# Gilles Aubert's research while affiliated with French National Centre for Scientific Research and other places

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## Publications (178)

Several numerical algorithms have been developed in the literature and employed for curves reconstruction. However, these techniques are developed within the discrete setting, namely the super-resolved image is defined on a finer grid than the observed images. Conversely, off-the-grid (or gridless) optimisation does not rely on a fine grid and offe...

Gridless sparse spike reconstruction is a rather new research field with significant results for the super-resolution problem, where we want to retrieve fine-scale details from a noisy and filtered acquisition. To tackle this problem, we are interested in optimisation under some prior, typically the sparsity i.e., the source is composed of spikes....

Super-resolution fluorescence microscopy overcomes blurring arising from light diffraction, allowing the reconstruction of fine scale details in biological structures. Standard methods come at the expense of long acquisition time and/or harmful effects on the biological sample, which makes the problem quite challenging for the imaging of body cells...

We focus on the minimization of the least square loss function under a k-sparse constraint encoded by a \(\ell _0\) pseudo-norm. This is a non-convex, non-continuous and NP-hard problem. Recently, for the penalized form (sum of the least square loss function and a \(\ell _0\) penalty term), a relaxation has been introduced which has strong results...

Among the many super-resolution techniques for microscopy, single-molecule localization microscopy methods are widely used. This technique raises the difficult question of precisely localizing fluorophores from a blurred, under-resolved, and noisy acquisition. In this work, we focus on the grid-based approach in the context of a high density of flu...

This paper is devoted to the analysis of necessary (not sufficient) optimality conditions for the \(\ell _0\)-regularized least-squares minimization problem. Such conditions are the roots of the plethora of algorithms that have been designed to cope with this NP-hard problem. Indeed, as global optimality is, in general, intractable, these algorithm...

We focus on the minimization of the least square loss function either under a $k$-sparse constraint or with a sparse penalty term. Based on recent results, we reformulate the $\ell_0$ pseudo-norm exactly as a convex minimization problem by introducing an auxiliary variable. We then propose an exact biconvex reformulation of the $\ell_2-\ell_0$ cons...

Single Molecule Localization Microscopy (SMLM) enables the acquisition of high-resolution images by alternating between activation of a sparse subset of fluorescent molecules present in a sample and localization. In this work, the localization problem is formulated as a constrained sparse approximation problem which is resolved by rewriting the $\e...

We focus on the problem of minimizing a least-squares loss function under a k-sparse constraint. We investigate the continuous relaxation approach as well as optimization algorithms to apply on Single Molecule Localization Microscopy.

Numerous nonconvex continuous penalties have been proposed to approach the 0 pseudonorm for optimization purpose. Apart from the theoretical results for convex 1 relaxation under restrictive hypotheses, only few works have been devoted to analyze the consistency, in terms of minimizers, between the 0-regularized least squares functional and relaxed...

Determining the receptive field of a visual sensory neuron is crucial to characterize the region of the visual field and the stimuli this neuron is sensitive to. We propose a new method to estimate receptive fields by a nonconvex variational approach, thus relaxing the simplifying and unrealistic assumption of convexity made by standard approaches....

Lemma 4.4 in [E. Soubies, L. Blanc-Feraud and G. Aubert, SIAM J. Imaging Sci., 8 (2015), pp. 1607-1639] is wrong for local minimizers of the continuous exact l(0) (CELO) functional. The argument used to conclude the proof of this lemma is not sufficient in the case of local minimizers. In this note, we supply a revision of this lemma where new resu...

Le problème d’optimisation l2 - l0 a fait l’objet de nombreux travaux ces dernières années du fait de son importance pour diverses applications en traitement du signal et de l’image. Dans cet article, nous nous intéressons à l’algorithme Iterative Hard Thresholding (IHT) qui minimise directement la fonction objectif l2 - l0, sans approximation de l...

Within the framework of the l0 regularized least squares problem, we focus, in this paper, on nonconvex continuous penalties approximating the l0 -norm. Such penalties are known to better promote sparsity than the l1 convex relaxation. Based on some results in one dimension and in the case of orthogonal matrices, we propose the Continuous Exact l0...

This paper presents a new way to address the NP-hard combinatorial l2-l0 problem by minimizing a continuous relaxed functional preserving the minimizers of the initial energy. We propose the Continuous Exact l0 penalty (CEL0), an approximation of the l0 norm leading to a tight continuous relaxation of the l2-l0 criteria whose global minimizers cont...

This paper is concerned with the computation of the topological gradient associated to a
fourth order Kirchhoff type partial differential equation and to a second order cost
function. This computation is motivated by fine structure detection in image analysis. The
study of the topological sensitivity is performed both in the cases of a circular
inc...

In this paper we propose a new variationnal method for segmenting/restoring images degraded by diverse noises and blurs. This method is based on the notion of topological gradient. First applied by [11] to restore images degraded by a Gaussian noise, we propose here to extend the segmen- tation/restoration process for possibly blurred images con- t...

Super-resolution microscopy techniques allow to overstep the diffraction limit of conventional optics. Theses techniques are very promising since they give access to the visualisation of finer structures which is of fundamental importance in biology. In this paper we deal with Multiple-Angle Total Internal Reflection Mi- croscopy (MA-TIRFM) which a...

In this report, we are interested in blind restoration of optical images that are degraded by a space-variant (SV) blur and corrupted with Poisson noise. For example, blur variation is due to refractive index mismatch in three dimensional fluorescence microscopy or due to atmospheric turbulence in astrophysical images. In our work, the SV Point Spr...

Dans cette note, on décrit une nouvelle approche pour la détection de structures fines dans une image. Cette approche est basée sur le calcul du gradient topologique associé à une fonction coût définie à partir des dérivées secondes d'une régularisation des données (éventuellement bruitées). Cette régularisation est obtenue via la résolution d'une...

We propose a new model for the reconstruction of biological structures using Multiple-Angle Total Internal Reflection Fluorescence Microscopy (MA-TIRFM). This recent microscopy technique allows the visualization of sub-cellular structures around the plasma membrane which is of fundamental importance in the comprehension of exchanges mechanisms of t...

This paper is concerned with the computation of the skeleton of a shape Ω included in ℝ2. We show some connections between the Euclidean distance function d to ∂Ω and the solution u of the Poisson problem Δu(x)=−1 if x is in Ω and u(x)=0 if x is on ∂Ω. This enables us to propose a new and fast algorithm to compute an approximation of the skeleton o...

We are interested in blind image restoration in confocal laser scanning microscopy (CLSM). Two challenging problems in this imaging system are considered: First, spherical aberrations due to refractive index mismatch leads to a depth variant (DV) blur. Second, low illumination leads to a signal dependent Poisson noise. In addition, the DV point spr...

We are interested in blind restoration of 3D confocal microscopy images. One challenging problem in this system is the depth-variant (DV) blur due to refractive index mismatch. In our work, we simplify the problem by approximating the DV point spread function (PSF) by a convex combination of a set of space-invariant (SI) PSFs. We show that each SI...

We are interested in blind restoration of 3D confocal microscopy images. One challenging problem in this system is the depth-variant (DV) blur due to refractive index mismatch. In our work, we simplify the problem by approximating the DV point spread function (PSF) by a convex combination of a set of space-invariant (SI) PSFs. We show that each SI...

The paper is concerned with the analysis of a new variational model to restore point-like and curve-like singularities in biological images. To this aim we investigate the variational properties of a suitable energy which governs these pathologies. Finally in order to realize numerical experiments we minimize, in the discrete setting, a regularized...

Purpose
– The inverse problem in the eddy current (EC) imaging of metallic parts is an ill‐posed problem. The purpose of the paper is to compare the performances of regularized algorithms to estimate the 3D geometry of a surface breaking defect.
Design/methodology/approach
– The forward problem is solved using a mesh‐free semi‐analytical model, th...

We propose a new variational method to restore point-like and curvelike singularities in 2-D images. As points and open curves are fine structures, they are difficult to restore by means of first order derivative operators computed in the noisy image. In this paper we propose to use the Laplacian operator of the observed intensity, since it becomes...

We analyze the illumination invariance of the level lines of an image. We show that if the scene surface has Lambertian reflectance and the light is directed, then a necessary condition for the level lines to be illumination invariant is that the 3D scene be developable and that its albedo satisfies some geometrical constraints. We then show that t...

The aim of this paper is to provide a rigorous variational formulation for
the detection of points in 2-d biological images. To this purpose
we introduce a new functional whose minimizers give the points we want to detect. Then we define an approximating sequence of functionals for
which we prove the Γ-convergence to the initial one.

In this paper, we focus on statistical region-based active contour models where image features (e.g. intensity) are random
variables whose distribution belongs to some parametric family (e.g. exponential) rather than confining ourselves to the special
Gaussian case. In the framework developed in this paper, we consider the general case of region-ba...

We propose a new variational model to locate points in 2-dimensional biological images. To this purpose we introduce a suitable functional whose minimizers are given by the points we want to detect. In order to provide numerical experiments we replace this energy with a sequence of a more treatable functionals by means of the notion of \Gamma-conve...

The goal of this paper is to develop an algorithm for extracting point features from sequences of aerial infrared images. We propose an efficient method for the detection of threats in a sequence of infrared images by looking for a trajectory which optimizes a regularized criterion. The regularity is introduced by a new concept of total curvature w...

We prove a new relaxation result for an anisotropic functional preserving point-like and curve-like singularities in image processing

In this chapter, we focus on statistical region-based active contour models where the region descriptor is chosen as the probability density function of an image feature (e.g. intensity) inside the region. Image features are then considered as random variables whose distribution may be either parametric, and then belongs to the exponential family,...

We propose a new variational method to isolate points in biological images. As points are fine structures they are difficult to detect by derivative operators computed in the noisy image. In this paper we propose to compute a vector field from the observed intensity so that its divergence explodes at points. As the image could contains spots but al...

In this work, we propose novel results for the optimization of divergences within the framework of region-based active contours.
We focus on parametric statistical models where the region descriptor is chosen as the probability density function (pdf)
of an image feature (e.g. intensity) inside the region and the pdf belongs to the exponential famil...

In this paper, we show that minimization problems involving sublinear regularizing terms are ill-posed, in general, although numerical experiments in image processing give very good results. The energies studied here are inspired by image restoration and image decomposition. Rewriting the nonconvex sublinear regularizing terms as weighted total var...

In this paper we propose a rigorous framework for texture image segmentation relying on region-based active contours (RBAC) and sparse texture representation. Such representations allow to efficiently describe a texture by transforming it in a dictionary of appropriate waveforms (atoms) where the texture representation coefficients are concentrated...

We propose an algorithm that equalizes the contrast of grayscale image pairs to simplify the task of change detection. To ensure robustness of the detection under different illumination conditions, some authors recently proposed algorithms that compare the level lines of the images. We show - using ideas from the ldquoshape from shadingrdquo commun...

The aim of this report is to provide a variational formulation for the detection of points in $2$-d biological images. To this purpose we introduce a new functional of the calculus of variation whose minimizers gives the points we want to detect. Then we build an approximating sequence of functional, for which we prove the $\Gamma$-convergence to t...

We asked whether the recent characterization of Sobolev spaces by Bourgain, Brezis and Mironescu (2001) could be useful to solve variational problems on W 1,p (Ω). For this, we in-troduced a sequence of functionals so that the semi-norm is approximated by an integral operator involving a differential quotient and a radial mollifier. Then, for the a...

This paper presents new algorithms to minimize total variation and more generally $l^1$-norms under a general convex constraint. The algorithms are based on a recent advance in convex optimization proposed by Yurii Nesterov. Depending on the regularity of the data fidelity term, we solve either a primal problem, either a dual problem. First we show...

This paper deals with video segmentation based on motion and spatial information. Classically, the motion term is based on a motion compensation error (MCE) between two consecutive frames. Defining a motion-based energy as the integral of a function of the MCE over the object domain implicitly results in making an assumption on the MCE distribution...

In this paper, we focus on statistical region-based active contour models where image features (e.g. intensity) are random variables whose distribution belongs to some parametric family (e.g. exponential) rather than confining ourselves to the special Gaussian case. Using shape derivation tools, our effort focuses on constructing a general expressi...

In this paper, we propose to combine formally noise and shape priors in region-based active contours. On the one hand, we use the general framework of exponential family as a prior model for noise. On the other hand, translation and scale invariant Legendre moments are considered to incorporate the shape prior (e.g. fidelity to a reference shape)....

In this report we propose a new variational method to isolate points in a $2$-dimensional image. To this purpose we introduce a suitable functional whose minimizers are given by the points we want to detect. Then in order to provide numerical experiments we approximate this energy by means of a sequence of more treatable functionals by using a $\Ga...

This paper focuses on the problem,of multiplicative noise removal. We draw our inspiration from the modeling of speckle noise. By using a MAP estimator, we can derive a functional whose minimizer corresponds to the denoised image we want to recover. Although the functional is not convex, we prove the existence of a minimizer and we show the capabil...

A variational approach to image or video segmentation consists in defining an energy depending on local or global image characteristics, the minimum of which being reached for objects of interest. This study focuses on energies written as an integral on a domain of a function which can depend on this domain. The derivative of the energy with respec...

This paper deals with the denoising of SAR images. We draw our inspiration from the modeling of multiplicative speckle noise.
By using a MAP estimator, we propose a functional whose minimizer corresponds to the denoised image we want to recover. Although
the functional is not convex, we prove the existence of a minimizer. Then we study a semi-discr...

Our goal in this paper is to propose a new algorithm for the detection and the completion of thin filaments in noisy and blurred 2D or 3D images. The detection method is based on the construction of a 3D vector field whose singularities (vorticity points) correspond to the filaments. The completion is then obtained by solving a Ginzburg-Landau syst...

This paper deals with video segmentation based on motion and spatial information. Classically, the nucleus of the motion term
is the motion compensation error (MCE) between two consecutive frames. Defining a motion-based energy as the integral of a
function of the MCE over the object domain implicitly results in making an assumption on the MCE dist...

We propose a new unifying method for solving variational problems defined on the Sobolev spaces $W^{1,p}(\Omega)$ or on the space of functions of bounded variations $BV(\Omega)$ ($\Omega\subset\R^N$). The method is based on a recent new characterization of these spaces by Bourgain, Brezis and Mironescu (2001), where norms can be approximated by a s...

Active contours are adapted to image segmentation by energy minimization. The energies often exhibit local minima, requiring regularization. Such an a priori can be expressed as a shape prior and used in two main ways: (1) a shape prior energy is combined with the segmentation energy into a trade-off between prior compliance and accuracy or (2) the...

In this paper, we focus on statistical region-based active contour models where image features (e.g. intensity) are random variables whose distribution belongs to some parametric family (e.g. exponential) rather than confining ourselves to the special Gaussian case. Using shape derivation tools, our effort focuses on constructing a general expressi...

We show the Γ-convergence of a family of discrete functionals to the Mumford and Shah image segmentation functional. The functionals of the family are constructed by modifying the elliptic approximating functionals proposed by Ambrosio and Tortorelli. The quadratic term of the energy related to the edges of the segmentation is replaced by a nonconv...

We developed a method to segment a vector field according to a vector homogeneity defined by a class of field lines, e.g., straight lines, circles... Simultaneously a so-called dominant parameter of the segmented region is determined. The proposed methodology,consists in deriving a domain,energy from the general form of equation of a class of field...

In this paper, we propose to focus on the segmentation of vectorial features (e.g. vector fields or color intensity) using region-based active contours. We search for a domain that minimizes a criterion based on homogeneity measures of the vectorial features. We choose to evaluate, within each region to be segmented, the average quantity of informa...

Our goal in this paper is to give algorithms for minimizing generic regularizing functionals under a $l^\infty$-constraint. We show that many classical models using total variation can be stated under this formalism. Among others are the Rudin-Oscher-Fatemi model, the BV-l1 model, BV-$l^\infty$ model and Meyer's cartoon +texture decomposition model...

Partial differential equations (PDEs) and variational methods were introduced into image processing about fifteen years ago. Since then, intensive research has been carried out. The goals of this book are to present a variety of image analysis applications, the precise mathematics involved and how to discretize them. Thus, this book is intended for...

In this chapter, we propose to concentrate on the research of an optimal domain with regards to a global criterion including region and boundary functionals. A local shape minimizer is obtained through the evolution of a deformable domain in the direction of the shape gradient. Shape derivation tools, coming from shape optimization theory, allow us...

Image inpainting refers to techniques which allow to fill in a gap $\Omega$ given the intensities around it. In this paper, we focus on geometric image inpainting for which several PDE based models have been proposed. Most of them rely on a transport or/and a diffusion equation of the intensity inside $\Omega$. The direction of the reconstructed is...

In this paper, we propose a new mathematical model for detecting in an image singularities of codi- mension greater than or equal to two. This means we want to detect isolated points in a 2-D image or points and curves in a 3-D image. We drew one's inspiration from Ginzburg-Landau (G-L) models which have proved their efficiency for modeling many ph...

This article is a companion paper of a previous work where we have
developed the numerical analysis of a variational model first introduced by Rudin et al. and revisited by Meyer for
removing the noise and capturing textures in an image. The basic idea in this model is
to decompose an image f into two components (u + v) and then to search for (u,v)...

This article is concerned with the problem of finding a regular and homogeneous partition of a Lipschitz open set of R 2 . This type of problem occurs in image classification which consists in assigning a label (that is a class or a phase) to each pixel of an observed image. Such a problem can numerically be solved by using a level set formulation....

- L'objectif est ici la segmentation de régions homogènes en utilisant des caractéristiques vectorielles. L'application visée est la segmentation des régions en mouvement en utilisant les vecteurs mouvement. Afin de caractériser l'homogénéité de la région en utilisant toutes les composantes des vecteurs mouvement, nous estimons puis minimisons l'en...

We construct an algorithm to split an image into a sum u + v of a bounded variation component and a component containing the textures and the noise. This decomposition is inspired from a recent work of Y. Meyer. We find this decomposition by minimizing a convex functional which depends on the two variables u and v, alternately in each variable. Eac...

This paper deals with video and image segmentation using region based active contours. We propose to search for an optimal domain with regards to a criterion based on information measures such as entropy of mutual information. We use a general derivation framework based on the notion of shape gradient. This general derivation is applied to criteria...

We present an anisotropic diffusion equation designed to restore interferometric images. It has two main purposes. the first is to preserve the structures and discontinuities formed by the fringes. The second is to incorporate the noise modeling which is specific to this kind of images. Besides we show that our model formalizes previous related wor...

We devise a new method to remove occlusions in an image by using its level-lines. We take into account the error in the computation of their orientation by introducing a field of probabilities for the level-lines orientations. We use second order partial differential equations for this field and the image to interpolate in the occluded part.

The paper deals with video and image segmentation using region based active contours. We consider the problem of segmentation through the minimization of a new criterion based on information theory. We first propose to derive a general criterion based on the probability density function using the notion of shape gradient. This general derivation is...

This paper deals with image and video segmentation using active contours. The proposed variational approach is based on a criterion featuring a shape prior allowing free-form deformation. The shape prior is defined as a functional of the distance between the active contour and a contour of reference. We develop the complete differentiation of this...

Among image restoration literature, there are mainly two kinds of approach. One is based on a process over image wavelet coefficients, as wavelet shrinkage for denoising. The other one is based on a process over image gradient. In order to get an edge-preserving regularization, one usually assume that the image belongs to the space of functions of...

Image and sequence segmentation of a the segmentation task are discussed from the point of view of optimizing the segmentation criterion. Such a segmentation criterion involves so-called (boundary and region) descriptors, which, in general, may depend on their respective boundaries or regions. This dependency must be taken into account when one is...

The purpose of this paper is to show the theoretical soundness of a variational method proposed in image processing for supervised classification. Based on works developed for phase transitions in fluid mechanics, the classification is obtained by minimizing a sequence of functionals. The method provides an image composed of homogeneous regions wit...

A supervised classification model based on a variational approach is presented. This model is specifically devoted to textured images. We want to get a partition of an image, composed of texture regions separated by regular interfaces. Each kind of texture defines a class. We use a wavelet packet transform to analyze the textures, characterized by...

We consider the problem of image segmentation using active contours through the minimization of an energy criterion involving both region and boundary functionals. These functionals are derived through a shape derivative approach instead of classical calculus of variation. The equations can be elegantly derived without converting the region integra...

In this paper, we address the problem of tracking video objects through several frames. We present a new tracking algorithm using the active contour theory framework. Our first contribution is to define a new tracking criterion combining geometrical and regionbased features. Our second contribution is to provide the di#erentiation of the criterion...

This article deals with image and video segmentation using active contours. The proposed variational approachs is based on a criterion combining geometric prior and statistical features computed on the inside region of the contours. The geometric prior involves a free form deformation from a reference contour as opposed to a parametric transformati...

We consider the problem of image segmentation through the minimization of an energy criterion involving both region and boundary functionals. We study the derivation of these functionals using the notion of shape derivative. From the derivative, we deduce the evolution equation of an active contour that will make it evolve towards a minimum of the...