Hilbert transform pairs of biorthogonal wavelet bases

Abstract

The forming of Hilbert transform pairs of biorthogonal wavelet bases of two-band filter banks is studied in this paper. We first derive necessary and sufficient conditions on the scaling filters that render two Hilbert transform pairs: one decomposition pair and one reconstruction pair. We show that the Hilbert transform pairs are achieved if and only if the decomposition scaling filter of one filter bank is half-sample delayed from that of the other filter bank; and the reconstruction scaling filter of the former is half-sample advanced from that of the latter. Hilbert transform pairs of wavelet bases are also characterized by equivalent relationships on the wavelet filters and the scaling functions associated with the two filter banks. An illustrative example is provided.

The dual-tree complex wavelet transform

“…The necessary and sufficient conditions for the biorthogonal case were proven in [121]. To understand intuitively why the half-sample delay condition leads to a nearly shift-invariant wavelet transform, note that the halfsample delay condition is equivalent to uniformly oversampling the low-pass signal at each scale by 2:1, thus largely avoiding the aliasing due to the low-pass downsamplers [53]- [55].…”

“…(Algorithms for designing an orthonormal wavelet basis to match a specified signal class are described, for example, in [20].) Unfortunately, this will sometimes result in g 0 (n) being substantially longer than h 0 (n) (but see [105] and [121] for relatively short g 0 (n)). By jointly designing h 0 (n) and g 0 (n), we can obtain a pair of filters of equal (or near-equal) length, where both are relatively short.…”

Designing Hilbert-Pair of wavelets: Recent progress and future trends

“…However, exactly realizable filters (using direct convolution (FIR) or recursive difference equations (IIR)) cannot satisfy (8) exactly due to the presence of the half-sample delay factor e −jω/2 . In practice (8) can only be approximated and the Hilbert transform relationship is also approximate as well, ie. approximate Hilbert-Pair.…”

The fractional Hilbert transform and dual-tree Gabor-like wavelet analysis

“…The Correspondence: {kunal.chaudhury,michael.unser}@epfl.ch. This work was partly supported by the Swiss National Science Foundation under grant 200020-109415. crucial observation that the dual-tree wavelets form an approximate Hilbert transform (HT) pair was made by Selesnick [3]; this consequently reduced the design of different flavors of dual-tree wavelets to the construction of new HT pairs of wavelets [3,4,5]. We refer the reader to the excellent tutorial [6] on the design and application of the dual-tree transform.…”