Dissertations / Theses: 'Wavelets (Mathematics)'

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Colthurst, Thomas. "Multidimensional wavelets." Thesis, Massachusetts Institute of Technology, 1997. http://hdl.handle.net/1721.1/43934.

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Kutyniok, Gitta. "Affine density in wavelet analysis /." Berlin [u.a.] : Springer, 2007. http://www.gbv.de/dms/ilmenau/toc/529512874.PDF.

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Hua, Xinhou. "Dynamical systems and wavelets." Thesis, University of Ottawa (Canada), 2002. http://hdl.handle.net/10393/6143.

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The first part of this thesis is concerned with Bakers Conjecture (1984) which says that two permutable transcendental entire functions have the same Julia set. To this end, we shall exhibit that two permutable transcendental entire functions of a certain type have the same Julia set. So far, this is the best result to the conjecture. The second part relates to Newton's method to find zeros of functions. We shall look for the locations of the limits of the iterating sequence of the relaxed Newton function on its wandering domains. A relaxed Newton function with corresponding properties is constructed. The third part relates to the dynamics of ordinary differential equations and inverse problems. Given a target solution, we shall construct second-order differential equations with Legendre polynomial basis to approximate the target solution. An algorithm and numerical solutions are provided. Examples show that the approximations we have found are much better than the known results obtained by means of first-order differential equations. We shall also discuss approximation using a wavelet basis. MATLAB is used to compute the numerical results. In the fourth part, we deal with variational problems in signal and image processing. For a given signal or image represented by a function, we shall provide a good approximation to the function, which minimizes a given functional.

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Karoui, Abderrazek. "Multidimensional wavelets and applications." Thesis, University of Ottawa (Canada), 1995. http://hdl.handle.net/10393/9492.

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In this thesis, one- as well as multi-dimensional biorthogonal wavelet filters are designed and used for the construction of compactly supported wavelet bases. In particular, an adaptation of the McClellan transformation is used to design nonseparable 2-D biorthogonal wavelet bases. Some examples of 2-D biorthogonal wavelet filters are given in the case of the quincunx sampling lattice. Some theoretical and technical results known in the one-dimensional case have been generalized to the n-dimensional case. This generalization leads to a better understanding of the theory and design of multidimensional biorthogonal wavelets. An important part of the thesis devoted to the design of fast discrete wavelet transforms. The main ingredient of the algorithms is the use of a one-point quadrature formula for approximating the nest coefficients of the signals together with a suitable design and implementation of symmetric biorthogonal filters. Special attention is given to the case where the signals have sharp transition points. In this case, a smoothing process has been used to obtain an accurate reconstruction of the signal.

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Bowman, Christopher 1969. "Pattern formation and wavelets." Diss., The University of Arizona, 1997. http://hdl.handle.net/10150/288741.

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This thesis is a collection of results associated with pattern formation, and consists of several novel results. A multi-scale analysis is carried out near the lasing bifurcation on equations which model the free carrier semiconductor laser. This analysis produces an amplitude equation which resembles the Swift-Hohenberg equation derived for the simpler two level laser, but with extra terms arising from the more complicated semiconductor system. New results are also presented in the analysis of phase equations for patterns, showing that defects are weak solutions of the phase diffusion equation, and that the Gaussian curvature of the phase surface condenses onto point and line defects. This latter fact allows for considerable simplification of the phase diffusion equation, and this analysis is presented as well. Finally, and most importantly, an algorithm is presented, based on the continuous wavelet transform, for the extraction of local phase and amplitude information from roll patterns. This algorithm allows a precise detection of phase grain boundaries and point defects, as well as the computation of soft modes like the mean flow. Several tests are conducted on numerically generated signals to demonstrate the applicability and precision of the algorithm. The algorithm is then applied to actual experimental convection patterns, and conclusions about the nature of the wave director field in such patterns are presented.

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Pelletier, Emile. "Instrument de-synthesis using wavelets." Thesis, University of Ottawa (Canada), 2005. http://hdl.handle.net/10393/27008.

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Our point of departure is the concept of 'additive synthesis', which is the traditional explanation for the individual of 'timbre' or 'colour' of the sound of the various musical instruments. When an instrument sounds a note, one hears the note as if by itself, but this is not what is physically happening. What is in fact occurring is a complex waveform featuring a collection of harmonic frequencies, referred to as the spectrum. A synthesizer attempts to imitate the sound of a particular instrument by replicating the amplitudes of its harmonics. We use the term 'de-synthesis' to refer to the inverse procedure, computerized instrument identification. We describe an experiment that we designed and executed with M ATLAB to explore the hypothesis that a computer will be able to recognize an instrument by its characteristic timbre. The idea of applying wavelets to analyze music comes naturally since music consists of sound waves, and wavelets are wave shaped functions. We propose a mathematical model that can take certain musical instrument's attack and decay features into account that utilizes Malvar wavelets: Super Malvar wavelets. Super wavelets are superpositions of ordinary wavelets in some linear combination that can be treated as a wavelet in itself. (Abstract shortened by UMI.)

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Shen, Jianhong 1971. "Asymptotics of wavelets and filters." Thesis, Massachusetts Institute of Technology, 1998. http://hdl.handle.net/1721.1/47469.

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Hunter, Karin M. "Interpolatory refinable functions, subdivision and wavelets." Thesis, Stellenbosch : University of Stellenbosch, 2005. http://hdl.handle.net/10019.1/1156.

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Thesis (DSc (Mathematical Sciences))--University of Stellenbosch, 2005. Subdivision is an important iterative technique for the efficient generation of curves and surfaces in geometric modelling. The convergence of a subdivision scheme is closely connected to the existence of a corresponding refinable function. In turn, such a refinable function can be used in the multi-resolutional construction method for wavelets, which are applied in many areas of signal analysis.

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Sablik, Mathieu. "Wavelets in Abstract Hilbert Space." Thesis, Uppsala University, Department of Mathematics, 2000. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-122553.

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Yu, Lu. "Wavelets on hierarchical trees." Diss., University of Iowa, 2016. https://ir.uiowa.edu/etd/2302.

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Signals on hierarchical trees can be viewed as a generalization of discrete signals of length 2^N. In this work, we extend the classic discrete Haar wavelets to a Haar-like wavelet basis that works for signals on hierarchical trees. We first construct a specific wavelet basis and give its inverse and normalized transform matrices. As analogue to the classic case, operators and wavelet generating functions are constructed for the tree structure. This leads to the definition of multiresolution analysis on a hierarchical tree. We prove the previously selected wavelet basis is an orthogonal multiresolution. Classification of all possible wavelet basis that generate an orthogonal multiresolution is then given. In attempt to find more efficient encoding and decoding algorithms, we construct a second wavelet basis and show that it is also an orthogonal multiresolution. The encoding and decoding algorithms are given and their time complexity are analyzed. In order to link change of tree structure and encoded signal, we define weighted hierarchical tree, tree cut and extension. It is then shown that a simply relation can be established without the need for global change of the transform matrix. Finally, we apply thresholding to the transform and give an upper bound of error.

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Zheng, Ellen Yanqing. "A comparative study of wavelets and multiwavelets." Thesis, University of Ottawa (Canada), 1996. http://hdl.handle.net/10393/9651.

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In this thesis, some basic concepts and theorems are studied, leading to the cascade algorithm which is the most usual approximating method used in the construction of wavelets. By considering the symmetry property of scalar wavelets, Lawton's complex-valued scalar wavelets are studied and some recent results are implemented in the theory of complex-valued scalar wavelets. Another important part of this thesis is the study of multiwavelets. Some comparisons are made among real-valued scalar wavelets, complex-valued scalar wavelets and real-valued multiwavelets. A special contribution is the figures of different kinds of wavelets and some numerical results.

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Tomas, Brian. "Theory and application of frequency selective wavelets /." Thesis, Connect to this title online; UW restricted, 1992. http://hdl.handle.net/1773/5755.

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Navarro, Jaime. "The Continuous Wavelet Transform and the Wave Front Set." Thesis, University of North Texas, 1993. https://digital.library.unt.edu/ark:/67531/metadc277762/.

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In this paper I formulate an explicit wavelet transform that, applied to any distribution in S^1(R^2), yields a function on phase space whose high-frequency singularities coincide precisely with the wave front set of the distribution. This characterizes the wave front set of a distribution in terms of the singularities of its wavelet transform with respect to a suitably chosen basic wavelet.

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Rohwer, Birgit. "A multiresolutional approach to the construction of spline wavelets." Thesis, Stellenbosch : Stellenbosch University, 2000. http://hdl.handle.net/10019.1/51580.

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Thesis (MSc) -- University of Stellenbosch, 2000. ENGLISH ABSTRACT: In this thesis we study a wavelet construction procedure based on a multiresolutional method, before specializing to the case of spline wavelets. First, we introduce and analyze the concepts of scaling functions and their duals, after which we analyze the multiresolutional analysis (MM) which they generate. The advantages of orthonormality in scaling functions are pointed out and discussed. Following the methods which were introduced in two standard texts of Chui, we next show how a minimally supported wavelet and its dual can be explicitly constructed from a given MM, thereby yielding an orthogonal decomposition of the space of square-(Lebesgue)integrable functions on the real line. We show that our method applied to orthonormal scaling functions also yields orthonormal wavelets, including as a special case the Daubechies wavelet. General decomposition and reconstruction algorithms are explicitly formulated, and the importance of the vanishing moments of a wavelet in practical applications is shown. We next introduce and analyze cardinal B-splines, in particular showing that these functions are refinable, and that they satisfy the criteria of Riesz stability. Thus the cardinal B-spline is an admissible choice for a scaling function, so that the previously developed wavelet construction procedure based on a MM yields an explicit formula for the minimally supported B-spline wavelet. The corresponding vanishing moment order is calculated, and the resulting ability of the B-spline wavelet to detect singularities in a given function is demonstrated by means of a numerical example. Finally, we develop an explicit procedure for the construction of minimally supported B-spline wavelets on a bounded interval. This method, as developed in work by de Villiers and Chui, is then compared with a previous boundary wavelet construction method introduced in work by Chui and Quak. AFRIKAANSE OPSOMMING: In hierdie tesis bestudeer ons 'n golfie konstruksieprosedure wat gebaseer is op 'n multiresolusiemetode, voordat ons spesialiseer na die geval van latfunksie-golfies. Eerstens word die konsepte van skaalfunksies en hulle duale bekendgestel en geanaliseer, waarna ons die multiresolusie analise (MM) wat sodoende gegenereer word, analiseer. Die voordeel van ortonormaliteit by skaalfunksies word uitgewys en bespreek. Deur die metodes te volg wat bekendgestel is in twee standaardtekste van Chui, wys ons vervolgens hoe 'n minimaal-gesteunde golfie en die duaal daarvan eksplisiet gekonstrueer kan word vanuit 'n gegewe MM, en daarmee 'n ortogonale dekomposisie van die ruimte van kwadraties-(Lebesgue)integreerbare funksies op die reële lyn lewer. Ons wys dat ons metode toegepas op ortonormale skaalfunksies ook ortonormale golfies oplewer, insluitende as 'n spesiale geval die Daubechies golfie. Algemene dekomposisie en rekonstruksie algoritmes word eksplisiet geformuleer, en die belangrikheid in praktiese toepassings van 'n golfie met die nulmomenteienskap word aangetoon. Vervolgens word kardinale B-Iatfunksies bekendgestel, en word daar in die besonder aangetoon dat hierdie funksies verfynbaar is, en dat hulle aan die Rieszstabiliteit vereiste voldoen. Dus is die kardinale B-Iatfunksie 'n toelaatbare keuse vir 'n skaalfunksie, sodat die golfie konstruksieprosedure gebaseer op 'n MM, soos vantevore ontwikkel, 'n eksplisiete formule vir die minimaal-gesteunde Blatfunksiegolfie oplewer. Die ooreenkomstige nulmomentorde word bereken, en die gevolglike vermoë van 'n B-Iatfunksiegolfie om singulariteite in 'n gegewe funksie raak te sien en uit te wys word gedemonstreer deur middel van 'n numeriese voorbeeld. Laastens ontwikkelons 'n eksplis.iete prosedure vir die konstruksie van minimaal-gesteunde B-Iatfunksiegolfies op 'n begrensde interval. Hierdie metode, soos ontwikkel in werk deur de Villiers en Chui, word dan vergelyk met 'n vorige randgolfie konstruksie wat bekendgestel is in werk deur Chui en Quak.

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Leach, Sandie Patricia. "Density conditions on Gabor frames." Thesis, Available online, Georgia Institute of Technology, 2004:, 2003. http://etd.gatech.edu/theses/available/etd-04082004-180257/unrestricted/leach%5Fsandie%5Fp%5F200312%5Fms.pdf.

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Hoover, Kenneth R. "Dimension functions of rationally dilated wavelets /." view abstract or download file of text, 2007. http://proquest.umi.com/pqdweb?did=1400959361&sid=1&Fmt=2&clientId=11238&RQT=309&VName=PQD.

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Thesis (Ph. D.)--University of Oregon, 2007. Typescript. Includes vita and abstract. Includes bibliographical references (leaves 80-83) and index. Also available for download via the World Wide Web; free to University of Oregon users.

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鍾鈞鎂 and Jun-mei Zhong. "Application of wavelets in image compression." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2000. http://hub.hku.hk/bib/B42575667.

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Zhong, Jun-mei. "Application of wavelets in image compression." Click to view the E-thesis via HKUTO, 2000. http://sunzi.lib.hku.hk/hkuto/record/B42575667.

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Sze, Chuen-kan, and 施泉根. "On framelets and their applications: a discrete approach." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2004. http://hub.hku.hk/bib/B29803937.

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張英傑 and Ying-kit Alan Cheung. "Some results in wavelet theory and their applications." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1997. http://hub.hku.hk/bib/B31215130.

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黃永樑 and Wing-leung Wong. "Some results on biorthogonal wavelet matrices and their applications." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2000. http://hub.hku.hk/bib/B42575370.

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Wong, Wing-leung. "Some results on biorthogonal wavelet matrices and their applications." Click to view the E-thesis via HKUTO, 2000. http://sunzi.lib.hku.hk/hkuto/record/B42575370.

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Cheung, Ying-kit Alan. "Some results in wavelet theory and their applications /." Hong Kong : University of Hong Kong, 1997. http://sunzi.lib.hku.hk/hkuto/record.jsp?B19102677.

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Cao, Jiansheng. "Construction of piecewise linear wavelets." [Johnson City, Tenn. : East Tennessee State University], 2002. http://etd-submit.etsu.edu/etd/theses/available/etd-0715102-142035/unrestricted/CaoJ071802a.pdf.

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Whitcher, Brandon. "Assessing nonstationary time series using wavelets /." Thesis, Connect to this title online; UW restricted, 1998. http://hdl.handle.net/1773/8957.

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Moubandjo, Desiree V. "Polynomial containment in refinement spaces and wavelets based on local projection operators." Thesis, Stellenbosch : Stellenbosch University, 2007. http://hdl.handle.net/10019.1/16418.

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Jacobs, Denise Anne. "Multiwavelets in higher dimensions." Diss., Georgia Institute of Technology, 2001. http://hdl.handle.net/1853/28780.

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Struble, Dale William. "Wavelets on manifolds and multiscale reproducing kernel Hilbert spaces." Related electronic resource:, 2007. http://proquest.umi.com/pqdweb?did=1407687581&sid=1&Fmt=2&clientId=3739&RQT=309&VName=PQD.

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Shi, Fangmin. "Wavelet transforms for stereo imaging." Thesis, University of South Wales, 2002. https://pure.southwales.ac.uk/en/studentthesis/wavelet-transforms-for-stereo-imaging(65abb68f-e30b-4367-a3a8-b7b3df85f566).html.

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Stereo vision is a means of obtaining three-dimensional information by considering the same scene from two different positions. Stereo correspondence has long been and will continue to be the active research topic in computer vision. The requirement of dense disparity map output is great demand motivated by modern applications of stereo such as three-dimensional high-resolution object reconstruction and view synthesis, which require disparity estimates in all image regions. Stereo correspondence algorithms usually require significant computation. The challenges are computational economy, accuracy and robustness. While a large number of algorithms for stereo matching have been developed, there still leaves the space for improvement especially when a new mathematical tool such as wavelet analysis becomes mature. The aim of the thesis is to investigate the stereo matching approach using wavelet transform with a view to producing efficient and dense disparity map outputs. After the shift invariance property of various wavelet transforms is identified, the main contributions of the thesis are made in developing and evaluating two wavelet approaches (the dyadic wavelet transform and complex wavelet transform) for solving the standard correspondence problem. This comprises an analysis of the applicability of dyadic wavelet transform to disparity map computation, the definition of a waveletbased similarity measure for matching, the combination of matching results from different scales based on the detectable minimum disparity at each scale and the application of complex wavelet transform to stereo matching. The matching method using the dyadic wavelet transform is through SSD correlation comparison and is in particular detailed. A new measure using wavelet coefficients is defined for similarity comparison. The approach applying a dual tree of complex wavelet transform to stereo matching is formulated through phase information. A multiscale matching scheme is applied for both the matching methods. Imaging testing has been made with various synthesised and real image pairs. Experimental results with a variety of stereo image pairs exhibit a good agreement with ground truth data, where available, and are qualitatively similar to published results for other stereo matching approaches. Comparative results show that the dyadic wavelet transform-based matching method is superior in most cases to the other approaches considered.

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Tieng, Quang Minh. "Wavelet transform based techniques for the recognition of objects in images." Thesis, Queensland University of Technology, 1996.

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Despite its short history, the wavelet transform has found application in a remarkable diversity of disciplines: Mathematics, Physics, Numerical Analysis,Signal Processing and others. In this thesis, we explore applications of this transform in image analysis and devise several algorithms for recognising objects in an image of a scene. Five different algorithms, consisting of representations and matching techniques, have been proposed for handling different kinds of objects in different situations.

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梁鴻鈞 and Hung-kwan Leung. "Multi-rank wavelet filters." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2001. http://hub.hku.hk/bib/B31224714.

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Leung, Hung-kwan. "Multi-rank wavelet filters." Hong Kong : University of Hong Kong, 2001. http://sunzi.lib.hku.hk/hkuto/record.jsp?B23242395.

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Westra, Seth Pieter Civil &amp Environmental Engineering Faculty of Engineering UNSW. "Probabilistic forecasting of multivariate seasonal reservoir inflows: accounting for spatial and temporal variability." Awarded by:University of New South Wales. Civil & Environmental Engineering, 2007. http://handle.unsw.edu.au/1959.4/40630.

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Hydrological variables such as rainfall and streamfiow vary at a range of temporal scales, from short term (diurnal and seasonal) to the inter annual time scales associated with the El Nino - Southern Oscillation (ENSO) and Indian Ocean Dipole (IOD) phenomena, to even longer time scales such as those linked to the Pacific (inter-) Decadal Oscillation (PDO). This temporal variability poses a significant challenge to hydrologists and water resource managers, since a failure to take such variability into account can lead to an underestimation of the likelihood of droughts and sequences of above average rainfall, which in turn has important implications for the design and operation of reservoirs for hydroelectricity generation, irrigation and municipal water supply. Understanding and accounting for this variability through well designed prediction systems is thus an important part of improving the planning, management and operation of complex water resources systems. This thesis outlines the application of two statistical techniques: wavelets and independent component analysis, to identify sources of hydrological variability, and then use this information to probabilistically generate multivariate seasonal forecasts or develop extended synthetic sequences of hydrological time series. The research is divided into four main parts. The first part outlines an application of the method of wavelets to analyse sources of Australian rainfall variability, and shows that there are coherent regions of variability in addition to the ENSO phenomenon that should be considered when developing seasonal forecasts. The second part examines the capability of three component extraction techniques: principal component analysis (PCA), Varimax and independent component analysis (ICA), in identifying and interpreting modes of variability in the global sea surface temperature dataset. The third part outlines a new technique that uses ICA to factorise multivariate reservoir inflow time series into a set of independent univariate time series, so that univariate methods can be used to develop multivariate synthetic sequences and probabilistic seasonal forecasts. Finally, the fourth part synthesises the previous three parts by demonstrating a wavelets- and correlation-based methodology for assessing sources of climate variability, and then using ICA to generate probabilistic multivariate seasonal forecasts of reservoir inflows that form part of Sydney's water supply system.

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Donovan, George C. "Fractal functions, splines, and wavelet." Diss., Georgia Institute of Technology, 1995. http://hdl.handle.net/1853/30411.

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盧子峰 and Tsz-fung Lo. "Wavelet-based head-related transfer function analysis for audiology." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1998. http://hub.hku.hk/bib/B31237472.

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Roberson, Dawnlee June. "Correlation and wavelet analysis of the surface electromyogram, the electroneurogram and generated force /." Digital version accessible at:, 1998. http://wwwlib.umi.com/cr/utexas/main.

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Lo, Tsz-fung. "Wavelet-based head-related transfer function analysis for audiology /." Hong Kong : University of Hong Kong, 1998. http://sunzi.lib.hku.hk/hkuto/record.jsp?B19712224.

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Hong, Tao. "Object recognition with features from complex wavelets." Thesis, University of Cambridge, 2012. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.610239.

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Leung, King Tai. "Super-resolution image reconstruction based on wavelet-estimation : development and theoretical framework." HKBU Institutional Repository, 2009. http://repository.hkbu.edu.hk/etd_ra/993.

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Li, Zheng. "Approximation to random process by wavelet basis." View abstract/electronic edition; access limited to Brown University users, 2008. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3318378.

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Lutz, Steven S. "Hokua – A Wavelet Method for Audio Fingerprinting." Diss., CLICK HERE for online access, 2009. http://contentdm.lib.byu.edu/ETD/image/etd3247.pdf.

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Mufti, Muid Ur-Rahman. "Fault detection and identification using fuzzy wavelets." Diss., Georgia Institute of Technology, 1995. http://hdl.handle.net/1853/16472.

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Tabb, Jeremiah R. "Using wavelets and principle components analysis to model data from simulated sheet forming processes." Thesis, Georgia Institute of Technology, 2000. http://hdl.handle.net/1853/10146.

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Betaneli, Dmitri 1970. "Wavelets and PDEs : the improvement of computational performance using multi-resolution analysis." Thesis, Massachusetts Institute of Technology, 1998. http://hdl.handle.net/1721.1/10132.

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Lavrik, Ilya A. "Novel wavelet-based statistical methods with applications in classification, shrinkage, and nano-scale image analysis." Available online, Georgia Institute of Technology, 2006, 2006. http://etd.gatech.edu/theses/available/etd-11162005-131744/.

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Thesis (Ph. D.)--Industrial and Systems Engineering, Georgia Institute of Technology, 2006. Huo, Xiaoming, Committee Member ; Heil, Chris, Committee Member ; Wang, Yang, Committee Member ; Hayter, Anthony, Committee Member ; Vidakovic, Brani, Committee Chair.

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Pun, Ka-shun Carson, and 潘加信. "New design and realization techniques for perfect reconstruction two-channel filterbanks and wavelets bases." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2002. http://hub.hku.hk/bib/B31226632.

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Yu, Xiaojiang Gabardo Jean-Pierre. "Wavelet sets, integral self-affine tiles and nonuniform multiresolution analyses." *McMaster only, 2005.

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Ng, Brian Walter. "Wavelet based image texture segementation using a modified K-means algorithm." Title page, table of contents and abstract only, 2003. http://web4.library.adelaide.edu.au/theses/09PH/09phn5759.pdf.

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"August, 2003" Bibliography: p. 261-268. In this thesis, wavelet transforms are chosen as the primary analytical tool for texture analysis. Specifically, Dual-Tree Complex Wavelet Transform is applied to the texture segmentation problem. Several possibilities for feature extraction and clustering steps are examined, new schemes being introduced and compared to known techniques.

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Garantziotis, Anastasios. "A wavelet-based prediction technique for concealment of loss-packet effects in wireless channels." Thesis, Monterey, Calif. : Springfield, Va. : Naval Postgraduate School ; Available from National Technical Information Service, 2002. http://library.nps.navy.mil/uhtbin/hyperion-image/02Jun%5FGarantziotis.pdf.

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Thesis (M.S. in Electrical Engineering)--Naval Postgraduate School, June 2002. Thesis advisor(s): Murali Tummala, Robert Ives. Includes bibliographical references (p. 89-90). Also available online.

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Ahiati, Veroncia Sitsofe. "Cardinal spline wavelet decomposition based on quasi-interpolation and local projection." Thesis, Stellenbosch : University of Stellenbosch, 2009. http://hdl.handle.net/10019.1/2580.

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Thesis (MSc (Mathematics))--University of Stellenbosch, 2009. Wavelet decomposition techniques have grown over the last two decades into a powerful tool in signal analysis. Similarly, spline functions have enjoyed a sustained high popularity in the approximation of data. In this thesis, we study the cardinal B-spline wavelet construction procedure based on quasiinterpolation and local linear projection, before specialising to the cubic B-spline on a bounded interval. First, we present some fundamental results on cardinal B-splines, which are piecewise polynomials with uniformly spaced breakpoints at the dyadic points Z/2r, for r ∈ Z. We start our wavelet decomposition method with a quasi-interpolation operator Qm,r mapping, for every integer r, real-valued functions on R into Sr m where Sr m is the space of cardinal splines of order m, such that the polynomial reproduction property Qm,rp = p, p ∈ m−1, r ∈ Z is satisfied. We then give the explicit construction of Qm,r. We next introduce, in Chapter 3, a local linear projection operator sequence {Pm,r : r ∈ Z}, with Pm,r : Sr+1 m → Sr m , r ∈ Z, in terms of a Laurent polynomial m solution of minimally length which satisfies a certain Bezout identity based on the refinement mask symbol Am, which we give explicitly. With such a linear projection operator sequence, we define, in Chapter 4, the error space sequence Wr m = {f − Pm,rf : f ∈ Sr+1 m }. We then show by solving a certain Bezout identity that there exists a finitely supported function m ∈ S1 m such that, for every r ∈ Z, the integer shift sequence { m(2 · −j)} spans the linear space Wr m . According to our definition, we then call m the mth order cardinal B-spline wavelet. The wavelet decomposition algorithm based on the quasi-interpolation operator Qm,r, the local linear projection operator Pm,r, and the wavelet m, is then based on finite sequences, and is shown to possess, for a given signal f, the essential property of yielding relatively small wavelet coefficients in regions where the support interval of m(2r · −j) overlaps with a Cm-smooth region of f. Finally, in Chapter 5, we explicitly construct minimally supported cubic B-spline wavelets on a bounded interval [0, n]. We also develop a corresponding explicit decomposition algorithm for a signal f on a bounded interval. ii Throughout Chapters 2 to 5, numerical examples are provided to graphically illustrate the theoretical results.

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Dissertations / Theses on the topic 'Wavelets (Mathematics)'

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