Linear independence of a finite set of time-frequency shifts

This paper introduces an open conjecture in time-frequency analysis on the linear independence of a finite set of time-frequency shifts of a given L² function. Firstly, background and motivation for the conjecture are provided. Secondly, the main progress of this linear independence in the past twenty five years is reviewed. Finally, the partial results of the high dimensional case and other cases for the conjecture are briefly presented.     

Enhancing signal-to-noise ratio and resolution in low-field NMR relaxation measurements using post-acquisition digital filters

The traditional way to enhance signal-to-noise ratio (SNR) of nuclear magnetic resonance (NMR) signals is to increase the number of scans. However, this procedure increases the measuring time that can be prohibitive for some applications. Therefore, we have tested the use of several post-acquisition digital filters to enhance SNR up to one order of magnitude in time domain NMR (TD-NMR) relaxation measurements. The procedures were studied using continuous wave free precession (CWFP-T₁) signals, acquired with very low flip angles that contain six times more noise than the Carr–Purcell–Meiboom–Gill (CPMG) signal of the same sample and experimental time. Linear (LI) and logarithmic (LO) data compression, low-pass infinity impulse response (LP), Savitzky–Golay (SG), and wavelet transform (WA) post-acquisition filters enhanced the SNR of the CWFP-T₁ signals by at least six times. The best filters were LO, SG, and WA that have high enhancement in SNR without significant distortions in the ILT relaxation distribution data. Therefore, it was demonstrated that these post-acquisition digital filters could be a useful way to denoise CWFP-T₁, as well as CPMG noisy signals, and consequently reducing the experimental time. It was also demonstrated that filtered CWFP-T₁ method has the potential to be a rapid and nondestructive method to measure fat content in beef and certainly in other meat samples.     

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.     

非正态变异下的非线性轮廓异常点识别方法研究

在质量控制领域,非线性轮廓异常点识别问题是重点研究问题之一。本文综合运用了小波分析、数据深度、聚类分析等数据分析处理技术,提出了一种新的非正态变异的异常点识别方法。文章通过仿真分析技术,将新方法χ²与控制图方法进行性能对比,结果证实新方法能够以更高的准确率和稳定性识别异常点,表现出更好的异常点识别性能。最后将新方法应用于木板垂直密度轮廓实例对新方法进行验证,分析结果表明本方法能够有效识别出异常轮廓数据。     

A classification of continuous wavelet transforms in dimension three

This paper presents a full catalogue, up to conjugacy and subgroups of finite index, of all matrix groups $H<\mathrm{GL}(3,\mathbb{R})$ that give rise to a continuous wavelet transform with associated irreducible quasi-regular representation. For each group in this class, coorbit theory allows to consistently define spaces of sparse signals, and to construct atomic decompositions converging simultaneously in a whole range of these spaces. As an application of the classification, we investigate the existence of compactly supported admissible vectors and atoms for the groups.     

An adaptive wavelet space‐time SUPG method for hyperbolic conservation laws

This article concerns with incorporating wavelet bases into existing streamline upwind Petrov‐Galerkin (SUPG) methods for the numerical solution of nonlinear hyperbolic conservation laws which are known to develop shock solutions. Here, we utilize an SUPG formulation using continuous Galerkin in space and discontinuous Galerkin in time. The main motivation for such a combination is that these methods have good stability properties thanks to adding diffusion in the direction of streamlines. But they are more expensive than explicit semidiscrete methods as they have to use space‐time formulations. Using wavelet bases we maintain the stability properties of SUPG methods while we reduce the cost of these methods significantly through natural adaptivity of wavelet expansions. In addition, wavelet bases have a hierarchical structure. We use this property to numerically investigate the hierarchical addition of an artificial diffusion for further stabilization in spirit of spectral diffusion. Furthermore, we add the hierarchical diffusion only in the vicinity of discontinuities using the feature of wavelet bases in detection of location of discontinuities. Also, we again use the last feature of the wavelet bases to perform a postprocessing using a denosing technique based on a minimization formulation to reduce Gibbs oscillations near discontinuities while keeping other regions intact. Finally, we show the performance of the proposed combination through some numerical examples including Burgers’, transport, and wave equations as well as systems of shallow water equations.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2062–2089, 2017     

Wavelet-based estimator for the hurst parameters of fractional brownian sheet

It is proposed a class of statistical estimators H =(H_1,…,H_d) for the Hurst parameters H =(H_1,…,H_d) of fractional Brownian field via multi-dimensional wavelet analysis and least squares,which are asymptotically normal.These estimators can be used to detect self-similarity and long-range dependence in multi-dimensional signals,which is important in texture classification and improvement of diffusion tensor imaging(DTI) of nuclear magnetic resonance(NMR).Some fractional Brownian sheets will be simulated and the simulated data are used to validate these estimators.We find that when H_i ≥ 1/2,the estimators are accurate,and when H_i 1/2,there are some bias.     

分数阶尺度函数的分解与重构算法

引入分数阶多分辨分析与分数阶尺度函数的概念.运用时频分析方法与分数阶小波变换,研究了分数阶正交小波的构造方法,得到分数阶正交小波存在的充要条件.给出分数阶尺度函数与小波的分解与重构算法,算法比经典的尺度函数与小波的分解与重构算法更具有一般性.