The optimal support-regularity-phase envelope and admissible parameter space of compactly supported orthonormal wavelet filters (Daubechies, 1988)

Daubechies' 1988 construction of orthonormal bases of compactly supported wavelets parametrizes every finite orthonormal low-pass filter with N vanishing moments by an odd residual polynomial R (with a positivity constraint) followed by a Riesz spectral factorization, and exhibits the single minimal choice R = 0 with minimal phase, whose regularity grows only linearly (with non-sharp constants) in the support width. The sharp asymptotic regularity of the standard Daubechies family is known (Lau and Sun, 2000, answering Cohen and Daubechies, 1996), and existence of the multiband and multidimensional compactly supported orthonormal constructions is established (Bi-Dai-Sun, Cohen-Daubechies, Ayache, Lai and Karoui). The open problem is to determine the sharp tradeoff between support width and smoothness over all compactly supported orthonormal wavelets (the value of the optimal asymptotic ratio, whether it is attained, and by which filters), and to give an effective characterization of the admissible set of R and factorization choices together with the extremal filters for regularity, phase/asymmetry and localization.

Background

A wavelet basis of $L^2(\mathbb{R})$ is an orthonormal basis built by dilates and translates $\psi_{jk}(x)=2^{j/2}\psi(2^j x-k)$ of a single mother wavelet $\psi$. It is generated by a multiresolution analysis (MRA): an increasing chain of closed subspaces approximating $L^2(\mathbb{R})$ at successive dyadic scales, spanned by the integer translates of one scaling function $\phi$ satisfying a two-scale (refinement) equation $\phi(x)=\sqrt{2}\sum_n h(n)\phi(2x-n)$. The finite sequence $h(n)$ is the low-pass filter; its symbol $m_0(\xi)=2^{-1/2}\sum_n h(n)e^{in\xi}$ must satisfy the quadrature-mirror (Smith-Barnwell) condition $|m_0(\xi)|^2+|m_0(\xi+\pi)|^2=1$. Before 1988 one had to choose: the Haar basis is compactly supported and orthonormal but discontinuous; Meyer's basis is $C^\infty$ and orthonormal but not compactly supported, so its discrete filters have infinite length; the Battle-Lemarie spline bases are smooth and orthonormal but only with non-compact scaling functions. Daubechies (1988) closed this gap: for every integer $N$ she constructed a filter with $2N$ taps, giving an MRA whose scaling function and wavelet are compactly supported, orthonormal, and of arbitrarily high regularity (Hölder smoothness exponent growing linearly with $N$, i.e. with the support width $[0,2N-1]$), with the wavelet having $N$ vanishing moments ($\int x^j\psi(x)dx=0$ for $j<N$, i.e. $m_0$ has a zero of order $N$ at $\xi=\pi$). The structural core of the construction is a complete parametrization of finite orthonormal filters with $N$ vanishing moments: writing $m_0(\xi)=\big[\tfrac12(1+e^{i\xi})\big]^N Q(e^{i\xi})$, one must have

$$|Q(e^{i\xi})|^2=P_N\big(\sin^2\tfrac{\xi}{2}\big)+\sin^{2N}\tfrac{\xi}{2}\,R\big(\tfrac12\cos\xi\big),$$

where $P_N(y)=\sum_{j=0}^{N-1}\binom{N-1+j}{j}y^j$ is fixed by the vanishing-moment condition and $R$ is an odd polynomial, subject to positivity of $|Q|^2$ on the unit circle and a supremum bound $\sup_{\xi}|Q(e^{i\xi})|<2^{N-1}$ used for regularity. A positive trigonometric polynomial $|Q|^2$ is then split into a factor $Q$ by the Riesz spectral-factorization lemma, which admits finitely many combinatorial choices (which member of each reciprocal pair or quadruplet of zeros of $|Q|^2$ is kept); Daubechies took the minimal-phase choice, the spectral factor whose zeros all lie inside the unit disk, which minimizes the phase delay of the filter. Her explicit family uses the smallest admissible choice $R\equiv 0$ and minimal phase, and her regularity estimates for it are lower bounds with conservative constants (Table 2 of the paper), explicitly not sharp. Regularity of a refinable function is governed by the spectral radius of the transition operator associated with the refinement mask (the operator whose iterates describe how the mask's spectrum concentrates under dyadic dilation; the tool later systematized by Cohen and Daubechies, 1996), and, for the standard $R\equiv 0$ family, by the values of $|Q|$ along the dyadic orbit of the frequency $\xi=2\pi/3$ (doubling permutes $2\pi/3$ and $4\pi/3$, so this periodic orbit is the bottleneck for regularity and is the reason the sharp asymptotics of Lau and Sun involve $|Q_N(2\pi/3)|$); the phase of the factorization governs the symmetry/asymmetry and time-domain localization of the wavelet (compactly supported orthonormal $\phi$ can never be symmetric except for Haar, so phase can only be optimized approximately, as in the least-asymmetric and coiflet variants; coiflets are variants in which the scaling function itself is given vanishing moments). Daubechies' 1992 book and 1993 'Variations on a Theme' consolidated and varied these design choices. What remains unresolved in the design space opened by the 1988 paper is the quantitative limits of the support-regularity tradeoff and the effective description of the admissible parameter space and its extremals.

Problem Statement

Determine the quantitative performance envelope of Daubechies-type compactly supported orthonormal wavelet constructions, and characterize the extremal filters that attain it. Concretely, for finite low-pass filters satisfying $|m_0(\xi)|^2+|m_0(\xi+\pi)|^2=1$ with $N$ vanishing moments, parametrized by the odd residual polynomial $R$ in $|Q(e^{i\xi})|^2=P_N(\sin^2(\xi/2))+\sin^{2N}(\xi/2)R(\frac{1}{2}\cos\xi)$ together with the Riesz spectral-factorization zero selection: (1) give an effective, constructive description of the full admissible set of $(R, \text{factorization})$ pairs and of how support width, Hölder/Sobolev regularity, and phase/asymmetry of the resulting scaling function and wavelet vary over it; (2) determine the sharp optimal tradeoff between support width and regularity over all compactly supported orthonormal wavelets, namely the value of the optimal asymptotic ratio of smoothness exponent to support width, whether that ratio is attained or only approached, and which filters attain or approach it; and (3) determine the analogous sharp regularity constants and extremal constructions for M-band decimations and multidimensional lattices, where existence of arbitrarily regular compactly supported orthonormal bases is known. Parts (1) and (2) concern the dyadic one-dimensional case; part (3) concerns the M-band and multidimensional analogues. The description of the admissible set in (1) is to be effective: specified by finitely many checkable conditions together with an explicit construction of the admissible pairs, not an existence statement alone.

References

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