Operator-like wavelets with application to functional magnetic resonance imaging

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.