Abstract
This chapter describes how to generate families of biorthogonal wavelet transforms in spaces of periodic signals using prediction p-filters originating from discrete-time and discrete splines. The transforms are generated by the lifting scheme (Sweldens (Wavelet applications in signal and image processing III, vol 2569, 1995, [7]), Sweldens (Appl Comput Harmon Anal 3:186–200, 1996, [8]), Sweldens (SIAM J Math Anal 29:511–546, 1997, [9]), see also Sect. 7.1 of this volume). The discrete-time wavelets related to those transforms are (anti)symmetric, well localized in time domain and have flat spectra. These families comprise wavelets with any number of local discrete vanishing moments (LDVMs). The transforms are implemented in a fast way using the FFT by critically sampled perfect reconstruction periodic filter banks.
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Notes
- 1.
In order to obtain more symmetric formulas, we write the updating p-filter as \(\frac{1}{2}\mathbf {f}_{\mathfrak {u}}\)
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Averbuch, A.Z., Neittaanmäki, P., Zheludev, V.A. (2019). Biorthogonal Wavelet Transforms Originating from Discrete and Discrete-Time Splines. In: Spline and Spline Wavelet Methods with Applications to Signal and Image Processing. Springer, Cham. https://doi.org/10.1007/978-3-319-92123-5_8
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DOI: https://doi.org/10.1007/978-3-319-92123-5_8
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