Time-frequency localization in transforms, subbands, and wavelets: a critical review

Abstract

In many still-image and video processing applications, the time-frequency localization properties of the decomposition technique are an important consideration. While bandwidth compression of the image requires operators with good localization in frequency, spatial features such as edge preservation demand a high degree of localization in time (or the spatial variable). These requirements compete with each other and one is secured at the expense of the other. The classical "uncertainty" principle in the continuous-time domain provides the back drop for this trade-off. Our purpose is to review recent extensions of this principle to the discrete-time case and to develop optimum wave forms. We review common features of block transforms, subband filter banks, and wavelets, and demonstrate how the discrete uncertainty can be used to evaluate these decomposition methods. In particular, we evaluate the trade-off between localization in time and in frequency for several proposed signal decomposition structures.

Empirical Tests for Evaluation of Multirate Filter Bank Parameters

“…. , M −1 with an ideal value of 0.5 expected (in the limit as K → ∞) for the time frequency uncertainty [2]. The empirical estimate υ = [.503, .555, .513, .555, .502] confirmed this expectation.…”

“…All of the empirical evaluation tests introduced here, including the Kronecker delta error (Section 2.3), M -shift biorthogonality error (Section 2.4), M -shift orthogonality error (Section 2.5), time domain centers and moments (Section 2.7), vanishing moments numbers (Section 2.8), time domain regularity (Section 2.9), and frequency domain selectivity (Section 2.10), have been devised independently by the author with the exception of the M -band reconstruction error (Section 2.6) and time frequency uncertainty (Section 2.11) which have been implemented as modifications and extensions of those in [5] and [2], respectively. These evaluation tests have been demonstrated and validated (Section 3) on a wide variety of filter banks with coefficients either generated by computational algorithms or obtained from published tables.…”

“…Finally, let υ ≡ [υ m ] = tfu(F) (1.68) denote the time frequency uncertainty for all filter bands in the filter bank. Note that the expressions introduced here apply to all bands of the filter bank including the highpass band omitted by Haddad et al [2], and that these expressions differ from theirs with respect to the DFTindependent normalization used for all bands, as well as the integrands and intervals of integration used for the bandpass and highpass bands.…”

“…For all discrete time filters, υ ≥ 0.5 with the optimal value 0.5 approached by binomial filters that tend to Gaussian functions in the limit [2]. Finally, let υ ≡ [υ m ] = tfu(F) (1.68) denote the time frequency uncertainty for all filter bands in the filter bank.…”

“…Using the names and acronyms summarized here, these filters include the Daubechies Real Biorthogonal Balanced Regular (DRBBR) and Complex Orthogonal Most Symmetric (DCOMS) [12,16], Rioul Real Orthogonal Most Selective (RROMS) [8], Heller Real Orthogonal M -band K-regular (HROMK) [3], Sherlock Real Orthogonal 2-band 1-regular [9] parameterized with Random Angles (SRORA), Hermite Real Nonorthogonal Symmetric Binomial (HRNSB) [2], and a wide assortment of filter banks available from Nayebi et al [5]. The filter banks chosen as test examples were selected to represent a variety of different classes with contrasting features.…”