Abstract
The Lifting Scheme introduced in (Sweldens, Appl. Comput. Harmon. Anal. 3(2), 186–200 (1996) and Sweldens, SIAM J. Math. Anal. 29(2), 511–546 (1997).) [3, 4] is a method that constructs bi-orthogonal wavelet transforms of signals and provides their efficient implementation. The main feature of the lifting scheme is that all the constructions are derived directly in the spatial domain and therefore can be custom designed to more general and irregular settings such as non-uniformly spaced data samples and bounded intervals. In this chapter, we outline the lifting scheme and describe how to use the local quasi-interpolating splines, introduced in Chap. 6, for the construction of wavelet transforms of non-equally sampled signals and real-time implementation of signals’ transforms in situation when samples arrive one after another at random times. On arrival of new samples, only a couple of adjacent transform coefficients are updated in a way that no boundary effects occur.
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Notes
- 1.
Recall that for cubic splines, the nodes’ grid t coincides with the sampling grid g.
References
A.Z. Averbuch, P. Neittaanmäki, V.A. Zheludev, Spline and Spline Wavelet Methods with Applications to Signal and Image Processing, Non-periodic Splines, vol II (Springer, Berlin, 2015)
C.M. Brislawn, Classification of nonexpansive symmetric extension transforms for multirate filter banks. Appl. Comput. Harmon. Anal. 3(4), 337–357 (1996)
W. Sweldens, The lifting scheme: a custom-design construction of biorthogonal wavelets. Appl. Comput. Harmon. Anal. 3(2), 186–200 (1996)
W. Sweldens, The lifting scheme: a construction of second generation wavelets. SIAM J. Math. Anal. 29(2), 511–546 (1997)
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Averbuch, A.Z., Neittaanmäki, P., Zheludev, V.A. (2019). Spline-Based Wavelet Transforms. 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_7
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DOI: https://doi.org/10.1007/978-3-319-92123-5_7
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