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**About this book :- **
**From Fourier Analysis to Wavelets ** written by
** Jonas Gomes, Luiz Velho **.

**Book Detail :- **
** Title: ** From Fourier Analysis to Wavelets
** Edition: **
** Author(s): ** Jonas Gomes, Luiz Velho
** Publisher: **
** Series: **
** Year: **
** Pages: ** 210
** Type: ** PDF
** Language: ** English
** ISBN: **
** Country: ** US
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**Book Contents :- **
**From Fourier Analysis to Wavelets ** written by
** Jonas Gomes, Luiz Velho **
cover the following topics.
'
**1. Introduction**

Computational Mathematics, Abstraction Levels, Relation Between the Abstraction Levels, Functions and Computational Mathematics, Representation and Reconstruction of Functions, Specification of Functions, What is the Relation with Graphics?, Description of Graphical Objects, Where do Wavelets Fit?, Function Representation Using Wavelets, Multiresolution Representation, About these Notes, Comments and References, Bibliography
**2. Function Representation and Reconstruction**

Representing Functions, The Representation Operator, Basis Representation, Complete Orthonormal Representation, Representation by Frames, Riesz Basis Representation, Representation by Projection, Galerkin Representation, Reconstruction, Point Sampling and Interpolation, Piecewise Constant Reconstruction, Piecewise Linear Reconstruction, Multiresolution Representation, Representation by Dictionaries, Redundancy in the Representation, Wavelets and Function Representation, Comments and References, Bibliography

3. The Fourier Transform

Analyzing Functions, Fourier Series, Fourier Transform, Spatial and Frequency Domain, A Pause to Think, Frequency Analysis, Fourier Transform and Filtering, Fourier Transform and Function Representation, Fourier Transform and Point Sampling, The Theorem of Shannon-Whittaker, Point Sampling and Representation by Projection, Point Sampling and Representation Coefficients, Comments and References, Bibliography
**4. Windowed Fourier Transform**

A Walk in The Physical Universe, The Windowed Fourier Transform, Invertibility of ˜f(t, ?), Image of the Windowed Fourier Transform, WFT and Function Representation, Time-frequency Domain, The Uncertainty Principle, Atomic Decomposition, WFT and Atomic Decomposition, Comments and References, Bibliography
**5. The Wavelet Transform**

The Wavelet Transform, Inverse of the Wavelet Transform, Image of the Wavelet Transform, Filtering and the Wavelet Transform, The Discrete Wavelet Transform, Function Representation, Comments and References, Bibliography
**6. Multiresolution Representation**

The Concept of Scale, Scale Spaces, A Remark About Notation, Multiresolution Representation, A Pause to Think, Multiresolution Representation and Wavelets, A Pause... to See the Wavescape, Two Scale Relation, Comments and References, Bibliography
**7. The Fast Wavelet Transform**

Multiresolution Representation and Recursion, Two-Scale Relations and Inner , Wavelet Decomposition and Reconstruction, Decomposition, Reconstruction, The Fast Wavelet Transform Algorithm, Forward Transform, Inverse Transform, Complexity Analysis of the Algorithm, Boundary Conditions, Comments and References, Bibliography

**8. Filter Banks and Multiresolution**

Two-Channel Filter Banks, Filter Banks and Multiresolution Representation, Discrete Multiresolution Analysis, Pause to Review, Reconstruction Bank, Computational Complexity, Comments and Referen, Bibliography
**9. Constructing Wavelets**

Wavelets in the Frequency Domain, The Relations of fˆ with m0, The Relations of ?ˆ with m1, Characterization of m0, Characterization of m1, Orthonormalization Method, A Recipe, Piecewise Linear Multiresolution, Shannon Multiresolution Analysis, Where to Go Now?, Comments and References, Bibliography
**10. Wavelet Design**

Synthesizing Wavelets from Filters, Conjugate Mirror Filters, Conditions for m0, Strategy for Computing m0, Analysis of P, Factorization of P, Example (Haar Wavelet), Properties of Wavelets, Orthogonality, Support of f and ?, Vanishing Moments and Polynomial Reproduction, Regularity, Symmetry or Linear Phase, Other Properties, Classes of Wavelets, Orthogonal Wavelets, Biorthogonal Wavelets, Comments and References, Bibliography
**11. Orthogonal Wavelets**

The Polynomial P, P as a Product Polynomial, P and the Halfband Property, The Expression of P, The Factorization of P, Analysis of P, Examples of Orthogonal Wavelets, Daubechies Extremal Phase Wavelets, Minimal Phase Orthogonal Wavelets, Coiflets, Comments and References, Bibliography
**12. Biorthogonal Wavelets**

Biorthogonal Multiresolution Analysis and Filters, Biorthogonal Basis Functions, Biorthogonality and Filters, Fast Biorthogonal Wavelet Transform, Filter Design Framework for Biorthogonal Wavelets, Perfect Reconstruction Filter Banks, Conjugate Quadrature Filters, The Polynomial P and Wavelet Design, Factorization of P for Biorthogonal Filters, Symmetric Biorthogonal Wavelets, B-Spline Wavelets, Wavelets with Closer Support Width, Biorthogonal Bases Closer to Orthogonal Bases, Comments and References, Bibliography
**13. Directions and Guidelines**

History and Motivation, A Look Back, Extending the Basic Wavelet Framework, Studying Functions on other Domains, Defining other Time-Frequency Decompositions, Solving Mathematical Problems, Applications of Wavelets, Comments and References, Bibliography
**A. Systems and Filters**

A.1 Systems and Filters, Spatial Invariant Linear Systems, Other Characteristics, Discretization of Systems, Discrete Signals, Discrete Systems, Upsampling and Downsampling Operators, Filter Banks, Comments and References, Bibliography
**B. The Z Transform**

B.1 The Z Transform, Some Properties, Transfer Function, The Variable z and Frequency, Subsampling Operations, Downsampling in the Frequency Domain, Upsampling in the Frequency Domain, Upsampling after Downsampling, Comments and References, Bibliography

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