Numerical issues when using wavelets

Abstract

Article Outline: Article Outline: Notations Glossary Definition of the Subject Introduction The Continuous Wavelet Transform The (Bi-)Orthogonal Wavelet Transform The Lifting Scheme The Undecimated Wavelet Transform The 2D Isotropic Undecimated Wavelet Transform Designing Non-orthogonal Filter Banks Iterative Reconstruction Future Directions Bibliography Notations: For a real discrete-time filter whose impulse response is is its time-reversed version. The hat notation will be used for the Fourier transform of square-integrable signals. For a filter h, its z-transform iswritten H(z).The convolution product of two signals l2(ℤ) in will be written∗. For the octave band wavelet representation, analysis(respectively, synthesis) filters are denoted h and g (respectively, and). The scaling and wavelet functions used for theanalysis (respectively, synthesis) are denoted and φ(φ(x/2)=∑k h[k]φ(x-k), x ∈R and k ∈ ℤ and ψ(ψ(x/2)=∑k g[k]φ(x-k), x∈ ℝ and k ∈ ℤ (respectively, φ and ψ).We also define the scaled dilated and translated version of φ at scale j and position k as φj,k(x)=2-jφ(2-jx-k), and similarly for ψ, φ and ψ.