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 of wavelets has been established, based on the number and direction of its pulses. Wavelets are imbued with specific properties that make them useful for signal processing. For example, a wavelet could be created to have a frequency of Middle C and a short duration of roughly one tenth of a second. If this wavelet were to be convolved with a signal created from the recording of a melody, then the resulting signal would be useful for determining when the Middle C note appeared in the song. Mathematically, a wavelet correlates with a signal if a portion of the signal is similar. Correlation is at the core of many practical wavelet applications. As a mathematical tool, wavelets can be used to extract information from many different kinds of data, including – but not l

Operator-like wavelets with application to functional magnetic resonance imaging

Ildar Khalidov

We introduce a new class of wavelets that behave like a given differential operator L. Our construction is inspired by the derivative-like behavior of classical wavelets. Within our framework, the wavelet coefficients of a signal y are the samples of a smoothed version of L{y}. For a linear system characterized by an operator equation L{y} = x, the operator-like wavelet transform essentially deconvolves the system output y and extracts the "innovation" signal x. The main contributions of the thesis include: Exponential-spline wavelets. We consider the system L described by a linear differential equation and build wavelets that mimic the behavior of L. The link between the wavelet and the operator is an exponential B-spline function; its integer shifts span the multiresolution space. The construction that we obtain is non-stationary in the sense that the wavelets and the scaling functions depend on the scale. We propose a generalized version of Mallat's fast filterbank algorithm with scale-dependent filters to efficiently perform the decomposition and reconstruction in the new wavelet basis. Activelets in fMRI. As a practical biomedical imaging application, we study the problem of activity detection in event-related fMRI. For the differential system that links the measurements and the stimuli, we use a linear approximation of the balloon/windkessel model for the hemodynamic response. The corresponding wavelets (we call them activelets) are specially tuned for temporal fMRI signal analysis. We assume that the stimuli are sparse in time and extract the activity-related signal by optimizing a criterion with a sparsity regularization term. We test the method with synthetic fMRI data. We then apply it to a high-resolution fMRI retinotopy dataset to demonstrate its applicability to real data. Operator-like wavelets. Finally, we generalize the operator-like wavelet construction for a wide class of differential operators L in multiple dimensions. We give conditions that L must satisfy to generate a valid multiresolution analysis. We show that Matérn and polyharmonic wavelets are particular cases of our construction.

EPFL2009

Operator-like wavelets with application to functional magnetic resonance imaging

Ildar Khalidov

EPFL2009

Wavelet analysis of interference patterns and signals

Operator-like wavelets with application to functional magnetic resonance imaging

Sound modeling by means of harmonic-band wavelets: new results and experiments

Operator-like wavelets with application to functional magnetic resonance imaging

Wavelet analysis of interference patterns and signals

Sound modeling by means of harmonic-band wavelets: new results and experiments