Least and most disjoint root sets for Daubechies wavelets

Abstract

A new set of wavelet filter families has been added to the systematized collection of Daubechies wavelets. This new set includes complex and real, orthogonal and biorthogonal, least and most disjoint families defined using constraints derived from the principle of separably disjoint root sets in the complex z-domain. All of the new families are considered to be constraint selected without a search and without any evaluation of filter properties such as time-domain regularity or frequency-domain selectivity. In contrast, the older families in the collection are considered to be search optimized for extremal properties. Some of the new families are demonstrated to be equivalent to some of the older families, thereby obviating the necessity for any search in their computation. A library that displays images of all filter families in the collection is available at <www.toolsmiths.com>.

Empirical Tests for Evaluation of Multirate Filter Bank Parameters

“…Tables 1.1, 1.2, 1.5, and 1.4 demonstrated respectively examples of real biorthogonal 2-band, complex orthogonal 2-band, real orthogonal 4-band, and real nonorthogonal 5-band filter banks for which observed values were consistent with expected values for parameters. These examples, and many others in [14,16,18] and elsewhere, serve to validate the evaluation tests. However, Tables 1.6 and 1.3 and Figure 1.1 all demonstrated examples with significant discrepancies between analytical design and experimental evaluation parameters.…”

“…For example, if a filter is claimed to be orthogonal, then the orthogonality test is the one that is relevant. Finally, empirical tests can be used to evaluate for equivalent performance of filter families with respect to certain characteristics within given tolerances [18,19]. Tables 1.1, 1.2, 1.5, and 1.4 demonstrated respectively examples of real biorthogonal 2-band, complex orthogonal 2-band, real orthogonal 4-band, and real nonorthogonal 5-band filter banks for which observed values were consistent with expected values for parameters.…”

“…These rankings are based on the assumption that the order of machine precision corresponds to 10 −16 ≤ e ≤ 10 −15 . Additional examples validating and demonstrating the filter bank evaluation tests elaborated here can be found in [14,16,18]. However, the brief survey of examples that have been analyzed and reported here has demonstrated that papers in the published literature contain errors of transmission, implementation, and interpretation.…”

“…Naming conventions established in [12] and further elaborated in [16,18] for the filter banks there have been extended to apply to the various other filter banks tested here. Briefly, each 2-band family of filters has been named with an identifying acronym followed by the parameters (N a , N s ; K a , K s ) in the biorthogonal cases and by (N ; K) in the orthogonal cases where N is the number of filter coefficients, K is the number of filter roots at z = −1, and the subscripts · a and · s refer to the analysis and synthesis filter banks, respectively.…”

Constraint-selected and search-optimized families of Daubechies wavelet filters computable by spectral factorization

“…Significant advantages of the spectral factorization approach include its generalizability to many different classes and families of wavelets, its suitability for easily interpretable visual displays, and thus its practicality in pedagogy. All of the complex orthogonal, real orthogonal, and real biorthogonal families of the Daubechies class computable by spectral factorization and constructed with a single unifying computational algorithm have been studied experimentally in the systematized collection developed by Taswell [10][11][12][15][16][17] over a wide range of vanishing moment numbers and filter lengths.…”