The power cepstrum applied to multi-peaked wavelets

The use of the power cepstrum method for echo detection is discussed with respect to signals wherein the basic wavelet itself may have an echo-like structure.     

Semi-transparent slab and kernel substitution method: review and determination of the optimal coefficients

The kernel substitution method permits to find an analytical expression of the temperature within a semi-transparent medium. But a question still remains: which coefficients must be used in the substitute kernel? This paper presents a synthesis to answer this question. A study presents several methods on a test case to determine the coefficients. The values obtained are then compared with those commonly used in the literature.     

The use of ultrasonic guided waves and wavelets analysis in pipe inspection

One of the main applications of guided waves is for pipe inspection, since its one-dimensional geometry allows inspecting long distances in a short time. As with most configurations (frequency, thickness) ultrasonic waves with many modes of propagation are generated, recent research focuses mostly on the generation and reception of specific modes, mainly by means of sophisticated arrays of transducers. In this paper this problem is addressed with a different approach, by processing ultrasonic signals with a low signal/noise ratio acquired with a single transducer in an pulse–echo configuration. In order to improve the evaluation of results, frequency bandpass filters and wavelet analysis were tested. Results showed that even when very noisy signals are utilized, signal processing improve the signal/noise (S/N) ratio up to 12 dB approximately and enhance the analysis of the results, thus demonstrating its usefulness.     

Use of wavelets in potential scattering problems

Application of the Daubechies compact support wavelets to problems of nonrelativistic potential scattering described by integral Lippmann-Schwinger equation is discussed. Structure of wavelet representation of various physical operators is investigated. It is shown that for a special class of potentials wavelets enable sparse approximation of the kernel of the Lippmann-Schwinger equation. Constraints for such potentials are derived. This paper is dedicated to Prof. J. Bičák on the occasion of his 60th birthday.     

The optimal faucet

The production of small fluid droplets relies on an instability of solutions to the Young-Laplace equation. We ask whether smaller droplets can be produced by changing the shape of the nozzle. At a given critical pressure, the circular nozzle actually produces the largest droplet. The droplet volume can be decreased by up to 18% using a triangular nozzle with stretched corners.     

Tail wavelets in merger of binary compact objects

We present a model for tail wavelets, a phenomenon known as "echo" in the literature. The tail wavelet may appear in signal reconnaissances in the merger of binary compact objects, including black holes and neutron stars. We show that the dark matter surrounding the compact objects lead to a speculated tail wavelet following the main gravitational wave(GW). We demonstrate that the radiation pressure of the main wave is fully capable of pushing away the surrounding matter to some altitude, and splashing down of the matter excites the tail wavelet after ringing down of the main wave. We illustrate this concept in a simplified model, where numerical estimations are conducted on the specific distribution of dark matter outside the black hole horizon and the threshold values in accordance with observations. We study the full back reaction of the surrounding dark matter to the metric and find that the effect on to the tail wavelets is insignificant. We reveal the fine difference between the tail wavelets of a dressed and a bare black hole. We demonstrate that the tail wavelet can appear as a natural phenomenon in the frame of general relativity, without invoking modified gravities or quantum effects.     

Tunnel effect in the motion of deformable wavelets

In a previous paper it was shown that the tunnel effect does not occur in the motion of an undeformable wavelet which we obtained by solving the Schrödinger equation and interpreted as representing a single particle. In the present supplementary paper we solve the same Schrödinger equation while modifying one of the restrictive conditions assumed previously, and obtain wavelets which deform in force fields. Tunnel effects are seen in the motion of these deformable wavelets. An apparent inaccuracy pointed out mathematically in these solutions is shown to be insignificant under conditions which are feasible in the physical sense.     

The heat equation with singular coefficients

The small time asymptotics of the kernel ofe ^?tH is defined and derived for \(H = \frac{{d^2 }}{{dx^2 }} + \frac{\kappa }{{x^2 }}\) on ?¹. Lemmas on singular asymptotics in the sense of distributions are formulated and used. The results are applied to derive an index formula on ?¹.     

Wavelet transform for Fresnel-transformed mother wavelets

In this paper, we propose the so-called continuous Fresnel-wavelet combinatorial transform which means that the mother wavelet undergoes the Fresnel transformation. This motivation can let the mother-wavelet-state itself vary from |ψ> to F_r,s⁺ |ψ> , except for variation within the family of dilations and translations. The Parseval's equality, admissibility condition and inverse transform of this continuous Fresnel-wavelet combinatorial transform are analysed. By taking certain parameters and using the admissibility condition of this continuous Fresnel-wavelet combinatorial transform, we obtain some mother wavelets. A comparison between the newly found mother wavelets is presented.     

Abel inversion using Legendre wavelets expansion

A new method is presented for reconstruction of the radially distributed emissivity from the line-of-sight projected intensity. The method is based on approximating the projected intensity profile by Legendre wavelets. The coefficients of the approximation are computed using the inner product of Legendre wavelets and the intensity profile. The emissivity profile is then obtained by the combination of the functions related to Legendre wavelets. The method is more accurate and noise resistant than other methods when applied to experimental data, and there is no need for a complete noise filtering of the intensity data before applying the inversion.     

Wavelet transform for Fresnel-transformed mother wavelets

In this paper, we propose the so-called continuous Fresnel-wavelet combinatorial transform which means that the mother wavelet undergoes the Fresnel transformation. This motivation can let the mother-wavelet-state itself vary from |ψ> to F_r,s⁺ |ψ> , except for variation within the family of dilations and translations. The Parseval's equality, admissibility condition and inverse transform of this continuous Fresnel-wavelet combinatorial transform are analysed. By taking certain parameters and using the admissibility condition of this continuous Fresnel-wavelet combinatorial transform, we obtain some mother wavelets. A comparison between the newly found mother wavelets is presented.     

The vacuum-vacuum amplitude and Bogoliubov coefficients

We consider the problem of fixing the phases of Bogoliubov coefficients in quantum electrodynamics such that the vacuum-vacuum amplitude can be expressed via them. For a constant electric field and particles with spins of 0 and 1/2, this is done starting from the definition of these coefficients. Using the symmetry etween electric and magnetic fields, we extend the result to a constant electromagnetic field. It turns out that for a constant magnetic field, it is necessary to distinguish the in-and out-states, although they differ only by a phase factor. For a spin-1 particle with a gyromagnetic of ratio g=2, this approach fails and we reconsider the problem using the proper-time method.     

基于小波变换的变分辨率频谱的实现

将FFT与小波理论相结合,本文提出了用变频率分辨率频谱描述信号的思想,系统地介绍了该方法的原理,提出了实现算法.从仿真对比研究的结果看出,该方法比传统的信号分析方法更为优越。     

Undersampled MRI reconstruction with patch-based directional wavelets

Compressed sensing has shown great potential in reducing data acquisition time in magnetic resonance imaging (MRI). In traditional compressed sensing MRI methods, an image is reconstructed by enforcing its sparse representation with respect to a preconstructed basis or dictionary. In this paper, patch-based directional wavelets are proposed to reconstruct images from undersampled k-space data. A parameter of patch-based directional wavelets, indicating the geometric direction of each patch, is trained from the reconstructed image using conventional compressed sensing MRI methods and incorporated into the sparsifying transform to provide the sparse representation for the image to be reconstructed. A reconstruction formulation is proposed and solved via an efficient alternating direction algorithm. Simulation results on phantom and in vivo data indicate that the proposed method outperforms conventional compressed sensing MRI methods in preserving the edges and suppressing the noise. Besides, the proposed method is not sensitive to the initial image when training directions.     

Emerging applications of wavelets: A review

Although most of its popular applications have been in discrete-time signal processing for over two decades, wavelet transform theory offers a methodology to generate continuous-time compact support orthogonal filter banks through the design of discrete-time finite length filter banks with multiple time and frequency resolutions. In this paper, we first highlight inherently built-in approximation errors of discrete-time signal processing techniques employing wavelet transform framework. Then, we present an overview of emerging analog signal processing applications of wavelet transform along with its still active research topics in more matured discrete-time processing applications. It is shown that analog wavelet transform is successfully implemented in biomedical signal processing for design of low-power pacemakers and also in ultra-wideband (UWB) wireless communications. The engineering details of analog circuit implementation for these continuous-time wavelet transform applications are provided for further studies. We expect a flurry of new research and technology development activities in the coming years utilizing still promising and almost untapped analog wavelet transform and multiresolution signal representation techniques.     

The isotope dependence of diatomic Dunham coefficients

The isotope dependence of the Dunham vibration-rotation coefficients Y_kl of a diatomic molecule is studied. Rovibronic interactions between different electronic states are taken into account by transformation to an effective vibration-rotation Hamiltonian for each electronic state. This contains modified vibrational and rotational reduced masses as well as the adiabatic correction to the potential energy. The effects of these contributions on the vibration-rotation energies are expressed in terms of two functions and for each atom i. The resultant formula for Y_kl is Y_kl=μ_c^−(k+2l)/2U_kl{1+m_eΔ_kl^a/M_a+m_eΔ_kl^b/M_b+O(m_e²/M_i²)}, where U_kl, Δ_kl^a, and Δ_kl^b are isotopically invariant, M_a and M_b are the atomic masses, and μ_c = M_aM_b/(M_a + M_b − Cm_e) is the atomic reduced mass, modified by the molecular charge number C for charged species. The U_kl with l ≥ 2 can be calculated from those with l = 0 and 1. The corrections U_klΔ_klⁱ are related to the functions and and to the Dunham corrections. Recent data for the CO molecule are discussed, and it is suggested that some large Δ_klⁱ values are associated with accidentally small U_kl values, since the size of U_klΔ_klⁱ is not directly related to that of U_kl.     

Causal nature of the motion of wavelets

The significance of the wavelets reported on previously is seen to lie in the causal nature of their motion. Some remarks are presented in order to bring out this significance.Formerly: Polytechnic Institute of New York.