Journal articles: 'Wavelets'

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Battle, Guy. "Osiris wavelets and Set wavelets." Journal of Applied Mathematics 2004, no. 6 (2004): 495–528. http://dx.doi.org/10.1155/s1110757x04404070.

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An alternative to Osiris wavelet systems is introduced in two dimensions. The basic building blocks are continuous piecewise linear functions supported on equilateral triangles instead of on squares. We refer to wavelets generated in this way as Set wavelets. We introduce a Set wavelet system whose homogeneous mode density is2/5. The system is not orthonormal, but we derive a positive lower bound on the overlap matrix.

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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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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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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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ASHUROV, RAVSHAN. "CONVERGENCE OF THE CONTINUOUS WAVELET TRANSFORMS ON THE ENTIRE LEBESGUE SET OF Lp-FUNCTIONS." International Journal of Wavelets, Multiresolution and Information Processing 09, no. 04 (July 2011): 675–83. http://dx.doi.org/10.1142/s0219691311004262.

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The almost everywhere convergence of wavelets transforms of Lp-functions under minimal conditions on wavelets is well known. But this result does not provide any information about the exceptional set (of Lebesgue measure zero), where convergence does not hold. In this paper, under slightly stronger conditions on wavelets, we prove convergence of wavelet transforms everywhere on the entire Lebesgue set of Lp-functions. On the other hand, practically all the wavelets, including Haar and "French hat" wavelets, used frequently in applications, satisfy our conditions. We also prove that the same conditions on wavelets guarantee the Riemann localization principle in L1 for the wavelet transforms.

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ZHAN, YINWEI, and HENK J. A. M. HEIJMANS. "NON-SEPARABLE 2D BIORTHOGONAL WAVELETS WITH TWO-ROW FILTERS." International Journal of Wavelets, Multiresolution and Information Processing 03, no. 01 (March 2005): 1–18. http://dx.doi.org/10.1142/s0219691305000713.

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In the literature 2D (or bivariate) wavelets are usually constructed as a tensor product of 1D wavelets. Such wavelets are called separable. However, there are various applications, e.g. in image processing, for which non-separable 2D wavelets are prefered. In this paper, we investigate the class of compactly supported orthonormal 2D wavelets that was introduced by Belogay and Wang.2 A characteristic feature of this class of wavelets is that the support of the corresponding filter comprises only two rows. We are concerned with the biorthogonal extension of this kind of wavelets. It turns out that the 2D wavelets in this class are intimately related to some underlying 1D wavelet. We explore this relation in detail, and we explain how the 2D wavelet transforms can be realized by means of a lifting scheme, thus allowing an efficient implementation. We also describe an easy way to construct wavelets with more rows and shorter columns.

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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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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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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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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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Gao, Li, Tao Zhao, and Yun Peng Liu. "Suppressing PD’s Narrow Band Noise in the PD Monitoring Frequency Band Using Complex Wavelet Packet Transform." Applied Mechanics and Materials 521 (February 2014): 352–57. http://dx.doi.org/10.4028/www.scientific.net/amm.521.352.

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By combining the method of construct wavelet packet and complex wavelets, complex wavelets packet is constructed with the same amplitude-frequency characteristics of corresponding real wavelet packet and the same phase spectrum of corresponding complex wavelets, and then give the specific method of constructing complex wavelets packet, evaluate the effect of original PD simulative signals direct reconstruction after decomposition by using Normalized Correlation Coefficient (NCC) and Variational Trend Parameter (VTP) also and the relative error of the amplitude, verify the capability of restore Non-stationary Oscillating Partial Discharge Signals effectively and the correctness of the complex wavelets packets construction. At last, complex wavelets packet is used to denoise PD admixture simulative and measured signals with different frequency and strength noises, overcome the difficults of more loss using hardware filter and bad effect using wavelets packet transformation when narrow band noise in the monitoring frequency band.

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Sahu, PK, and S. Saha Ray. "Comparison on wavelets techniques for solving fractional optimal control problems." Journal of Vibration and Control 24, no. 6 (July 25, 2016): 1185–201. http://dx.doi.org/10.1177/1077546316659611.

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This paper presents efficient numerical techniques for solving fractional optimal control problems (FOCP) based on orthonormal wavelets. These wavelets are like Legendre wavelets, Chebyshev wavelets, Laguerre wavelets and Cosine And Sine (CAS) wavelets. The formulation of FOCP and properties of these wavelets are presented. The fractional derivative considered in this problem is in the Caputo sense. The performance index of FOCP has been considered as function of both state and control variables and the dynamic constraints are expressed by fractional differential equation. These wavelet methods are applied to reduce the FOCP as system of algebraic equations by applying the method of constrained extremum which consists of adjoining the constraint equations to the performance index by a set of undetermined Lagrange multipliers. These algebraic systems are solved numerically by Newton's method. Illustrative examples are discussed to demonstrate the applicability and validity of the wavelet methods.

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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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Olphert, Sean, and Stephen C. Power. "Higher Rank Wavelets." Canadian Journal of Mathematics 63, no. 3 (June 1, 2011): 689–720. http://dx.doi.org/10.4153/cjm-2011-012-1.

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Abstract A theory of higher rank multiresolution analysis is given in the setting of abelian multiscalings. This theory enables the construction, from a higher rank MRA, of finite wavelet sets whose multidilations have translates forming an orthonormal basis in L2(ℝd). While tensor products of uniscaled MRAs provide simple examples we construct many nonseparable higher rank wavelets. In particular we construct Latin square wavelets as rank 2 variants of Haar wavelets. Also we construct nonseparable scaling functions for rank 2 variants of Meyer wavelet scaling functions, and we construct the associated nonseparable wavelets with compactly supported Fourier transforms. On the other hand we show that compactly supported scaling functions for biscaled MRAs are necessarily separable.

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Hadas-Dyduch, Monika. "Efficiency of Authored Mixed Prediction Model with Application to the Labor Market." Engineering Management Research 7, no. 1 (February 27, 2018): 46. http://dx.doi.org/10.5539/emr.v7n1p46.

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The aim of this article is to evaluate the prediction of time series using a model containing wavelets. The research hypothesis is: “Models that take into account wavelets are an effective tool for predicting employment”. To verify the hypothesis, an original model was devised. The model is based on wavelet analysis with Daubechies wavelets and an exponential alignment model. The exponential alignment model been appropriately modified by the introduction of wavelet functions. The results obtained show that a model that partially includes wavelets is an effective tool in the prediction and analysis of employment

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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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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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Cattani, Carlo. "Shannon Wavelets Theory." Mathematical Problems in Engineering 2008 (2008): 1–24. http://dx.doi.org/10.1155/2008/164808.

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Shannon wavelets are studied together with their differential properties (known as connection coefficients). It is shown that the Shannon sampling theorem can be considered in a more general approach suitable for analyzing functions ranging in multifrequency bands. This generalization coincides with the Shannon wavelet reconstruction ofL2(ℝ)functions. The differential properties of Shannon wavelets are also studied through the connection coefficients. It is shown that Shannon wavelets areC∞-functions and their any order derivatives can be analytically defined by some kind of a finite hypergeometric series. These coefficients make it possible to define the wavelet reconstruction of the derivatives of theCℓ-functions.

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VYAS, APARNA, and RAJESHWARI DUBEY. "NON-MSF WAVELETS FROM SIX INTERVAL MSF WAVELETS." International Journal of Wavelets, Multiresolution and Information Processing 09, no. 03 (May 2011): 375–85. http://dx.doi.org/10.1142/s021969131100416x.

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In this article, we obtain two classes of non-MSF wavelets by considering two different classes of six-interval wavelet sets provided by Arcozzi, Behera and Madan (J. Geom. Anal.13 (2003) 557–579). Out of these classes one is countable, the non-MSF wavelets of which are non-MRA and the other one is uncountable, the non-MSF wavelets of which are MRA. The set of all non-MSF MRA wavelets of the latter class is shown to be pathconnected.

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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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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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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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Naik, Ameya K., and Raghunath S. Holambe. "A unified framework for the design of low-complexity wavelet filters." International Journal of Wavelets, Multiresolution and Information Processing 15, no. 06 (November 2017): 1750054. http://dx.doi.org/10.1142/s0219691317500540.

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An outline is presented for construction of wavelet filters with compact support. Our approach does not require any extensive simulations for obtaining the values of design variables like other methods. A unified framework is proposed for designing halfband polynomials with varying vanishing moments. Optimum filter pairs can then be generated by factorization of the halfband polynomial. Although these optimum wavelets have characteristics close to that of CDF 9/7 (Cohen-Daubechies-Feauveau), a compact support may not be guaranteed. Subsequently, we show that by proper choice of design parameters finite wordlength wavelet construction can be achieved. These hardware friendly wavelets are analyzed for their possible applications in image compression and feature extraction. Simulation results show that the designed wavelets give better performances as compared to standard wavelets. Moreover, the designed wavelets can be implemented with significantly reduced hardware as compared to the existing wavelets.

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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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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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Ahmadi, H., G. Dumont, F. Sassani, and R. Tafreshi. "Performance of Informative Wavelets for Classification and Diagnosis of Machine Faults." International Journal of Wavelets, Multiresolution and Information Processing 01, no. 03 (September 2003): 275–89. http://dx.doi.org/10.1142/s0219691303000189.

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This paper deals with an application of wavelets for feature extraction and classification of machine faults in a real-world machine data analysis environment. We have utilized informative wavelet algorithm to generate wavelets and subsequent coefficients that are used as feature variables for classification and diagnosis of machine faults. Informative wavelets are classes of functions generated from a given analyzing wavelet in a wavelet packet decomposition structure in which for the selection of best wavelets, concepts from information theory, i.e. mutual information and entropy are utilized. Training data are used to construct probability distributions required for the computation of the entropy and mutual information. In our data analysis, we have used machine data acquired from a single cylinder engine under a series of induced faults in a test environment. The objective of the experiment was to evaluate the performance of the informative wavelet algorithm for the accuracy of classification results using a real-world machine data and to examine to what extent the results were influenced by different analyzing wavelets chosen for data analysis. Accuracy of classification results as related to the correlation structure of the coefficients is also discussed in the paper.

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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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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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HOU, YU. "A COMPACTLY SUPPORTED, SYMMETRICAL AND QUASI-ORTHOGONAL WAVELET." International Journal of Wavelets, Multiresolution and Information Processing 08, no. 06 (November 2010): 931–40. http://dx.doi.org/10.1142/s0219691310003900.

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Based on the wavelet theory and optimization method, a class of single wavelets with compact support, symmetry and quasi-orthogonality are designed and constructed. Some mathematical properties of the wavelets, such as orthogonality, linear phase property and vanishing moments and so on, are studied. A speech compression experiment is implemented in order to investigate the performance of signal reconstruction and speech compression for the proposed wavelets. Comparison with some conventional wavelets shows that the proposed wavelets have a very good performance of signal reconstruction and speech compression.

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Zhao, Jian Tang, and Hong Lin Guo. "Constructive Approaches to Biorthogonal Finitely Supported Ternary Wavelets and Wavelet Wraps." Advanced Materials Research 219-220 (March 2011): 504–7. http://dx.doi.org/10.4028/www.scientific.net/amr.219-220.504.

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In this paper, the notion of biorthogonal two-directional shortly supported wavelets with poly-scale is developed. A new method for designing two-directional biorthogonal wavelets is proposed. The existence of shortly supported biorthogonal vector-valued wavelets associated with a pair of biorthogonal compactly supported vector-valued scaling functions is investigated. A novel constructive method for designing a sort of biorthogonal vector-valued wavelet wraps is presented and their biorthogonality traits are characerized. Two biorthogonality formulas regarding these wavelet wraps are established.

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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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Cattani, Carlo, and Aleksey Kudreyko. "Application of Periodized Harmonic Wavelets towards Solution of Eigenvalue Problems for Integral Equations." Mathematical Problems in Engineering 2010 (2010): 1–8. http://dx.doi.org/10.1155/2010/570136.

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This article deals with the application of the periodized harmonic wavelets for solution of integral equations and eigenvalue problems. The solution is searched as a series of products of wavelet coefficients and wavelets. The absolute error for a general case of the wavelet approximation was analytically estimated.

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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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Fu, Shengyu, B. Muralikrishnan, and J. Raja. "Engineering Surface Analysis With Different Wavelet Bases." Journal of Manufacturing Science and Engineering 125, no. 4 (November 1, 2003): 844–52. http://dx.doi.org/10.1115/1.1616947.

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Traditional surface texture analysis involves filtering surface profiles into different wavelength bands commonly referred to as roughness, waviness and form. The primary motivation in filtering surface profiles is to map each band to the manufacturing process that generated the part and the intended functional performance of the component. Current trends in manufacturing are towards tighter tolerances and higher performance standards that require close monitoring of the process. Thus, there is a need for finer bandwidths for process mapping and functional correlation. Wavelets are becoming increasingly popular tools for filtering profiles in an efficient manner into multiple bands. While they have broadly been demonstrated as having superior performance and capabilities than traditional filtering, fundamental issues such as choice of wavelet bases have remained unaddressed. The major contribution of this paper is to present the differences between wavelets in terms of the transmission characteristics of the associated filter banks, which is essential for surface analysis. This paper also reviews fundamental mathematics of wavelet theory necessary for applying wavelets to surface texture analysis. Wavelets from two basic categories—orthogonal wavelet bases and biorthogonal wavelet bases are studied. The filter banks corresponding to the wavelets are compared and multiresolution analysis on surface profiles is performed to highlight the applicability of this technique.

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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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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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Mittal, R. C., and Sapna Pandit. "Quasilinearized Scale-3 Haar wavelets-based algorithm for numerical simulation of fractional dynamical systems." Engineering Computations 35, no. 5 (July 2, 2018): 1907–31. http://dx.doi.org/10.1108/ec-09-2017-0347.

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Purpose The main purpose of this work is to develop a novel algorithm based on Scale-3 Haar wavelets (S-3 HW) and quasilinearization for numerical simulation of dynamical system of ordinary differential equations. Design/methodology/approach The first step in the development of the algorithm is quasilinearization process to linearize the problem, and then Scale-3 Haar wavelets are used for space discretization. Finally, the obtained system is solved by Gauss elimination method. Findings Some numerical examples of fractional dynamical system are considered to check the accuracy of the algorithm. Numerical results show that quasilinearization with Scale-3 Haar wavelet converges fast even for small number of collocation points as compared of classical Scale-2 Haar wavelet (S-2 HW) method. The convergence analysis of the proposed algorithm has been shown that as we increase the resolution level of Scale-3 Haar wavelet error goes to zero rapidly. Originality/value To the best of authors’ knowledge, this is the first time that new Haar wavelets Scale-3 have been used in fractional system. A new scheme is developed for dynamical system based on new Scale-3 Haar wavelets. These wavelets take less time than Scale-2 Haar wavelets. This approach extends the idea of Jiwari (2015, 2012) via translation and dilation of Haar function at Scale-3.

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Rakowski, Waldemar. "Application of Cubic Box Spline Wavelets in the Analysis of Signal Singularities." International Journal of Applied Mathematics and Computer Science 25, no. 4 (December 1, 2015): 927–41. http://dx.doi.org/10.1515/amcs-2015-0066.

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Abstract In the subject literature, wavelets such as the Mexican hat (the second derivative of a Gaussian) or the quadratic box spline are commonly used for the task of singularity detection. The disadvantage of the Mexican hat, however, is its unlimited support; the disadvantage of the quadratic box spline is a phase shift introduced by the wavelet, making it difficult to locate singular points. The paper deals with the construction and properties of wavelets in the form of cubic box splines which have compact and short support and which do not introduce a phase shift. The digital filters associated with cubic box wavelets that are applied in implementing the discrete dyadic wavelet transform are defined. The filters and the algorithme à trous of the discrete dyadic wavelet transform are used in detecting signal singularities and in calculating the measures of signal singularities in the form of a Lipschitz exponent. The article presents examples illustrating the use of cubic box spline wavelets in the analysis of signal singularities.

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HUANG, YONGDONG, and ZHENGXING CHENG. "PARAMETRIZATION OF COMPACTLY SUPPORTED TRIVARIATE ORTHOGONAL WAVELET FILTER." International Journal of Wavelets, Multiresolution and Information Processing 05, no. 04 (July 2007): 627–39. http://dx.doi.org/10.1142/s0219691307001938.

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Multivariate wavelets analysis is a powerful tool for multi-dimensional signal processing, but tensor product wavelets have a number of drawbacks. In this paper, we give an algorithm of parametric representation compactly supported trivariate orthogonal wavelet filter, which simplifies the study of trivariate orthogonal wavelet. Four examples are also given to demonstrate the method.

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Singh, Ashok Kumar, and Hemant Bhate. "Stochastic wavelets from minimizers of an uncertainty principle: An example." International Journal of Wavelets, Multiresolution and Information Processing 18, no. 06 (July 31, 2020): 2050046. http://dx.doi.org/10.1142/s0219691320500460.

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This paper proposes a method through which a family of wavelets can be obtained. This is done by choosing each member based on a random variable. The method is preferred in situations where a single mother wavelet proves inadequate and an evolving sequence of mother wavelets is needed but a priori the next member in the sequence is uncertain. The adopted approach is distinct from the way spatiotemporal wavelets are used or even the way stochastic processes have been studied using spatiotemporal wavelets.

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Jansen, Maarten. "Non-equispaced B-spline wavelets." International Journal of Wavelets, Multiresolution and Information Processing 14, no. 06 (November 2016): 1650056. http://dx.doi.org/10.1142/s0219691316500569.

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This paper has three main contributions. The first is the construction of wavelet transforms from B-spline scaling functions defined on a grid of non-equispaced knots. The new construction extends the equispaced, biorthogonal, compactly supported Cohen–Daubechies–Feauveau wavelets. The new construction is based on the factorization of wavelet transforms into lifting steps. The second and third contributions are new insights on how to use these and other wavelets in statistical applications. The second contribution is related to the bias of a wavelet representation. It is investigated how the fine scaling coefficients should be derived from the observations. In the context of equispaced data, it is common practice to simply take the observations as fine scale coefficients. It is argued in this paper that this is not acceptable for non-interpolating wavelets on non-equidistant data. Finally, the third contribution is the study of the variance in a non-orthogonal wavelet transform in a new framework, replacing the numerical condition as a measure for non-orthogonality. By controlling the variances of the reconstruction from the wavelet coefficients, the new framework allows us to design wavelet transforms on irregular point sets with a focus on their use for smoothing or other applications in statistics.

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ZENG, LI, RUI MA, JIANYUAN HUANG, and P. R. HUNZIKER. "THE CONSTRUCTION OF 2D ROTATIONALLY INVARIANT WAVELETS AND THEIR APPLICATION IN IMAGE EDGE DETECTION." International Journal of Wavelets, Multiresolution and Information Processing 06, no. 01 (January 2008): 65–82. http://dx.doi.org/10.1142/s0219691308002227.

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Construction of rotationally invariant 2D wavelets is important in image processing, but is difficult. In this paper, the discrete form of a 2D rotationally invariant wavelet is constructed by back-projection from a 1D symmetrical wavelet. Such rotationally invariant 2D wavelets allow effective edge detection in any direction. These wavelets are combined with the 2D directional wavelets for the use in non-maximum suppression edge detection. The resulting binary edges are characterized by finer contours, differential detection characteristics and noise robustness compared to other edge detectors in various test images. In particular, where fine binary edges in noisy images are required, this novel approach compares favorably to the classical methods of Canny and Mallat with detection of more edges thanks to the implicit denoising properties and the full rotational invariance of the method.

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GRIVET-TALOCIA, STEFANO, and ANITA TABACCO. "WAVELETS ON THE INTERVAL WITH OPTIMAL LOCALIZATION." Mathematical Models and Methods in Applied Sciences 10, no. 03 (April 2000): 441–62. http://dx.doi.org/10.1142/s0218202500000252.

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This paper introduces a novel construction of wavelets on the unit interval. With this construction, explicit upper bounds for the length of the modified border wavelets filters can be given. This insures a good localization of the border wavelets when a triangular biorthogonalization scheme is employed. The resulting wavelet bases are then well-suited for the adaptive solution of partial differential equations.

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TÜRÜKI, TURGHUNJAN ABDUKIRIM, MUHAMMAD HUSSAIN, KOICHI NIIJIMA, and SHIGERU TAKANO. "THE DYADIC LIFTING SCHEMES AND THE DENOISING OF DIGITAL IMAGES." International Journal of Wavelets, Multiresolution and Information Processing 06, no. 03 (May 2008): 331–51. http://dx.doi.org/10.1142/s0219691308002380.

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The dyadic lifting schemes, which generalize Sweldens lifting schemes, have been proposed for custom-design of dyadic and bi-orthogonal wavelets and their duals. Starting with dyadic wavelets and exploiting the control provided in the form of free parameters, one can custom-design dyadic as well as bi-orthogonal wavelets adapted to a particular application. To validate the usefulness of the schemes, two construction methods have been proposed for designing dyadic wavelet filters with higher number of vanishing moments; using these design techniques, spline dyadic wavelet filters have been custom-designed for denoising of digital images, which exhibit enhanced denoising effects.

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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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BHATT, GHANSHYAM, and FRITZ KEINERT. "COMPLETION OF MULTIVARIATE WAVELETS." International Journal of Wavelets, Multiresolution and Information Processing 05, no. 03 (May 2007): 485–500. http://dx.doi.org/10.1142/s0219691307001860.

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The construction of wavelet functions from known scaling functions is called the 'completion problem'. Completion algorithms exist for univariate wavelets, including multiwavelets. For multivariate wavelets, however, completion is not always possible. We present a new algorithm (a generalization of a method of Lai) which works in many cases.

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Pothisarn, Chaichan, Jittiphong Klomjit, Atthapol Ngaopitakkul, Chaiyan Jettanasen, Dimas Anton Asfani, and I. Made Yulistya Negara. "Comparison of Various Mother Wavelets for Fault Classification in Electrical Systems." Applied Sciences 10, no. 4 (February 11, 2020): 1203. http://dx.doi.org/10.3390/app10041203.

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This paper presents a comparative study on mother wavelets using a fault type classification algorithm in a power system. The study aims to evaluate the performance of the protection algorithm by implementing different mother wavelets for signal analysis and determines a suitable mother wavelet for power system protection applications. The factors that influence the fault signal, such as the fault location, fault type, and inception angle, have been considered during testing. The algorithm operates by applying the discrete wavelet transform (DWT) to the three-phase current and zero-sequence signal obtained from the experimental setup. The DWT extracts high-frequency components from the signals during both the normal and fault states. The coefficients at scales 1–3 have been decomposed using different mother wavelets, such as Daubechies (db), symlets (sym), biorthogonal (bior), and Coiflets (coif). The results reveal different coefficient values for the different mother wavelets even though the behaviors are similar. The coefficient for any mother wavelet has the same behavior but does not have the same value. Therefore, this finding has shown that the mother wavelet has a significant impact on the accuracy of the fault classification algorithm.

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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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Sharif, I., and S. Khare. "Comparative Analysis of Haar and Daubechies Wavelet for Hyper Spectral Image Classification." ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XL-8 (November 28, 2014): 937–41. http://dx.doi.org/10.5194/isprsarchives-xl-8-937-2014.

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With the number of channels in the hundreds instead of in the tens Hyper spectral imagery possesses much richer spectral information than multispectral imagery. The increased dimensionality of such Hyper spectral data provides a challenge to the current technique for analyzing data. Conventional classification methods may not be useful without dimension reduction pre-processing. So dimension reduction has become a significant part of Hyper spectral image processing. This paper presents a comparative analysis of the efficacy of Haar and Daubechies wavelets for dimensionality reduction in achieving image classification. Spectral data reduction using Wavelet Decomposition could be useful because it preserves the distinction among spectral signatures. Daubechies wavelets optimally capture the polynomial trends while Haar wavelet is discontinuous and resembles a step function. The performance of these wavelets are compared in terms of classification accuracy and time complexity. This paper shows that wavelet reduction has more separate classes and yields better or comparable classification accuracy. In the context of the dimensionality reduction algorithm, it is found that the performance of classification of Daubechies wavelets is better as compared to Haar wavelet while Daubechies takes more time compare to Haar wavelet. The experimental results demonstrate the classification system consistently provides over 84% classification accuracy.

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Journal articles on the topic 'Wavelets'

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