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Publication# Operator-like wavelets with application to functional magnetic resonance imaging

Résumé

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.

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In this thesis, we study two distinct problems.
The first problem consists of studying the linear system of partial differential equations which consists of taking a k-form, and applying the exterior derivative 'd' to it and add the wedge product with a 1-form 'a'. The study of this differential operator is linked to the study of the multiplication by a two form, that is the system of linear equations where we take a k-form and apply the exterior wedge product by 'da', the exterior derivative of 'a'. We establish links between the partial differential equation and the linear system.
The second problem is a generalization of the symmetric gradient and the curl equation. The equation of a symmetric gradient consists of taking a vector field, apply the gradient and then add the transpose of the gradient, whereas in the curl equation we subtract the transpose of the gradient. Both can be seen as an equation of the form A * grad u + (grad u)t * A, where A is a symmetric matrix for the case of the symmetric gradient and skew symmetric for the curl equation. We generalize to the case where A verifies no symmetry assumption and more significantly add a Dirichlet condition on the boundary.

Decoding visual cognition from non-invasive measurements of brain activity has shown valuable applications. Vision-based Brain-Computer Interfaces (BCI) systems extend from spellers to database search and spatial navigation. Despite the high performance of these systems, they are limited to the laboratory environment. Several attempts were made to improve the quality of life of patients with severe neurodegenerative disorders by helping them acquire more independence. Bringing vision-based BCI to everyday life would benefit a wide range of population from handicapped to healthy people. However, real-life applications impose additional challenges on obtaining neural signals with a good signal-to-noise ratio. The natural environments are rich, dynamic and ambiguous that challenges the visual perception and, therefore, the decoding of underlying neural correlates. The limited attentional resources require to explore and visually sample the environment by freely moving eyes and fixating on the relevant objects. Moreover, everyday activities are rarely isolated leading to a mixture and complex interactions between neural correlates in the measured signal.
In this thesis, I explore various aspects of decoding brain activity by developing new protocols for realistic scenarios with challenging visual tasks. The tasks include free visual exploration, dynamic scenes, and multiple tasks, for example, recognition of a billboard category during simulated driving. The major efforts have been directed towards synchronous decoding of Event-Related Potentials (ERP) acquired by the means of electroencephalography (EEG) and Eye-tracking technology, which allowed the extraction of Eye-Fixation Related Potentials (EFRP). Additionally, I investigate the neural correlates of perceptual decision making which is tightly linked to visual recognition. The perceptual challenges can lead to higher temporal variability of obtained ERP which was also addressed from the decoding perspective.
The obtained results, firstly, provide new insights on how to tackle the temporal variability of ERP. Secondly, they raise new questions on studying and decoding neural correlates of perceptual decision making. And thirdly, they show the feasibility of decoding visual recognition in a simulated realistic scenario of car driving. Although, as expected, I found a lower performance compared to classical ERP protocols, these findings hold promise and raise new questions to be investigated in order to improve the quality of decoding visual cognition for everyday application.