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1
de Macedo, Isadora A. S., and José Jadsom S. de Figueiredo. "On the seismic wavelet estimative and reflectivity recovering based on linear inversion: Well-to-seismic tie on a real data set from Viking Graben, North Sea." GEOPHYSICS 85, no. 5 (September 1, 2020): D157—D165. http://dx.doi.org/10.1190/geo2019-0183.1.
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Tying seismic data to well data is critical in reservoir characterization. In general, the main factors controlling a successful seismic well tie are an accurate time-depth relationship and a coherent wavelet estimate. Wavelet estimation methods are divided into two major groups: statistical and deterministic. Deterministic methods are based on using the seismic trace and the well data to estimate the wavelet. Statistical methods use only the seismic trace and generally require assumptions about the wavelet’s phase or a random process reflectivity series. We have compared the estimation of the wavelet for seismic well tie purposes through least-squares minimization and zero-order quadratic regularization with the results obtained from homomorphic deconvolution. Both methods make no assumption regarding the wavelet’s phase or the reflectivity. The best-estimated wavelet is used as the input to sparse-spike deconvolution to recover the reflectivity near the well location. The results show that the wavelets estimated from both deconvolutions are similar, which builds our confidence in their accuracy. The reflectivity of the seismic section is recovered according to known stratigraphic markers (from gamma-ray logs) present in the real data set from the Viking Graben field, Norway.
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Knapp, Ralph W. "Energy distribution in wavelets and implications on resolving power." GEOPHYSICS 58, no. 1 (January 1993): 39–46. http://dx.doi.org/10.1190/1.1443350.
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The suite of a wavelet is defined as being all wavelets that share a common amplitude spectrum and total energy but differ in phase spectra. Within a suite there are also classes of wavelets. A wavelet class has a common amplitude envelope and energy distribution. As such, it includes all wavelets that differ by only a constant‐angle phase shift. Of all wavelets within suite, the zero‐phase wavelet has the minimum energy envelope width; its energy is confined to minimum time dispersion. Therefore, the zero‐phase wavelet has maximum resolving power within the suite. Because a zero‐phase wavelet shares its amplitude envelope with a class of wavelets that differ by only a constant phase shift, all wavelets of the class also have maximum resolving power within the suite. The most familiar of these is the quadrature‐phase wavelet (90‐degree phase shift). Use of the complex trace results in an evaluation of the total energy, both potential and kinetic, of the wavelet signal. Assuming the wavelet signal is the output of a velocity geophone, partial energy represents only kinetic energy. Total energy better represents wavelet energy propagating through the earth. Use of partial energy (real signal only) applies a bias that favors the zero‐phase wavelets with respect to others of its class despite identical energy distribution. This bias is corrected when the wavelet envelope is used in the evaluation rather than wavelet trace amplitude. On a wiggle‐trace seismic section (amplitude display) a zero‐phase wavelet maintains a detectability advantage in the presence of noise because of a slightly greater amplitude; however, the advantage is lost in complex trace sections (energy displays) because both reflection strength and instantaneous frequency are independent of a constant phase shift in the wavelet. These sections are identical whether the wavelet is zero‐phase, quadrature‐phase or any other constant phase value, i.e., a wavelet within the zero‐phase class. (This does not imply that reflection strength sections should replace wiggle trace ones, only that they have advantages in the solution of some problems.)
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Abuhamdia, Tariq, Saied Taheri, and John Burns. "Laplace wavelet transform theory and applications." Journal of Vibration and Control 24, no. 9 (May 11, 2017): 1600–1620. http://dx.doi.org/10.1177/1077546317707103.
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This study introduces the theory of the Laplace wavelet transform (LWT). The Laplace wavelets are a generalization of the second-order under damped linear time-invariant (SOULTI) wavelets to the complex domain. This generalization produces the mother wavelet function that has been used as the Laplace pseudo wavelet or the Laplace wavelet dictionary. The study shows that the Laplace wavelet can be used to transform signals to the time-scale or time-frequency domain and can be retrieved back. The properties of the new generalization are outlined, and the characteristics of the companion wavelet transform are defined. Moreover, some similarities between the Laplace wavelet transform and the Laplace transform arise, where a relation between the Laplace wavelet transform and the Laplace transform is derived. This relation can be beneficial in evaluating the wavelet transform. The new wavelet transform has phase and magnitude, and can also be evaluated for most elementary signals. The Laplace wavelets inherit many properties from the SOULTI wavelets, and the Laplace wavelet transform inherits many properties from both the SOULTI wavelet transform and the Laplace transform. In addition, the investigation shows that both the LWT and the SOULTI wavelet transform give the particular solutions of specific related differential equations, and the particular solution of these linear time-invariant differential equations can in general be written in terms of a wavelet transform. Finally, the properties of the Laplace wavelet are verified by applications to frequency varying signals and to vibrations of mechanical systems for modes decoupling, and the results are compared with the generalized Morse and Morlet wavelets in addition to the short time Fourier transform’s results.
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SHUKLA, NIRAJ K. "NON-MSF A-WAVELETS FROM A-WAVELET SETS." International Journal of Wavelets, Multiresolution and Information Processing 11, no. 01 (January 2013): 1350002. http://dx.doi.org/10.1142/s0219691313500021.
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Generalizing the result of Bownik and Speegle [Approximation Theory X: Wavelets, Splines and Applications, Vanderbilt University Press, pp. 63–85, 2002], we provide plenty of non-MSF A-wavelets with the help of a given A-wavelet set. Further, by showing that the dimension function of the non-MSF A-wavelet constructed through an A-wavelet set W coincides with the dimension function of W, we conclude that the non-MSF A-wavelet and the A-wavelet set through which it is constructed possess the same nature as far as the multiresolution analysis is concerned. Some examples of non-MSF d-wavelets and non-MSF A-wavelets are also provided. As an illustration we exhibit a pathwise connected class of non-MSF non-MRA wavelets sharing the same wavelet dimension function.
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KING, EMILY J. "SMOOTH PARSEVAL FRAMES FOR L2(ℝ) AND GENERALIZATIONS TO L2(ℝd)." International Journal of Wavelets, Multiresolution and Information Processing 11, no. 06 (November 2013): 1350047. http://dx.doi.org/10.1142/s0219691313500471.
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Wavelet set wavelets were the first examples of wavelets that may not have associated multiresolution analyses. Furthermore, they provided examples of complete orthonormal wavelet systems in L2(ℝd) which only require a single generating wavelet. Although work had been done to smooth these wavelets, which are by definition discontinuous on the frequency domain, nothing had been explicitly done over ℝd, d > 1. This paper, along with another one cowritten by the author, finally addresses this issue. Smoothing does not work as expected in higher dimensions. For example, Bin Han's proof of existence of Schwartz class functions which are Parseval frame wavelets and approximate Parseval frame wavelet set wavelets does not easily generalize to higher dimensions. However, a construction of wavelet sets in [Formula: see text] which may be smoothed is presented. Finally, it is shown that a commonly used class of functions cannot be the result of convolutional smoothing of a wavelet set wavelet.
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ZENG, LI, JIQIANG GUO, and CHENCHENG HUANG. "THE BACK-PROJECTION METHOD FOR CONSTRUCTING 3D NON-TENSOR PRODUCT MOTHER WAVELETS AND THE APPLICATION IN IMAGE EDGE DETECTION." International Journal of Wavelets, Multiresolution and Information Processing 10, no. 03 (May 2012): 1250026. http://dx.doi.org/10.1142/s0219691312500269.
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In this paper, a non-tensor product method for constructing three-dimension (3D) mother wavelets by back-projecting two dimension (2D) mother wavelets is presented. We have proved that if a 2D mother wavelet satisfies certain conditions, the back-projection of the 2D mother wavelet is a 3D mother wavelet. And the construction instances of 3D Mexican-hat wavelet and 3D Meyer wavelet are given. These examples imply that we can get some new 3D mother wavelets from known 1D or 2D mother wavelets by using back-projecting method. This method inaugurates a new approach for constructing non-tensor product 3D wavelet. In addition, the non-tensor product 3D Mexican-hat wavelet is used for detecting the edge of two 3D images in our experimental section. Compared with the Mallat's maximum wavelet module approach which uses 3D directional wavelets, experimental results show it can obtain better outcome especial for the edge which the orientation is not along the coordinate axis. Furthermore, the edge is more fine, and the computational cost is much smaller. The non-tensor product mother wavelets constructed by using the method of this paper also can be widely used for compression, filtering and denoising of 3D images.
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Phinyomark, Angkoon, Chusak Limsakul, and Pornchai Phukpattaranont. "Optimal Wavelet Functions in Wavelet Denoising for Multifunction Myoelectric Control." ECTI Transactions on Electrical Engineering, Electronics, and Communications 8, no. 1 (August 1, 2009): 43–52. http://dx.doi.org/10.37936/ecti-eec.201081.172001.
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Wavelet analysis is one of the most important methods for analyzing the surface Electromyography (sEMG) signal. The aim of this study was to investigate the wavelet function that is optimum to identify and denoise the sEMG signal for multifunction myoelectric control. This study is motivated by the fact that there is no universal mother wavelet that is suitable for all types of signal. The right wavelet function becomes to achieve the optimal performance. In this study, the optimal wavelets are evaluated in term of mean square error of two criterions, namely denoising and reconstruction. Fifty-three wavelet functions are used to perform an iterative denoising and reconstruction on different noise levels that are added in sEMG signals. In addition, various possible decomposition levels and types of wavelets in the denoising procedure are tested. The results show that the best mother wavelets for tolerance of noise in denoising are the first order of Daubechies, BioSplines, and ReverseBior but the classification results are not recommended. The fifth order of Coiflet is the best wavelet in perfect reconstruction point of view. Various families can be used except the third order of BiorSplines and Discrete Meyer are not recommended to use. Suitable number of decomposition levels is four and optimal wavelets are independent of wavelet denoising algorithms.
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TODA, HIROSHI, ZHONG ZHANG, and TAKASHI IMAMURA. "PERFECT-TRANSLATION-INVARIANT CUSTOMIZABLE COMPLEX DISCRETE WAVELET TRANSFORM." International Journal of Wavelets, Multiresolution and Information Processing 11, no. 04 (July 2013): 1360003. http://dx.doi.org/10.1142/s0219691313600035.
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The theorems, giving the condition of perfect translation invariance for discrete wavelet transforms, have already been proven. Based on these theorems, the dual-tree complex discrete wavelet transform, the 2-dimensional discrete wavelet transform, the complex wavelet packet transform, the variable-density complex discrete wavelet transform and the real-valued discrete wavelet transform, having perfect translation invariance, were proposed. However, their customizability of wavelets in the frequency domain is limited. In this paper, also based on these theorems, a new type of complex discrete wavelet transform is proposed, which achieves perfect translation invariance with high degree of customizability of wavelets in the frequency domain.
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Gulai, A. V., and V. M. Zaitsev. "INTELLIGENT TECHNOLOGY OF WAVELET ANALYSIS OF VIBRATION SIGNALS." Doklady BGUIR, no. 7-8 (December 29, 2019): 101–8. http://dx.doi.org/10.35596/1729-7648-2019-126-8-101-108.
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During solution of engineering problems of machinery dynamics a need of revealing the harmonic components often arises in the narrow timing gate. This requires the use of wavelet-transformation oscillation methods and introduction of intelligent systems to hardware and software used in the experiment. The wavelet is considered as a short in time signal functional window, which has its internal structure in the form of a fading wavelike burst, and it is characterized by a scale of display of certain events in the field of the signal frequency spectrum, as well as and by time axis shifts. Complex-functioned continuous functions of real arguments (Daubechies wavelets, Gaussian wavelets, MHat-wavelets), complex-valued functions of real arguments (Morlet and Paul wavelets), as well as real discrete functions (HААRT- and FHat-wavelets) are used as wavelet functions. The wavelet analysis method of vibration signals is disclosed at acoustic diagnostics of machines and mechanisms. Digital implementation of discrete indications of wavelets with the subsequent visualization of results in the form of scalotons is the mathematical basis of the algorithm for procession of vibration signals. It has been suggested that engineering analysis and reconstruction of signals should be implemented by means of directed and reverse continuous wavelet conversions, which are discrete by arguments. The structural and functional scheme of the multichannel system of the intelligent wavelet analysis of vibration signals in machines has been considered. The intelligent system for study of vibration signals makes it possible to form the totality of photographic parameters, when scalotons are calculated by wavelet functions. An example of experimental implementation of the wavelet conversion method of vibration signals parameters is shown. Results of scalotons calculation are shown, when MHat-wavelet and DOG-wavelet are used.
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Gao, Jing Li, and Shi Hui Cheng. "The Traits of Canonical Banach Frames Generated by Multiple Scaling Functions and Applications in Applied Materials." Advanced Materials Research 684 (April 2013): 663–66. http://dx.doi.org/10.4028/www.scientific.net/amr.684.663.
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Frame theory has become a popular subject in scientific research during the past twenty years. In our study we use generalized multiresolution analyses in with dilation factor 4. We describe, in terms of the underlying multiresolution structure, all generalized multiresolution analyses Parseval frame wavelets all semi-orthogonal Parseval frame wavelets in . We show that there exist wavelet frame generated by two functions which have good dual wavelet frames, but for which the canonical dual wavelet frame does not consist of wavelets, according to scaling functions. That is to say, the canonical dual wavelet frame cannot be generated by the translations and dilations of a single function. Traits of tight wavelet frames are presented.
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Zhang, Fanchang, and Nanying Lan. "Seismic-gather wavelet-stretching correction based on multiwavelet decomposition algorithm." GEOPHYSICS 85, no. 5 (July 10, 2020): V377—V384. http://dx.doi.org/10.1190/geo2018-0835.1.
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Normal moveout correction is crucial in seismic data processing, but it generates a wavelet-stretching effect, especially on the larger offset or incident-angle seismic data. Wavelet stretching reduces the dominant frequency of seismic data. The greater the incident angle or offset, the lower dominant the frequency becomes. This is an unfavorable effect to amplitude variation with offset analysis. Therefore, we have introduced a wavelet stretching correction method based on the multiwavelet decomposition (MWD) algorithm. First, it decomposes the near-offset pilot trace and all the far-offset seismic traces in the same gather into a series of wavelets via the MWD algorithm. Then, the dominant frequencies of wavelets in the far-offset seismic traces are replaced by those corresponding wavelets in the pilot trace. Finally, the wavelets after the stretching correction are used to reconstruct the seismic trace. The model and field-data processing results show that this method can not only effectively reduce the wavelet stretching effect but it can also maintain the amplitude of each wavelet as invariant during the stretching correction procedure. Because only the frequencies of the decomposed wavelets are used, and no inverse wavelet operators is introduced, the wavelet stretching correction method does not distort the amplitude information.
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Romanchak, V. M. "Local transformations with a singular wavelet." Informatics 17, no. 1 (March 29, 2020): 39–46. http://dx.doi.org/10.37661/1816-0301-2020-17-1-39-46.
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The paper considers a local wavelet transform with a singular basis wavelet. The problem of nonparametric approximation of a function is solved by the use of the sequence of local wavelet transforms. Traditionally believed that the wavelet should have an average equal to zero. Earlier, the author considered singular wavelets when the average value is not equal to zero. As an example, the delta-shaped functions, participated in the estimates of Parzen – Rosenblatt and Nadara – Watson, were used as a wavelet. Previously, a sequence of wavelet transforms for the entire numerical axis and finite interval was constructed for singular wavelets. The paper proposes a sequence of local wavelet transforms, a local wavelet transform is defined, the theorems that formulate the properties of a local wavelet transform are proved. To confirm the effectiveness of the algorithm an example of approximating the function by use of the sum of discrete local wavelet transforms is given.
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Low, Yin Fen, and Rosli Besar. "Optimal Wavelet Filters for Medical Image Compression." International Journal of Wavelets, Multiresolution and Information Processing 01, no. 02 (June 2003): 179–97. http://dx.doi.org/10.1142/s0219691303000128.
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Recently, the wavelet transform has emerged as a cutting edge technology, within the field of image compression research. The basis functions of the wavelet transform are known as wavelets. There are a variety of different wavelet functions to suit the needs of different applications. Among the most popular wavelets are Haar, Daubechies, Coiflet and Biorthogonal, etc. The best wavelets (functions) for medical image compression are widely unknown. The purpose of this paper is to examine and compare the difference in impact and quality of a set of wavelet functions (wavelets) to image quality for implementation in a digitized still medical image compression with different modalities. We used two approaches to the measurement of medical image quality: objectively, using peak signal to noise ratio (PSNR) and subjectively, using perceived image quality. Finally, we defined an optimal wavelet filter for each modality of medical image.
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Zhang, Xinming, Jiaqi Liu, and Ke'an Liu. "A Wavelet Galerkin Finite-Element Method for the Biot Wave Equation in the Fluid-Saturated Porous Medium." Mathematical Problems in Engineering 2009 (2009): 1–18. http://dx.doi.org/10.1155/2009/142384.
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A wavelet Galerkin finite-element method is proposed by combining the wavelet analysis with traditional finite-element method to analyze wave propagation phenomena in fluid-saturated porous medium. The scaling functions of Daubechies wavelets are considered as the interpolation basis functions to replace the polynomial functions, and then the wavelet element is constructed. In order to overcome the integral difficulty for lacking of the explicit expression for the Daubechies wavelets, a kind of characteristic function is introduced. The recursive expression of calculating the function values of Daubechies wavelets on the fraction nodes is deduced, and the rapid wavelet transform between the wavelet coefficient space and the wave field displacement space is constructed. The results of numerical simulation demonstrate that the method is effective.
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Willmore, Ben, Ryan J. Prenger, Michael C. K. Wu, and Jack L. Gallant. "The Berkeley Wavelet Transform: A Biologically Inspired Orthogonal Wavelet Transform." Neural Computation 20, no. 6 (June 2008): 1537–64. http://dx.doi.org/10.1162/neco.2007.05-07-513.
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We describe the Berkeley wavelet transform (BWT), a two-dimensional triadic wavelet transform. The BWT comprises four pairs of mother wavelets at four orientations. Within each pair, one wavelet has odd symmetry, and the other has even symmetry. By translation and scaling of the whole set (plus a single constant term), the wavelets form a complete, orthonormal basis in two dimensions. The BWT shares many characteristics with the receptive fields of neurons in mammalian primary visual cortex (V1). Like these receptive fields, BWT wavelets are localized in space, tuned in spatial frequency and orientation, and form a set that is approximately scale invariant. The wavelets also have spatial frequency and orientation bandwidths that are comparable with biological values. Although the classical Gabor wavelet model is a more accurate description of the receptive fields of individual V1 neurons, the BWT has some interesting advantages. It is a complete, orthonormal basis and is therefore inexpensive to compute, manipulate, and invert. These properties make the BWT useful in situations where computational power or experimental data are limited, such as estimation of the spatiotemporal receptive fields of neurons.
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Song, Yu Yang, Rong Li, Ming Yan Li, and Wen Hui Zhang. "Relationship between Scale and Period and its Ecological Applications in Wavelet Analysis." Advanced Materials Research 726-731 (August 2013): 4252–57. http://dx.doi.org/10.4028/www.scientific.net/amr.726-731.4252.
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The relations between the scales and periods of Mexican Hat (Mexh) and Morlet (Morl) wavelets have been deduced. Based on these relations, variances, coefficients, and power spectra of these two wavelets’ original and eco-used wavelets are compared and analyzed theoretically and experimentally for the distribution pattern of Haloxylon ammodendron Bunge population in Gurban Tonggut desert, China. The research shows that: (1) Mexh and Morl eco-used wavelets can be simultaneously used to describe the distribution period of Haloxylon population and to study the same phenomenon by combining these two wavelet advantages. (2) The primary period value identified using Mexh eco-used wavelet than using its original wavelet is closer to the true one, while Morl eco-used wavelet helps find all changes in the period earlier. (3) For the same wavelet function, with its period enlarging, its primary period can be found in a smaller scale, inversely found later.
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Toda, Hiroshi, Zhong Zhang, and Takashi Imamura. "Practical design of perfect-translation-invariant real-valued discrete wavelet transform." International Journal of Wavelets, Multiresolution and Information Processing 12, no. 04 (July 2014): 1460005. http://dx.doi.org/10.1142/s0219691314600054.
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The real-valued tight wavelet frame having perfect translation invariance (PTI) has already proposed. However, due to the irrational-number distances between wavelets, its calculation amount is very large. In this paper, based on the real-valued tight wavelet frame, a practical design of a real-valued discrete wavelet transform (DWT) having PTI is proposed. In this transform, all the distances between wavelets are multiples of 1/4, and its transform and inverse transform are calculated fast by decomposition and reconstruction algorithms at the sacrifice of a tight wavelet frame. However, the real-valued DWT achieves an approximate tight wavelet frame.
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Lal, Shyam, and Harish Yadav. "Approximation of functions belonging to Hölder’s class and solution of Lane-Emden differential equation using Gegenbauer wavelets." Filomat 37, no. 12 (2023): 4029–45. http://dx.doi.org/10.2298/fil2312029l.
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In this paper, a very new technique based on the Gegenbauer wavelet series is introduced to solve the Lane-Emden differential equation. The Gegenbauer wavelets are derived by dilation and translation of an orthogonal Gegenbauer polynomial. The orthonormality of Gegenbauer wavelets is verified by the orthogonality of classical Gegenbauer polynomials. The convergence analysis of Gegenbauer wavelet series is studied in H?lder?s class. H?lder?s class H?[0,1) and H?[0,1) of functions are considered, H?[0,1) class consides with classical H?lder?s class H?[0, 1) if ?(t) = t?, 0 < ? ? 1. The Gegenbauer wavelet approximations of solution functions of the Lane-Emden differential equation in these classes are determined by partial sums of their wavelet series. In briefly, four approximations E(1) 2k?1,0, E(1) 2k?1,M, E(2) 2k?1,0, E(2) 2k?1,M of solution functions of classes H?[0, 1), H?[0, 1) by (2k?1, 0)th and (2k?1,M)th partial sums of their Gegenbauer wavelet expansions have been estimated. The solution of the Lane-Emden differential equation obtained by the Gegenbauer wavelets is compared to its solution derived by using Legendre wavelets and Chebyshev wavelets. It is observed that the solutions obtained by Gegenbauer wavelets are better than those obtained by using Legendre wavelets and Chebyshev wavelets, and they coincide almost exactly with their exact solutions. This is an accomplishment of this research paper in wavelet analysis.
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SHUKLA, N. K., and G. C. S. YADAV. "CONSTRUCTING NON-MSF WAVELETS FROM GENERALIZED JOURNÉ WAVELET SETS." Analysis and Applications 09, no. 02 (April 2011): 225–33. http://dx.doi.org/10.1142/s0219530511001820.
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Dai and Larson [Mem. Amer. Math. Soc.134 (1998), no. 640] obtained a family of wavelet sets using the Journé wavelet set. In this paper, we expand this family and call its members to be generalized Journé wavelet sets. Furthermore, with the help of these wavelet sets, we provide a class of non-MSF wavelets which includes the one constructed by Vyas [Bull. Polish Acad. Sci. Math.57 (2009) 33–40]. Most of these non-MSF wavelets are found to be non-MRA.
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Tkacheshak, N. V., and М. І. Horbiichuk. "INVESTIGATION OF EXHAUSTING PHENOMENA OF GAS TURBINE ENGINE ON THE BASIS OF WAVELET ANALYSIS." Scientific Bulletin of Ivano-Frankivsk National Technical University of Oil and Gas, no. 2(45) (November 27, 2018): 24–33. http://dx.doi.org/10.31471/1993-9965-2018-2(45)-24-33.
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The method for research of a gas turbine engine (GTE) surging phenomena based on wavelet analysis was developed. GTE DG-90 was chosen for research. A wavelet decomposition of the air pressure signal was carried out using a high pressure compressor and Dobeski, Symlet, Coffle and Meyer's discrete wavelet. The advantages and disadvantages of filtering properties of these wavelets were graphically represented in the form of amplitude-frequency characteristics. Based on the results obtained, a wavelet transform parameter selection scheme based on the analysis of the frequency response of wavelet filters was constructed, and the optimum sampling time of the wavelet filtering of the investigated signal was calculated for determination of the propagation and rotational breakdown in the GTE turbocompressor. According to this scheme, among the wavelets of Dobeski, Simlet, Koefleta, and Meyer's discrete wavelet, for studying the air pressure signal at the compressor during bursting processes, the first were selected by the Dobeches and Symmetes wavelets of the 2nd order according to their filtration rate. Considering high speed and characteristics of the description of the occurrence of excite phenomena by means of a fast Fourier transform to the distribution of the wavelet coefficients of the air pressure signal by turbocompressor, for the diagnosis of unstable flows occurring in the gas path of the gas turbine trajectory (surging and rotary breakdown), the Dobeski wavelet was selected 2nd order. At the same time, to monitor the wavelet coefficients behavior, the detail of the investigated signal is necessary at the 7th level of decomposition. Thus, the obtained results allow to carry out further analysis of breakdown processes using low-order wavelets.
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Newland, D. E. "Wavelet Analysis of Vibration: Part 2—Wavelet Maps." Journal of Vibration and Acoustics 116, no. 4 (October 1, 1994): 417–25. http://dx.doi.org/10.1115/1.2930444.
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Wavelet maps provide a graphical picture of the frequency composition of a vibration signal. This paper, which is Part 2 of a pair, describes their construction and properties. In the case of harmonic wavelets, there are close similarities between wavelet maps and sonograms. A range of practical examples illustrate how the wavelet method may be applied to vibration analysis and some of its advantages.
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Stepanov, Andrey. "Polynomial, Neural Network, and Spline Wavelet Models for Continuous Wavelet Transform of Signals." Sensors 21, no. 19 (September 26, 2021): 6416. http://dx.doi.org/10.3390/s21196416.
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In this paper a modified wavelet synthesis algorithm for continuous wavelet transform is proposed, allowing one to obtain a guaranteed approximation of the maternal wavelet to the sample of the analyzed signal (overlap match) and, at the same time, a formalized representation of the wavelet. What distinguishes this method from similar ones? During the procedure of wavelets’ synthesis for continuous wavelet transform it is proposed to use splines and artificial neural networks. The paper also suggests a comparative analysis of polynomial, neural network, and wavelet spline models. It also deals with feasibility of using these models in the synthesis of wavelets during such studies like fine structure of signals, as well as in analysis of large parts of signals whose shape is variable. A number of studies have shown that during the wavelets’ synthesis, the use of artificial neural networks (based on radial basis functions) and cubic splines enables the possibility of obtaining guaranteed accuracy in approaching the maternal wavelet to the signal’s sample (with no approximation error). It also allows for its formalized representation, which is especially important during software implementation of the algorithm for calculating the continuous conversions at digital signal processors and microcontrollers. This paper demonstrates the possibility of using synthesized wavelet, obtained based on polynomial, neural network, and spline models, during the performance of an inverse continuous wavelet transform.
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Jurado, F., and S. Lopez. "A wavelet neural control scheme for a quadrotor unmanned aerial vehicle." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, no. 2126 (July 9, 2018): 20170248. http://dx.doi.org/10.1098/rsta.2017.0248.
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Wavelets are designed to have compact support in both time and frequency, giving them the ability to represent a signal in the two-dimensional time–frequency plane. The Gaussian, the Mexican hat and the Morlet wavelets are crude wavelets that can be used only in continuous decomposition. The Morlet wavelet is complex-valued and suitable for feature extraction using the continuous wavelet transform. Continuous wavelets are favoured when high temporal and spectral resolution is required at all scales. In this paper, considering the properties from the Morlet wavelet and based on the structure of a recurrent high-order neural network model, a novel wavelet neural network structure, here called a recurrent Morlet wavelet neural network, is proposed in order to achieve a better identification of the behaviour of dynamic systems. The effectiveness of our proposal is explored through the design of a decentralized neural backstepping control scheme for a quadrotor unmanned aerial vehicle. The performance of the overall neural identification and control scheme is verified via simulation and real-time results. This article is part of the theme issue ‘Redundancy rules: the continuous wavelet transform comes of age’.
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Bayjja, M., G. Alsharahi, M. Aghoutane, and N. A. Touhami. "Comparison of Wavelet Packet and Wavelet in Solving Arbitrary Array of Parallel Wires Integral Equations in Electromagnetics." Advanced Electromagnetics 9, no. 3 (November 23, 2020): 8–14. http://dx.doi.org/10.7716/aem.v9i3.1487.
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In this paper, wavelets transformation (WT) and wavelet packet transformation (WPT) are used in solving, by the method of moments, a semicircular array of parallel wires electric field integral equation. First, the integral equation is solved by applying the direct method of moments via point-matching procedure, results in a linear system with a dense matrix. Therefore, wavelet transformation and wavelet packet transformation are used to sparsify the impedance matrix, using two categories of wavelets functions, Biorthogonal (bior2.2) and Orthogonal (db4) wavelets. The far-field scattering patterns and the comparison between wavelets transformation and wavelet packet transformation in term number of zeros in impedance matrix and CPU Time reduction are presented. Numerical results are presented to identify which technique is best suited to solve such scattering electromagnetic problems and compared with published results.
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Apolonio, Felipe A., Daniel H. T. Franco, and Fábio N. Fagundes. "A Note on Directional Wavelet Transform: Distributional Boundary Values and Analytic Wavefront Sets." International Journal of Mathematics and Mathematical Sciences 2012 (2012): 1–11. http://dx.doi.org/10.1155/2012/758694.
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By using a particular class of directional wavelets (namely, the conical wavelets, which are wavelets strictly supported in a proper convex cone in thek-space of frequencies), in this paper, it is shown that a tempered distribution is obtained as a finite sum of boundary values of analytic functions arising from the complexification of the translational parameter of the wavelet transform. Moreover, we show that for a given distributionf∈𝒮′(ℝn), the continuous wavelet transform offwith respect to a conical wavelet is defined in such a way that the directional wavelet transform offyields a function on phase space whose high-frequency singularities are precisely the elements in the analytic wavefront set off.
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Hsu, Pai-Hui. "EVALUATING THE INITIALIZATION METHODS OF WAVELET NETWORKS FOR HYPERSPECTRAL IMAGE CLASSIFICATION." ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XLI-B7 (June 17, 2016): 83–89. http://dx.doi.org/10.5194/isprs-archives-xli-b7-83-2016.
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The idea of using artificial neural network has been proven useful for hyperspectral image classification. However, the high dimensionality of hyperspectral images usually leads to the failure of constructing an effective neural network classifier. To improve the performance of neural network classifier, wavelet-based feature extraction algorithms can be applied to extract useful features for hyperspectral image classification. However, the extracted features with fixed position and dilation parameters of the wavelets provide insufficient characteristics of spectrum. In this study, wavelet networks which integrates the advantages of wavelet-based feature extraction and neural networks classification is proposed for hyperspectral image classification. Wavelet networks is a kind of feed-forward neural networks using wavelets as activation function. Both the position and the dilation parameters of the wavelets are optimized as well as the weights of the network during the training phase. The value of wavelet networks lies in their capabilities of optimizing network weights and extracting essential features simultaneously for hyperspectral images classification. In this study, the influence of the learning rate and momentum term during the network training phase is presented, and several initialization modes of wavelet networks were used to test the performance of wavelet networks.
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Hsu, Pai-Hui. "EVALUATING THE INITIALIZATION METHODS OF WAVELET NETWORKS FOR HYPERSPECTRAL IMAGE CLASSIFICATION." ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XLI-B7 (June 17, 2016): 83–89. http://dx.doi.org/10.5194/isprsarchives-xli-b7-83-2016.
Full textAbstract:
The idea of using artificial neural network has been proven useful for hyperspectral image classification. However, the high dimensionality of hyperspectral images usually leads to the failure of constructing an effective neural network classifier. To improve the performance of neural network classifier, wavelet-based feature extraction algorithms can be applied to extract useful features for hyperspectral image classification. However, the extracted features with fixed position and dilation parameters of the wavelets provide insufficient characteristics of spectrum. In this study, wavelet networks which integrates the advantages of wavelet-based feature extraction and neural networks classification is proposed for hyperspectral image classification. Wavelet networks is a kind of feed-forward neural networks using wavelets as activation function. Both the position and the dilation parameters of the wavelets are optimized as well as the weights of the network during the training phase. The value of wavelet networks lies in their capabilities of optimizing network weights and extracting essential features simultaneously for hyperspectral images classification. In this study, the influence of the learning rate and momentum term during the network training phase is presented, and several initialization modes of wavelet networks were used to test the performance of wavelet networks.
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DEBNATH, LOKENATH, and SARALEES NADARAJAH. "POPULAR WAVELET MODELS." International Journal of Wavelets, Multiresolution and Information Processing 05, no. 04 (July 2007): 655–66. http://dx.doi.org/10.1142/s0219691307001951.
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The modern approach for wavelets imposes a Bayesian prior model on the wavelet coefficients to capture the sparseness of the wavelet expansion. The idea is to build flexible probability models for the marginal posterior densities of the wavelet coefficients. In this note, we derive exact expressions for two popular models for the marginal posterior density. We also illustrate the superior performance of these models over the standard models for wavelet coefficients.
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Li, Qiufu, and Linlin Shen. "Neuron segmentation using 3D wavelet integrated encoder–decoder network." Bioinformatics 38, no. 3 (October 14, 2021): 809–17. http://dx.doi.org/10.1093/bioinformatics/btab716.
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Abstract Motivation 3D neuron segmentation is a key step for the neuron digital reconstruction, which is essential for exploring brain circuits and understanding brain functions. However, the fine line-shaped nerve fibers of neuron could spread in a large region, which brings great computational cost to the neuron segmentation. Meanwhile, the strong noises and disconnected nerve fibers bring great challenges to the task. Results In this article, we propose a 3D wavelet and deep learning-based 3D neuron segmentation method. The neuronal image is first partitioned into neuronal cubes to simplify the segmentation task. Then, we design 3D WaveUNet, the first 3D wavelet integrated encoder–decoder network, to segment the nerve fibers in the cubes; the wavelets could assist the deep networks in suppressing data noises and connecting the broken fibers. We also produce a Neuronal Cube Dataset (NeuCuDa) using the biggest available annotated neuronal image dataset, BigNeuron, to train 3D WaveUNet. Finally, the nerve fibers segmented in cubes are assembled to generate the complete neuron, which is digitally reconstructed using an available automatic tracing algorithm. The experimental results show that our neuron segmentation method could completely extract the target neuron in noisy neuronal images. The integrated 3D wavelets can efficiently improve the performance of 3D neuron segmentation and reconstruction. Availabilityand implementation The data and codes for this work are available at https://github.com/LiQiufu/3D-WaveUNet. Supplementary information Supplementary data are available at Bioinformatics online.
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Dixit, M. M., C. P. Pandey, and Pratima Devi. "Wavelet frames in Sobolev space over locally compact abelian group." Boletim da Sociedade Paranaense de Matemática 42 (May 8, 2024): 1–11. http://dx.doi.org/10.5269/bspm.65621.
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In this paper we construct wavelet frames for continuous and discrete wavelets on Sobolev space over abelian group. A necessary condition and sufficient conditions for wavelet frames in Sobolev space over Locally CompactAbelian Group are given. Moreover some important properties of continuous wavelet transform and corresponding wavelet Frames have been discussed
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Zhang, Xi, and Noriaki Fukuda. "Lossy to lossless image coding based on wavelets using a complex allpass filter." International Journal of Wavelets, Multiresolution and Information Processing 12, no. 04 (July 2014): 1460002. http://dx.doi.org/10.1142/s0219691314600029.
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Wavelet-based image coding has been adopted in the international standard JPEG 2000 for its efficiency. It is well-known that the orthogonality and symmetry of wavelets are two important properties for many applications of signal processing and image processing. Both can be simultaneously realized by the wavelet filter banks composed of a complex allpass filter, thus, it is expected to get a better coding performance than the conventional biorthogonal wavelets. This paper proposes an effective implementation of orthonormal symmetric wavelet filter banks composed of a complex allpass filter for lossy to lossless image compression. First, irreversible real-to-real wavelet transforms are realized by implementing a complex allpass filter for lossy image coding. Next, reversible integer-to-integer wavelet transforms are proposed by incorporating the rounding operation into the filtering processing to obtain an invertible complex allpass filter for lossless image coding. Finally, the coding performance of the proposed orthonormal symmetric wavelets is evaluated and compared with the D-9/7 and D-5/3 biorthogonal wavelets. It is shown from the experimental results that the proposed allpass-based orthonormal symmetric wavelets can achieve a better coding performance than the conventional D-9/7 and D-5/3 biorthogonal wavelets both in lossy and lossless coding.
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Xu, Yong Fan. "The Characteristics of Orthogonal Wavelet Frames and Canonical Frames and Applications in Material Science." Advanced Materials Research 721 (July 2013): 741–44. http://dx.doi.org/10.4028/www.scientific.net/amr.721.741.
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Wavelet analysis has become a popular subject in scientific research during the past twenty years. We show that there exist wavelet frame generated by two functions which have good dual wavelet frames, but for which the canonical dual wavelet frame does not consist of wavelets, according to scaling functions. That is to say, the canonical dual wavelet frame cannot be generated by the translations and dilations of a single function. Traits of tight wavelet frames are presented.
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OTHMANI, MOHAMED, WAJDI BELLIL, CHOKRI BEN AMAR, and ADEL M. ALIMI. "A NEW STRUCTURE AND TRAINING PROCEDURE FOR MULTI-MOTHER WAVELET NETWORKS." International Journal of Wavelets, Multiresolution and Information Processing 08, no. 01 (January 2010): 149–75. http://dx.doi.org/10.1142/s0219691310003353.
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This paper deals with the features of a new wavelet network structure founded on several mother wavelets families. This new structure is similar to the classic wavelets network but it admits some differences eventually. The wavelet network basically uses the dilations and translations versions of only one mother wavelet to construct the network, but the new one uses several mother wavelets and the objective is to maximize the probability of selection of the best wavelets. Two methods are presented to assist the training procedure of this new structure. On one hand, we have an optimal selection technique that is based on an improved version of the Orthogonal Least Squares method; on the other, the Generalized Cross-Validation method to determine the number of wavelets to be selected for every mother wavelet. Some simulation results are reported to demonstrate the performance and the effectiveness of the new structure and the training procedure for function approximation in one and two dimensions.
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Ashurov, Ravshan. "On the almost-everywhere convergence of the continuous wavelet transforms." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 142, no. 6 (November 27, 2012): 1121–29. http://dx.doi.org/10.1017/s030821051000123x.
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Almost-everywhere convergence of wavelet transforms of Lp-functions under minimal conditions on wavelets was proved by Rao et al. in 1994. However, results on convergence almost everywhere do not provide any information regarding the exceptional set (of Lebesgue measure zero), where convergence does not hold. We prove that if a wavelet ψ satisfies a single additional condition xψ(x) ∈ L1 (R), then, instead of almost-everywhere convergence, we have a more sophisticated result, i.e. convergence of wavelet transforms everywhere on the entire Lebesgue set of Lp-functions. For example, wavelets with compact support, used frequently in applications, obviously satisfy this extra condition. Moreover, we prove that our conditions on wavelets ensure the Riemann localization principle in L1 for the wavelet transforms.
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KUMARI, R. SHANTHA SELVA, S. BHARATHI, and V. SADASIVAM. "QRS COMPLEX DETECTION USING OPTIMAL DISCRETE WAVELET." International Journal of Computational Intelligence and Applications 08, no. 02 (June 2009): 97–109. http://dx.doi.org/10.1142/s1469026809002576.
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Wavelet transform has emerged as a powerful tool for time frequency analysis of complex nonstationary signals such as the electrocardiogram (ECG) signal. In this paper, the design of good wavelets for cardiac signal is discussed from the perspective of orthogonal filter banks. Optimum wavelet for ECG signal is designed and evaluated based on perfect reconstruction conditions and QRS complex detection. The performance is evaluated by using the ECG records from the MIT-BIH arrhythmia database. In the first step, the filter coefficients (optimum wavelet) is designed by reparametrization of filter coefficients. In the second step, ECG signal is decomposed to three levels using the optimum wavelet and reconstructed. From the reconstructed signal, the range of error signal is calculated and it is compared with the performance of other suitable wavelets already available in the literature. The optimum wavelet gives the maximum error range as 10-14–10-11 which is better than that of other wavelets existing in the literature. In the third step, the baseline wandering is removed from the ECG signal for better detection of QRS complex. The optimum wavelet detects all R peaks of all records. That is using optimum wavelet 100% sensitivity and positive predictions are achieved. Based on the performance, it is confirmed that optimum wavelet is more suitable for ECG signal.
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Zhang, Ming, Zhuo Ma, and Min Xuan Zhang. "FPGA Implementation of Rational Symmetric Biorthogonal 11-9 Wavelet Transform." Applied Mechanics and Materials 182-183 (June 2012): 1791–95. http://dx.doi.org/10.4028/www.scientific.net/amm.182-183.1791.
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Coefficients of most existing wavelets are irrational, and it costs much hardware resources when implementing on FPGA, which is inefficient especially in embedded system. Some rational wavelets can overcome this deficiency by elaborate design. Motivated by previous works on rational wavelets, we establish a hardware structure for rational 1-D symmetric biorthogonal 11-9 wavelet and implement it on Xilinx FPGA XC3S500E. The experiment reveals that the area in slices of rational 1-D 11-9 wavelet is less than 1/2 of the pipelined 9-7 wavelet when implementing on FPGA.
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Fang, Lanting, Lenan Wu, and Yudong Zhang. "A Novel Demodulation System Based on Continuous Wavelet Transform." Mathematical Problems in Engineering 2015 (2015): 1–9. http://dx.doi.org/10.1155/2015/513849.
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Considering the problem of EBPSK signal demodulation, a new approach based on the wavelet scalogram using continuous wavelet transform is proposed. Our system is twofold: an adaptive wavelet construction method that replaces manual selection existing wavelets method and, on the other hand, a nonlinear demodulation system based on image processing and pattern classification is proposed. To evaluate the performance of the adaptive wavelet and compare the performance of the proposed system with the existing systems, a series of comprehensive simulation experiments is conducted under the environment of uniform white noise, colored noise, and additive white Gaussian noise channel, respectively. Simulation results of different wavelets show that the system using adaptive wavelet has lower bit error rate (BER). Moreover, simulation results of several systems show that the BER of the proposed system is the lowest among all systems, such as amplitude detection, integral detection, and some continuous wavelet transform systems (specific scales and times and maximum lines). In a word, the adaptive wavelet construction proposed in this paper yields superior performances compared with the manual selection, and the proposed system has better performances than the existing systems. Index terms are signal demodulation, adaptive wavelet, continuous wavelet transform, and BER.
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Wang, Jun Qiu, and Jian Guo Wang. "The Characterization of a Pair of Canonical Frames Generated by Several Compactly Supported Functions." Key Engineering Materials 439-440 (June 2010): 1135–40. http://dx.doi.org/10.4028/www.scientific.net/kem.439-440.1135.
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Wavelet analysis has become a popular subject in scientific research during the past twenty years. We show that there exist wavelet frame generated by two functions which have good dual wavelet frames, but for which the canonical dual wavelet frame does not consist of wavelets, according to scaling functions. That is to say, the canonical dual wavelet frame cannot be generated by the translations and dilations of a single function.
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Li, Rui Ming, Liang Gong, and Qi Xiong. "Construction Method of Perfect Reconstruction Condition-Based Biorthogonal Wavelet." Applied Mechanics and Materials 416-417 (September 2013): 1305–8. http://dx.doi.org/10.4028/www.scientific.net/amm.416-417.1305.
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This paper proposes a general construction method of biorthogonal wavelet based on perfect reconstruction condition. With the certain filter length and vanishing moment, it can educe the biorthogonal wavelets filter coefficient by solving equations. Thereafter, this method constructs 5/3 wavelet, CDF9/7 wavelet and 9/7 wavelet with simple coefficient, which applies well to hardware, for JPEG2000.
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Garcia-Castro, Oscar Federico, Luis Enrique Ramos-Velasco, Rodolfo Garcia-Rodriguez, Mario Alejandro Vega-Navarrete, Enrique Escamilla-Hernández, and Luz Noe Oliva-Moreno. "Estudio comparativo de controladores PID WaveNet-IIR aplicado a un helicóptero de 2 GDL." Pädi Boletín Científico de Ciencias Básicas e Ingenierías del ICBI 10, Especial5 (November 11, 2022): 36–42. http://dx.doi.org/10.29057/icbi.v10iespecial5.10067.
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En aplicaciones de control inteligente, uno de los problemas es determinar el número de capas y de neuronas en cada capa. Esto se vuelve más complejo con ciertos tipos de redes neuronales como las basadas en wavelets donde las traslaciones y dilataciones son parámetros adicionales. Este artículo presenta un estudio comparativo para determinar el tipo de wavelet y número de neuronas que muestran el mejor desempeño para controlar un helicóptero Quanser de dos grados de libertad (GDL). Se presenta un controlador tipo PID-WaveNet-IIR el cuál se compone de controladores PID discretos en el tiempo, con ganancias auto-sintonizables por una red neuronal de base radial cuyas funciones de activación son wavelets y un par de filtros de respuesta de impulso infinito (IIR) para ``podar'' algunas neuronas. Por medio de simulaciones numéricas, usando LabVIEW, se presenta el desempeño del sistema en lazo cerrado para: diferentes condiciones de operación, tipos de familia wavelets donde se fijan valores mínimos de los errores de seguimiento, tipo de wavelet, el número de neuronas de la red y número de coeficientes de adelanto y atraso de los filtros IIR.
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Kuznetsov, Nikolay A. "METHOD FOR CONSTRUCTING NONLINEAR WAVELET CODE TO ENSURE DATA INTEGRITY IN COMMUNICATION CHANNELS." T-Comm 15, no. 2 (2021): 26–32. http://dx.doi.org/10.36724/2072-8735-2021-15-2-26-32.
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A method for constructing a nonlinear wavelet code (NVC) to ensure data integrity in communication channels, taking into account current threats to information security in a modern dynamic stochastic environment, is proposed. A special place among the methods of combating threats to the integrity of information is occupied by noise-resistant encoding. The article presents a computationally effective method for ensuring data integrity in communication channels by using nonlinear transformations and wavelets. The approximation of the wavelet transform refers to the division of the signal into approximating and detailing components. Continuous and discrete wavelet transforms are widely used [2] for signal analysis in modern communication channels. The set of functions defining the wavelet transform belongs to the space of square-integrable functions on a straight line and provides a necessary condition for constructing constructions of nonlinear codes based on the theory of wavelet decomposition. As is known, in the process of wavelet analysis, the signal is decomposed along the orthogonal basis formed by shifts of the wavelet function. A distinctive feature of this approach is that convolution of the signal with wavelets allows us to identify the characteristic features of the signal in the area of localization of these wavelets. To perform computational calculations, you need a set of scaling function coefficients and a wavelet. The wavelet transform matrix depends on the coefficients of the scaling function. The results presented in the article describe a new approach to ensuring data integrity in communication channels using nvcs. A computational example is presented.
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Sun, Song Zhen, and Yi Guo. "Study of Periodic Frames and Trivariate Tight Wavelet Frames and Applications in Materials Engineering." Advanced Materials Research 1079-1080 (December 2014): 878–81. http://dx.doi.org/10.4028/www.scientific.net/amr.1079-1080.878.
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It is shown that there exists a frame wavelet with fast decay in the time domain and compact support in the frequency domain generating a wavelet system whose canonical dual frame cannot be generated by an arbitrary number of generators. We show that there exist wavelet frame generated by two functions which have good dual wavelet frames, but for which the canonical dual wavelet frame does not consist of wavelets, according to scaling functions.
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Tang, Shoufeng, Minming Tong, and Xinmin He. "The Optimum Wavelet Base of Wavelet Analysis in Coal Rock Microseismic Signals." Advances in Mechanical Engineering 6 (January 1, 2014): 537415. http://dx.doi.org/10.1155/2014/537415.
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Coal rock rupture microseismic signal is characterized by time-varying, nonstationary, unpredictability, and transient property. Wavelet transform is an important method in microseismic signals processing. However, different wavelet bases yield different results when analyzing the same signal. To study the comparability of different wavelet bases in analyzing microseismic signals, the current paper uses the microseismic signals released from coal rock bursting as the research subject. Through the analysis of the properties of commonly used wavelet basis functions and the characteristics of coal rock microseismic signals, the current study found that Coiflet and Symlet wavelets are suitable for analyzing coal rock microseismic signals. Sym 8 and Coif 2 wavelets were found to be suitable for analyzing and denoising coal rock microseismic signals. After Sym 8 wavelet denoising, signal-to-noise ratio (SNR) and the root mean square error were 30.4184 and 1.3109 E–07, respectively. After Coif 2 wavelet denoising, the SNR and the root mean square error values were 35.2176 and 1.0312 E–07, respectively. The results will aid in the analysis and extraction of coal rock microseismic signals.
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Ahmad, Owais. "Characterization of tight wavelet frames with composite dilations in L2(Rn)." Publications de l'Institut Math?matique (Belgrade) 113, no. 127 (2023): 121–29. http://dx.doi.org/10.2298/pim2327121a.
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Tight wavelet frames are different from the orthonormal wavelets because of redundancy. By sacrificing orthonormality and allowing redundancy, the tight wavelet frames become much easier to construct than the orthonormal wavelets. Guo, Labate, Lim, Weiss, and Wilson [Electron. Res. Announc. Am. Math. Soc. 10 (2004), 78-87] introduced the theory of wavelets with composite dilations in order to provide a framework for the construction of waveforms defined not only at various scales and locations but also at various orientations. In this paper, we provide the characterization of composite wavelet system to be tight frame for L2(Rn).
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Kathuria, Leena, Shashank Goel, and Nikhil Khanna. "Fourier–Boas-Like Wavelets and Their Vanishing Moments." Journal of Mathematics 2021 (March 6, 2021): 1–7. http://dx.doi.org/10.1155/2021/6619551.
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In this paper, we propose Fourier–Boas-Like wavelets and obtain sufficient conditions for their higher vanishing moments. A sufficient condition is given to obtain moment formula for such wavelets. Some properties of Fourier–Boas-Like wavelets associated with Riesz projectors are also given. Finally, we formulate a variation diminishing wavelet associated with a Fourier–Boas-Like wavelet.
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Barnes, A. E. "Instantaneous frequency and amplitude at the envelope peak of a constant‐phase wavelet." GEOPHYSICS 56, no. 7 (July 1991): 1058–60. http://dx.doi.org/10.1190/1.1443115.
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Robertson and Nogami (1984) have shown that the instantaneous frequency at the peak of a zero‐phase Ricker wavelet is exactly equal to that wavelet’s average Fourier spectral frequency weighted by its amplitude spectrum. Bodine (1986) gave an example which shows this is also true for constant‐phase bandpass wavelets. Here I prove that this holds for any constant‐phase wavelet. I then develop an equation expressing this quantity as a function of propagation time through an attenuating medium. A corresponding equation is derived for the amplitude of the envelope peak. Taken together, these may aid in the analysis of seismic data as suggested by Robertson and Nogami (1984), Bodine (1986), and Robertson and Fisher (1988).
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Zeelan Basha, CMAK, K. M. Sricharan, Ch Krishna Dheeraj, and R. Ramya Sri. "A Study on Wavelet Transform Using Image Analysis." International Journal of Engineering & Technology 7, no. 2.32 (May 31, 2018): 94. http://dx.doi.org/10.14419/ijet.v7i2.32.13535.
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The wavelet transforms have been in use for variety of applications. It is widely being used in signal analysis and image analysis. There have been lot of wavelet transforms for compression, decomposition and reconstruction of images. Out of many transforms Haar wavelet transform is the most computationally feasible wavelet transform to implement. The wave analysis technique has an understandable impact on the removal of noise within the signal. The paper outlines the principles and performance of wavelets in image analysis. Compression performance and decomposition of images into different layers have been discussed in this paper. We used Haar distinct wavelet remodel (HDWT) to compress the image. Simulation of wavelet transform was done in MATLAB. Simulation results are conferred for the compression with Haar rippling with totally different level of decomposition. Energy retention and PSNR values are calculated for the wavelets. Result conjointly reveals that the extent of decomposition will increase beholding of the photographs goes on decreasing however the extent of compression is incredibly high. Experimental results demonstrate the effectiveness of the Haar wavelet transform in energy retention in comparison to other wavelet transforms.
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Xia, Chunxu, and Chunguang Liu. "Identification and Representation of Multi-Pulse Near-Fault Strong Ground Motion Using Adaptive Wavelet Transform." Applied Sciences 9, no. 2 (January 12, 2019): 259. http://dx.doi.org/10.3390/app9020259.
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In order to identify the horizontal seismic motion owning the largest pulse energy, and represent the dominant pulse-like component embedded in this seismic motion, we used the adaptive wavelet transform algorithm in this paper. Fifteen candidate mother wavelets were evaluated to select the optimum wavelet based on the similarities between the candidate mother wavelet and the target seismic motion, evaluated by the minimum cross variance. This adaptive choosing algorithm for the optimum mother wavelet was invoked before identifying both the horizontal direction owning the largest pulse energy and every dominant pulse, which provides the optimum mother wavelet for the continuous wavelet transform. Each dominant pulse can be represented by its adaptively selected optimum mother wavelet. The results indicate that the identified multi-pulse component fits well with the seismic motion. In most cases, mother wavelets in one multi-pulse seismic motion were different from each other. For the Chi-Chi event (1999-Sep-20 17:47:16 UTC, Mw = 7.6), 62.26% of the qualified pulse-like earthquake motions lay in the horizontal direction ranging from ±15° to ±75°. The Daubechies 6 (db6) mother wavelet was the most frequently used type for both the first and second pulse components.
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Ilango, S. Sudhakar, and V. Seenivasagam. "Wavelet Based Image Compression Using Soft Computing Techniques." Applied Mechanics and Materials 573 (June 2014): 477–82. http://dx.doi.org/10.4028/www.scientific.net/amm.573.477.
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The Wavelet Transform is a multi-resolution transform, that is, it allows a form of time–frequency analysis (or translation–scale in wavelet speak). When using the Fourier transform the result is a very precise analysis of the frequencies contained in the signal, but no information on when those frequencies occurred. The wavelet based image compression algorithms are used widely compared with other conventional compression algorithms. The wavelet coding based on the coefficient selection and sub band level. In this paper we have used two wavelets such as spherical and geometric wavelets. The spherical representation is a hierarchical description of how total energy gets distributed within each wavelet sub band.In the proposed method, we used fuzzy quantization technique for coefficient selection in the spherical wavelet. The other scheme introduces binary space partitioning scheme and geometric wavelet, where the existing pruning method of binary space partitioning is replaced by the genetic algorithm. We had another experiment with geometric wavelet with Artificial Bee Colony (ABC) algorithm. The experimental results for all the three methods are discussed in this paper. The advantages of these methods are the improved PSNR values at high and medium bit rates.
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BAHRI, MAWARDI, and ECKHARD S. M. HITZER. "CLIFFORD ALGEBRA Cl3,0-VALUED WAVELET TRANSFORMATION, CLIFFORD WAVELET UNCERTAINTY INEQUALITY AND CLIFFORD GABOR WAVELETS." International Journal of Wavelets, Multiresolution and Information Processing 05, no. 06 (November 2007): 997–1019. http://dx.doi.org/10.1142/s0219691307002166.
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In this paper, it is shown how continuous Clifford Cl3,0-valued admissible wavelets can be constructed using the similitude group SIM(3), a subgroup of the affine group of ℝ3. We express the admissibility condition in terms of a Cl3,0 Clifford Fourier transform and then derive a set of important properties such as dilation, translation and rotation covariance, a reproducing kernel, and show how to invert the Clifford wavelet transform of multivector functions. We invent a generalized Clifford wavelet uncertainty principle. For scalar admissibility constant, it sets bounds of accuracy in multivector wavelet signal and image processing. As concrete example, we introduce multivector Clifford Gabor wavelets, and describe important properties such as the Clifford Gabor transform isometry, a reconstruction formula, and an uncertainty principle for Clifford Gabor wavelets.