Publications by Thomas Jean Debarre

TV-based spline reconstruction with Fourier measurements: Uniqueness and convergence of grid-based methods

Julien René Pierre Fageot, Thomas Jean Debarre, Quentin Alain Denoyelle

We study the problem of recovering piecewise-polynomial periodic functions from their low-frequency information. This means that we only have access to possibly corrupted versions of the Fourier samples of the ground truth up to a maximum cutoff frequency ...

ELSEVIER2023

Sparsest Univariate Learning Models Under Lipschitz Constraint

Michaël Unser, Shayan Aziznejad, Thomas Jean Debarre

Beside the minimizationof the prediction error, two of the most desirable properties of a regression scheme are stability and interpretability. Driven by these principles, we propose continuous-domain formulations for one-dimensional regression problems. I ...

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC2022

Coupled Splines for Sparse Curve Fitting

Michaël Unser, Shayan Aziznejad, Thomas Jean Debarre

We formulate as an inverse problem the construction of sparse parametric continuous curve models that fit a sequence of contour points. Our prior is incorporated as a regularization term that encourages rotation invariance and sparsity. We prove that an op ...

2022

On the uniqueness of solutions for the basis pursuit in the continuum

Julien René Pierre Fageot, Thomas Jean Debarre

This paper studies the continuous-domain inverse problem of recovering Radon measures on the one-dimensional torus from low-frequency Fourier coefficients, where K (c) is the cutoff frequency. Our approach consists in minimizing the total-variation norm am ...

2022

Variational Methods For Continuous-Domain Inverse Problems: the Quest for the Sparsest Solution

Thomas Jean Debarre

The 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 ...

EPFL2022

Sparsest piecewise-linear regression of one-dimensional data

Michaël Unser, Julien René Pierre Fageot, Thomas Jean Debarre, Quentin Alain Denoyelle

We study the problem of one-dimensional regression of data points with total-variation (TV) regularization (in the sense of measures) on the second derivative, which is known to promote piecewise-linear solutions with few knots. While there are efficient a ...

Elsevier2021

Continuous-Domain Formulation of Inverse Problems for Composite Sparse-Plus-Smooth Signals

Michaël Unser, Shayan Aziznejad, Thomas Jean Debarre

We present a novel framework for the reconstruction of 1D composite signals assumed to be a mixture of two additive components, one sparse and the other smooth, given a finite number of linear measurements. We formulate the reconstruction problem as a cont ...

2021

Hessian Splines For Scanning Transmission X-Ray Microscopy

Michaël Unser, Thomas Jean Debarre

Scanning transmission X-ray microscopy (STXM) produces images in which each pixel value is related to the measured attenuation of an X-ray beam. In practice, the location of the illuminated region does not exactly match the desired uniform pixel grid. This ...

IEEE2020

Sparse Dictionaries for Continuous-Domain Inverse Problems

Michaël Unser, Shayan Aziznejad, Thomas Jean Debarre

We study 1D continuous-domain inverse problems for multicomponent signals. The prior assumption on these signals is that each component is sparse in a different dictionary specified by a regularization operators. We introduce a hybrid regularization functi ...

2019

Pocket guide to solve inverse problems with GlobalBioIm

Michaël Unser, Daniel Sage, Laurène Donati, Thanh-An Michel Pham, Ferréol Arnaud Marie Soulez, Michael Thompson McCann, Emmanuel Emilien Louis Soubies, Thomas Jean Debarre

GlobalBioIm is an open-source MATLAB (R) library for solving inverse problems. The library capitalizes on the strong commonalities between forward models to standardize the resolution of a wide range of imaging inverse problems. Endowed with an operator-al ...

2019

Solving Continuous-Domain Problems Exactly with Multiresolution B-Splines

Michaël Unser, Julien René Pierre Fageot, Harshit Gupta, Thomas Jean Debarre

We propose a discretization method for continuous-domain linear inverse problems with multiple-order total-variation (TV) regularization. It is based on a recent result that proves that such inverse problems have sparse polynomial-spline solutions. Our met ...

IEEE2019

Hybrid-Spline Dictionaries for Continuous-Domain Inverse Problems

Michaël Unser, Shayan Aziznejad, Thomas Jean Debarre

We study one-dimensional continuous-domain inverse problems with multiple generalized total-variation regularization, which involves the joint use of several regularization operators. Our starting point is a new representer theorem that states that such in ...

2019

B-Spline-Based Exact Discretization of Continuous-Domain Inverse Problems With Generalized TV Regularization

Michaël Unser, Julien René Pierre Fageot, Harshit Gupta, Thomas Jean Debarre

We study continuous-domain linear inverse problems with generalized total-variation (gTV) regularization, expressed in terms of a regularization operator L. It has recently been proved that such inverse problems have sparse spline solutions, with fewer jum ...

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC2019