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Person# Ildar Khalidov

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Wavelet

A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy

Differential operator

In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract

Basis function

In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a linear combination of basis functions, just as

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Ildar Khalidov, Michaël Unser, John Paul Ward

The connection between derivative operators and wavelets is well known. Here we generalize the concept by constructing multiresolution approximations and wavelet basis functions that act like Fourier multiplier operators. This construction follows from a stochastic model: signals are tempered distributions such that the application of a whitening (differential) operator results in a realization of a sparse white noise. Using wavelets constructed from these operators, the sparsity of the white noise can be inherited by the wavelet coefficients. In this paper, we specify such wavelets in full generality and determine their properties in terms of the underlying operator.

Ildar Khalidov, Michaël Unser, John Paul Ward

The connection between derivative operators and wavelets is well known. Here we generalize the concept by constructing multiresolution approximations and wavelet basis functions that act like Fourier multiplier operators. This construction follows from a stochastic model: signals are tempered distributions such that the application of a whitening (differential) operator results in a realization of a sparse white noise. Using wavelets constructed from these operators, the sparsity of the white noise can be inherited by the wavelet coefficients. In this paper, we specify such wavelets in full generality and determine their properties in terms of the underlying operator.

Ildar Khalidov, François Lazeyras, Michaël Unser, Dimitri Nestor Alice Van De Ville

We propose a new framework to extract the activity-related component in the BOLD functional magnetic resonance imaging (fMRI) signal. As opposed to traditional fMRI signal analysis techniques, we do not impose any prior knowledge of the event timing. Instead, our basic assumption is that the activation pattern is a sequence of short and sparsely distributed stimuli, as is the case in slow event-related fMRI.