Fast directional continuous spherical wavelet transform algorithms

We describe the construction of a spherical wavelet analysis through the inverse stereographic projection of the Euclidean planar wavelet framework, introduced originally by Antoine and Vandergheynst and developed further by Wiaux et al. Fast algorithms for performing the directional continuous wavelet analysis on the unit sphere are presented. The fast directional algorithm, based on the fast spherical convolution algorithm developed by Wandelt and Gorski, provides a saving of O(sqrt(Npix)) over a direct quadrature implementation for Npix pixels on the sphere, and allows one to perform a directional spherical wavelet analysis of a 10^6 pixel map on a personal computer.Comment: 10 pages, 3 figures, replaced to match version accepted by IEEE Trans. Sig. Pro

Solar hard X-ray imaging by means of Compressed Sensing and Finite Isotropic Wavelet Transform

This paper shows that compressed sensing realized by means of regularized deconvolution and the Finite Isotropic Wavelet Transform is effective and reliable in hard X-ray solar imaging. The method utilizes the Finite Isotropic Wavelet Transform with Meyer function as the mother wavelet. Further, compressed sensing is realized by optimizing a sparsity-promoting regularized objective function by means of the Fast Iterative Shrinkage-Thresholding Algorithm. Eventually, the regularization parameter is selected by means of the Miller criterion. The method is applied against both synthetic data mimicking the Spectrometer/Telescope Imaging X-rays (STIX) measurements and experimental observations provided by the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI). The performances of the method are compared with the results provided by standard visibility-based reconstruction methods. The results show that the application of the sparsity constraint and the use of a continuous, isotropic framework for the wavelet transform provide a notable spatial accuracy and significantly reduce the ringing effects due to the instrument point spread functions

Turn-to-turn fault diagnosis of an Induction motor by the analysis of Transient and Steady state Stator current

This work proposes an online diagnosis of turn-to-turn stator winding fault of an induction motor through the combine use of Wavelet Transform (WT) and Fast Fourier Transform (FFT). Both steady state and non-stationary transient part of motor stator currents are assessed for detection of this inter turn short circuit fault. First non-stationary part is assessed by formation of contour of the coefficients of Continuous Wavelet Transform (CWT) of stator current – result shows significant change of amplitude at certain frequencies. Secondly, the supply frequency is filtered off from the steady part of motor current. Then on this filtered signal (i) Fast Fourier Transform (FFT) is performed where from spectrums are observed at different percentage of inter turn short condition and (ii) performing Discrete Wavelet Transform (DWT) detailed wave energy has been calculated using Parseval’s theorem. Work has been performed both under load and no-load condition of the motor. The proposed method has been validated in a laboratory prototype. Results indicate that the proposed technique is suitable for real-time application Keywords: CWT, DWT, FFT, fault diagnosis, induction motor, inter-turn short, wave energ

Applications of a fast, continuous wavelet transform

A fast, continuous, wavelet transform, based on Shannon`s sampling theorem in frequency space, has been developed for use with continuous mother wavelets and sampled data sets. The method differs from the usual discrete-wavelet approach and the continuous-wavelet transform in that, here, the wavelet is sampled in the frequency domain. Since Shannon`s sampling theorem lets us view the Fourier transform of the data set as a continuous function in frequency space, the continuous nature of the functions is kept up to the point of sampling the scale-translation lattice, so the scale-translation grid used to represent the wavelet transform is independent of the time- domain sampling of the signal under analysis. Computational cost and nonorthogonality aside, the inherent flexibility and shift invariance of the frequency-space wavelets has advantages. The method has been applied to forensic audio reconstruction speaker recognition/identification, and the detection of micromotions of heavy vehicles associated with ballistocardiac impulses originating from occupants` heart beats. Audio reconstruction is aided by selection of desired regions in the 2-D representation of the magnitude of the transformed signal. The inverse transform is applied to ridges and selected regions to reconstruct areas of interest, unencumbered by noise interference lying outside these regions. To separate micromotions imparted to a mass-spring system (e.g., a vehicle) by an occupants beating heart from gross mechanical motions due to wind and traffic vibrations, a continuous frequency-space wavelet, modeled on the frequency content of a canonical ballistocardiogram, was used to analyze time series taken from geophone measurements of vehicle micromotions. By using a family of mother wavelets, such as a set of Gaussian derivatives of various orders, features such as the glottal closing rate and word and phrase segmentation may be extracted from voice data

Vibration-based fault diagnostic of a spur gearbox

This paper presents comparative studies of Fast Fourier Transform (FFT), Short Time Fourier Transform (STFT) and Continuous Wavelet Transform (CWT) as several advanced time-frequency analysis methods for diagnosing an early stage of spur gear tooth failure. An incipient fault of a chipped tooth was investigated in this work using vibration measurements from a spur gearbox test rig. Time Synchronous Averaging was implemented for the analysis to enhance the clarity of fault feature from the gear of interest. Based on the experimental results and analysis, it was shown that FFT method could identify the location of the faulty gear with sufficient accuracy. On the other hand, Short Time Fourier Transform method could not provide the angular location information of the faulty gear. It was found that the Continuous Wavelet Transform method offered the best representation of angle-frequency representation. It was not only able to distinguish the difference between the normal and faulty gearboxes from the joint angle-frequency results but could also provide an accurate angular location of the faulty gear tooth in the gearbox

Cubic Spline Wavelet Bases of Sobolev Spaces and Multilevel Interpolation

AbstractIn this paper, a semi-orthogonal cubic spline wavelet basis of homogeneous Sobolev spaceH20(I) is constructed, which turns out to be a basis of the continuous spaceC0(I). At the same time, the orthogonal projections on the wavelet subspaces inH20(I) are extended to the interpolating operators on the corresponding wavelet subspaces inC0(I). A fast discrete wavelet transform (FWT) for functions inC0(I) is also given, which is different from the pyramid algorithm and easy to perform using a parallel algorithm. Finally, it is shown that the singularities of a function can be traced from its wavelet coefficients, which provide an adaptive approximation scheme allowing us to reduce the operation time in computation

Diagnosa Kerusakan Roda Gigi Dengan Sinyal Getaran

Artikel ini membahas deteksi dan diagnosa kerusakan roda gigi dengan menggunakan sinyal getaran. Deteksi dan diagnosa kerusakan roda gigi dilakukan dengan memakai metode pendekatan waktu-frekuensi (time- frequency) dan analisis transformasi wavelet kontinyu (continuous wavelet transform) kemudian hasilnya dibandingkan dengan penggunaan metode analisis cepstrum. Studi eksperimen dilakukan dengan pengujian roda gigi pada kondisi normal, aus dan patah satu gigi. Sinyal getaran diakuisisi dari test-rig menggunakan tiga buah akselerometer, sedangkan data putaran poros diambil dengan sensor tachometer optik. Hasil eksperimen menunjukkan bahwa analisis sinyal domain frekuensi dengan fast-fourier transfiorm (FFT) kurang sensitif terhadap kondisi roda gigi aus maupun patah. Namun demikian, metode short-time fourier transform (STFT) dapat memonitor adanya kerusakan pada roda gigi. Metode transformasi wavelet terbukti cukup baik untuk mendeteksi adanya cacat atau kerusakan pada roda gigi. Pada penelitian ini, metode wavelet dilakukan pada sinyal getaran roda gigi yang sebelumnya sudah dilakukan proses time synchronous averaging (TSA)

Intelligent sequence stratigraphy through a wavelet-based decomposition of well log data

Identification of sequence boundaries is an important task in geological characterization of gas reservoirs. In this study, a continuous wavelet transform (CWT) approach is applied to decompose gamma ray and porosity logs into a set of wavelet coefficients at varying scales. A discrete wavelet transform (DWT) is utilized to decompose well logs into smaller frequency bandwidths called Approximations (A) and Details (D). The methodology is illustrated by using a case study from the Ilam and upper Sarvak formations in the Dezful embayment, southwestern Iran. Different graphical visualization techniques of the continuous wavelet transform results allowed a better understanding of the main sequence boundaries. Using the DWT, maximum flooding surface was successfully recognised from both highest frequency and low frequency contents of signals. There is a sharp peak in all A&D corresponding to the maximum flooding surface (MFS), which can specifically be seen in fifth Approximation (a5), fifth Detail (d5), fourth Detail (d4) and third Detail (d3) coefficients. Sequence boundaries were best recognised from the low frequency contents of signals, especially the fifth Approximation (a5). Normally, the troughs of the fifth Approximation correspond to sequence boundaries where higher porosities developed in the Ilam and upper Sarvak carbonate rocks. Through hybridizing both CWT and DWT coefficient a more effective discrimination of sequence boundaries was achieved. The results of this study show that wavelet transform is a successful, fast and easy approach for identification of the main sequence boundaries from well log data. There is a good agreement between core derived system tracts and those derived from decomposition of well logs by using the wavelet transform approach