Journal articles: 'Wavelets (Mathematics)'

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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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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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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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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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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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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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Benedetto, John J., Michael W. Frazier, and Bruno Torrésani. "Wavelets: Mathematics and Applications." Physics Today 47, no. 11 (November 1994): 90–91. http://dx.doi.org/10.1063/1.2808703.

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Dremin, I. M. "Wavelets: Mathematics and applications." Physics of Atomic Nuclei 68, no. 3 (March 2005): 508–20. http://dx.doi.org/10.1134/1.1891202.

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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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Jiang, Zhuhan, and Xiling Guo. "A note on the extension of a family of biorthogonal Coifman wavelet systems." ANZIAM Journal 46, no. 1 (July 2004): 111–20. http://dx.doi.org/10.1017/s1446181100013717.

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AbstractWavelet systems with a maximum number of balanced vanishing moments are known to be extremely useful in a variety of applications such as image and video compression. Tian and Wells recently created a family of such wavelet systems, called the biorthogonal Coifman wavelets, which have proved valuable in both mathematics and applications. The purpose of this work is to establish along with direct proofs a very neat extension of Tian and Wells' family of biorthogonal Coifman wavelets by recovering other “missing” members of the biorthogonal Coifman wavelet systems.

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Gu, Qing, and Deguang Han. "Super-Wavelets and Decomposable Wavelet Frames." Journal of Fourier Analysis and Applications 11, no. 6 (November 1, 2005): 683–96. http://dx.doi.org/10.1007/s00041-005-5005-x.

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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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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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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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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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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.

18

Sharma, Vikram, and P. Manchanda. "WAVELET PACKETS ASSOCIATED WITH NONUNIFORM MULTIRESOLUTION ANALYSIS ON POSITIVE HALF LINE." Asian-European Journal of Mathematics 06, no. 01 (March 2013): 1350007. http://dx.doi.org/10.1142/s1793557113500071.

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Gabardo and Nashed [Nonuniform multiresolution analysis and spectral pairs, J. Funct. Anal.158 (1998) 209–241] introduced the Nonuniform multiresolution analysis (NUMRA) whose translation set is not a group. Farkov [Orthogonal p-wavelets on ℝ+, in Proc. Int. Conf. Wavelets and Splines (St. Petersburg State University, St. Petersburg, 2005), pp. 4–26] studied multiresolution analysis (MRA) on positive half line and constructed associated wavelets. Meenakshi et al. [Wavelets associated with Nonuniform multiresolution analysis on positive half line, Int. J. Wavelets, Multiresolut. Inf. Process.10(2) (2011) 1250018, 27pp.] studied NUMRA on positive half line and proved the analogue of Cohen's condition for the NUMRA on positive half line. We construct the associated wavelet packets for such an MRA and study its properties.

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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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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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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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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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Heinlein, Peter. "Discretizing continuous wavelet transforms using integrated wavelets." Applied and Computational Harmonic Analysis 14, no. 3 (May 2003): 238–56. http://dx.doi.org/10.1016/s1063-5203(03)00005-8.

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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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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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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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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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Paul, Tuhin Utsab, Sayantan Chakraborty, Saikat Mukhopadhyay, and Dr Samir K. Bandhyopadhyay. "A new insight into Wavelet Transforms using the concepts of Frame Theory." INTERNATIONAL JOURNAL OF COMPUTERS & TECHNOLOGY 12, no. 9 (March 14, 2014): 3860–74. http://dx.doi.org/10.24297/ijct.v12i9.2826.

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Wavelet analysis being a relatively new subject of study is being explored, all around the globe, using various mathematical tools, currently available. This paper is a humble attempt to provide a comprehensive study of the same, by means of exhaustive mathematical analysis. Since frame theory has been established as a standard notion in applied mathematics, so it was used as the analytical tool to explain the formation, purpose and use of wavelets. The theoretical explanation follows the mathematical analysis, which is an attempt to give picture the â€˜theoremsâ€™, â€˜definitionsâ€™ and the â€˜lemmasâ€™.

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JIANG, YINGCHUN, and YOUMING LIU. "INTERPOLATORY CURL-FREE WAVELETS AND APPLICATIONS." International Journal of Wavelets, Multiresolution and Information Processing 05, no. 05 (September 2007): 843–58. http://dx.doi.org/10.1142/s0219691307002075.

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Divergence-free and curl-free vector wavelets have many applications for the analysis and numerical simulation of incompressible flows and certain electromagnetic phenomena. Because special domains are required in those applications, Bittner and Urban constructed interpolating divergence-free multi-wavelets based on cubic Hermite splines in 2005. In this paper, we construct interpolating curl-free multi-wavelets and give two wavelet estimates for a class of vector Besov spaces.

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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.

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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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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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Bhat, Mohd Younus. "Dual wavelets associated with nonuniform MRA." Tamkang Journal of Mathematics 50, no. 2 (June 30, 2018): 119–32. http://dx.doi.org/10.5556/j.tkjm.50.2019.2646.

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A generalization of Mallats classical multiresolution analysis, based on thetheory of spectral pairs, was considered in two articles by Gabardo and Nashed. In thissetting, the associated translation set is no longer a discrete subgroup of R but a spectrumassociated with a certain one-dimensional spectral pair and the associated dilation is aneven positive integer related to the given spectral pair. In this paper, we construct dualwavelets which are associated with Nonuniform Multiresolution Analysis. We show thatif the translates of the scaling functions of two multiresolution analyses are biorthogonal,then the associated wavelet families are also biorthogonal. Under mild assumptions onthe scaling functions and the wavelets, we also show that the wavelets generate Rieszbases

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DeVore, Ronald A., and Bradley J. Lucier. "Wavelets." Acta Numerica 1 (January 1992): 1–56. http://dx.doi.org/10.1017/s0962492900002233.

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The subject of ‘wavelets’ is expanding at such a tremendous rate that it is impossible to give, within these few pages, a complete introduction to all aspects of its theory. We hope, however, to allow the reader to become sufficiently acquainted with the subject to understand, in part, the enthusiasm of its proponents toward its potential application to various numerical problems. Furthermore, we hope that our exposition can guide the reader who wishes to make more serious excursions into the subject. Our viewpoint is biased by our experience in approximation theory and data compression; we warn the reader that there are other viewpoints that are either not represented here or discussed only briefly. For example, orthogonal wavelets were developed primarily in the context of signal processing, an application upon which we touch only indirectly. However, there are several good expositions (e.g. Daubechies (1990) and Rioul and Vetterli (1991)) of this application. A discussion of wavelet decompositions in the context of Littlewood-Paley theory can be found in the monograph of Frazieret al. (1991). We shall also not attempt to give a complete discussion of the history of wavelets. Historical accounts can be found in the book of Meyer (1990) and the introduction of the article of Daubechies (1990). We shall try to give sufficient historical commentary in the course of our presentation to provide some feeling for the subject's development.

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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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Verma, Amit, and Diksha Tiwari. "On some computational aspects of Hermite & Haar wavelets on a class of nonlinear singular BVPs." Applicable Analysis and Discrete Mathematics, no. 00 (2021): 20. http://dx.doi.org/10.2298/aadm191123020v.

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We propose a new class of SBVPs which deals with exothermic reactions. We also propose four computationally stable methods to solve singular nonlinear BVPs by using Hermite wavelet collocation which are coupled with Newton's quasilinearization and Newton-Raphson method. We compare the results which are obtained by using Hermite wavelets with the results obtained by using Haar wavelets. The efficiency of these methods are verified by applying these four methods on Lane-Emden equations. Convergence analysis is also presented.

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Abeyratne, M. K., W. Freeden, and C. Mayer. "Multiscale deformation analysis by Cauchy-Navier wavelets." Journal of Applied Mathematics 2003, no. 12 (2003): 605–45. http://dx.doi.org/10.1155/s1110757x03206033.

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A geoscientifically relevant wavelet approach is established for the classical (inner) displacement problem corresponding to a regular surface (such as sphere, ellipsoid, and actual earth surface). Basic tools are the limit and jump relations of (linear) elastostatics. Scaling functions and wavelets are formulated within the framework of the vectorial Cauchy-Navier equation. Based on appropriate numerical integration rules, a pyramid scheme is developed providing fast wavelet transform (FWT). Finally, multiscale deformation analysis is investigated numerically for the case of a spherical boundary.

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Cohen, Albert, Ingrid Daubechies, and Pierre Vial. "Wavelets on the Interval and Fast Wavelet Transforms." Applied and Computational Harmonic Analysis 1, no. 1 (December 1993): 54–81. http://dx.doi.org/10.1006/acha.1993.1005.

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Liu, Youming, and Xiaochen Zeng. "Strong Lp convergence of wavelet deconvolution density estimators." Analysis and Applications 16, no. 02 (February 5, 2018): 183–208. http://dx.doi.org/10.1142/s0219530517500154.

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Using compactly supported wavelets, Giné and Nickl [Uniform limit theorems for wavelet density estimators, Ann. Probab. 37(4) (2009) 1605–1646] obtain the optimal strong [Formula: see text] convergence rates of wavelet estimators for a fixed noise-free density function. They also study the same problem by spline wavelets [Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections, Bernoulli 16(4) (2010) 1137–1163]. This paper considers the strong [Formula: see text] convergence of wavelet deconvolution density estimators. We first show the strong [Formula: see text] consistency of our wavelet estimator, when the Fourier transform of the noise density has no zeros. Then strong [Formula: see text] convergence rates are provided, when the noises are severely and moderately ill-posed. In particular, for moderately ill-posed noises and [Formula: see text], our convergence rate is close to Giné and Nickl’s.

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Cattani, Carlo, and Luis M. Sánchez Ruiz. "Discrete differential operators in multidimensional Haar wavelet spaces." International Journal of Mathematics and Mathematical Sciences 2004, no. 44 (2004): 2347–55. http://dx.doi.org/10.1155/s0161171204307234.

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We consider a class of discrete differential operators acting on multidimensional Haar wavelet basis with the aim of finding wavelet approximate solutions of partial differential problems. Although these operators depend on the interpolating method used for the Haar wavelets regularization and the scale dimension space, they can be easily used to define the space of approximate wavelet solutions.

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Izuki, Mitsuo. "Wavelets and Modular Inequalities in Variable 𝐿𝑝 Spaces." gmj 15, no. 2 (June 2008): 281–93. http://dx.doi.org/10.1515/gmj.2008.281.

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Abstract The aim of this paper is to characterize variable 𝐿𝑝 spaces 𝐿𝑝(·)() using wavelets with proper smoothness and decay. We obtain conditions for the wavelet characterizations of 𝐿𝑝(·)() with respect to the norm estimates and modular inequalities.

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Sahu, P. K., and S. Saha Ray. "A New Bernoulli Wavelet Method for Numerical Solutions of Nonlinear Weakly Singular Volterra Integro-Differential Equations." International Journal of Computational Methods 14, no. 03 (April 13, 2017): 1750022. http://dx.doi.org/10.1142/s0219876217500220.

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In this paper, Bernoulli wavelet method has been developed to solve nonlinear weakly singular Volterra integro-differential equations. Bernoulli wavelets are generated by dilation and translation of Bernoulli polynomials. The properties of Bernoulli wavelets and Bernoulli polynomials are first presented. The present wavelet method reduces these integral equations to a system of nonlinear algebraic equations and again these algebraic systems have been solved numerically by Newton’s method. Convergence analysis of the present method has been discussed in this paper. Some illustrative examples have been demonstrated to show the applicability and accuracy of the present method.

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Ďuriš, Viliam, Vladimir I. Semenov, and Sergey G. Chumarov. "Wavelets and digital filters designed and synthesized in the time and frequency domains." Mathematical Biosciences and Engineering 19, no. 3 (2022): 3056–68. http://dx.doi.org/10.3934/mbe.2022141.

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<abstract> <p>The relevance of the problem under study is due to the fact that the comparison is made for wavelets constructed in the time and frequency domains. The wavelets constructed in the time domain include all discrete wavelets, as well as continuous wavelets based on derivatives of the Gaussian function. This article discusses the possibility of implementing algorithms for multiscale analysis of one-dimensional and two-dimensional signals with the above-mentioned wavelets and wavelets constructed in the frequency domain. In contrast to the discrete wavelet transform (Mallat algorithm), the authors propose a multiscale analysis of images with a multiplicity of less than two in the frequency domain, that is, the scale change factor is less than 2. Despite the fact that the multiplicity of the analysis is less than 2, the signal can be represented as successive approximations, as with the use of discrete wavelet transform. Reducing the multiplicity allows you to increase the depth of decomposition, thereby increasing the accuracy of signal analysis and synthesis. At the same time, the number of decomposition levels is an order of magnitude higher compared to traditional multi-scale analysis, which is achieved by progressive scanning of the image, that is, the image is processed not by rows and columns, but by progressive scanning as a whole. The use of the fast Fourier transform reduces the conversion time by four orders of magnitude compared to direct numerical integration, and due to this, the decomposition and reconstruction time does not increase compared to the time of multiscale analysis using discrete wavelets.</p> </abstract>

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Rzeszotnik, Ziemowit, and Darrin Speegle. "On wavelets interpolated from a pair of wavelet sets." Proceedings of the American Mathematical Society 130, no. 10 (May 8, 2002): 2921–30. http://dx.doi.org/10.1090/s0002-9939-02-06416-x.

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Shumilov, Boris M. "Construction of an Effective Preconditioner for the Even-odd Splitting of Cubic Spline Wavelets." WSEAS TRANSACTIONS ON MATHEMATICS 20 (December 28, 2021): 717–28. http://dx.doi.org/10.37394/23206.2021.20.76.

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In this study, the method for decomposing splines of degree m and smoothness C^m-1 into a series of wavelets with zero moments is investigated. The system of linear algebraic equations connecting the coefficients of the spline expansion on the initial scale with the spline coefficients and wavelet coefficients on the embedded scale is obtained. The originality consists in the application of some preconditioner that reduces the system to a simpler band system of equations. Examples of applying the method to the cases of first-degree spline wavelets with two first zero moments and cubic spline wavelets with six first zero moments are presented. For the cubic case after splitting the system into even and odd rows, the resulting matrix acquires a seven-diagonals form with strict diagonal dominance, which makes it possible to apply an effective sweep method to its solution

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MAHARAJ, ELIZABETH ANN. "USING WAVELETS TO COMPARE TIME SERIES PATTERNS." International Journal of Wavelets, Multiresolution and Information Processing 03, no. 04 (December 2005): 511–21. http://dx.doi.org/10.1142/s0219691305000993.

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In this paper, a procedure for comparing the patterns of time series using wavelets is developed. Randomization tests based on the ratio of the sum of squared wavelet coefficients of pairs of time series at different scales are used. A simulation study using pairs of different types of time series using the Haar and Daubechies wavelets is carried out. The results reveal that the tests perform fairly well at scales where there are a sufficient number of wavelet coefficients. The tests are applied to a set of financial time series.

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Manbir Kaur and Inderdeep Singh. "Comprehensive review of numerical schemes based on Hermite wavelets." World Journal of Advanced Research and Reviews 15, no. 3 (September 30, 2022): 240–47. http://dx.doi.org/10.30574/wjarr.2022.15.3.0908.

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Differential and integral equations are encountered in many applications of science and engineering and many mathematical models have also been formulated in terms of these equations. Due to some shortcomings of the already existing numerical methods, researchers are making efforts to find more efficient alternatives for obtaining solutions of many practical and physical problems giving rise to differential or integral equations. As a result, wavelet methods have found their way for the numerical solution of the resulting different kinds of equations. So this review paper intends to provide the great utility, accuracy and employability of Hermite wavelets to address situations of various areas of applied mathematics, physics, biology, optimal control systems, communication theory, queuing theory, medicine and many other scientific and engineering problems.

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EHLER, MARTIN. "COMPACTLY SUPPORTED MULTIVARIATE, PAIRS OF DUAL WAVELET FRAMES OBTAINED BY CONVOLUTION." International Journal of Wavelets, Multiresolution and Information Processing 06, no. 02 (March 2008): 183–208. http://dx.doi.org/10.1142/s0219691308002306.

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In this paper, we present a construction of compactly supported multivariate pairs of dual wavelet frames. The approach is based on the convolution of two refinable distributions. We obtain smooth wavelets with any preassigned number of vanishing moments. Their underlying refinable function is fundamental. In the examples, we obtain symmetric wavelets with small support from optimal refinable functions, i.e. the refinable function has minimal mask size with respect to smoothness and approximation order of its generated multiresolution analysis. The wavelet system has maximal approximation order with respect to the underlying refinable function.

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Mahapatra, Prasadini, and Divya Singh. "Scaling sets and generalized scaling sets on Cantor dyadic group." International Journal of Wavelets, Multiresolution and Information Processing 18, no. 04 (February 28, 2020): 2050019. http://dx.doi.org/10.1142/s0219691320500198.

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Scaling and generalized scaling sets determine wavelet sets and hence wavelets. In real case, wavelet sets were proved to be an important tool for the construction of MRA as well as non-MRA wavelets. However, any result related to scaling/generalized scaling sets is not available in case of locally compact abelian groups. This paper gives a characterization of scaling sets and its generalized version along with relevant examples in dual Cantor dyadic group [Formula: see text]. These results can further be generalized to arbitrary locally compact abelian groups.

Journal articles on the topic 'Wavelets (Mathematics)'

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