Wavelets Made Easy

A Algorithms for Wavelet Transforms.- 1 Haar’s Simple Wavelets.- 1.0 Introduction.- 1.1 Simple Approximation.- 1.2 Approximation with Simple Wavelets.- 1.2.1 The Basic Haar Wavelet Transform.- 1.2.2 Significance of the Basic Haar Wavelet Transform.- 1.2.3 Shifts and Dilations of the Basic Haar Transform.- 1.3 The Ordered Fast Haar Wavelet Transform.- 1.3.1 Initialization.- 1.3.2 The Ordered Fast Haar Wavelet Transform.- 1.4 The In-Place Fast Haar Wavelet Transform.- 1.4.1 In-Place Basic Sweep.- 1.4.2 The In-Place Fast Haar Wavelet Transform.- 1.5 The In-Place Fast Inverse Haar Wavelet Transform.- 1.6 Examples.- 1.6.1 Creek Water Temperature Analysis.- 1.6.2 Financial Stock Index Event Detection.- 2 Multidimensional Wavelets and Applications.- 2.0 Introduction.- 2.1 Two-Dimensional Haar Wavelets.- 2.1.1 Two-Dimensional Approximation with Step Functions.- 2.1.2 Tensor Products of Functions.- 2.1.3 The Basic Two-Dimensional Haar Wavelet Transform.- 2.1.4 Two-Dimensional Fast Haar Wavelet Transform.- 2.2 Applications of Wavelets.- 2.2.1 Noise Reduction.- 2.2.2 Data Compression.- 2.2.3 Edge Detection.- 2.3 Computational Notes.- 2.3.1 Fast Reconstruction of Single Values.- 2.3.2 Operation Count.- 2.4 Examples.- 2.4.1 Creek Water Temperature Compression.- 2.4.2 Financial Stock Index Image Compression.- 2.4.3 Two-Dimensional Diffusion Analysis.- 2.4.4 Three-Dimensional Diffusion Analysis.- 3 Algorithms for Daubechies Wavelets.- 3.0 Introduction.- 3.1 Calculation of Daubechies Wavelets.- 3.2 Approximation of Samples with Daubechies Wavelets.- 3.2.1 Approximate Interpolation.- 3.2.2 Approximate Averages.- 3.3 Extensions to Alleviate Edge Effects.- 3.3.1 Zigzag Edge Effects from Extensions by Zeros.- 3.3.2 Medium Edge Effects from Mirror Reflections.- 3.3.3 Small Edge Effects from Smooth Periodic Extensions.- 3.4 The Fast Daubechies Wavelet Transform.- 3.5 The Fast Inverse Daubechies Wavelet Transform.- 3.6 Multidimensional Daubechies Wavelet Transforms.- 3.7 Examples.- 3.7.1 Hangman Creek Water Temperature Analysis.- 3.7.2 Financial Stock Index Image Compression.- B Basic Fourier Analysis.- 4 Inner Products and Orthogonal Projections.- 4.0 Introduction.- 4.1 Linear Spaces.- 4.1.1 Number Fields.- 4.1.2 Linear Spaces.- 4.1.3 Linear Maps.- 4.2 Projections.- 4.2.1 Inner Products.- 4.2.2 Gram—Schmidt Orthogonalization.- 4.2.3 Orthogonal Projections.- 4.3 Applications of Orthogonal Projections.- 4.3.1 Application to Three-Dimensional Computer Graphics.- 4.3.2 Application to Ordinary Least-Squares Regression.- 4.3.3 Application to the Computation of Functions.- 4.3.4 Applications to Wavelets.- 5 Discrete and Fast Fourier Transforms.- 5.0 Introduction.- 5.1 The Discrete Fourier Transform (DFT).- 5.1.1 Definition and Inversion.- 5.1.2 Unitary Operators.- 5.2 The Fast Fourier Transform (FFT).- 5.2.1 The Forward Fast Fourier Transform.- 5.2.2 The Inverse Fast Fourier Transform.- 5.2.3 Interpolation by the Inverse Fast Fourier Transform.- 5.2.4 Bit Reversal.- 5.3 Applications of the Fast Fourier Transform.- 5.3.1 Noise Reduction Through the Fast Fourier Transform.- 5.3.2 Convolution and Fast Multiplication.- 5.4 Multidimensional Discrete and Fast Fourier Transforms.- 6 Fourier Series for Periodic Functions.- 6.0 Introduction.- 6.1 Fourier Series.- 6.1.1 Orthonormal Complex Trigonometric Functions.- 6.1.2 Definition and Examples of Fourier Series.- 6.1.3 Relation Between Series and Discrete Transforms.- 6.1.4 Multidimensional Fourier Series.- 6.2 Convergence and Inversion of Fourier Series.- 6.2.1 The Gibbs—Wilbraham Phenomenon.- 6.2.2 Piecewise Continuous Functions.- 6.2.3 Convergence and Inversion of Fourier Series.- 6.2.4 Convolutions and Dirac’s “Function” ?.- 6.2.5 Uniform Convergence of Fourier Series.- 6.3 Periodic Functions.- C Computation and Design of Wavelets.- 7 Fourier Transforms on the Line and in Space.- 7.0 Introduction.- 7.1 The Fourier Transform.- 7.1.1 Definition and Examples of the Fourier Transform.- 7.2 Convolutions and Inversion of the Fourier Transform.- 7.3 Approximate Identities.- 7.3.1 Weight Functions.- 7.3.2 Approximate Identities.- 7.3.3 Dirac Delta (?) Function.- 7.4 Further Features of the Fourier Transform.- 7.4.1 Algebraic Features of the Fourier Transform.- 7.4.2 Metric Features of the Fourier Transform.- 7.4.3 Uniform Continuity of Fourier Transforms.- 7.5 The Fourier Transform with Several Variables.- 7.6 Applications of Fourier Analysis.- 7.6.1 Shannon’s Sampling Theorem.- 7.6.2 Heisenberg’s Uncertainty Principle.- 8 Daubechies Wavelets Design.- 8.0 Introduction.- 8.1 Existence, Uniqueness, and Construction.- 8.1.1 The Recursion Operator and Its Adjoint.- 8.1.2 The Fourier Transform of the Recursion Operator.- 8.1.3 Convergence of Iterations of the Recursion Operator.- 8.2 Orthogonality of Daubechies Wavelets.- 8.3 Mallat’s Fast Wavelet Algorithm.- 9 Signal Representations with Wavelets.- 9.0 Introduction.- 9.1 Computational Features of Daubechies Wavelets.- 9.1.1 Initial Values of Daubechies’ Scaling Function.- 9.1.2 Computational Features of Daubechies’ Function.- 9.1.3 Exact Representation of Polynomials by Wavelets.- 9.2 Accuracy of Signal Approximation by Wavelets.- 9.2.1 Accuracy of Taylor Polynomials.- 9.2.2 Accuracy of Signal Representations by Wavelets.- 9.2.3 Approximate Interpolation by Daubechies’ Function.- D Directories.- Acknowledgments.- Collection of Symbols.