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Ondelette

thumb|Ondelette de Daubechies d'ordre 2.
Une ondelette est une fonction à la base de la décomposition en ondelettes, décomposition similaire à la transformée de Fourier à court terme, utilisée dans l

Algorithme

thumb|Algorithme de découpe d'un polygone quelconque en triangles (triangulation).
Un algorithme est une suite finie et non ambiguë d'instructions et d’opérations permettant de résoudre une classe de

Méthode expérimentale

Les méthodes expérimentales scientifiques consistent à tester la validité d'une hypothèse, en reproduisant un phénomène (souvent en laboratoire) et en faisant varier un paramètre. Le paramètre que l

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Eric Bezzam, Paul Hurley, Sepand Kashani, Matthieu Martin Jean-André Simeoni, Martin Vetterli

Fourier transforms are an often necessary component in many computational tasks, and can be computed efficiently through the fast Fourier transform (FFT) algorithm. However, many applications involve an underlying continuous signal, and a more natural choice would be to work with e.g. the Fourier series (FS) coefficients in order to avoid the additional overhead of translating between the analog and discrete domains. Unfortunately, there exists very little literature and tools for the manipulation of FS coefficients from discrete samples. This paper introduces a Python library called pyFFS for efficient FS coefficient computation, convolution, and interpolation. While the libraries SciPy and NumPy provide efficient functionality for discrete Fourier transform coefficients via the FFT algorithm, pyFFS addresses the computation of FS coefficients through what we call the fast Fourier series (FFS). Moreover, pyFFS includes an FS interpolation method based on the chirp Z-transform that can make it more than an order of magnitude faster than the SciPy equivalent when one wishes to perform interpolation. GPU support through the CuPy library allows for further acceleration, e.g. an order of magnitude faster for computing the 2-D FS coefficients of 1000 x 1000 samples and nearly two orders of magnitude faster for 2-D interpolation. As an application, we discuss the use of pyFFS in Fourier optics. pyFFS is available as an open source package at https://github.com/imagingofthings/pyFFS, with documentation at https://pyffs.readthedocs.io.

2022The goal of this thesis is to study continuous-domain inverse problems for the reconstruction of sparse signals and to develop efficient algorithms to solve such problems computationally. The task is to recover a signal of interest as a continuous function from a finite number of measurements. This problem being severely ill-posed, we choose to favor sparse reconstructions. We achieve this by formulating an optimization problem with a regularization term involving the total-variation (TV) norm for measures. However, such problems often lead to nonunique solutions, some of which, contrary to expectations, may not be sparse. This requires particular care to assert that we reach a desired sparse solution.Our contributions are divided into three parts. In the first part, we propose exact discretization methods for large classes of TV-based problems with generic measurement operators for one-dimensional signals. Our methods are based on representer theorems that state that our problems have spline solutions. Our approach thus consists in performing an exact discretization of the problems in spline bases, and we propose algorithms which ensure that we reach a desired sparse solution. We then extend this approach to signals that are expressed as a sum of components with different characteristics. We either consider signals whose components are sparse in different bases or signals whose first component is sparse, and the other is smooth. In the second part, we consider more specific TV-based problems and focus on the identification of cases of uniqueness. Moreover, in cases of nonuniqueness, we provide a precise description of the solution set, and more specifically of the sparsest solutions. We then leverage this theoretical study to design efficient algorithms that reach such a solution. In this line, we consider the problem of interpolating one-dimensional data points with second-order TV regularization. We also study this same problem with an added Lipschitz constraint to favor stable solutions. Finally, we consider the problem of the recovery of periodic splines with low-frequency Fourier measurements, which we prove to always have a unique solution.In the third and final part, we apply our sparsity-based frameworks to various real-world problems. Our first application is a method for the fitting of sparse curves to contour data. Finally, we propose an image-reconstruction method for scanning transmission X-ray microscopy.

Despite a long history of research in motor control, the exact mechanism of how the brain communicates with the the invertebrate ventral nerve cord (VNC) and the vertebrate spinal cord on a single neuron basis remains largely elusive.Drosophila melanogaster is an ideal model organism to address the role of individual neurons, thanks to its stereotyped nervous system.Descending neurons (DNs), interneurons in the brain that project to the invertebrate ventral nerve cord or vertebrate spinal cord, have been shown to elicit specific behaviors upon optogenetic activation in Drosophila.These results suggest that behavior is controlled by sparse sets of command neurons, each eliciting a particular behavior.However, it remains to be seen how many DNs are involved in the control of naturalistic behaviors and what other information they might encode.In order to avoid commands leading to physically impossible or destabilizing actions, the brain has to be aware of the current behavior state.Ascending neurons (ANs), interneurons in the invertebrate ventral nerve cord or vertebrate spinal cord projecting to the brain, are likely conveying this information to the brain.To which degree of fidelity and in what way ANs encode behavior state remains unclear.To address these questions, we developed a dissection approach that allows functional imaging of ANs and DNs in the cervical connective.This dissection gives us optical access to the cervical connective and parts of the VNC in behaving adult Drosophila such that a two-photon fluorescence microscope can be used to image neural activity via calcium indicators.We began by imaging a new library of sparse split GAL4 driver lines targeting ANs.Our recordings from 247 genetically identifiable ANs revealed neurons encoding walking, resting, turning, eye grooming, and foreleg movements.Anatomical characterization of these ANs showed that they predominantly project to two brain regions: the anterior ventrolateral protocerebrum (AVLP) and the gnathal ganglion (GNG).Our results suggest that ANs encoding the motion of the animal with respect to the surrounding environment (self-motion) predominantly project to the AVLP, an integrative sensory hub. ANs projecting to the GNG mostly encode discrete actions.We then proceeded to image populations of DNs.The majority of DNs we recorded encoded walking behavior, with smaller fractions encoding resting and head grooming.Some of these neurons are likely to drive walking, but we also identified neurons encoding walking speed and turning.This suggests that a core set of DNs drives a behavior whose features can be modulated by additional DNs.Besides encoding behaviors, some DNs encode sensory information, including the presence of odors and deflection of the antennae.