Butterworth wavelet transforms derived from discrete interpolatory splines: recursive implementation

Abstract

In the paper we present a new family of biorthogonal wavelet transforms and the related library of biorthogonal symmetric waveforms. For the construction we used the interpolatory discrete splines which enabled us to design a library of perfect reconstruction ÿlter banks. These ÿlter banks are related to Butterworth ÿlters. The construction is performed in a "lifting" manner. The di erence from the conventional lifting scheme is that the transforms of a signal are performed via recursive ÿltering with the use of IIR ÿlters. These ÿlters have linear phase property and the basic waveforms are symmetric. The ÿlters allow fast cascade or parallel implementation. We present explicit formulas for construction of wavelets with arbitrary number of vanishing moments. In addition, these ÿlters yield perfect frequency resolution. The proposed scheme is based on interpolation and, as such, it involves only samples of signals and it does not require any use of quadrature formulas. ?

Interpolatory subdivision schemes with infinite masks originated from splines

“…The following proposition, which was established in [3], characterizes the structure of the denominator D r (z) : Proposition 2.4 ([3]). If r = 2p + 1, then the following representation holds:…”

“…Together with the polynomial splines we explore the so-called interpolatory discrete splines as a source for devising refinement masks [3,11]. The derived masks are related to the Butterworth filters, which are commonly used in signal processing [12].…”

“…But, due to rational structure of the symbols, this obstacle could be circumvented by employing recursive filtering [3,13]. As a result, the computational complexity implementing these schemes is even lower than the complexity of implementation of schemes with finite masks, which have comparable properties.…”

Image Compression Using Spline Based Wavelet Transforms

“…We describe the discrete splines construction in [7,8]. In this case explicit formulas for the transforms with any number of vanishing moments are established.…”

“…Details of construction and proofs of formulated propositions can be found in [8]. The rest of the paper is organized as follows.…”