Wavelet phd thesis

Wavelet phd thesis

As was emphasized earlier, we employ only the best and most Wavelet Phd Thesis proficient academic writers. All of our writing experts have an academic degree and broad expertise in scholarly writing, which allows them to deliver superb essay help online/10() Comparison of wavelet transformations to enhance convolutional neural network performance in brain tumor segmentation. This paper was extracted from the PhD thesis in Medical Informatics of The application of wavelet and ridgelet transforms in pattern recognition is still in its infancy; while their use in denoising has been a very hot topic in recent years. The aim of this thesis is to study these two important problems. In the area of pattern recognition, we develop a handwritten numeral recognition descriptor using multi-wavelets and neural networks

Appendices - Wavelet Based Approximation Schemes for Singular Integral Equations

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Download Free PDF, wavelet phd thesis. An Investigation of Wavelet Bases for Grid-Based Multi-Scale Simulations Final Report, Tech, wavelet phd thesis. Rep …, Roy Baty. Download Download PDF Full PDF Package Download Full PDF Package This Paper. A short summary of this paper. An Investigation of Wavelet Bases for Grid-Based Multi-Scale Simulations. SAND Unlimited Release Printed November An Investigation of Wavelet Bases for1 Grid-Based Multi-Scale Simulations Final Report Mark A.

Christon, Timothy G. Trucano and Joe R. Roach and David E. Womble Applied Mathematics Department Thomas E. Voth Thermal Sciences Department Sandia National Laboratories P. Box Albuquerque, NM 1 Keywords and Phrases: wavelets, multi-wavelets, reproducing kernel, multi-resolution. The reproducing kernel particle method has been found to yield extremely accurate phase characteristics for hyperbolic problems while providing a convenient framework for multi-scale analyses.

Wing Kam Liu at Northwestern University, Prof, wavelet phd thesis. Douglas P. Hardin wavelet phd thesis Vanderbilt University, and Prof.

Peter M. Massopust at Sam Houston University. In particular, the contributions by Prof. The principal investigator also wishes to thank Philip M. Gresho at Lawrence Livermore National Laboratory for his helpful sug- gestions, wavelet phd thesis. Special thanks go to Thomas M. Smith and Allen C. Robinson for reading the preliminary version of this manuscript.

In addition, the princi- pal investigator would like to acknowledge the support of the Computation, Wavelet phd thesis and Math Center at Sandia National Laboratories in this en- deavor.

Linear Finite Element. ue is the interpolant of the exact solution, and uh is the discrete solution. Node num- bers correspond to the insertion of multi-scale DOF.

The objective for wavelet decompositions is to represent a function in terms of multiple scales of resolution { particularly functions that are wavelet phd thesis in both time and frequency or space and wave number. kwhere 2j x?

Wavelets are based on the application of trans- lation x! kand dilation x! Here, j is the dilation parameter and is understood to range over all the scales of interest, while k indicates the translation and ranges over all possible integer trans- lates shifts.

The basic idea behind the relationship hinges on a multi-resolution analysis whose goal is to break down the original L2 IR space into a sequence of nested subspaces. In the nested sequence of spaces, the V0 subspace lies in the V1 subspace, and so, any function in V0 may be expressed in terms of the scaling functions in V1. Equation 1. Example 1 Translation and Dilation Two examples of translation and dilation are shown in Figure 1.

Unlike the constant function, the hat function is not orthogonal to its integer translates and requires an orthogonalization procedure for use as a scaling function in the wavelet phd thesis spline Battle-Lemarie' construction. Wjand Wj? Here, Pj f represents the projection of a function, wavelet phd thesis, f onto the space Vjand Qj f represents the projection of f onto the wavelet space, wavelet phd thesis, Wj.

In Eq. At each level of resolution, ja function in L2 IR may be represented in terms of the scaling functions at scale jor in terms of scaling functions and wavelets at scale j? This is shown as the Scale-1 projection in Figure 1. Subsequent projections onto increasingly larger spaces yields more accu- rate representations of the original sine function. For exam- ple, to move from Scale-2 to Scale-3, the Q2 f x projection may be computed directly using the Haar wavelet rather than the box functions at Scale-3, i, wavelet phd thesis.

This brief overview of scaling functions, wavelets and multi-resolution analysis has been presented as background for the chapters that follow in this report. Additional details on wavelet construction and multi-resolution analysis may be found in Strang,88,89 Daubechies,35,36 Mallat,74 Meyer,77 Strichartz,93 Massopust,75,76 Williams, Graphs,44 Jawerth,54 and Chui. Meyer77 points out that there are seven primary origins for wavelets that date from around with the Haar wavelet dating back to A brief overview of some of the current literature on wavelet based ap- proaches to solving is presented in the subsequent sections.

and Dahmen. As a cau- 30 33 tionary word, the literature survey presented wavelet phd thesis is undoubtedly incomplete. This representa- tion yielded a numerical solution strategy where data compression was built into the method.

In this work, wavelet phd thesis, the computational complexity of the time- integration scheme was shown to be directly dependent on the rate of data compression. Cai and Wang 13, 14used a cubic spline wavelet with a dis- crete wavelet transform and collocation in their method for solving PDEs. A similar ap- proach has been taken by Vasilyev and Paolucciwhere the 98, 99 computational cost of their method has been demonstrated to be on the or- der of the total number of collocation points and independent of the spatial dimension of the problem.

This is undoubtedly due to the generality that the Galerkin method provides, and the ease with which alternative bases may be imple- mented and tested. In this section, only a subset of the past work is reviewed, beginning in Glowinski et al. In addition, it was suggested that wavelet bases lend themselves naturally to multi-level so- lution methods, but that the extension to multiple dimensions is non-trivial. Although a wavelet phd thesis approach is outlined, scaling functions are used for the test and trial functions, and no direct use of the wavelets in the solution algorithm is apparent.

Latto and Tenenbaum also used Daubachies wavelets D6 in 61 36 the solution of equation with a Galerkin procedure. In this work, only the Daubechies scaling functions were used in the solution method.

InWells et al. Xu and Shann also used Daubachies wavelets in the solution of one-dimensional elliptic problems using orthogonality of the wavelets in the construction of an iterative solution strategy, wavelet phd thesis.

Here, only Dirichlet boundary conditions were considered and a change of basis was used to reduce the operations count during the iterative solution procedure.

The relationship between the change of basis and a hierarchical basis see Yserentant is illustrated in this paper. Biorthogonality refers to the fact that a scaling function and its dual are used as generators for the multi-resolution analysis with orthogonality between the primary and dual basis, wavelet phd thesis. Dahlke and Kunoth, also in22formulated a Galerkin method using biorthogonal wavelets and a two- grid solution strategy that made use of the wavelets for the construction of the restriction and prolongation operations.

Qian and Weiss used Daubachies scaling functions to solve a 84 Helmholtz equation in two dimensions. The use of wavelets has also wavelet phd thesis in multi-level schemes and pre- conditioners. An early example of this is demonstrated by Rieder et al. A later paper by Rieder et al. Work by Glowinski et al. The work by Amaratunga and Williams used Daubachies D6, 1 D8, D10, and D12 scaling functions with a Galerkin procedure to generate solutions to a Helmholtz equation on a periodic domain in one spatial domain.

In this work application to elliptic problems in a unit square is demonstrated. However, no assessment of the computational complexity or numerical performance of these elements was made. Here, a multi-scale decom- position of the Schur complement of the discrete system is suggested for time-dependent problems.

In related work, Kunoth developed a class of multi-level precondi- 59 tioners for elliptic boundary value problems. InPerrier and Charton 82 demonstrated the solution of the incompressible, time-dependent, Navier- Stokes equations using a wavelet-Galerkin method.

Wavelet phd thesis re- cently, Weiss has used the wavelet-Galerkin method for the study of enstrophy transfers in two-dimensional turbulence in simple geometries. Simple examples of these el- ements consist of the linear hat-function and its associated sombrero wavelet.

The treatment of boundary conditions { especially for more traditional wavelets, wavelet phd thesis, i. The application of wavelets that are constructed to be biorthogonal in the sense of a weighted inner product was demonstrated by Sweldens in 94 In Sweldens work, it was suggested that a similar idea could be used for more general operators such as the weak-form of the Helmholtz equation.

Chapters 3 and 4 follow this general line of reasoning. Work on element-by-element construction methods has been reported by Dahmen and Stevenson with a focus on wavelets for unstructured 34 grids that yield uniformly bounded condition numbers for elliptic operators.

As evidenced by the brief review of this wavelet-Galerkin literature, the work in the early s relied heavily on Daubechies wavelets. The strict use of orthogonality has been relaxed yielding biorthogonal and semi-orthogonal 32 wavelets. to Vm.

Theoretical Framework and Conceptual Framework in Your Doctoral Dissertation with Dr. Guy E. White

, time: 9:12

Appendices - Wavelet Based Approximation Schemes for Singular Integral Equations

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