Showing papers on "Harmonic wavelet transform published in 2012"
TL;DR: This paper constructs translation-invariant operators on L 2 .R d /, which are Lipschitz-continuous to the action of diffeomorphisms, and extendsScattering operators are extended on L2 .G/, where G is a compact Lie group, and are invariant under theaction of G.
Abstract: This paper constructs translation-invariant operators on L 2 .R d /, which are Lipschitz-continuous to the action of diffeomorphisms. A scattering propagator is a path-ordered product of nonlinear and noncommuting operators, each of which computes the modulus of a wavelet transform. A local integration defines a windowed scattering transform, which is proved to be Lipschitz-continuous to the action of C 2 diffeomorphisms. As the window size increases, it converges to a wavelet scattering transform that is translation invariant. Scattering coefficients also provide representations of stationary processes. Expected values depend upon high-order moments and can discriminate processes having the same power spectrum. Scattering operators are extended on L 2 .G/, where G is a compact Lie group, and are invariant under the action of G. Combining a scattering on L 2 .R d / and on L 2 .SO.d// defines a translation- and rotation-invariant scattering on L 2 .R d /. © 2012 Wiley Periodicals, Inc.
941 citations
TL;DR: In this paper, a novel fractional wavelet transform (FRWT) is proposed in order to rectify the limitations of the WT and the FRFT in the time-frequency domain.
Abstract: The wavelet transform (WT) and the fractional Fourier transform (FRFT) are powerful tools for many applications in the field of signal processing. However, the signal analysis capability of the former is limited in the time-frequency plane. Although the latter has overcome such limitation and can provide signal representations in the fractional domain, it fails in obtaining local structures of the signal. In this paper, a novel fractional wavelet transform (FRWT) is proposed in order to rectify the limitations of the WT and the FRFT. The proposed transform not only inherits the advantages of multiresolution analysis of the WT, but also has the capability of signal representations in the fractional domain which is similar to the FRFT. Compared with the existing FRWT, the novel FRWT can offer signal representations in the time-fractional-frequency plane. Besides, it has explicit physical interpretation, low computational complexity and usefulness for practical applications. The validity of the theoretical derivations is demonstrated via simulations.
123 citations
TL;DR: A novel time-frequency transform, called spline-kernelled chirplet transform (SCT), is proposed, which is particularly powerful for the strongly nonlinear frequency-modulated signals.
Abstract: The conventional time-frequency analysis (TFA) methods, including continuous wavelet transform, short-time Fourier transform, and Wigner-Ville distribution, have played important roles in analyzing nonstationary signals. However, they often show less capability in dealing with nonstationary signals with time-varying frequency due to the bad energy concentration in the time-frequency plane. On the other hand, by introducing an extra transform kernel that matches the instantaneous frequency of the signal, parameterized TFA methods show powerful ability in characterizing time-frequency patterns of nonstationary signals with time-varying frequency. In this paper, a novel time-frequency transform, called spline-kernelled chirplet transform (SCT), is proposed. By introducing a frequency-rotate operator and a frequency-shift operator constructed with spline kernel function, the SCT is particularly powerful for the strongly nonlinear frequency-modulated signals. In addition, an effective algorithm is developed to estimate the parameters of transform kernel in the SCT. The capabilities of the SCT and parameter estimation algorithm are validated by their applications for numerical signals and a set of vibration signal collected from a rotor test rig.
116 citations
TL;DR: A novel frequency-shifting wavelet decomposition via the Hilbert transform is introduced for PQ analysis and can be used for estimating power quantities accurately and for detecting flickers.
Abstract: The wavelet transform, the S-transform, the Gabor transform, and the Wigner distribution function are popular techniques for power quality (PQ) analysis in electrical power systems. They are mainly used to identify power harmonics and power disturbances and to estimate power quantities in the presence of nonstationary power components such as root-mean-square values and total harmonic distortions. Recently, the Hilbert-Huang transform has been also used in PQ analysis. These techniques have proven to be useful in PQ analysis; however, their performances depend on the types of PQ events. In this paper, a novel frequency-shifting wavelet decomposition via the Hilbert transform is introduced for PQ analysis. The proposed algorithm overcomes the spectra leakage problem in the discrete wavelet packet transform and can be used for estimating power quantities accurately and for detecting flickers. The effectiveness of the proposed algorithm was verified by computer simulations and experimental tests.
113 citations
TL;DR: The proposed continuous wavelet transform for non-stationary vibration measurement by distributed vibration sensor based on phase optical time-domain reflectometry (OTDR) can give simultaneously the frequency and time information of the vibration event.
Abstract: We propose the continuous wavelet transform for non-stationary vibration measurement by distributed vibration sensor based on phase optical time-domain reflectometry (OTDR). The continuous wavelet transform approach can give simultaneously the frequency and time information of the vibration event. Frequency evolution is obtained by the wavelet ridge detection method from the scalogram of the continuous wavelet transform. In addition, a novel signal processing algorithm based on the global wavelet spectrum is used to determine the location of vibration. Distributed vibration measurements of 500Hz and 500Hz to 1kHz sweep events over 20 cm fiber length are demonstrated using a single mode fiber.
96 citations
Book•
08 Mar 2012TL;DR: In this paper, a systematic treatment of orthogonal transform methods for signal processing, data analysis and communications is presented, with a discussion of how each transform method can be applied to real-world experimental problems.
Abstract: A systematic, unified treatment of orthogonal transform methods for signal processing, data analysis and communications, this book guides the reader from mathematical theory to problem solving in practice. It examines each transform method in depth, emphasizing the common mathematical principles and essential properties of each method in terms of signal decorrelation and energy compaction. The different forms of Fourier transform, as well as the Laplace, Z-, Walsh–Hadamard, Slant, Haar, Karhunen–Loeve and wavelet transforms, are all covered, with discussion of how each transform method can be applied to real-world experimental problems. Numerous practical examples and end-of-chapter problems, supported by online Matlab and C code and an instructor-only solutions manual, make this an ideal resource for students and practitioners alike.
94 citations
04 Apr 2012
TL;DR: Researchers has concentrated on continuous wavelet transform (CWT) that gives more reliable and detailed time-scale representation rather than the classical short time Fourier transform (STFT) giving a time-frequency representation.
Abstract: The wavelet transform (WT) a powerful tool of signal and image processing that have been successfully used in many scientific fields such as signal processing, image compression, computer graphics, and pattern recognition (Daubechies 1990; Lewis and Knowles 1992; Do and Vetterli 2002; Meyer, Averbuch et al. 2002; Heric and Zazula 2007). On contrary the traditional Fourier Transform, the WT is particularly suitable for the applications of nonstationary signals which may instantaneous vary in time (Daubechies 1990; Mallat and Zhang 1993; Akay and Mello 1998). It is crucial to analyze the time-frequency characteristics of the signals which classified as non-stationary or transient signals in order to understand the exact features of such signals (Rioul and Vetterli 1991; Ergen, Tatar et al. 2010). For this reason, firstly, researchers has concentrated on continuous wavelet transform (CWT) that gives more reliable and detailed time-scale representation rather than the classical short time Fourier transform (STFT) giving a time-frequency representation (Jiang 1998; Qian and Chen 1999).
80 citations
TL;DR: In this article, the authors discuss the pros and cons of the three phase demodulation techniques on phase calculation at discontinuities and/or slopes, and compare the performance of Fourier transform, Windowed Fourier Transform (WFT), and Wavelet Transform (WT).
73 citations
TL;DR: The windowed Fourier transform is generalized to the windowed linear canonical transform by substituting the Fouriertransform kernel with the linear canonicaltransform kernel in the windowing Fourier Transform definition.
73 citations
13 Dec 2012
TL;DR: In this article, a continuous wavelet transform (CWT) analysis was used to find feature vectors for switching voltage transients for non-intrusive load monitoring (NILM) is presented and compared with the previously used short time Fourier transform (STFT).
Abstract: Continuous Wavelet Transform (CWT) analysis to find feature vectors for switching voltage transients for Non-Intrusive Load Monitoring (NILM) is presented and discussed, and compared with the previously used short time Fourier transform (STFT). The feature vectors computed from both CWT and STFT were used to train Support Vector Machines (SVMs) that identify the connection or disconnection of appliances for a NILM system. Experimental results show that the CWT analysis based on the complex Morlet wavelet improves classification accuracy as compared to the analysis based on STFT. More importantly, a 20× reduction of the vector size requirement is shown, thus greatly lowering computational requirements. It can be expected that commercial transient-based NILM will be based upon the CWT methods shown here.
71 citations
TL;DR: In this article, an inversion-based algorithm for computing the time-frequency analysis of reflection seismograms using constrained least-squares spectral analysis is formulated and applied to modeled seismic waveforms and real seismic data.
Abstract: An inversion-based algorithm for computing the time-frequency analysis of reflection seismograms using constrained least-squares spectral analysis is formulated and applied to modeled seismic waveforms and real seismic data. The Fourier series coefficients are computed as a function of time directly by inverting a basis of truncated sinusoidal kernels for a moving time window. The method resulted in spectra that have reduced window smearing for a given window length relative to the discrete Fourier transform irrespective of window shape, and a time-frequency analysis with a combination of time and frequency resolution that is superior to the short time Fourier transform and the continuous wavelet transform. The reduction in spectral smoothing enables better determination of the spectral characteristics of interfering reflections within a short window. The degree of resolution improvement relative to the short time Fourier transform increases as window length decreases. As compared with the continu...
Book•
06 Jul 2012
TL;DR: In this article, Fourier Transforms of Stable Signals, Fourier series of Locally Stable Periodic Signals and Pointwise Convergence of Fourier Series are discussed.
Abstract: A. FOURIER ANALYSIS IN L1 Fourier Transforms of Stable Signals / Fourier Series of Locally Stable Periodic Signals / Pointwise Convergence of Fourier Series B. SIGNAL PROCESSING Filtering / Sampling / Digital Signal Processing / Subband Coding C. FOURIER ANALYSIS IN L2 Hilbert Spaces / Complete Orthonormal Systems / Fourier Transforms of Finite Energy Signals / Fourier Series of Finite Power Periodic Signals D. WAVELET ANALYSIS The Windowed Fourier Transform / The Wavelet Transform / Wavelet Orthonormal Expansions / Construction of a MRA / Smooth Multiresolution Analysis
TL;DR: It is shown that the EPWT leads, for a suitable choice of the pathways, to optimal N-term approximations for piecewise Holder continuous functions with singularities along curves.
Abstract: The Easy Path Wavelet Transform (EPWT),20 has recently been proposed by one of the authors as a tool for sparse representations of bivariate functions from discrete data, in particular from image data. The EPWT is a locally adaptive wavelet transform. It works along pathways through the array of function values and it exploits the local correlations of the given data in a simple appropriate manner. In this paper, we show that the EPWT leads, for a suitable choice of the pathways, to optimal N-term approximations for piecewise Holder continuous functions with singularities along curves.
TL;DR: The FrFT provides a new fundamental perspective on the nature and role of group-velocity dispersion in Fourier domain OCT and it is shown how the optimized fractional Fourier transform order parameter can be used to extract the group velocity dispersion coefficient of a material.
Abstract: We address numerical dispersion compensation based on the use of the fractional Fourier transform (FrFT). The FrFT provides a new fundamental perspective on the nature and role of group-velocity dispersion in Fourier domain OCT. The dispersion induced by a 26 mm long water cell was compensated for a spectral bandwidth of 110 nm, allowing the theoretical axial resolution in air of 3.6 μm to be recovered from the dispersion degraded point spread function. Additionally, we present a new approach for depth dependent dispersion compensation based on numerical simulations. Finally, we show how the optimized fractional Fourier transform order parameter can be used to extract the group velocity dispersion coefficient of a material.
TL;DR: The phase or instantaneous frequency of fringe patterns with spatial or temporal carriers can be recovered from the wavelet ridge, a path that follows the maximum modulus of the CWT.
TL;DR: A multiresolution algorithm to perform the flaglet transform rapidly, while capturing all information at each wavelet scale in the minimal number of samples on the ball is introduced, which achieves floating-point precision.
Abstract: We develop an exact wavelet transform on the three-dimensional ball (i.e. on the solid sphere), which we name the flaglet transform. For this purpose we first construct an exact transform on the radial half-line using damped Laguerre polynomials and develop a corresponding quadrature rule. Combined with the spherical harmonic transform, this approach leads to a sampling theorem on the ball and a novel three-dimensional decomposition which we call the Fourier-Laguerre transform. We relate this new transform to the well-known Fourier-Bessel decomposition and show that band-limitedness in the Fourier-Laguerre basis is a sufficient condition to compute the Fourier-Bessel decomposition exactly. We then construct the flaglet transform on the ball through a harmonic tiling, which is exact thanks to the exactness of the Fourier-Laguerre transform (from which the name flaglets is coined). The corresponding wavelet kernels are well localised in real and Fourier-Laguerre spaces and their angular aperture is invariant under radial translation. We introduce a multiresolution algorithm to perform the flaglet transform rapidly, while capturing all information at each wavelet scale in the minimal number of samples on the ball. Our implementation of these new tools achieves floating-point precision and is made publicly available. We perform numerical experiments demonstrating the speed and accuracy of these libraries and illustrate their capabilities on a simple denoising example.
TL;DR: A new approach for image encryption based on the real-valuedness of the reality-preserving multiple-parameter fractional Fourier transform and the decorrelation property of chaotic maps is proposed in order to meet the requirements of the secure image transmission.
TL;DR: A double image encryption method is proposed by utilizing discrete multiple-parameter fractional Fourier transform and chaotic maps to encrypt two original images scrambled by one chaotic map with one original image as its phase.
TL;DR: In this article, an overview is given of several recent generalizations of the Fourier transform, related to either the Lie algebra sl_2 or the Lie superalgebra osp(1|2).
Abstract: In this review, an overview is given of several recent generalizations of the Fourier transform, related to either the Lie algebra sl_2 or the Lie superalgebra osp(1|2). In the former case, one obtains scalar generalizations of the Fourier transform, including the fractional Fourier transform, the Dunkl transform, the radially deformed Fourier transform and the super Fourier transform. In the latter case, one has to use the framework of Clifford analysis and arrives at the Clifford-Fourier transform and the radially deformed hypercomplex Fourier transform. A detailed exposition of all these transforms is given, with emphasis on aspects such as eigenfunctions and spectrum of the transform, characterization of the integral kernel and connection with various special functions.
TL;DR: In this article, the use of the continuous wavelet transform for modal identification from ambient vibration tests is presented, which is based on the equivalent formulation between the free response and the correlation function of a linear mechanical system under stationary stochastic excitation.
TL;DR: In this paper, the authors present techniques for implementing a fast algorithm for implementing odd-type DCTs and DSTs, which include a mapping between the real-valued data sequence to an intermediate sequence to be used as an input to a DFT.
Abstract: This disclosure presents techniques for implementing a fast algorithm for implementing odd-type DCTs and DSTs. The techniques include the computation of an odd-type transform on any real-valued sequence of data (e.g., residual values in a video coding process or a block of pixel values of an image coding process) by mapping the odd-type transform to a discrete Fourier transform (DFT). The techniques include a mapping between the real-valued data sequence to an intermediate sequence to be used as an input to a DFT. Using this intermediate sequence, an odd-type transform may be achieved by calculating a DFT of odd size. Fast algorithms for a DFT may be then be used, and as such, the odd-type transform may be calculated in a fast manner
TL;DR: In this article, a numerical experiment was conducted to demonstrate the shortcomings of commonly-used data processing techniques in traffic engineering (i.e., averaging, moving averaging, second-order difference, oblique cumulative curve, and short-time Fourier transform).
Abstract: Serving as a powerful tool for extracting localized variations in non-stationary signals, applications of wavelet transforms (WTs) in traffic engineering have been introduced; however, lacking in some important theoretical fundamentals. In particular, there is little guidance provided on selecting an appropriate WT across potential transport applications. This research described in this paper contributes uniquely to the literature by first describing a numerical experiment to demonstrate the shortcomings of commonly-used data processing techniques in traffic engineering (i.e., averaging, moving averaging, second-order difference, oblique cumulative curve, and short-time Fourier transform). It then mathematically describes WT’s ability to detect singularities in traffic data. Next, selecting a suitable WT for a particular research topic in traffic engineering is discussed in detail by objectively and quantitatively comparing candidate wavelets’ performances using a numerical experiment. Finally, based on several case studies using both loop detector data and vehicle trajectories, it is shown that selecting a suitable wavelet largely depends on the specific research topic, and that the Mexican hat wavelet generally gives a satisfactory performance in detecting singularities in traffic and vehicular data.
TL;DR: This paper proposes a fast and simple online encoding by the application of pseudorandom downsampling of the 2-D fast Fourier transform to video frames and proves that the AMP method can be rewritten as a forward-backward splitting algorithm.
Abstract: In this paper, we apply compressed sensing (CS) to video compression. CS techniques exploit the observation that one needs much fewer random measurements than given by the Shannon-Nyquist sampling theory to recover an object if this object is compressible (i.e., sparse in the spatial domain or in a transform domain). In the CS framework, we can achieve sensing, compression, and denoising simultaneously. We propose a fast and simple online encoding by the application of pseudorandom downsampling of the 2-D fast Fourier transform to video frames. For offline decoding, we apply a modification of the recently proposed approximate message passing (AMP) algorithm. The AMP method has been derived using the statistical concept of “state evolution,” and it has been shown to considerably accelerate the convergence rate in special CS-decoding applications. We shall prove that the AMP method can be rewritten as a forward-backward splitting algorithm. This new representation enables us to give conditions that ensure convergence of the AMP method and to modify the algorithm in order to achieve higher robustness. The success of reconstruction methods for video decoding also essentially depends on the chosen transform, where sparsity of the video signals is assumed. We propose incorporating the 3-D dual-tree complex wavelet transform that possesses sufficiently good directional selectivity while being computationally less expensive and less redundant than other directional 3-D wavelet transforms.
01 Jan 2012
TL;DR: In this correspondence paper, biometrics is chosen as the primary application; and hence, a new technique is proposed for securing fingerprints during communication and transmission over insecure channel, i.e., fractional random wavelet transform (FrRnWT).
Abstract: In this correspondence paper, the wavelet transform, which is an important tool in signal and image processing, has been generalized by coalescing wavelet transform and fractional random transform. The new transform, i.e., fractional random wavelet transform (FrRnWT) inherits the excellent mathematical properties of wavelet transform and fractional random transform. Possible applications of the proposed transform are in biometrics, image compression, image transmission, transient signal processing, etc. In this correspondence paper, biometrics is chosen as the primary application; and hence, a new technique is proposed for securing fingerprints during communication and transmission over insecure channel.
TL;DR: The concept of fractional wavelet packet transform is explored with its application in digital watermarking and a reliable watermark extraction algorithm is developed for the extraction of watermark from the distorted image.
Abstract: In this study, the concept of fractional wavelet packet transform is explored with its application in digital watermarking. The core idea of the proposed watermarking scheme is to decompose an image via fractional wavelet packet transform and then a reference image is created by changing the positions of all frequency sub-bands at each level with respect to some rule which is secret and only known to the owner/creator. For embedding, the reference image is segmented into non-overlapping blocks and modify its singular values with the watermark singular values. Finally, a reliable watermark extraction algorithm is developed for the extraction of watermark from the distorted image. The feasibility of this method and its robustness against different kind of attacks are verified by computer simulations.
07 Jan 2012
TL;DR: The synthesis results show that the computation for calculating the 32-point Fast Fourier transform is efficient in terms of speed.
Abstract: The Fast Fourier Transform (FFT) is one of the rudimentary operations in field of digital signal and image processing. Some of the very vital applications of the fast fourier transform include Signal analysis, Sound filtering, Data compression, Partial differential equations, Multiplication of large integers, Image filtering etc. Fast Fourier transform (FFT) is an efficient implementation of the discrete Fourier transform (DFT). This paper concentrates on the development of the Fast Fourier Transform (FFT), based on Decimation-In-Time (DIT) domain, Radix-2 algorithm, this paper uses VHDL as a design entity, and their Synthesis by Xilinx Synthesis Tool on Vertex kit has been done. The input of Fast Fourier transform has been given by a PS2 KEYBOARD using a test bench and output has been displayed using the waveforms on the Xilinx Design Suite 12.1. The synthesis results show that the computation for calculating the 32-point Fast Fourier transform is efficient in terms of speed.
01 Jul 2012-Graphical Models \/graphical Models and Image Processing \/computer Vision, Graphics, and Image Processing
TL;DR: This paper systematically studies the well-known Mexican hat wavelet on manifold geometry, including its derivation, properties, transforms, and applications, and proposes a fast method to compute convolutions by Fourier transform.
Abstract: This paper systematically studies the well-known Mexican hat wavelet (MHW) on manifold geometry, including its derivation, properties, transforms, and applications. The MHW is rigorously derived from the heat kernel by taking the negative first-order derivative with respect to time. As a solution to the heat equation, it has a clear initial condition: the Laplace-Beltrami operator. Following a popular methodology in mathematics, we analyze the MHW and its transforms from a Fourier perspective. By formulating Fourier transforms of bivariate kernels and convolutions, we obtain its explicit expression in the Fourier domain, which is a scaled differential operator continuously dilated via heat diffusion. The MHW is localized in both space and frequency, which enables space-frequency analysis of input functions. We defined its continuous and discrete transforms as convolutions of bivariate kernels, and propose a fast method to compute convolutions by Fourier transform. To broaden its application scope, we apply the MHW to graphics problems of feature detection and geometry processing.
TL;DR: In this article, a new representation of the Schrodinger operator of a free particle by using the short-time Fourier transform was proposed, and its applications were described. But this representation is restricted to the case of a single particle.
Abstract: We propose a new representation of the Schrodinger operator of a free particle by using the short-time Fourier transform and give its applications.
01 Oct 2012
TL;DR: A novel multifocus color image fusion algorithm based on the quaternion wavelet transform (QWT) is proposed in this paper, aiming at solving the image blur problem.
Abstract: A novel multifocus color image fusion algorithm based on the quaternion wavelet transform (QWT) is proposed in this paper, aiming at solving the image blur problem. The proposed method uses a multiresolution analysis procedure based on the quaternion wavelet transform. The performance of the proposed fusion scheme is assessed by some experiments, and the experimental results show that the proposed method is effective and performs better than the existing fusion methods.