Orthogonal Matched Wavelets with Vanishing Moments: A Sparsity Design Approach

Karel, Joël; Peeters, Ralf

doi:10.1007/s00034-017-0716-1

Cited by 10 publications

(13 citation statements)

References 28 publications

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“…We limited our approach to orthogonal wavelets to ensure that a Parseval’s relation exists in ( 4 ), i.e., ensuring equivalence between the ℓ 2 fit criterion in the wavelet domain and in the time domain. Within this setting, designed orthogonal wavelets could be used to tailor this method to specific situations [ 17 , 20 ]. This creates more freedom than using a fixed (semi-)physiological model that cannot be adopted to pathological situations, as in [ 42 ].…”

Section: Discussionmentioning

confidence: 99%

“…In case that the stationary wavelet transform is used, the matrix W is no longer orthogonal. However, if the used wavelet filter is orthogonal, the energy is weighted by a factor 2 j with increasing j as the scale becomes coarser [ 19 , 20 ].…”

Section: Methodsmentioning

confidence: 99%

“…Concerning the choice of a suitable wavelet, electrograms can be sparsely represented with the orthogonal Daubechies-2 wavelet (2 vanishing moments, filter length 4) [ 19 ]. If desired, one may also design a more dedicated orthogonal wavelet by using the approach of [ 17 , 20 ]. The Online Supplement (Online Figure 3 ) illustrates this.…”

Section: Methodsmentioning

confidence: 99%

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Wavelet-promoted sparsity for non-invasive reconstruction of electrical activity of the heart

Cluitmans

Karel

Bonizzi

et al. 2018

Med Biol Eng Comput

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We investigated a novel sparsity-based regularization method in the wavelet domain of the inverse problem of electrocardiography that aims at preserving the spatiotemporal characteristics of heart-surface potentials. In three normal, anesthetized dogs, electrodes were implanted around the epicardium and body-surface electrodes were attached to the torso. Potential recordings were obtained simultaneously on the body surface and on the epicardium. A CT scan was used to digitize a homogeneous geometry which consisted of the body-surface electrodes and the epicardial surface. A novel multitask elastic-net-based method was introduced to regularize the ill-posed inverse problem. The method simultaneously pursues a sparse wavelet representation in time-frequency and exploits correlations in space. Performance was assessed in terms of quality of reconstructed epicardial potentials, estimated activation and recovery time, and estimated locations of pacing, and compared with performance of Tikhonov zeroth-order regularization. Results in the wavelet domain obtained higher sparsity than those in the time domain. Epicardial potentials were non-invasively reconstructed with higher accuracy than with Tikhonov zeroth-order regularization (p < 0.05), and recovery times were improved (p < 0.05). No significant improvement was found in terms of activation times and localization of origin of pacing. Next to improved estimation of recovery isochrones, which is important when assessing substrate for cardiac arrhythmias, this novel technique opens potentially powerful opportunities for clinical application, by allowing to choose wavelet bases that are optimized for specific clinical questions. Graphical AbstractThe inverse problem of electrocardiography is to reconstruct heart-surface potentials from recorded bodysurface electrocardiograms (ECGs) and a torso-heart geometry. However, it is ill-posed and solving it requires additional constraints for regularization. We introduce a regularization method that simultaneously pursues a sparse wavelet representation in time-frequency and exploits correlations in space. Our approach reconstructs epicardial (heart-surface) potentials with higher accuracy than common methods. It also improves the reconstruction of recovery isochrones, which is important when assessing substrate for cardiac arrhythmias. This novel technique opens potentially powerful opportunities for clinical application, by allowing to choose wavelet bases that are optimized for specific clinical questions. Electronic supplementary materialThe online version of this article (10.1007/s11517-018-1831-2) contains supplementary material, which is available to authorized users.

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Section: Discussionmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

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Wavelet-promoted sparsity for non-invasive reconstruction of electrical activity of the heart

Cluitmans

Karel

Bonizzi

et al. 2018

Med Biol Eng Comput

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“…In [8], the problem of designing the compactly supported orthogonal wavelets for finitelength signals was firstly addressed. In [9], a novel approach was presented to design orthogonal wavelets matched to a signal with compact support and vanishing moments.…”

Section: Introductionmentioning

confidence: 99%

A study on optimally constructed compactly supported orthogonal wavelet filters

Fan

Pan

et al. 2022

COMSIS J

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Compactly supported orthogonal wavelet filters are extensively applied to the analysis and description of abrupt signals in fields such as multimedia. Based on the application of an elementary method for compactly supported orthogonal wavelet filters and the construction of a system of nonlinear equations for filter coefficients, we design compactly supported orthogonal wavelet filters, in which both the scaling and wavelet functions have many vanishing moments, by approximately solving the system of nonlinear equations. However, when solving such a system about filter coefficients of compactly supported wavelets, the most widely used method, the Newton Iteration method, cannot converge to the solution if the selected initial value is not near the exact solution. For such, we propose optimization algorithms for the Gauss-Newton type method that expand the selection range of initial values. The proposed method is optimal and promising when compared to other works, by analyzing the experimental results obtained in terms of accuracy, iteration times, solution speed, and complexity.

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“…In [5] a parameterization was developed for orthogonal scalar wavelets, which were matched to a prototype signal. This approach was further expanded in [6,7] by using a parameterization based on lossless systems. In that parameterization, the polyphase filter associated with the orthogonal wavelet is recursively constructed as the transfer matrix of a lossless system using Schur interpolation theory [8] with rotation matrices and elementary delay operators.…”

Section: Introductionmentioning

confidence: 99%

The Design of Matched Balanced Orthogonal Multiwavelets

Karel

Steenkiste

Peeters

2022

Front. Appl. Math. Stat.

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The theory of orthogonal multiwavelets offers enhanced flexibility for signal processing applications and analysis by employing multiple waveforms simultaneously, rather than a single one. When implementing them with polyphase filter banks, it has been recognized that balanced vanishing moments are needed to prevent undesirable artifacts to occur, which otherwise compromise the interpretation and usefulness of the multiwavelet analysis. In the literature, several such balanced orthogonal multiwavelets have been constructed and published; but however useful, their choice is still limited. In this work we present a full parameterization of the space of all orthogonal multiwavelets with two balanced vanishing moments (of orders 0 and 1), for arbitrary given multiplicity and degree of the polyphase filter. This allows one to search for matching multiwavelets for a given application, by optimizing a suitable design criterion. We present such a criterion, which is sparsity-based and useful for detection purposes, which we illustrate with an example from electrocardiographic signal analysis. We also present explicit conditions to build in a third balanced vanishing moment (of order 2), which can be used as a constraint together with the earlier parameterization. This is demonstrated by constructing a balanced orthogonal multiwavelet of multiplicity three, having three balanced vanishing moments, but this approach can easily be employed for arbitrary multiplicity.

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Orthogonal Matched Wavelets with Vanishing Moments: A Sparsity Design Approach

Cited by 10 publications

References 28 publications

Wavelet-promoted sparsity for non-invasive reconstruction of electrical activity of the heart

Wavelet-promoted sparsity for non-invasive reconstruction of electrical activity of the heart

A study on optimally constructed compactly supported orthogonal wavelet filters

The Design of Matched Balanced Orthogonal Multiwavelets

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