Showing papers on "Fourier transform published in 2013"
TL;DR: The mathematical analysis of wavelet scattering networks explains important properties of deep convolution networks for classification.
Abstract: A wavelet scattering network computes a translation invariant image representation which is stable to deformations and preserves high-frequency information for classification. It cascades wavelet transform convolutions with nonlinear modulus and averaging operators. The first network layer outputs SIFT-type descriptors, whereas the next layers provide complementary invariant information that improves classification. The mathematical analysis of wavelet scattering networks explains important properties of deep convolution networks for classification. A scattering representation of stationary processes incorporates higher order moments and can thus discriminate textures having the same Fourier power spectrum. State-of-the-art classification results are obtained for handwritten digits and texture discrimination, with a Gaussian kernel SVM and a generative PCA classifier.
1,337 citations
TL;DR: Experimental results demonstrate the state-of-the-art denoising performance of BM4D, and its effectiveness when exploited as a regularizer in volumetric data reconstruction.
Abstract: We present an extension of the BM3D filter to volumetric data. The proposed algorithm, BM4D, implements the grouping and collaborative filtering paradigm, where mutually similar d -dimensional patches are stacked together in a (d+1) -dimensional array and jointly filtered in transform domain. While in BM3D the basic data patches are blocks of pixels, in BM4D we utilize cubes of voxels, which are stacked into a 4-D “group.” The 4-D transform applied on the group simultaneously exploits the local correlation present among voxels in each cube and the nonlocal correlation between the corresponding voxels of different cubes. Thus, the spectrum of the group is highly sparse, leading to very effective separation of signal and noise through coefficient shrinkage. After inverse transformation, we obtain estimates of each grouped cube, which are then adaptively aggregated at their original locations. We evaluate the algorithm on denoising of volumetric data corrupted by Gaussian and Rician noise, as well as on reconstruction of volumetric phantom data with non-zero phase from noisy and incomplete Fourier-domain (k-space) measurements. Experimental results demonstrate the state-of-the-art denoising performance of BM4D, and its effectiveness when exploited as a regularizer in volumetric data reconstruction.
748 citations
TL;DR: In this paper, the authors cover the principle of dispersive Fourier transformation and its implementation in diverse applications, such as optical rogue waves and rare cancer cells in blood, as well as their application in real-time instrumentation and measurement.
Abstract: It's challenging to measure non-repetitive events in real time in the field of instrumentation and measurement. Dispersive Fourier transformation is an emerging method that permits capture of rare events, such as optical rogue waves and rare cancer cells in blood. This Review article covers the principle of dispersive Fourier transformation and its implementation in diverse applications.
745 citations
Book•
02 Dec 2013
TL;DR: In this article, a general theory of data mapping and distribution theory is presented, along with the Fourier Transform of Causal Functions (FTF) for dimensionless variables.
Abstract: Preface.- Multi-Dimensional Seismic Inversion.- The One- Dimensional Inverse Problem.- Inversion in Higher Dimensions.- Large-Wavenumber Fourier Imaging.- Inversion in Heterogeneous Media.- Two-and-One Half Dimensional Inversion.- The General Theory of Data Mapping.- Distribution Theory.- The Fourier Transform of Causal Functions.- Dimensional vs. Dimensionless Variables.- An Example of Ill-Posedness.-An Elementary Introduction to Ray Theory.- Author Index.- Subject Index.
482 citations
Book•
14 Jun 2013
TL;DR: In this article, the Fourier Transform and Dirac Delta Function are used to measure the properties of an ultrasonic NDE with models, and the Stationary Phase Method (SPM) is used to scale the model-based defect sizing.
Abstract: 1. An Ultrasonic System. 2. Linear Systems and the Fourier Transform. 3. Fundamentals. 4. Propagation of Bulk Waves. 5. Reciprocal Theorem and Other Integral Relations. 6. Reflection and Refraction of Bulk Waves. 7. Propagation of Surface and Plate Waves. 8. Ultrasonic Transducer Radiation. 9. Materials Attenuation and Efficiency Factors. 10. Flaw Scattering. 11. Transducer Reception Process. 12. Ultrasonic Measurement Models. 13. Near-Field Measurement Models. 14. Quantitative Ultrasonic NDE with Models. 15. Model-Based Flaw Sizing. Appendixes: A. Fourier Transform. B. Dirac Delta Function. C. Basic Notations and Concepts. D. Hilbert Transform. E. Stationary Phase Method. F. Properties of Ellipsoids. Index.
314 citations
23 Jun 2013
TL;DR: In this paper, a fast convolutional sparse coding algorithm with globally optimal sub problems and super-linear convergence is proposed for sparse coding with signal processing and augmented Lagrange methods.
Abstract: Sparse coding has become an increasingly popular method in learning and vision for a variety of classification, reconstruction and coding tasks. The canonical approach intrinsically assumes independence between observations during learning. For many natural signals however, sparse coding is applied to sub-elements ( i.e. patches) of the signal, where such an assumption is invalid. Convolutional sparse coding explicitly models local interactions through the convolution operator, however the resulting optimization problem is considerably more complex than traditional sparse coding. In this paper, we draw upon ideas from signal processing and Augmented Lagrange Methods (ALMs) to produce a fast algorithm with globally optimal sub problems and super-linear convergence.
277 citations
TL;DR: The quaternionic Fourier transform applied to quaternion fields is treated and properties useful for applications are investigated and wide-ranging non-commutative multivector FT generalizations of the QFT are arrived at.
Abstract: We treat the quaternionic Fourier transform (QFT) applied to quaternion fields and investigate QFT properties useful for applications. Different forms of the QFT lead us to different Plancherel theorems. We relate the QFT computation for quaternion fields to the QFT of real signals. We research the general linear ($GL$) transformation behavior of the QFT with matrices, Clifford geometric algebra and with examples. We finally arrive at wide-ranging non-commutative multivector FT generalizations of the QFT. Examples given are new volume-time and spacetime algebra Fourier transformations.
201 citations
TL;DR: The proposed method for classification of fault and prediction of degradation of components and machines in manufacturing system and the result indicates its higher efficiency and effectiveness comparing to traditional methods.
Abstract: This paper proposes a method for classification of fault and prediction of degradation of components and machines in manufacturing system. The analysis is focused on the vibration signals collected from the sensors mounted on the machines for critical components monitoring. The pre-processed signals were decomposed into several signals containing one approximation and some details using Wavelet Packet Decomposition and, then these signals are transformed to frequency domain using Fast Fourier Transform. The features extracted from frequency domain could be used to train Artificial Neural Network (ANN). Trained ANN could predict the degradation (Remaining Useful Life) and identify the fault of the components and machines. A case study is used to illustrate the proposed method and the result indicates its higher efficiency and effectiveness comparing to traditional methods.
196 citations
TL;DR: A review of non-model based methodologies applied to diagnosis of Proton Exchange Membrane Fuel Cell (PEMFC) system is presented and hybrid approaches resulting from integration of different methods are believed to be promising.
184 citations
TL;DR: Intrinsic oxygen-vacancy defects, which are formed on TiO2(001) and TiO 2(101) surfaces by ultraviolet (UV) light irradiation and at elevated temperatures, are found to be most effective in anchoring the SO2 gas molecules to theTiO2 surfaces.
Abstract: Fixation of SO2 molecules on anatase TiO2 surfaces with defects have been investigated by first-principles density functional theory (DFT) calculations and in situ Fourier transform infrared (FTIR) ...
172 citations
TL;DR: Time-frequency analysis plays a significant role in seismic data processing and interpretation and demonstrates that this method promises higher spectral-spatial resolution than the short-time Fourier transform or wavelet transform.
Abstract: Time-frequency analysis plays a significant role in seismic data processing and interpretation. Complete ensemble empirical mode decomposition decomposes a seismic signal into a sum of oscillatory components, with guaranteed positive and smoothly varying instantaneous frequencies. Analysis on synthetic and real data demonstrates that this method promises higher spectral-spatial resolution than the short-time Fourier transform or wavelet transform. Application on field data thus offers the potential of highlighting subtle geologic structures that might otherwise escape unnoticed.
TL;DR: In this paper, the coherent dynamic of phonons in crystalline materials was observed using X-ray free-electron (XFED) laser pulses, which could provide insight into low energy collective excitations in solids and how they interact at a microscopic level to determine the material's macroscopic properties.
Abstract: Femtosecond pulses from X-ray free-electron lasers offer a powerful method for observing the coherent dynamic of phonons in crystalline materials, it is now shown. This time-resolved spectroscopic tool could provide insight into low-energy collective excitations in solids and how they interact at a microscopic level to determine the material’s macroscopic properties.
TL;DR: A robust watermark strategy for quantum images that embeds the watermark image into the fourier coefficients of the quantum carrier image, which will not affect the carrier image’s visual effect.
Abstract: We present a robust watermark strategy for quantum images. The watermark image is embedded into the fourier coefficients of the quantum carrier image, which will not affect the carrier image's visual effect. Before being embedded into the carrier image, the watermark image is preprocessed to be seemingly meaningless using quantum circuit, which further ensures the security of the watermark image. The properties of fourier transform ensure that the watermark embedded in the carrier image resists the unavoidable noise and cropping.
Dissertation•
20 Mar 2013
TL;DR: In this paper, a nonlinear Fourier transform (NFT) is proposed for data transmission over integrable channels such as optical fibers, where pulse propagation is governed by the nonlinear Schr\"odinger (NLS) equation.
Abstract: The central objective of this thesis is to suggest and develop one simple, unified method for communication over optical fiber
networks, valid for all values of dispersion and nonlinearity parameters, and for a single-user channel or a multiple-user network. The method is based on the nonlinear Fourier transform (NFT), a powerful tool in soliton theory and exactly solvable models 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 thesis,
this observation is exploited for data transmission over integrable channels such as optical fibers, where pulse propagation is
governed by the nonlinear Schr\"odinger (NLS) 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 spectrum of the signal. Just as the (ordinary) Fourier transform converts a linear convolutional channel into a number of parallel scalar channels, the nonlinear Fourier transform converts a nonlinear dispersive channel described by a \emph{Lax convolution} into a number of parallel scalar channels. Since, in the spectral coordinates the NLS equation is
multiplicative, users of a network can operate in independent nonlinear frequency bands with no deterministic inter-channel
interference. 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 thesis lays the foundations of such a nonlinear frequency-division multiplexing system.%%%%PhD
TL;DR: In this article, the Cauchy problem for the spatially one-dimensional distributed order diffusion-wave equation is considered and a representation of the fundamental solution in the transform domain is obtained by employing the technique of the Fourier and Laplace transforms.
Abstract: In this paper, the Cauchy problem for the spatially one-dimensional distributed order diffusion-wave equation
$$\int_0^2 {p(\beta )D_t^\beta u(x,t)d\beta } = \frac{{\partial ^2 }} {{\partial x^2 }}u(x,t) $$
is considered. Here, the time-fractional derivative Dtβ is understood in the Caputo sense and p(β) is a non-negative weight function with support somewhere in the interval [0, 2]. By employing the technique of the Fourier and Laplace transforms, a representation of the fundamental solution of the Cauchy problem in the transform domain is obtained. The main focus is on the interpretation of the fundamental solution as a probability density function of the space variable x evolving in time t. In particular, the fundamental solution of the time-fractional distributed order wave equation (p(β) ≡ 0, 0 ≤ β < 1) is shown to be non-negative and normalized. In the proof, properties of the completely monotone functions, the Bernstein functions, and the Stieltjes functions are used.
TL;DR: The fractional order parameters, α and β, and the entropy measure, were found to provide excellent contrast between white and gray matter and to give results that were sensitive to the type of diffusion experiment performed.
TL;DR: In order to reduce the influence of Heisenberg' s uncertainty, it is proposed that different signal components are windowed by different Gaussian windows, which brings better adaption and flexibility.
Abstract: This paper proposes a real-time power quality disturbances (PQDs) classification by using a hybrid method (HM) based on S-transform (ST) and dynamics (Dyn). Classification accuracy and run time are mainly considered in our work. The HM firstly uses Dyn to identify the location of the signal components in the frequency spectrum yielded by Fourier transform, and uses inverse Fourier transform to only some of the signal components. Then features from Fourier transform, ST, and Dyn are selected, and a decision tree is used to classify the types of PQD. In order to reduce the influence of Heisenberg' s uncertainty, we proposed that different signal components are windowed by different Gaussian windows, which brings better adaption and flexibility. By the HM, run time of the application has been greatly reduced with satisfactory classification accuracy. Finally, a DSP-FPGA based hardware platform is adopted to test the run time and correctness of the proposed method under real standard signals. Field signal tests have also presented. Both simulations and experiments validate the feasibility of the new method.
TL;DR: Spectroscopic analysis of infrared-resonant antenna probes for tip-enhanced optical microscopy corroborates their functionality as resonant antennas and verifies the broad tunability, and experimentally demonstrates high-performance mid-infrared nanoimaging of molecular absorption.
Abstract: We report the development of infrared-resonant antenna probes for tip-enhanced optical microscopy. We employ focused-ion-beam machining to fabricate high-aspect ratio gold cones, which replace the standard tip of a commercial Si-based atomic force microscopy cantilever. Calculations show large field enhancements at the tip apex due to geometrical antenna resonances in the cones, which can be precisely tuned throughout a broad spectral range from visible to terahertz frequencies by adjusting the cone length. Spectroscopic analysis of these probes by electron energy loss spectroscopy, Fourier transform infrared spectroscopy, and Fourier transform infrared near-field spectroscopy corroborates their functionality as resonant antennas and verifies the broad tunability. By employing the novel probes in a scattering-type near-field microscope and imaging a single tobacco mosaic virus (TMV), we experimentally demonstrate high-performance mid-infrared nanoimaging of molecular absorption. Our probes offer excellent...
TL;DR: A sub-aperture correlation based numerical phase correction method for interferometric full field imaging systems provided the complex object field information can be extracted without the need of any adaptive optics, spatial light modulators (SLM) and additional cameras.
Abstract: This paper proposes a sub-aperture correlation based numerical phase correction method for interferometric full field imaging systems provided the complex object field information can be extracted. This method corrects for the wavefront aberration at the pupil/ Fourier transform plane without the need of any adaptive optics, spatial light modulators (SLM) and additional cameras. We show that this method does not require the knowledge of any system parameters. In the simulation study, we consider a full field swept source OCT (FF SSOCT) system to show the working principle of the algorithm. Experimental results are presented for a technical and biological sample to demonstrate the proof of the principle.
TL;DR: A novel method is presented for the parallelization of electromagnetic pseudo-spectral solvers that requires only local FFTs and exchange of local guard cell data between neighboring regions, by taking advantage of the properties of DFTs, the linearity of Maxwell's equations and the finite speed of light.
TL;DR: A novel gray-level image encryption/decryption scheme is proposed, which is the first time that the double random-phase encoding technique is generalized to quantum scenarios and paves the way for introducing more optical information processing techniques into quantum scenarios.
Abstract: A novel gray-level image encryption/decryption scheme is proposed, which is based on quantum Fourier transform and double random-phase encoding technique. The biggest contribution of our work lies in that it is the first time that the double random-phase encoding technique is generalized to quantum scenarios. As the encryption keys, two phase coding operations are applied in the quantum image spatial domain and the Fourier transform domain respectively. Only applying the correct keys, the original image can be retrieved successfully. Because all operations in quantum computation must be invertible, decryption is the inverse of the encryption process. A detailed theoretical analysis is given to clarify its robustness, computational complexity and advantages over its classical counterparts. It paves the way for introducing more optical information processing techniques into quantum scenarios.
TL;DR: In this paper, the authors considered a class of functions composed of waveforms that repeat nearly periodically, and for which the instantaneous frequency can be given a rigorous meaning, and they showed that Synchrosqueezing can be used to determine instantaneous frequency of functions in this class, even if the waveform is not harmonic, thus generalizing earlier results for cosine wave functions.
TL;DR: A novel acquisition and processing method that enables single snapshot wide field imaging of optical properties in the Spatial Frequency Domain is described, which makes use of a Fourier transform performed on a single image and processing in the frequency space to extract two spatial frequency images at once.
Abstract: A novel acquisition and processing method that enables single snapshot wide field imaging of optical properties in the Spatial Frequency Domain (SFD) is described. This method makes use of a Fourier transform performed on a single image and processing in the frequency space to extract two spatial frequency images at once. The performance of the method is compared to the standard six image SFD acquisition method, assessed on tissue mimicking phantoms and in vivo. Overall both methods perform similarly in extracting optical properties.
TL;DR: F Fourier transform infrared spectro-microtomography is reported, a nondestructive three-dimensional imaging approach that reveals the distribution of distinctive chemical compositions throughout an intact biological or materials sample.
Abstract: Synchrotron-based Fourier transform infrared (FTIR) spectro-microtomography is a nondestructive, label-free imaging technique that allows chemical fingerprinting of intact, three-dimensional biological samples.
TL;DR: In this article, the authors considered the problem of the fast computation of forced periodic motions using the Euler equations and evaluated three methods: time domain, frequency domain, and harmonic balance method.
Abstract: Dynamic derivatives are used to represent the influence of the aircraft motion rates on the aerodynamic forces and moments needed for studies of flight dynamics. The use of computational fluid dynamics has potential to supplement costly wind-tunnel testing. The paper considers the problem of the fast computation of forced periodic motions using the Euler equations. Three methods are evaluated. The first is computation in the time domain, which provides the benchmark solution in the sense that the time-accurate solution is obtained. Two acceleration techniques in the frequency domain are compared. The first uses a harmonic solution of the linearized problem, referred to as the linear frequency-domain approach. The second uses the harmonic balance method, which approximates the nonlinear problem using a number of Fourier modes. These approaches are compared for the ability to predict dynamic derivatives and for computational cost. The NACA 0012 aerofoil and the DLR-F12 passenger jet wind-tunnel model are the test cases. Compared to time-domain simulations, an order of magnitude reduction in computational costs is achieved and satisfactory predictions are obtained for cases with a narrow frequency spectrum and moderate amplitudes using the frequency-domain methods.
TL;DR: In this paper, a joint amplitude and frequency demodulation method is proposed for fault diagnosis of planetary gearboxes by matching the dominant peaks in the envelope spectrum and the spectrum of instantaneous frequency with the theoretical characteristic frequencies of faulty gears.
TL;DR: In this article, the problem of finding the differentiability conditions for bilinear Fourier multipliers that are as small as possible to ensure the boundedness of the corresponding operators from products of Hardy spaces H 1 ×Hp2 to L, 1/p1 + 1/ p2 = 1 p, is considered.
Abstract: The problem of finding the differentiability conditions for bilinear Fourier multipliers that are as small as possible to ensure the boundedness of the corresponding operators from products of Hardy spaces H1 ×Hp2 to L, 1/p1 + 1/p2 = 1/p, is considered. The minimal conditions in terms of the product type Sobolev norms are given for the whole range 0 < p1, p2 ≤ ∞.
Book•
13 Sep 2013
TL;DR: In this article, flat space Fourier analysis on R = SL(2, R) was used to define fundamental domains for Discrete Subgroups (DSG) in the Poincare Upper Half-Plane.
Abstract: Chapter 1 Flat Space Fourier Analysis on R^m- 11 Distributions or Generalized Functions- 12 Fourier Integrals- 13 Fourier Series and the Poisson Summation Formula- 14 Mellin Transforms, Epstein and Dedekind Zeta Functions- 15 Finite Symmetric Spaces, Wavelets, Quasicrystals, Weyl's Criterion for Uniform Distribution- Chapter 2 A Compact Symmetric Space--The Sphere- 21 Fourier Analysis on the Sphere- 22 O(3) and R^3 The Radon Transform- Chapter 3 The Poincare Upper Half-Plane- 31 Hyperbolic Geometry- 32 Harmonic Analysis on H- 33 Fundamental Domains for Discrete Subgroups ? of G = SL(2, R)- 34 Modular of Automorphic Forms--Classical- 35 Automorphic Forms--Not So Classical--Maass Waveforms- 36 Modular Forms and Dirichlet Series Hecke Theory and Generalizations- References- Index
TL;DR: The Prony filter, together with its phasor estimates, provides instantaneous estimates of damping and frequency, corresponding to the first derivative of amplitude and phase, which are very useful to assess the power system stability.
Abstract: Prony's method can be used as a dynamic phasor estimator. It can be regarded as the adaptive approximation of its complex exponential signal model to the dynamic phasor of an oscillation over a finite time interval. Equipped with a closed signal model, it is possible to implement it in one cycle. In its first adaptive stage, it estimates the best damping and frequency for its signal model; and, in the second one, the best phasor over the considered time window. This paper compares the performance of Prony's method with that of the very wellknown one-cycle Fourier filter. With a higher flexibility, due to its adaptive nature, Prony estimates improve the Fourier ones under oscillation conditions. The Fourier filter can be considered as a static subclass of the Prony filters. With its static signal model, it is unable to accurately follow oscillations when the frequency fluctuates. Additionally, the Prony filter, together with its phasor estimates, provides instantaneous estimates of damping and frequency, corresponding to the first derivative of amplitude and phase, which are very useful to assess the power system stability. Finally, by being implemented in one-cycle windows, and its good rejection of the dc or exponentially attenuated components, it can also be used in protection applications.