## Framelets and Wavelets: Algorithms, Analysis, and Applications

Link to publisher webpage of the book

Maple Routines for all 1D framelets and wavelets
• fw1d: maple routines for handling all tasks of 1D real/complex-valued framelets/wavelets with any dilation factor, with any multiplicity, and with/without symmetry: constructing framelets/wavelets, computing smoothness of and plotting refinable scalar/vector functions and wavelets, etc.
• tutorialfw1d.mw: Tutorial for constructing real-valued refinable scalar/vector functions and wavelets with any dilation factor and with/without symmetry.
• tutorialfw1dComplex.mw: Tutorial for constructing complex-valued refinable scalar/vector functions and wavelets with any dilation factor and with/without (complex) symmetry.
• These maple routines in fw1d can be used to perform almost all tasks related to 1D wavelets/famelets and multiwavelets/multiframelets with any dilation factor and with/without symmetry. The maple routines can be used to produce all the examples in the book.
• fw1d is significantly revised and simplified from my old maple routines developed over many years. New routines are expected to be added to fw1d in the future. If you find any mistakes or have any comments, please send emails to: Bin Han at bhan@ualberta.ca

LaTex macros
• wavelet.sty: LaTex macros used for preparing the book.
• notation.pdf: Explanation of macros and symbols used in wavelet.sty

Send typos/mistakes of the book to: bhan@ualberta.ca (Thanks in advance)
Content of Book:
• Chapter 1: Discrete Framelet Transform.
1. Section 1.1: Perfect reconstruction of discrete framelet transforms
2. Section 1.2: Sparsity of discrete framelet transforms
3. Section 1.3: Multilevel discrete framelet transforms and stability
4. Section 1.4: The oblique extension principle (OEP)
5. Section 1.5: Discrete framelet transforms for signals on bounded intervals
6. Section 1.6: Discrete framelet transforms implemented in the frequency domain
7. Section 1.7: Exercises
• Chapter 2: Wavelet Filter Banks.
1. Section 2.1: Interpolatory filters and filters with linear-phase moments.
2. Section 2.2: Real orthogonal wavelet filter banks with minimal supports
3. Section 2.3: Real orthogonal wavelet filter banks with linear-phase moments
4. Section 2.4: Complex orthogonal wavelet filters with symmetry and minimal supports
5. Section 2.5: Complex orthogonal wavelet filters with symmetry and linear-phase moments
6. Section 2.6: Biorthogonal wavelet filter banks by CBC (coset by coset) algorithm
7. Section 2.7: Polyphase matrix and chain structure of biorthogonal wavelet filters
8. Section 2.8: Exercises
• Chapter 3: Framelet Filter Banks
1. Section 3.1: Properties of Laurent polynomials with symmetry
2. Section 3.2: Dual framelet filter banks with symmetry and two high-pass filters
3. Section 3.3: Tight framelet filter banks with symmetry and two high-pass filters
4. Section 3.4: Tight framelet filter banks with two high-pass filters
5. Section 3.5: Tight framelet filter banks with symmetry and three high-pass filters
6. Section 3.6: Existence of tight framelet filter banks with symmetry
7. Section 3.7: Exercises
• Chapter 4: Analysis of Affine Systems and Dual Framelets
1. Section 4.1: Frequency-based dual framelets and connections to filter banks
2. Section 4.2: Frames and bases in Hilbert spaces
3. Section 4.3: Nonhomogeneous and homogeneous affine systems in L_2(R)
4. Section 4.4: Shift-invariant subspaces of L_2(R)
5. Section 4.5: Refinable structure and multiresolution analysis
6. Section 4.6: Framelets and wavelets in Sobolev spaces
7. Section 4.7: Approximation by dual framelets and quasi-projection operators
8. Section 4.8: Frequency-based nonstationary dual framelets
9. Section 4.9: Periodic framelets and wavelets
10. Section 4.10: Exercises
• Chapter 5: Analysis of Refinable Vector Functions
1. Section 5.1: Distributional solutions to vector refinement equations
2. Section 5.2: Linear independence of integer shifts of compactly supported functions
3. Section 5.3: Stability of integer shifts of functions in L_p(R)
4. Section 5.4: Approximation using quasi-projection operators in L_p(R)
5. Section 5.5: Accuracy and approximation orders of shift-invariant subspaces of L_p(R)
6. Section 5.6: Convergence of cascade algorithms in Sobolev spcaes W^m_p(R)
7. Section 5.7: Express sm_p(a) using the p-norm joint spectral radius
8. Section 5.8: Smoothness of refinable functions and computation of sm_p(a)
9. Section 5.9: Cascade algorithms and refinable functions with perturbed filters
10. Section 5.10: Exercises
• Chapter 6: Framelets and Wavelets Derived from Refinable Functions
1. Section 6.1: Refinable functions having analytic expressions
2. Section 6.2: Refinable Hermite interpolants and hermite interpolatory filters
3. Section 6.3: Compactly supported refinable functions in H^\tau(R) with \tau\in R
4. Section 6.4: Framelets and wavelets in Sobolev spaces with filter banks
5. Section 6.5: Pairs of biorthogonal wavelet filters with increasing orders of sum rules
6. Section 6.6: Framelets/wavelets with filters of Hoelder class of exponential decay
7. Section 6.7: Smooth refinable duals and local linear independence
8. Section 6.8: Stability of discrete affine systems in the space l_2(R)
9. Section 6.9: Exercises
• Chapter 7: Applications of Framelets and Wavelets
1. Section 7.1: Multidimensional framelets and wavelets
2. Section 7.2: Multidiemsional cascade algorithms and refinable functions
3. Section 7.3: Subdivision schemes in computer graphics
4. Section 7.4: Directional tensor product complex tight framelets for image processing
5. Section 7.5: Framelets/wavelets on a fintie interval for numerical algorithms
6. Section 7.6: Fast multifrramelet transform and its balanced property
7. Section 7.7: Exercises