Optimizing time-frequency resolution of orthonormal wavelets

Dorize, Christian; Villemoes, Lars

doi:10.1109/icassp.1991.150802

Cited by 22 publications

(23 citation statements)

References 3 publications

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“…where [6,10]). The product of the time duration and frequency bandwidth is the area of the resolution cell of f .…”

Section: Biorthogonal Multiwavelets With Optimum Time-frequency Resolmentioning

confidence: 96%

“…The case L = 2 with S (2) (z) = 0 2 leads to optimal balanced biorthogonal multiwavelets 8,6 bo , 8,6 bo , while the case L = 2 with S (2) (z) = 0 2 gives 8,10…”

Section: Optimal Multifilter Banks Of Odd Filter Lengthmentioning

confidence: 98%

“…Let Biort N, N denote the optimal balanced multifilter bank N, N H bo , N, N G bo , N, N H bo , N, N G bo . Thus the multifilter banks considered are Biort 7,5 , Biort 7,9 , Biort 8,6 , Biort 8,10 . Since all these multifilter banks are obtained from balanced multiwavelets, no prefiltering is needed in their application (see [18]).…”

Section: Optimal Multifilter Banks Of Odd Filter Lengthmentioning

confidence: 99%

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Construction of Biorthogonal Multiwavelets Using the Lifting Scheme

Goh

Jiang

Xia

2000

Applied and Computational Harmonic Analysis

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Abstract-The lifting scheme has been found to be a flexible method for constructing scalar wavelets with desirable properties. Here it is extended to the construction of multiwavelets. It is shown that any set of compactly supported biorthogonal multiwavelets can be obtained from the Lazy matrix filters with a finite number of lifting steps. As an illustration of the general theory, compactly supported biorthogonal multiwavelets with optimum time-frequency resolution are constructed. In addition, experimental results of applying these multiwavelets to image compression are presented.

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“…where [6,10]). The product of the time duration and frequency bandwidth is the area of the resolution cell of f .…”

Section: Biorthogonal Multiwavelets With Optimum Time-frequency Resolmentioning

confidence: 96%

“…The case L = 2 with S (2) (z) = 0 2 leads to optimal balanced biorthogonal multiwavelets 8,6 bo , 8,6 bo , while the case L = 2 with S (2) (z) = 0 2 gives 8,10…”

Section: Optimal Multifilter Banks Of Odd Filter Lengthmentioning

confidence: 98%

Section: Optimal Multifilter Banks Of Odd Filter Lengthmentioning

confidence: 99%

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Construction of Biorthogonal Multiwavelets Using the Lifting Scheme

Goh

Jiang

Xia

2000

Applied and Computational Harmonic Analysis

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“…One of the main reasons for the great significance of the IWT in signal processing and image analysis over the classical bandpass filters is that the wavelet bandpass filter provides localized time (or space) information with the so-called zoom-in/zoom-out capability where the width of the time window is governed by the standard deviation (cf. [4,6]) for bandpass filtering can be used as one of the wavelet filter design criteria. Observe that in contrast with the well-known uncertainty principle…”

Section: Introductionmentioning

confidence: 99%

High-Order Orthonormal Scaling Functions and Wavelets Give Poor Time-Frequency Localization

Chui

Wang

1995

J Fourier Anal Appl

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For a fairly general class of orthonormal scaling functions and wavelets with regularity exponents n, we prove that the areas of the time-frequency windows tend to infinity as n → ∞. This class includes those of Battle-Lemarié and Daubechies. In addition, if the scaling functions have at least asymptotic linear phase, then we prove that they converge to the "sinc" function and their corresponding orthonormal wavelets converge to the "difference" of two sinc functions.

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“…6 One might wonder why the enterprise of minimizing the joint space-spatial frequency uncertainty, Uf,, is of interest to the vision community. The viewprint calculations of Klein and Levi provide some justification.…”

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confidence: 99%

Minimizing and maximizing the joint space-spatial frequency uncertainty of Gabor-like functions: comment

Klein

Beutter

1992

J. Opt. Soc. Am. A

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We dispute the claim that Hermite functions (similar to derivatives of Gaussians) minimize a joint uncertainty relation in space and spatial frequency. These functions are found to maximize rather than minimize the uncertainty of the class of functions consisting of an mth-order polynomial times a Gaussian.Daugman' argued that one of the beauties of Gabor functions is that they are compact in both space and spatial frequency since they minimize the Heisenberg uncertainty in space and spatial frequency. The goal is to minimize the product of the variance of the receptive field in space times its variance in spatial frequency. The second integral in Eq. (2) where Dm represents the mth derivative and a is the standard deviation of the Gaussian. These functions have been normalized:The orthonormality of the Hermite functions allows any function r(x) that is well behaved at x = 0 to be expanded in a series of Hermite functions:By using Hermite polynomial recursion relations, the joint uncertainty, Eq. (2), can be written in terms of the expansion coefficients, am,To get a clearer understanding of the joint uncertainty it is useful to restrict Eq. (6) to the case in which only two coefficients are nonvanishing. We can then make the general transformation: where the last term with the Kronecker delta contributes only if n -ml = 2.The joint uncertainty, U of Eq. (8) is plotted in Fig. 1 for m = 2 and n ranging from 0 to 5. The horizontal axis is , which goes from -iT/2 to +7r/ 2 . If In -ml • 2, the curves in Fig. 1 One might be tempted to say that at = 0, U is a maximum for n < m and a minimum for n > m. This holds

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Optimizing time-frequency resolution of orthonormal wavelets

Cited by 22 publications

References 3 publications

Construction of Biorthogonal Multiwavelets Using the Lifting Scheme

Construction of Biorthogonal Multiwavelets Using the Lifting Scheme

High-Order Orthonormal Scaling Functions and Wavelets Give Poor Time-Frequency Localization

Minimizing and maximizing the joint space-spatial frequency uncertainty of Gabor-like functions: comment

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