On the Hilbert Transform of Wavelets

Chaudhury, Kunal N.; Unser, Michaël

doi:10.1109/tsp.2010.2103072

Cited by 24 publications

(11 citation statements)

References 12 publications

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“…It was observed there that for a wavelet with large number of vanishing moments and a reasonably good decay, its Hilbert transform automatically has an ultra-fast decay. We extend the results in [3] first to the Riesz transform, and then to its higher-order counterparts. While the main principles are the same, we are required to use a different approach in the higher dimensional setting.…”

Section: Introductionmentioning

confidence: 76%

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Decay Properties of Riesz Transforms and Steerable Wavelets

Ward¹,

Chaudhury²,

Unser³

2013

SIAM J. Imaging Sci.

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The Riesz transform is a natural multi-dimensional extension of the Hilbert transform, and it has been the object of study for many years due to its nice mathematical properties. More recently, the Riesz transform and its variants have been used to construct complex wavelets and steerable wavelet frames in higher dimensions. The flip side of this approach, however, is that the Riesz transform of a wavelet often has slow decay. One can nevertheless overcome this problem by requiring the original wavelet to have sufficient smoothness, decay, and vanishing moments. In this paper, we derive necessary conditions in terms of these three properties that guarantee the decay of the Riesz transform and its variants, and as an application, we show how the decay of the popular Simoncelli wavelets can be improved by appropriately modifying their Fourier transforms. By applying the Riesz transform to these new wavelets, we obtain steerable frames with rapid decay.

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Section: Introductionmentioning

confidence: 76%

“…We expand upon the work in [3], where the intimate connection between the Hilbert transform (the onedimensional Riesz transform) and wavelets was investigated. It was observed there that for a wavelet with large number of vanishing moments and a reasonably good decay, its Hilbert transform automatically has an ultra-fast decay.…”

Section: Introductionmentioning

confidence: 99%

Decay Properties of Riesz Transforms and Steerable Wavelets

Ward¹,

Chaudhury²,

Unser³

2013

SIAM J. Imaging Sci.

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“…Several authors [23]- [27] showed that considerable enhancement can be achieved by using a pair of wavelet transforms where the wavelets form a Hilbert transform pair [23]. The dual-tree discrete wavelet transform was originally introduced by N. Kingsbury [23].…”

Section: E Hilbert Transform Pairs Of Wavelet Basesmentioning

confidence: 99%

Mathematical frameworks to assess the dynamics of cerebral autoregulation

Mahmic

Ibrahim

Tan

2012

2012 IEEE-EMBS Conference on Biomedical Engineering and Sciences

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This paper presents mathematical frameworks, with the purpose to identify the best approach for the assessment of dynamic cerebral autoregulation. The cerebral blood flow velocity (CBFV) was measured by transcranial Doppler (TCD) ultrasonograph and mean arterial blood pressure (MABP) was non-invasively measured using vascular unloading technique, in order to determine the status of autoregulation. The findings show that cerebral autoregulation is a non-stationary and non linear system, thus linear transfer function analysis is not a valid technique for studying the relationship between CBFV and MABP. While wavelet based analyses have shown better performance for the extraction of spontaneous oscillation in the cerebral autoregulation system.

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“…4 It is also known that the poor translation-invariance of standard wavelet bases can be improved by considering a pair of wavelet bases, whose mother wavelets are related through the Hilbert transform (see Refs. 5,12,13 and 15).…”

Section: Introductionmentioning

confidence: 99%

On Hilbert Transform of Gabor and Wilson Systems

Jarrah

Panwar

2014

Int. J. Wavelets Multiresolut Inf. Process.

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Given that the Gabor system {E mb Tnag} m,n∈Z is a Gabor frame for L 2 (R), a sufficient condition is obtained for the Gabor system {E mb TnaHg} m,n∈Z to be a Gabor frame, where Hg denotes the Hilbert transform of g ∈ L 2 (R). It is proved that the Hilbert transform operator and the frame operator for the Gabor Bessel sequence {E mb Tnag} m,n∈Z commute with each other under certain conditions. Also, a sufficient condition is obtained for the Wilson system {ψ k,n Hg} k∈Z n∈N 0 to be a Wilson frame given that {ψ k,n g} k∈Z n∈N 0 is a Wilson frame. Finally, we obtain conditions under which the Hilbert transform operator and the frame operator for the Wilson Bessel sequence {ψ k,n g} k∈Z n∈N 0 commute with each other.Example 3.1. Consider the function g = cos( 2πx 3 )χ [0, 3 4 ] ∈ L 2 (R), then it is easy to observe that {E mb T na g} m,n∈Z and {E mb T na Hg} m,n∈Z are both Gabor frames for 0 < a < 3 4 and 0 < b ≤ 4 3 . It is natural to raise the following general question.Problem: Let g ∈ L 2 (R) be such that {E mb T na g} m,n∈Z is a Gabor frame. Is {E mb T na Hg} m,n∈Z a Gabor frame?The answer to this problem is negative.Example 3.2. If g = χ [0,1] , then {E m T n g} m,n∈Z is a Gabor frame but {E m T n Hg} m,n∈Z is not a Gabor frame.

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On the Hilbert Transform of Wavelets

Cited by 24 publications

References 12 publications

Decay Properties of Riesz Transforms and Steerable Wavelets

Decay Properties of Riesz Transforms and Steerable Wavelets

Mathematical frameworks to assess the dynamics of cerebral autoregulation

On Hilbert Transform of Gabor and Wilson Systems

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