Design of orthonormal and overcomplete wavelet transforms based on rational sampling factors

Abstract

Most wavelet transforms used in practice are based on integer sampling factors. Wavelet transforms based on rational sampling factors offer in principle the potential for time-scale signal representations having a finer frequency resolution. Previous work on rational wavelet transforms and filter banks includes filter design methods and frequency domain implementations. We present several specific examples of Daubechies-type filters for a discrete orthonormal rational wavelet transform (FIR filters having a maximum number of vanishing moments) obtained using Gröbner bases. We also present the design of overcomplete rational wavelet transforms (tight frames) with FIR filters obtained using polynomial matrix spectral factorization.

Critically sampled discrete wavelet transforms with rational dilation factor of 3/2

“…Higher regularity order is possible but that comes with the expense of longer filters. The design approach has successfully yielded rational discrete wavelets of dilation factor of 3/2 which are critically sampled unlike to other designs in literature [5,7] which gives rise to overcomplete transforms. Additionally, though longer filters are required as compared to the literature designed filters [5,7] in which minimal length is achieved, the proposed the designs offer better flexibility in term of the filters' frequency response as stop-band, pass-band and transition band of the individual filters of the filter bank are freely adjustable.…”

“…Among these few designs, Blu's original design [4] achieved only one order of regularity imposed in the filters for a critically sampled orthonormal design and this algorithm is reported to diverge if more than one regularity order is required. On the other hand, Bayram's designs [5,7,8] have achieved arbitrary K regularity orders to the rational rate filter bank but these designs are achieved in the context of over-sampling rational rate filter banks. In this paper, an approach for designing a rational rate filter bank of sampling rate changes of (2/3,1/3) in a two-branch filter bank with multiple orders of regularity is proposed.…”

“…The frequency selectivity and regularity order of the low pass filters in the rational rate filter banks have big impact on the shift error of the obtained scaling and wavelet functions. However, only a few designs [4] [5,7,8] exist in the literature that successfully address the issue of imposing regularity factors in the rational rate filter bank designs. Among these few designs, Blu's original design [4] achieved only one order of regularity imposed in the filters for a critically sampled orthonormal design and this algorithm is reported to diverge if more than one regularity order is required.…”

M-RWTL: Learning Signal-Matched Rational Wavelet Transform in Lifting Framework

“…A number of RWT designs have been proposed in the literature including FIR (Finite Impulse Response) orthonormal rational filterbank design [12], overcomplete FIR RWT designs [23], [24], IIR (Infinite Impulse Response) rational filterbank design [25], biorthogonal FIR rational filterbank design [26], [27], frequency response masking technique based design of rational FIR filterbank [14], and complex rational filterbank design [28]. However, so far RWT designed and used in applications are meant to meet certain fixed requirements in the frequency domain or time-domain instead of learning the transform from a given signal of interest.…”

Resolution scalable image coding using rational wavelet transforms

“…Research work on the construction of wavelet filters for rational wavelet transforms is very scarce, and has not produced the kind of diversity in the choice of filters that is offered for dyadic wavelet transforms. However, significant advances have been made recently on the design of wavelet filters for rational wavelet transforms [15][16][17]. Encouraged by these recent developments, the research presented in the current document proposes to use An obvious challenge that arises from the stated objectives is how to derive a wavelet decomposition structure capable of producing native spatial resolutions of dyadic and non-dyadic nature.…”