Framelets and Wavelets: Algorithms, Analysis, and Applications

Link to publisher webpage of the book
Maple Routines for all 1D framelets and wavelets
LaTex macros
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