Pierre Weiss

Axel Flinth, Pierre Weiss

We study the solutions of infinite dimensional linear inverse problems over Banach spaces. The regularizer is defined as the total variation of a linear mapping of the function to recover, while the data fitting term is a near arbitrary convex function. The first contribution is about the solu-tion's structure: we show that under suitable assumptions, there always exist an m-sparse solution, where m is the number of linear measurements of the signal. Our second contribution is about the computation of the solution. While most existing works first discretize the problem, we show that exacts solutions of the infinite dimensional problem can be obtained by solving two consecutive finite dimensional convex programs. These results extend recent advances in the understanding of total-variation reg-ularized problems.

Paul Escande, Pierre Weiss

We consider a class of linear integral operators with impulse responses varying regularly in time or space. These operators appear in a large number of applications ranging from signal/image processing to biology. Evaluating their action on functions is a computation-ally intensive problem necessary for many practical problems. We analyze a technique called convolution-product expansion: the operator is locally approximated by a convolution, allowing to design fast numerical algorithms based on the fast Fourier transform. We design various types of expansions, provide their explicit rates of approximation and their complexity depending on the time varying impulse response smoothness. This analysis suggests novel wavelet based implementations of the method with numerous assets such as optimal approximation rates, low complexity and storage requirements as well as adaptivity to the kernels regularity. The proposed methods are an alternative to more standard procedures such as panel clustering, cross approximations, wavelet expansions or hierarchical matrices.

Paul Escande, Pierre Weiss

Image deblurring is a fundamental problem in imaging, usually solved with com-putationally intensive optimization procedures. We show that the minimization can be significantly accelerated by leveraging the fact that images and blur operators are compressible in the same orthogonal wavelet basis. The proposed methodology consists of three ingredients: i) a sparse approximation of the blur operator in wavelet bases, ii) a diagonal preconditioner and iii) an implementation on massively parallel architectures. Combing the three ingredients leads to acceleration factors ranging from 30 to 250 on a typical workstation. For instance, a 1024 x 1024 image can be deblurred in 0.15 seconds, which corresponds to real-time.

Clément Cazorla, Nathanaël Munier, Renaud Morin et al.

The most popular networks used for cell segmentation (e.g. Cellpose, Stardist, HoverNet,...) rely on a prediction of a distance map. It yields unprecedented accuracy but hinges on fully annotated datasets. This is a serious limitation to generate training sets and perform transfer learning. In this paper, we propose a method that still relies on the distance map and handles partially annotated objects. We evaluate the performance of the proposed approach in the contexts of frugal learning, transfer learning and regular learning on regular databases. Our experiments show that it can lead to substantial savings in time and resources without sacrificing segmentation quality. The proposed algorithm is embedded in a user-friendly Napari plugin.

Alban Gossard, Frédéric de Gournay, Pierre Weiss

We propose a novel learning based algorithm to generate efficient and physically plausible sampling patterns in MRI. This method has a few advantages compared to recent learning based approaches: i) it works off-the-grid and ii) allows to handle arbitrary physical constraints. These two features allow for much more versatility in the sampling patterns that can take advantage of all the degrees of freedom offered by an MRI scanner. The method consists in a high dimensional optimization of a cost function defined implicitly by an algorithm. We propose various numerical tools to address this numerical challenge.

Paul Escande, Pierre Weiss

Restoring images degraded by spatially varying blur is a problem encountered in many disciplines such as astrophysics, computer vision or biomedical imaging. One of the main challenges to perform this task is to design efficient numerical algorithms to approximate integral operators.We introduce a new method based on a sparse approximation of the blurring operator in the wavelet domain. This method requires $\mathcal{O}\left(N ε^{-d/M}\right)$ operations to provide $ε$-approximations, where $N$ is the number of pixels of a $d$-dimensional image and $M\geq 1$ is a scalar describing the regularity of the blur kernel. In addition, we propose original methods to define sparsity patterns when only the operators regularity is known.Numerical experiments reveal that our algorithm provides a significant improvement compared to standard methods based on windowed convolutions.

Nicolas Chauffert, Philippe Ciuciu, Jonas Kahn et al.

We consider the problem of projecting a probability measure $π$ on a set $\mathcal{M}\_N$ of Radon measures. The projection is defined as a solution of the following variational problem:\begin{equation*}\inf\_{μ\in \mathcal{M}\_N} \|h\star (μ- π)\|\_2^2,\end{equation*}where $h\in L^2(Ω)$ is a kernel, $Ω\subset \R^d$ and $\star$ denotes the convolution operator.To motivate and illustrate our study, we show that this problem arises naturally in various practical image rendering problems such as stippling (representing an image with $N$ dots) or continuous line drawing (representing an image with a continuous line).We provide a necessary and sufficient condition on the sequence $(\mathcal{M}\_N)\_{N\in \N}$ that ensures weak convergence of the projections $(μ^*\_N)\_{N\in \N}$ to $π$.We then provide a numerical algorithm to solve a discretized version of the problem and show several illustrations related to computer-assisted synthesis of artistic paintings/drawings.

Alexis Huck, François de Vieilleville, Pierre Weiss et al.

In this paper, we address the issue of hyperspectral pan-sharpening, which consists in fusing a (low spatial resolution) hyperspectral image HX and a (high spatial resolution) panchromatic image P to obtain a high spatial resolution hyperspectral image. The problem is addressed under a variational convex constrained formulation. The objective favors high resolution spectral bands with level lines parallel to those of the panchromatic image. This term is balanced with a total variation term as regularizer. Fit-to-P data and fit-to-HX data constraints are effectively considered as mathematical constraints, which depend on the statistics of the data noise measurements. The developed Alternating Direction Method of Multipliers (ADMM) optimization scheme enables us to solve this problem efficiently despite the non differentiabilities and the huge number of unknowns.