysis, and the numerical solution of differential equations. We present FINUFFT, an efficient parallel library for type 1 (nonuiform to uniform), type 2 (uniform to nonuniform), or type 3 (nonuniform to nonuniform) transforms, in dimensions 1, 2, or 3. It uses minimal RAM, requires no precomputation or plan steps, and has a simple interface to several languages. We perform the expensive spreading/interpolation between nonuniform points and the fine grid via a simple new kernel---the `exponential of semicircle' $e^{\beta \sqrt{1-x^2}}$ in $x\in[-1,1]$---in a cache-aware load-balanced multithreaded implementation. The deconvolution step requires the Fourier transform of the kernel, for which we propose efficient numerical quadrature. For types 1 and 2, rigorous error bounds asymptotic in the kernel width approach the fastest known exponential rate, namely that of the Kaiser--Bessel kernel. We benchmark against several popular CPU-based libraries, showing favorable speed and memory footprint, especially in three dimensions when high accuracy and/or clustered point distributions are desired.

Deep Learning Based Phase Reconstruction for Speaker Separation: A Trigonometric Perspective

Abstract: This study investigates phase reconstruction for deep learning based monaural talker-independent speaker separation in the short-time Fourier transform (STFT) domain. The key observation is that, for a mixture of two sources, with their magnitudes accurately estimated and under a geometric constraint, the absolute phase difference between each source and the mixture can be uniquely determined; in addition, the source phases at each time-frequency (T-F) unit can be narrowed down to only two candidates. To pick the right candidate, we propose three algorithms based on iterative phase reconstruction, group delay estimation, and phase-difference sign prediction. State-of-the-art results are obtained on the publicly available wsj0-2mix and 3mix corpus.

Stability of the Brascamp-Lieb constant and applications

Abstract: We prove that the best constant in the general Brascamp-Lieb inequality is a locally bounded function of the underlying linear transformations. As applications we deduce certain very general Fourier restriction, Kakeya-type, and nonlinear variants of the Brascamp-Lieb inequality which have arisen recently in harmonic analysis.

FNFT: A Software Library for Computing Nonlinear Fourier Transforms

Abstract: The conventional Fourier transform was originally developed in order to solve the heat equation, which is a standard example for a linear evolution equation. Nonlinear Fourier transforms (NFTs)1 are generalizations of the conventional Fourier transform that can be used to solve certain nonlinear evolution equations in a similar way (Ablowitz et al. 1974). An important difference to the conventional Fourier transform is that NFTs are equationspecific. The Korteweg-de Vries (KdV) equation (Gardner et al. 1967) and the nonlinear Schroedinger equation (NSE) (Shabat and Zakharov 1972) are two popular examples for nonlinear evolution equations that can be solved using appropriate NFTs.

Beyond Fourier Transform Infrared Spectroscopy: External Cavity Quantum Cascade Laser-Based Mid-infrared Transmission Spectroscopy of Proteins in the Amide I and Amide II Region.

Abstract: In this work, we present a setup for mid-IR measurements of the protein amide I and amide II bands in aqueous solution. Employing a latest generation external cavity-quantum cascade laser (EC-QCL) at room temperature in pulsed operation mode allowed implementing a high optical path length of 31 μm that ensures robust sample handling. By application of a data processing routine, which removes occasionally deviating EC-QCL scans, the noise level could be lowered by a factor of 4. The thereby accomplished signal-to-noise ratio is better by a factor of approximately 2 compared to research-grade Fourier transform infrared (FT-IR) spectrometers at equal acquisition times. Employing this setup, characteristic spectral features of three representative proteins with different secondary structures could be measured at concentrations as low as 1 mg mL-1. Mathematical evaluation of the spectral overlap confirms excellent agreement of the quantum cascade laser infrared spectroscropy (QCL-IR) transmission measurements with protein spectra acquired by FT-IR spectroscopy. The presented setup combines performance surpassing FT-IR spectroscopy with large applicable optical paths and coverage of the relevant spectral range for protein analysis. This holds high potential for future EC-QCL-based protein studies, including the investigation of dynamic secondary structure changes and chemometrics-based protein quantification in complex matrices.

Approximate Fast Graph Fourier Transforms via Multilayer Sparse Approximations

Abstract: The fast Fourier transform is an algorithm of paramount importance in signal processing as it allows to apply the Fourier transform in $\mathcal {O}(n \log n)$ instead of $\mathcal {O}(n^2)$ arithmetic operations. Graph signal processing is a recent research domain that generalizes classical signal processing tools, such as the Fourier transform, to situations where the signal domain is given by any arbitrary graph instead of a regular grid. Today, there is no method to rapidly apply graph Fourier transforms. In this paper, we propose a method to obtain approximate graph Fourier transforms that can be applied rapidly and stored efficiently. It is based on a greedy approximate diagonalization of the graph Laplacian matrix, carried out using a modified version of the famous Jacobi eigenvalues algorithm. The method is described and analyzed in detail, and then applied to both synthetic and real graphs, showing its potential.

Complex Variational Mode Decomposition for Slop-Preserving Denoising

Abstract: We have introduced a new decomposition method for seismic data, termed complex variational mode decomposition (VMD), and we have also designed a new filtering technique for random noise attenuation in seismic data by applying the VMD on constant-frequency slices in the frequency–offset ( $f$ – $x$ ) domain. The motivation behind this paper is to overcome the potential low performance of empirical mode decomposition (EMD) for energy preservation of the steeply dipping events when used for noise attenuation, and low resolution when used for signal decomposition. The VMD is proposed to decompose a signal into an ensemble of band-limited modes. For seismic data consisting of linear events, the constant-frequency slices of its $f$ – $x$ spectrum are exactly band-limited. The noise attenuation algorithm is summarized as follows. First, the Fourier transform is applied on the time axis of the 2-D seismic data. Next, the VMD is applied on each frequency slice of the $f$ – $x$ spectrum and the decomposed modes are combined to obtain the filtered frequency slice. Finally, an inverse Fourier transform is applied on the frequency axis of the $f$ – $x$ spectrum to obtain the denoised result. The resulting VMD-based noise attenuation method is equivalent to applying a Wiener filter on each decomposed mode, which is achieved during the decomposition progress. We also applied 2-D VMD on 3-D seismic data for denoising. Numerical results show that the proposed VMD-based method achieves a higher denoising quality than both the $f$ – $x$ deconvolution method and the EMD-based denoising method, especially for preserving the steep slopes.

Average decay of the Fourier transform of measures with applications

Abstract: Supported by the ERC grant 277778 and the MINECO grants MTM2013-41780-P, SEV-2015-0554 and MTM2017-85934-C3-1-P (Spain). The rst author has also been partially supported by FIRB 2012 "Dispersive dynamics: Fourier Analysis and Variational Methods" (Italy).

Random Fourier Features for Kernel Ridge Regression: Approximation Bounds and Statistical Guarantees

Abstract: Random Fourier features is one of the most popular techniques for scaling up kernel methods, such as kernel ridge regression. However, despite impressive empirical results, the statistical properties of random Fourier features are still not well understood. In this paper we take steps toward filling this gap. Specifically, we approach random Fourier features from a spectral matrix approximation point of view, give tight bounds on the number of Fourier features required to achieve a spectral approximation, and show how spectral matrix approximation bounds imply statistical guarantees for kernel ridge regression. Qualitatively, our results are twofold: on the one hand, we show that random Fourier feature approximation can provably speed up kernel ridge regression under reasonable assumptions. At the same time, we show that the method is suboptimal, and sampling from a modified distribution in Fourier space, given by the leverage function of the kernel, yields provably better performance. We study this optimal sampling distribution for the Gaussian kernel, achieving a nearly complete characterization for the case of low-dimensional bounded datasets. Based on this characterization, we propose an efficient sampling scheme with guarantees superior to random Fourier features in this regime.

Quantum linear systems algorithms: a primer

Abstract: The Harrow-Hassidim-Lloyd (HHL) quantum algorithm for sampling from the solution of a linear system provides an exponential speed-up over its classical counterpart. The problem of solving a system of linear equations has a wide scope of applications, and thus HHL constitutes an important algorithmic primitive. In these notes, we present the HHL algorithm and its improved versions in detail, including explanations of the constituent sub- routines. More specifically, we discuss various quantum subroutines such as quantum phase estimation and amplitude amplification, as well as the important question of loading data into a quantum computer, via quantum RAM. The improvements to the original algorithm exploit variable-time amplitude amplification as well as a method for implementing linear combinations of unitary operations (LCUs) based on a decomposition of the operators using Fourier and Chebyshev series. Finally, we discuss a linear solver based on the quantum singular value estimation (QSVE) subroutine.