Single molecule localization by ℓ2−ℓ0\ell_2-\ell_0 constrained optimization

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 $\ell_0$ pseudo-norm using an auxiliary term. In the preliminary experiments with the simulated ISBI datasets the algorithm yields as good results as the state-of-the-art in high-density molecule localization algorithms.Comment: In Proceedings of iTWIST'18, Paper-ID: 13, Marseille, France, November, 21-23, 201

Variational approximation for detecting point-like target problems

International audienceThe 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 Gamma-convergence to the initial one

New algorithm for solving variational problems in W^{1,p}\SO and BV\SO: Application to image restoration

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 sequence of integral operators involving a differential quotient and a suitable sequence of radial mollifiers. We use this characterization to define a variational formulation, for which existence, uniqueness and convergence of the solution is proved. The proposed approximation is valid for any $p$ and does not depend on the attach term. Implementation details are given and we show examples on the image restoration problem

ERRATUM: A Continuous Exact l0 penalty (CEL0) for least squares regularized problem

International audienceLemma 4.4 in [E. Soubies, L. Blanc-Féraud and G. Aubert, SIAM J. Imaging Sci., 8 (2015), pp. 1607-1639] is wrong for local minimizers of the CEL0 functional. The argument used to conclude the proof of this lemma is not sufficient in the case of local minimizers. In this note, we supplya revision of this Lemma where new results are established for local minimizers. Theorem 4.8 in that paper remains unchanged but the proof has to be rewritten according to the new version of the lemma. Finally, some remarks of this paper are also rewritten using the corrected lemma

DREAM²S: Deformable Regions Driven by an Eulerian Accurate Minimization Method for Image and Video Segmentation

This paper deals with image and video segmentation using active contours. We propose a general form for the energy functional related to region-based active contours. We compute the associated evolution equation using shape derivation tools and accounting for the evolving region-based terms. Then we apply this general framework to compute the evolution equation from functionals that include various statistical measures of homogeneity for the region to be segmented. Experimental results show that the determinant of the covariance matrix appears to be a very relevant tool for segmentation of homogeneous color regions. As an example, it has been successfully applied to face segmentation in real video sequences

Active contour segmentation with a parametric shape prior: Link with the shape gradient

International audienceActive 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 segmentation energy is minimized in the space defined by a parametric shape prior. Methods (1) require the tuning of a data-dependent balance parameter and methods (1) and (2) are often dedicated to a specific prior or contour representation, with the prior and segmentation aspects often meshed together, increasing complexity. A general framework for category (2) is proposed: it is independent of the prior and contour representations and it separates the prior and segmentation aspects. It relies on the relationship shown here between the shape gradient, the prior-induced admissible contour transformations, and the segmentation energy minimization

Variational approximation for detecting point-like target problems

International audienceThe 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 Gamma-convergence to the initial one

Variational approximation for a functional governing point-like singularities

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 the initial one