On the Hilbert transform of wavelets

A wavelet is a localized function having a prescribed number of vanishing moments. In this correspondence, we provide precise arguments as to why the Hilbert transform of a wavelet is again a wavelet. In particular, we provide sharp estimates of the localization, vanishing moments, and smoothness of the transformed wavelet. We work in the general setting of non-compactly supported wavelets. Our main result is that, in the presence of some minimal smoothness and decay, the Hilbert transform of a wavelet is again as smooth and oscillating as the original wavelet, whereas its localization is controlled by the number of vanishing moments of the original wavelet. We motivate our results using concrete examples.Comment: Appears in IEEE Transactions on Signal Processing, vol. 59, no. 4, pp. 1890-1894, 201

Wavelet denoising for P300 single-trial detection

National audienceTemplate-based analysis techniques are good candidates to robustly detect transient temporal graphic elements (e.g. event-related potential, k-complex, sleep spindles, vertex waves, spikes) in noisy and multi-sources electro-encephalographic signals. More specifically, we present the impact on a large dataset of a wavelet denoising to detect evoked potentials in a single-trial P300 speller. Using coiflets as a denoising process allows to obtain more stable accurracies for all subjects

Application of the wavelet transform to biomedical signals

The basic concepts and fundamentals of the wavelet signal representation were examined. The orthonormal wavelet was selected for this project after being compared to various types of wavelets. The orthonormal wavelet was chosen due to the equal time and frequency resolution exhibited in the wavelet coefficients. Programs were written in Matlab to implement the orthonormal wavelet in developing wavelet coefficients for a given signal. The programs include the conditions for an orthonormal wavelet in and which produce the wavelet filters g(n) and h(n). The wavelet filters were then incorporated into another program that applied Mallat\u27s multiresolution algorithm for a given signal. The resulting wavelet coefficients were obtained and interpreted. The orthonormal wavelet was applied to various types of biomedical signals. The wavelet transform was applied to motor evoked potentials (MEPs) created cortical magnetic stimulation. The wavelet was also applied to evoked potentials (EPs) and to various types of EKG signals. The wavelet representation exposed new ways of observing biomedical signals by bringing out details and structures not present in the original waveforms

Activation detection in fMRI data via multi-scale singularity detection

Detection of active areas in the brain by functional magnetic resonance imaging (fMRI) is a challenging problem in medical imaging. Moreover, determining the onset and end of activation signal at specific locations in 3-space can determined networks of temporal relationships required for brain mapping. We introduce a method for activation detection in fMRI data via wavelet analysis of singular features. We pose the problem of determining activated areas as singularity detection in the temporal domain. Overcomplete wavelet expansion at integer scales are used to trace wavelet modulus maxima across scales to construct maxima lines. Form these maxima lines, singularities in the signal are located corresponding to the onset and end of an activation signal. We present result for simulated phantom waveforms and clinical fMRI dat from human finger tapping experiments. Different levels of noise were added to two waveforms of phantom data. No assumptions about specific frequency and amplitude of an activation signal were made prior to analysis. Detection was reliable for modest levels of random noise, but less precise at higher levels. For clinical fMRI data, activation maps were comparable to those of existing standard techniques

On Shannon's formula and Hartley's rule: beyond the mathematical coincidence

In the information theory community, the following “historical“ statements are generally well accepted: (1) Hartley did put forth his rule twenty years before Shannon; (2) Shannon's formula as a fundamental tradeoff between transmission rate, bandwidth, and signal-to-noise ratio came out unexpected in 1948; (3) Hartley's rule is inexact while Shannon's formula is characteristic of the additive white Gaussian noise channel; (4) Hartley's rule is an imprecise relation that is not an appropriate formula for the capacity of a communication channel. We show that all these four statements are somewhat wrong. In fact, a careful calculation shows that “Hartley's rule“ in fact coincides with Shannon's formula. We explain this mathematical coincidence by deriving the necessary and sufficient conditions on an additive noise channel such that its capacity is given by Shannon's formula and construct a sequence of such channels that makes the link between the uniform (Hartley) and Gaussian (Shannon) channels.In the information theory community, the following “historical” statements are generally well accepted: (1) Hartley did put forth his rule twenty years before Shannon; (2) Shannon’s formula as a fundamental tradeoff between transmission rate, bandwidth, a16948921910FAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO2014/13835-6 ; 2013/25977-

Construction of Hilbert Transform Pairs of Wavelet Bases and Gabor-like Transforms

We propose a novel method for constructing Hilbert transform (HT) pairs of wavelet bases based on a fundamental approximation-theoretic characterization of scaling functions--the B-spline factorization theorem. In particular, starting from well-localized scaling functions, we construct HT pairs of biorthogonal wavelet bases of L^2(R) by relating the corresponding wavelet filters via a discrete form of the continuous HT filter. As a concrete application of this methodology, we identify HT pairs of spline wavelets of a specific flavor, which are then combined to realize a family of complex wavelets that resemble the optimally-localized Gabor function for sufficiently large orders. Analytic wavelets, derived from the complexification of HT wavelet pairs, exhibit a one-sided spectrum. Based on the tensor-product of such analytic wavelets, and, in effect, by appropriately combining four separable biorthogonal wavelet bases of L^2(R^2), we then discuss a methodology for constructing 2D directional-selective complex wavelets. In particular, analogous to the HT correspondence between the components of the 1D counterpart, we relate the real and imaginary components of these complex wavelets using a multi-dimensional extension of the HT--the directional HT. Next, we construct a family of complex spline wavelets that resemble the directional Gabor functions proposed by Daugman. Finally, we present an efficient FFT-based filterbank algorithm for implementing the associated complex wavelet transform.Comment: 36 pages, 8 figure