Some remarks on “On the windowed Fourier transform and wavelet transform of almost periodic functions,” by J.R. Partington and B. Ünalmı

J.R. Partington and B. Ünalmı consider in their 2001 paper [J.R. Partington, B. Ünalmı , Appl. Comput. Harmon. Anal. 10 (1) (2001) 45–60] the windowed Fourier transform and wavelet transform as tools for analyzing almost periodic signals. They establish Parseval-type identities and consider discretized versions of these transforms in order to construct generalized frame decompositions. We have found a gap in the construction of generalized frames in the windowed Fourier transform case; we comment on this gap and give an alternative proof. As for the wavelet transform case, in [J.R. Partington, B. Ünalmı , Appl. Comput. Harmon. Anal. 10 (1) (2001) 45–60] the generalized frame decomposition is done only for the simplest wavelet, the Haar wavelet; we show how to construct generalized frame decompositions for a wide family of wavelets.     

On the construction of wavelets on a bounded interval

This paper presents a general approach to a multiresolution analysis and wavelet spaces on the interval [–1, 1]. Our method is based on the Chebyshev transform, corresponding shifts and the discrete cosine transformation (DCT). For the wavelet analysis of given functions, efficient decomposition and reconstruction algorithms are proposed using fast DCT-algorithms. As examples for scaling functions and wavelets, polynomials and transformed splines are considered.     

Time–frequency analysis of bivariate signals

Many phenomena are described by bivariate signals or bidimensional vectors in applications ranging from radar to EEG, optics and oceanography. We show that an adequate quaternion Fourier transform permits to build relevant time–frequency representations of bivariate signals that naturally identify geometrical or polarization properties. First, a bivariate counterpart of the usual analytic signal of real signals is introduced, called the quaternion embedding of bivariate signals. Then two fundamental theorems ensure that a quaternion short term Fourier transform and a quaternion continuous wavelet transform are well defined and obey desirable properties such as conservation laws and reconstruction formulas. The resulting spectrograms and scalograms provide meaningful representations of both the time–frequency and geometrical/polarization content of the signal. Moreover the numerical implementation remains simply based on the use of FFT. A toolbox is available for reproducibility. Synthetic and real-world examples illustrate the relevance and efficiency of the proposed approach.     

Quantum -modules and generalized mirror transformations

In the previous paper [Hiroshi Iritani, Quantum D-modules and equivariant Floer theory for free loop spaces, Math. Z. 252 (3) (2006) 577–622], the author defined equivariant Floer cohomology for a complete intersection in a toric variety and showed that it is isomorphic to the small quantum D-module after a mirror transformation when the first Chern class c₁(M) of the tangent bundle is nef. In this paper, even when c₁(M) is not nef, we show that the equivariant Floer cohomology reconstructs the big quantum D-module under certain conditions on the ambient toric variety. The proof is based on a mirror theorem of Coates and Givental [T. Coates, A.B. Givental, Quantum Riemann — Roch, Lefschetz and Serre, Ann. of Math. (2) 165 (1) (2007) 15–53]. The reconstruction procedure here gives a generalized mirror transformation first observed by Jinzenji in low degrees [Masao Jinzenji, On the quantum cohomology rings of general type projective hypersurfaces and generalized mirror transformation, Internat. J. Modern Phys. A 15 (11) (2000) 1557–1595; Masao Jinzenji, Co-ordinate change of Gauss–Manin system and generalized mirror transformation, Internat. J. Modern Phys. A 20 (10) (2005) 2131–2156].     

Fractional and continuous wavelet transforms for the simultaneous spectral analysis of a binary mixture system

A combined application of the fractional wavelet transform-continuous wavelet transform to the quantitative resolution of the overlapping signals of olmesartan modoxomil and hydrochlorothiazide in a binary mixture was presented. In this work the recorded absorption signals of two compounds in the range 4–12 μg/mL olmesartan and 2–10 μg/mL hydrochlorothiazide were processed by the fractional transform approach and with the continuous wavelet transform, respectively. The proposed combined complex signal analysis method was validated by analyzing the synthetic binary mixtures of the related compound. By analyzing the statistical parameters we conclude that the new approach is suitable to be used for the quality control of the commercial samples of compounds.     

Plancherel Inversion as Unified Approach to Wavelet Transforms and Wigner Functions

We demonstrate that the Plancherel transform for Type-I groups provides one with a natural, unified perspective for the generalized continuous wavelet transform, on the one hand, and for a class of Wigner functions, on the other. We first prove that a Plancherel inversion formula, well known for Bruhat functions on the group, holds for a much larger class of functions. This result allows us to view the wavelet transform as essentially the inverse Plancherel transform. The wavelet transform of a signal is an L²-function on an appropriately chosen group while the Wigner function is defined on a coadjoint orbit of the group and serves as an alternative characterization of the signal, which is often used in practical applications. The Plancherel transform maps L²-functions on a group unitarily to fields of Hilbert-Schmidt operators, indexed by unitary irreducible representations of the group. The wavelet transform can essentially be looked upon as a restricted inverse Plancherel transform, while Wigner functions are modified Fourier transforms of inverse Plancherel transforms, usually restricted to a subset of the unitary dual of the group. Some known results on both Wigner functions and wavelet transforms, appearing in the literature from very different perspectives, are naturally unified within our approach. Explicit computations on a number of groups illustrate the theory. Communicated by Gian Michele Graf submitted 05/06/01, accepted: 19/09/02     

正交小波变换的向量空间算法

本文揭示了一个事实，小波不仅可构成Ｌ２空间中的正交基，小波分解与重构滤波还可产生Ｎ维空间中的正交基．在本文提出修改的小波变换算法之下，Ｎ点信号的小波变换等价于Ｎ维空间中的正交变换．用该算法进行信号或图象压缩，无需对信号或图象进行周期延拓，可严格地在Ｎ维空间中进行．     

Complex Wavelets for Shift Invariant Analysis and Filtering of Signals

This paper describes a form of discrete wavelet transform, which generates complex coefficients by using a dual tree of wavelet filters to obtain their real and imaginary parts. This introduces limited redundancy (2^m:1 for m-dimensional signals) and allows the transform to provide approximate shift invariance and directionally selective filters (properties lacking in the traditional wavelet transform) while preserving the usual properties of perfect reconstruction and computational efficiency with good well-balanced frequency responses. Here we analyze why the new transform can be designed to be shift invariant and describe how to estimate the accuracy of this approximation and design suitable filters to achieve this. We discuss two different variants of the new transform, based on odd/even and quarter-sample shift (Q-shift) filters, respectively. We then describe briefly how the dual tree may be extended for images and other multi-dimensional signals, and finally summarize a range of applications of the transform that take advantage of its unique properties.     

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].     

Unique reconstruction of band-limited signals by a Mallat-Zhong wavelet transform algorithm

We show that uniqueness and existence for signal reconstruction from multiscale edges in the Mallat and Zhong algorithm become possible if we restrict our signals to Paley-Wiener space, band-limit our wavelets, and irregularly sample at the wavelet transform (absolute) maxima—the edges—while possibly including (enough) extra points at each level. We do this in a setting that closely resembles the numerical analysis setting of Mallat and Zhong and that seems to capture something of the essence of their (practical) reconstruction method. Our work builds on a uniqueness result for reconstructing an L² signal from irregular sampling of its wavelet transform of Grochenig and the related work of Benedetto, Heller, Mallat, and Zhong. We show that the rate of convergence for this reconstruction algorithm is geometric and computable in advance. Finally, we consider the effect on the rate of convergence of not sampling enough local maxima.     

Unique Reconstruction of Band-limited Signals by a Mallat-Zhong Wavelet Transform Algorithm

We show that uniqueness and existence for signal reconstruction from multiscale edges in the Mallat and Zhong algorithm become possible if we restrict our signals to Paley-Wiener space, band-limit our wavelets, and irregularly sample at the wavelet transform (absolute) maxima—the edges—while possibly including (enough) extra points at each level. We do this in a setting that closely resembles the numerical analysis setting of Mallat and Zhong and that seems to capture something of the essence of their (practical) reconstruction method. Our work builds on a uniqueness result for reconstructing an L² signal from irregular sampling of its wavelet transform of Gröchenig and the related work of Benedetto, Heller, Mallat, and Zhong. We show that the rate of convergence for this reconstruction algorithm is geometric and computable in advance. Finally, we consider the effect on the rate of convergence of not sampling enough local maxima.     

The Theory of Multiresolution Analysis Frames and Applications to Filter Banks

The notion of a frame multiresolution analysis (FMRA) is formulated. An FMRA is a natural extension to affine frames of the classical notion of a multiresolution analysis (MRA). The associated theory of FMRAs is more complex than that of MRAs. A basic result of the theory is a characterization of frames of integer translates of a function φ in terms of the discontinuities and zero sets of a computable periodization of the Fourier transform of φ. There are subband coding filter banks associated with each FMRA. Mathematically, these filter banks can be used to construct new frames for finite energy signals. As with MRAs, the FMRA filter banks provide perfect reconstruction of all finite energy signals in any one of the successive approximation subspacesV_jdefining the FMRA. In contrast with MRAs, the perfect reconstruction filter bank associated with an FMRA can be narrow band. Because of this feature, in signal processing FMRA filter banks achieve quantization noise reduction simultaneously with reconstruction of a given narrow-band signal.     

Hyperbolic Wavelets and Multiresolution in H^{2}(\mathbb{T})

In signal processing and system identification for H²(\BbbT)H^{2}(\Bbb{T}) and H²(\BbbD)H^{2}(\Bbb{D}) the traditional trigonometric bases and trigonometric Fourier transform are replaced by the more efficient rational orthogonal bases like the discrete Laguerre, Kautz and Malmquist-Takenaka systems and the associated transforms. These bases are constructed from rational Blaschke functions, which form a group with respect to function composition that is isomorphic to the Blaschke group, respectively to the hyperbolic matrix group. Consequently, the background theory uses tools from non-commutative harmonic analysis over groups and the generalization of Fourier transform uses concepts from the theory of the voice transform. The successful application of rational orthogonal bases needs a priori knowledge of the poles of the transfer function that may cause a drawback of the method. In this paper we give a set of poles and using them we will generate a multiresolution in H²(\BbbT)H^{2}(\Bbb{T}) and H²(\BbbD)H^{2}(\Bbb{D}). The construction is an analogy with the discrete affine wavelets, and in fact is the discretization of the continuous voice transform generated by a representation of the Blaschke group over the space H²(\BbbT)H^{2}(\Bbb{T}). The constructed discretization scheme gives opportunity of practical realization of hyperbolic wavelet representation of signals belonging to H²(\BbbT)H^{2}(\Bbb{T}) and H²(\BbbD)H^{2}(\Bbb{D}) if we can measure their values on a given set of points inside the unit circle or on the unit circle. Convergence properties of the hyperbolic wavelet representation will be studied.     

Uniqueness and stable determination in the inverse Robin transmission problem with one electrostatic measurement

In this paper, we consider the inverse Robin transmission problem with one electrostatic measurement. We prove a uniqueness result for the simultaneous determination of the Robin parameter p, the conductivity k, and the subdomain D, when D is a ball. When D and k are fixed, we prove a uniqueness result and a directional Lipschitz stability estimate for the Robin parameter p. When p and k are fixed, we give an upper bound to the subdomain D. For the reconstruction purposes of the Robin parameter p, we set the inverse problem under an optimization form for a Kohn–Vogelius cost functional. We prove the existence and the stability of the optimization problem. Finally, we show some numerical experiments that agree with the theoretical considerations. Copyright © 2013 John Wiley & Sons, Ltd.     

基于Curvelet域高斯尺度混合模型的地震信号降噪方法

结合曲波变换和高斯尺度混合模型提出地震信号随机噪声压制方法.该方法首先运用曲波变换对含有随机噪声的地震信号进行分解,然后对各小波子带系数分别建立高斯尺度混合模型估计出原始地震信号所对应的小波系数,最后经曲波逆变换重构获得降噪处理后的地震信号.仿真地震信号和实际地震信号的实验结果均表明本文方法能够有效压制地震信号中的随机噪声干扰,较多地保留了有效信号.     

A note on the perturbation bound of the Drazin inverse

A perturbation bound for the Drazin inverse A^D with Ind(A+E)=1 has recently been developed. However, those upper bounds are not satisfied since it is not tight enough. In this paper, a sharper upper bounds for ||(A+E)^#−A^D|| with weaker conditions is derived. That new bound is also a generalization of a new general upper bound of the group inverse. We also derive a new expression of the Drazin inverse (A+E)^D with Ind(A+E)>1 and the corresponding upper bound of ||(A+E)^D−A^D|| in a special case. Numerical examples are given to illustrate the sharpness of the new bounds.     

On the limit of sampled signal extrapolation with a wavelet signal model ∗

The extrapolation of sampled signals from a given interval using a wavelet model with various sampling rates is examined in this research. We present sufficient conditions on signals and wavelet bases so that the discrete-time extrapolated signal converges to its continuous-time counterpart when the sampling rate goes to infinity. Thus, this work provides a practical procedure to implement continuous-time signal extrapolation, which is important in wideband radar and sonar signal processing, with a discrete one via carefully choosing the sampling rate and the wavelet basis. A numerical example is given to illustrate our theoretical result.     

The current state of the problem of interaction of acoustic beams with obstacles in a deformable medium

We analyze the state of the problem of the formation of radiated and scattered acoustic beams in application to the development of a methodology for studying the information aspects of hydroacoustics and nondestructive control. We discuss the problems of the selective generation of characteristic vibrations of elastic objects in a deformable medium using sharply directed acoustic impulses. We study the problem of posing and methods of solving a certain class of inverse problems of scattering theory.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 56–64.In conclusion we note the papers [7, 74, 100] connected with the traditional method of solving inverse problems-the selection method.     

Estimation of dimension functions of band-limited wavelets

The dimension function D_ψ of a band-limited wavelet ψ is bounded by n if its Fourier transform is supported in [−(2ⁿ⁺²/3)π,(2ⁿ⁺²/3)π]. For each and for each , 0<<δ=δ(n), we construct a wavelet ψ with supp

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such that D_ψ>n on a set of positive measure, which proves that [−(2ⁿ⁺²/3)π,(2ⁿ⁺²/3)π] is the largest symmetric interval for estimating the dimension function by n. This construction also provides a family of (uncountably many) wavelet sets each consisting of infinite number of intervals.     

一个二进小波变换像空间的再生核函数

1 引言 小波分析是结合泛函分析、应用数学、逼近论、调和分析、广义函数论等数学知识的结晶,具有深刻的理论意义和广泛的应用范围,被称为”数学显微镜”.基于其多分辨分析的特点以及在时、频两域都具有表征信号局部特征的功能,应用它可以解决许多Fourier变换不能解决的难题,为工程应用提供了一种新的、更有效的分析工具[1],由...