Single Wavelets Research Articles

Time-frequency super-resolution with superlets

Due to the Heisenberg–Gabor uncertainty principle, finite oscillation transients are difficult to localize simultaneously in both time and frequency. Classical estimators, like the short-time Fourier transform or the continuous-wavelet transform optimize either temporal or frequency resolution, or find a suboptimal tradeoff. Here, we introduce a spectral estimator enabling time-frequency super-resolution, called superlet, that uses sets of wavelets with increasingly constrained bandwidth. These are combined geometrically in order to maintain the good temporal resolution of single wavelets and gain frequency resolution in upper bands. The normalization of wavelets in the set facilitates exploration of data with scale-free, fractal nature, containing oscillation packets that are self-similar across frequencies. Superlets perform well on synthetic data and brain signals recorded in humans and rodents, resolving high frequency bursts with excellent precision. Importantly, they can reveal fast transient oscillation events in single trials that may be hidden in the averaged time-frequency spectrum by other methods.

Comparison between Orthogonal and Bi-Orthogonal Wavelets

Theoretical aspects and analysis of wavelets have applications in mathematical modeling, artificial neural networks, digital signal processing, and image processing and numerical methods. The term orthogonal deals with the mathematical part which covers a wide area of digital signal processing and image processing. An orthogonal wavelet generates the wavelet whose nature is orthogonal. This means an inverse or transpose wavelet transform is nothing but the adjoint of a wavelet transform. If this condition fails by missing orthogonality it may result in biorthogonal wavelets. Single scaling functions and single wavelets are generated but orthogonal wavelet filter bank. A biorthogonal wavelet associated with the wavelet transformation is invertible. There is no need that if it is invertible so it should be orthogonal. The biorthogonal wavelet allows maximum freedom in the case of designing the orthogonal wavelet. It also supports the construction of symmetric wavelet functions. In biorthogonal wavelets, as the name indicates, two scaling factors or functions are responsible for the generation of the various multi-resolutions on the basis of different wavelets. For the image and signal reconstruction purpose we need that wavelet. We get a better result in the presence of biorthogonal wavelets. In the present work, we analyzed the performance of orthogonal and biorthogonal wavelet filters for image processing. We test the image and observed that the filter coefficient and image quality for the orthogonal and biorthogonal wavelet. On the basis of performance analysis it is concluded that biorthogonal wavelets are better than orthogonal wavelets.

The simulation of far-field wavelets using frequency-domain air-gun array near-field wavelets

Air-gun arrays are used in marine-seismic exploration. Far-field wavelets in subsurface media represent the stacking of single air-gun ideal wavelets. We derived single air-gun ideal wavelets using near-field wavelets recorded from near-field geophones and then synthesized them into far-field wavelets. This is critical for processing wavelets in marineseismic exploration. For this purpose, several algorithms are currently used to decompose and synthesize wavelets in the time domain. If the traveltime of single air-gun wavelets is not an integral multiple of the sampling interval, the complex and error-prone resampling of the seismic signals using the time-domain method is necessary. Based on the relation between the frequency-domain phase and the time-domain time delay, we propose a method that first transforms the real near-field wavelet to the frequency domain via Fourier transforms; then, it decomposes it and composes the wavelet spectrum in the frequency domain, and then back transforms it to the time domain. Thus, the resampling problem is avoided and single air-gun wavelets and far-field wavelets can be reliably derived. The effect of ghost reflections is also considered, while decomposing the wavelet and removing the ghost reflections. Modeling and real data processing were used to demonstrate the feasibility of the proposed method.

Discrimination of Low-Frequency Tones Employs Temporal Fine Structure

An auditory neuron can preserve the temporal fine structure of a low-frequency tone by phase-locking its response to the stimulus. Apart from sound localization, however, much about the role of this temporal information for signal processing in the brain remains unknown. Through psychoacoustic studies we provide direct evidence that humans employ temporal fine structure to discriminate between frequencies. To this end we construct tones that are based on a single frequency but in which, through the concatenation of wavelets, the phase changes randomly every few cycles. We then test the frequency discrimination of these phase-changing tones, of control tones without phase changes, and of short tones that consist of a single wavelet. For carrier frequencies below a few kilohertz we find that phase changes systematically worsen frequency discrimination. No such effect appears for higher carrier frequencies at which temporal information is not available in the central auditory system.

Power Quality Events Classification Using MWT and MLP

The work presented uses multiwavelet because of its inherent property to resolve the signal better than all single wavelets. Multiwavelets are based on more than one scaling function. The proposed methodology utilizes an enhanced resolving capability of multiwavelet to recognize power system disturbances. The disturbance classification schema is performed with multiwavelet neural network (MWNN). It performs a feature extraction and a classification algorithm composed of a multiwavelet feature extractor based on norm entropy and a classifier based on a multi-layer perceptron. The performance of this classifier is evaluated by using total 1000 PQ disturbance signals which are generated the based model. The classification performance of different PQ disturbance using proposed algorithm is tested. The rate of average correct classification is about 99.65% for the different PQ disturbance signals and noisy disturbances.

PQ Event Classification Using Fuzzy Classifier

The work presented here uses multiwavelet because of its inherent property to resolve the signal better than all single wavelets. Multiwavelets are based on more than one scaling function. The proposed methodology utilizes an enhanced resolving capability of multiwavelet to recognize power quality events. PQ events classification scheme is performed using multiwavelet transform for feature extraction and fuzzy classifier for classification. In proposed algorithm,statistical features (.i.e. mean, standard deviation, variation etc.) and energy of the signal at different decomposition levels have been considered as feature vectors. The performance of fuzzy classifier has been evaluated by using total 1000 PQ disturbance signals which are generated using the based model. The classification performance of different PQ events using proposed algorithm has been tested. The rate of average correct classification is about 99.95% for the different PQ disturbance signals and noisy disturbances.

Multiwavelet transform based classification of PQ events

SUMMARYWavelets have been used extensively in the recent past. The work presented uses multiwavelet because of its inherent property to resolve the signal better than all single wavelets. Multiwavelets are based on more than one scaling function. The resolution is better compared to Daubechies (D4) and hence the event can be detected from lesser number of samples as compared to Daubechies (D4). The proposed methodology utilizes an enhanced resolving capability of multiwavelet to recognize power system disturbances. Dempster–Shafer (DS), an intelligent classifier, has been implemented and tested for various PQ events. Two sub‐classifiers, heuristic classifier and statistical classifier (χ2 distribution), have been used to support and strengthen the structural identification and temporal distribution factor of DS classifier. Results on various PQ events, such as voltage sag, voltage swell, outage, interruption, impulsive‐transient (IT), oscillatory‐transient (OT), noise, and notching, show that multiwavelets can detect and classify different PQ events efficiently and consistently. Copyright © 2011 John Wiley & Sons, Ltd.

A COMPACTLY SUPPORTED, SYMMETRICAL AND QUASI-ORTHOGONAL WAVELET

Based on the wavelet theory and optimization method, a class of single wavelets with compact support, symmetry and quasi-orthogonality are designed and constructed. Some mathematical properties of the wavelets, such as orthogonality, linear phase property and vanishing moments and so on, are studied. A speech compression experiment is implemented in order to investigate the performance of signal reconstruction and speech compression for the proposed wavelets. Comparison with some conventional wavelets shows that the proposed wavelets have a very good performance of signal reconstruction and speech compression.

Intrinsic s-elementary Parseval frame multiwavelets in [formula omitted

A countable family { x j } , j ∈ J , in a separable Hilbert space H, is a Parseval frame for H if ‖ f ‖ 2 = ∑ j ∈ J | 〈 f , x j 〉 | 2 holds for all f ∈ H . In the case that H = L 2 ( R d ) and the affine system { D A j T k ψ i | j ∈ Z , k ∈ Z d , 1 ⩽ i ⩽ n } obtained from a finite subset Ψ = { ψ 1 , ψ 2 , … , ψ n } of L 2 ( R d ) is a Parseval frame, Ψ is called an A-dilation Parseval frame multiwavelet (of length n). Here A stands for a d × d expansive matrix, and T, D A are the translation and A-dilation unitary operators acting on L 2 ( R d ) defined by ( T ℓ f ) ( t ) = f ( t − ℓ ) , ( D A f ) ( t ) = | det A | 1 2 f ( A t ) , ∀ f ∈ L 2 ( R d ) , ℓ ∈ Z d , t ∈ R d . In the special case that there exist disjoint measurable sets { E 1 , E 2 , … , E n } such that ψ ˆ i = 1 ( 2 π ) d / 2 χ E i for each i, Ψ is called an A-dilation s-elementary Parseval frame multiwavelet. A measurable set E is called a frame multiwavelet set of multiplicity m (under A-dilation) if E can be written as a disjoint union of measurable sets { E 1 , E 2 , … , E m } such that ψ ˆ i = 1 ( 2 π ) d / 2 χ E i defines a Parseval frame multiwavelet Ψ = { ψ 1 , ψ 2 , … , ψ m } , and that this cannot be done for any integer less than m. An A-dilation s-elementary Parseval frame multiwavelet with length m that is defined on a frame multiwavelet set of multiplicity m is said to be intrinsic. It is known that single A-dilation wavelets exist in L 2 ( R d ) for any expansive matrix A. In this paper, we show that for any d × d expansive matrix A and any given m ∈ N , the family of intrinsic A-dilation s-elementary Parseval frame multiwavelet with length m is not empty, and is path-connected under the norm topology of ( L 2 ( R d ) ) m . The same result holds for the family of all intrinsic A-dilation s-elementary Parseval frame multiwavelets of length m.

Tight frames and geometric properties of wavelet sets

A construction for providing single dyadic orthonormal wavelets in Euclidean space ℝd is given. It is called the general neighborhood mapping construction. The fact that one wavelet is sufficient to generate an orthonormal basis for L2(ℝd) is the critical issue. The validity of the construction is proved, and the construction is implemented computationally to provide a host of examples illustrating various geometrical properties of such wavelets in the spectral domain. Because of the inherent complexity of these single orthonormal wavelets, the method is applied to the construction of single dyadic tight frame wavelets, and these tight frame wavelets can be surprisingly simple in nature. The structure of the spectral domains of the wavelets arising from the general neighborhood mapping construction raises a basic geometrical question. There is also the question of whether or not the general neighborhood mapping construction gives rise to all single dyadic orthonormal wavelets. Results are proved giving partial answers to both of these questions.

AN EQUIVALENCE RELATION ON WAVELETS IN HIGHER DIMENSIONS

The action of translation operators on wavelet subspaces in higher dimensions is investigated. This action defines an equivalence relation on the set of single wavelets of $L^2(\mathbb R^n)$ associated with an arbitrary dilation matrix. The corresponding equivalence classes are characterized in terms of the support of the Fourier transform of the wavelets. Further, examples of wavelets in each of these classes are constructed. This construction shows the existence of wavelets for which the associated wavelet subspaces are invariant under various groups of translation operators.

Multiwavelets denoising using neighboring coefficients

Multiwavelets give better results than single wavelets for signal denoising. We study multiwavelet thresholding by incorporating neighboring coefficients. Experimental results show that this approach is better than the conventional approach, which only uses the term-by-term multiwavelet denoising. Also, it outperforms neighbor single wavelet denoising for some standard test signals and real-life images. This is an extension to Cai and Silverman's (see Sankhya: Ind. J. Stat. B, pt.2, vol.63, p.127-148, 2001) work.

Frames and Single Wavelets for Unitary Groups

AbstractWe consider a unitary representation of a discrete countable abelian group on a separable Hilbert space which is associated to a cyclic generalized frame multiresolution analysis. We extend Robertson’s theorem to apply to frames generated by the action of the group. Within this setup we use Stone’s theorem and the theory of projection valued measures to analyze wandering frame collections. This yields a functional analytic method of constructing a wavelet from a generalized frame multiresolution analysis in terms of the frame scaling vectors. We then explicitly apply our results to the action of the integers given by translations on L2(ℝ).

Averaging improves the quality of impedance stroke volume measurements during the head up tilt test.

To assess the improvement in quality following averaging data from two or more tilts in the stroke volume (SV) response curve during a head-up tilt test. Ten adult male subjects were tilted head-up to 55 degrees four times. They were kept head-up for 3-minutes with intervening rests of 3-minutes. Impedance measurements were recorded by the RheoCardioMonitor every 10-seconds. The percentage changes in impedance variables with tilting and coefficients of variability (CV) were calculated. The SV data were divided into four single tilt response curves, which were used to assess the variability of the data when these tilts were analyzed separately, in pairs or in triplets. Confidence intervals reflecting this variability were estimated. Head-up tilting resulted in a mean (SD) decrease in SV of 31 (11)%. Significant variability existed between SV readings (CV = 8 (2)%). Averaging improved the resolution of the wavelets and in most subjects (7 out of 10) allowed dynamic and static phases of the response to be identified. Upper confidence intervals (mean (range)) were reduced from 15 (10-23)% for single wavelets to 8 (3-10)% for pairs to 5 (3-8)% for triplets. Impedance measurements can be very variable, making the assessment of SV changes during a head-up tilt test difficult. By averaging the data from several tilts one can improved the quality of the SV wavelet sufficiently to identify important postural changes.

The construction of single wavelets in d-dimensions

Sets K in d-dimensional Euclidean space are constructed with the property that the inverse Fourier transform of the characteristic function 1 K is a single dyadic orthonormal wavelet. The construction is characterized by its generality in the procedure, by its computational implementation, and by its simplicity. The general case in which the inverse Fourier transforms of the characteristic functions 1K 1, ..., 1K L are a family of orthonormal wavelets is treated in [27].

Microlocal filtering with multiwavelets

Hyperfunctions in R n are intuitively considered as sums of boundary values of holomorphic functions defined in infinitesimal wedges in C n . Orthonormal multiwavelets, which are a generalization of orthonormal single wavelets, generate a multiresolution analysis by means of several scaling functions. Microlocal analysis is briefly reviewed and a multiwavelet system adapted to microlocal filtering is proposed. A rough estimate of the microlocal content of functions or signals is obtained from their multiwavelet expansions. A fast algorithm for multiwavelet microlocal filtering is presented and several numerical examples are considered.

On interpolation families of wavelet sets

The question of which groups are isomorphic to groups of interpolation maps for interpolation families of wavelet sets was raised by Dai and Larson. In this article it is shown that any finite group is isomorphic to a group of interpolation maps for some interpolation family of wavelet sets. A dyadic orthonormal (or orthogonal) wavelet is a function 4 E L2 (R) (Lebesgue measure), with the property that the set {20'(2't 1): n, l E 2} is an orthonormal basis for L2 (R). For certain measurable sets E C R the normalized characteristic function 72==7rXE is the Fourier transform of such a wavelet. There are several characterizations of such sets. In [DL] they are called wavelet sets, and the corresponding wavelet is called an s-elementary wavelet. In [FW],[HWW1],[HWW2] they are the support sets of MFS (minimal frequency supported) wavelets. In [DL], a method of operator-theoretic interpolation between certain special pairs of wavelets, and between single wavelets and special families of them is developed. The interpolation pairs of wavelet sets are extensively discussed and the method yields a new construction of a class of wavelets due to Meyer. An interpolation pair is the simplest case of an interpolation family (see definitions below). Affiliated to an interpolation family is a group (under composition) of measure preserving transformations of R called the interpolation maps. Problem E in [DL] asked for a characterization of all groups which are isomorphic to a group of interpolation maps for some interpolation family of wavelet sets. They showed, by example, that the cyclic groups 22 and 23 of order 2 and 3 could be realized in this way, but they proved no cases of other groups. The purpose of this paper is to prove that every finite group can be realized in this way. Before introducing the characterization of wavelet sets, we will need some more terminology. In [DL], two measurable sets E, F are called translation congruent modulo 2wr if there is a measurable bijection 0: E -> F such that s(s) s is an integral multiple of 2wr for each s E E. Analogously, two measurable sets G, H are called dilation congruent modulo 2 if there is a measurable bijection T: G -> H such that for each s E G there is an integer n depending on s, such that T (s) = 2s. We note in passing that if Ei and Fi are measurable subsets of R such that Ei and Received by the editors July 7, 1998 and, in revised form, November 24, 1998. 2000 Mathematics Subject Classification. Primary 42C40.

Single wavelets in n-dimensions

Under very minimal regularity assumptions, it can be shown that 2n−1 functions are needed to generate an orthonormal wavelet basis for L2(ℝn). In a recent paper by Dai et al. it is shown, by abstract means, that there exist subsets K of ℝn such that the single function ψ, defined by $$\hat \psi = \chi K$$ , is an orthonormal wavelet for L2(ℝn). Here we provide methods for construucting explicit examples of these sets. Moreover, we demonstrate that these wavelets do not behave like their one-dimensional couterparts.

Translation-invariant denoising using multiwavelets

Translation invariant (TI) single wavelet denoising was developed by Coifman and Donoho (1994), and they show that TI is better than non-TI single wavelet denoising. On the other other hand, Strela et al. (1994) have found that non-TI multiwavelet denoising gives better results than non-TI single wavelets. We extend Coifman and Donoho's TI single wavelet denoising scheme to multiwavelets. Experimental results show that TI multiwavelet denoising is better than the single case for soft thresholding.

Orthogonal multiwavelets with vanishing moments

A scaling function is the solution to a dilation equation Φ(<i>t</i>) = Σ<i>c</i>_<i>k</i>Φ(2<i>t</i> - <i>k</i>), in which the coefficients come from a low-pass filter. The coefficients in the wavelet W(<i>t</i>) = Σ<i>d_k</i>Φ(2<i>t</i> - <i>k</i>) come from a high-pass filter. When these coefficients are matrices, Φ and <b>W</b> are vectors: there are two or more scaling functions and an equal number of wavelets. By dilation and translation of the wavelets, we have an orthogonal basis <i>W_ijk</i> = <i>W_i</i>(2^<i>j</i><i>t</i> - <i>k</i>) for all functions of finite energy. These "multiwavelets" open new possibilities. They can be shorter, with more vanishing moments, than single wavelets. They can be symmetric, which is impossible for scalar wavelets (except for Haar's). We determine the conditions to impose on the matrix coefficients <i>c_k</i> in the design of multiwavelets, and we construct a new pair of piecewise linear orthogonal wavelets with two vanishing moments.