Numerical Issues When Using Wavelets

Notations For a real discrete-time filter whose impulse response is h[n], h̄[n] = h[−n], n ∈ Z is its timereversed version. The hat ˆ notation will be used for the Fourier transform of square-integrable signals. For a filter h, its z-transform is written H(z). The convolution product of two signals in l2(Z) will be written ∗. For the octave band wavelet representation, analysis (respectively, synthesis) filters are denoted h and g (respectively, h̃ and g̃). The scaling and wavelet functions used for the analysis (respectively, synthesis) are denoted φ (φ(x2 ) = ∑ k h[k]φ(x − k), x ∈ R and k ∈ Z) and ψ (ψ(x2 ) = ∑ k g[k]φ(x − k), x ∈ R and k ∈ Z) (respectively, φ̃ and ψ̃). 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, φ̃ and ψ̃. J.-L. Starck is with the CEA-Saclay, DAPNIA/SEDI-SAP, Service d’Astrophysique, F-91191 Gif sur Yvette, France J. Fadili is with the GREYC CNRS UMR 6072, Image Processing Group, ENSICAEN 14050, Caen Cedex, France