Extreme Value Analysis of Empirical Frame Coefficients and Implications for Denoising by Soft-Thresholding

Denoising by frame thresholding is one of the most basic and efficient methods for recovering a discrete signal or image from data that are corrupted by additive Gaussian white noise. The basic idea is to select a frame of analyzing elements that separates the data in few large coefficients due to the signal and many small coefficients mainly due to the noise \epsilon_n. Removing all data coefficients being in magnitude below a certain threshold yields a reconstruction of the original signal. In order to properly balance the amount of noise to be removed and the relevant signal features to be kept, a precise understanding of the statistical properties of thresholding is important. For that purpose we derive the asymptotic distribution of max_{\omega \in \Omega_n} || for a wide class of redundant frames (\phi_\omega^n: \omega \in \Omega_n}. Based on our theoretical results we give a rationale for universal extreme value thresholding techniques yielding asymptotically sharp confidence regions and smoothness estimates corresponding to prescribed significance levels. The results cover many frames used in imaging and signal recovery applications, such as redundant wavelet systems, curvelet frames, or unions of bases. We show that `generically' a standard Gumbel law results as it is known from the case of orthonormal wavelet bases. However, for specific highly redundant frames other limiting laws may occur. We indeed verify that the translation invariant wavelet transform shows a different asymptotic behaviour.Comment: [Content: 39 pages, 4 figures] Note that in this version 4 we have slightely changed the title of the paper and we have rewritten parts of the introduction. Except for corrected typos the other parts of the paper are the same as the original versions

HEAR: Approach for Heartbeat Monitoring with Body Movement Compensation by IR-UWB Radar

Further applications of impulse radio ultra-wideband radar in mobile health are hindered by the difficulty in extracting such vital signals as heartbeats from moving targets. Although the empirical mode decomposition based method is applied in recovering waveforms of heartbeats and estimating heart rates, the instantaneous heart rate is not achievable. This paper proposes a Heartbeat Estimation And Recovery (HEAR) approach to expand the application to mobile scenarios and extract instantaneous heartbeats. Firstly, the HEAR approach acquires vital signals by mapping maximum echo amplitudes to the fast time delay and compensating large body movements. Secondly, HEAR adopts the variational nonlinear chirp mode decomposition in extracting instantaneous frequencies of heartbeats. Thirdly, HEAR extends the clutter removal method based on the wavelet decomposition with a two-parameter exponential threshold. Compared to heart rates simultaneously collected by electrocardiograms (ECG), HEAR achieves a minimum error rate 4.6% in moving state and 2.25% in resting state. The Bland–Altman analysis verifies the consistency of beat-to-beat intervals in ECG and extracted heartbeat signals with the mean deviation smaller than 0.1 s. It indicates that HEAR is practical in offering clinical diagnoses such as the heart rate variability analysis in mobile monitoring

Three real-space discretization techniques in electronic structure calculations

A characteristic feature of the state-of-the-art of real-space methods in electronic structure calculations is the diversity of the techniques used in the discretization of the relevant partial differential equations. In this context, the main approaches include finite-difference methods, various types of finite-elements and wavelets. This paper reports on the results of several code development projects that approach problems related to the electronic structure using these three different discretization methods. We review the ideas behind these methods, give examples of their applications, and discuss their similarities and differences.Comment: 39 pages, 10 figures, accepted to a special issue of "physica status solidi (b) - basic solid state physics" devoted to the CECAM workshop "State of the art developments and perspectives of real-space electronic structure techniques in condensed matter and molecular physics". v2: Minor stylistic and typographical changes, partly inspired by referee comment

Computing continuous-time growth models with boundary conditions via wavelets.

This paper presents an algorithm for solving boundary value differential equations, which often arise in economics from the application of Pontryagin’s maximum principle. We propose a wavelet-collocation algorithm, study its convergence properties and illustrate how this approach can be applied to different economic problemsWavelets; Continuous-time growth models; Boundary value problems;

Wavelets and some of their applications

http://www.worldcat.org/oclc/3927123

Function spaces vs. Scaling functions: Some issues in image classification

Criteria based on the computation of fractal dimensions have been used in order to perform image analysis and classification; we show that such criteria often amount to deter- mine the regularity of the image in some classes of function spaces, and that looking for richer criteria naturally leads to the introduction of new classes of function spaces. We will investigate the properties of some of these classes, and show which type of additional information they yield for the initial image