Within the framework of the 0 regularized least squares problem, we focus, in this paper, on nonconvex continuous penalties approximating the 0-norm. Such penalties are known to better promote sparsity than the 1 convex relaxation. Based on some results in one dimension and in the case of orthogonal matrices, we propose the continuous exact 0 penalty (CEL0) leading to a tight continuous relaxation of the 2 − 0 problem. The global minimizers of the CEL0 functional contain the global minimizers of 2 − 0, and from each global minimizer of CEL0 one can easily identify a global minimizer of 2 − 0. We also demonstrate that from each local minimizer of the CEL0 functional, a local minimizer of 2 − 0 is easy to obtain. Moreover, some strict local minimizers of the initial functional are eliminated with the proposed tight relaxation. Then solving the initial 2 − 0 problem is equivalent, in a sense, to solving it by replacing the 0-norm with the CEL0 which provides better properties for the objective function in terms of minimization, such as the continuity and the convexity with respect to each direction of the standard R N basis, although the problem remains nonconvex. Finally, recent nonsmooth nonconvex algorithms are used to address this relaxed problem within a macro algorithm ensuring the convergence to a critical point of the relaxed functional which is also a (local) optimum of the initial problem.Key words. inverse problems, 0 regularization, sparse modeling, underdetermined linear systems, global minimizers, local minimizers, minimizer equivalence, continuous exact 0 penalty, nonconvex nonsmooth penalty AMS subject classifications. 15A29, 49J99, 90C26, 90C27, 94A08, 94A121. Introduction. In many applications, such as coding (to reduce data storage), compressed sensing (to recover a signal from fewer measurements), source separation, variable selection, and image decomposition, one aims to compute a sparse solution of an underdetermined linear system of equations. In other words, these problems search for an approximation of a signal as a linear combination of redundant dictionary atoms, also known as the synthesis approach. The underlying idea relies on the existence of a representation of the unknown signal involving only a few atoms of the dictionary. This can be modeled using sparsity constraints or penalties. The problem reads as a least squares loss function Ax − d 2 defined from a matrix A ∈ R M ×N and data d ∈ R M plus a sparsity prior usually provided by the

This paper showcases the theoretical and numerical performance of the Sliding Frank-Wolfe, which is a novel optimization algorithm to solve the BLASSO sparse spikes super-resolution problem.The BLASSO is a continuous (i.e. off-the-grid or grid-less) counterpart to the well-known 1 sparse regularisation method (also known as LASSO or Basis Pursuit). Our algorithm is a variation on the classical Frank-Wolfe (also known as conditional gradient) which follows a recent trend of interleaving convex optimization updates (corresponding to adding new spikes) with non-convex optimization steps (corresponding to moving the spikes). Our main theoretical result is that this algorithm terminates in a finite number of steps under a mild non-degeneracy hypothesis. We then target applications of this method to several instances of single molecule fluorescence imaging modalities, among which certain approaches rely heavily on the inversion of a Laplace transform. Our second theoretical contribution is the proof of the exact support recovery property of the BLASSO to invert the 1-D Laplace transform in the case of positive spikes. On the numerical side, we conclude this paper with an extensive study of the practical performance of the Sliding Frank-Wolfe on different instantiations of single molecule fluorescence imaging, including convolutive and non-convolutive (Laplace-like) operators. This shows the versatility and superiority of this method with respect to alternative sparse recovery technics.

Optical diffraction tomography relies on solving an inverse scattering problem governed by the wave equation. Classical reconstruction algorithms are based on linear approximations of the forward model (Born or Rytov), which limits their applicability to thin samples with low refractive-index contrasts. More recent works have shown the benefit of adopting nonlinear models. They account for multiple scattering and reflections, improving the quality of reconstruction. To reduce the complexity and memory requirements of these methods, we derive an explicit formula for the Jacobian matrix of the nonlinear Lippmann-Schwinger model which lends itself to an efficient evaluation of the gradient of the data-fidelity term. This allows us to deploy efficient methods to solve the corresponding inverse problem subject to sparsity constraints.

Single molecule localization microscopy has made great improvements in spatial resolution achieving performance beyond the diffraction limit by sequentially activating and imaging small subsets of molecules. Here, we present an algorithm designed for high-density molecule localization which is of a major importance in order to improve the temporal resolution of such microscopy techniques. We formulate the localization problem as a sparse approximation problem which is then relaxed using the recently proposed CEL0 penalty, allowing an optimization through recent nonsmooth nonconvex algorithms. Finally, performances of the proposed method are compared with one of the best current method for high-density molecules localization on simulated and real data.

A broad class of imaging modalities involve the resolution of an inverse-scattering problem. Among them, three-dimensional optical diffraction tomography (ODT) comes with its own challenges. These include a limited range of views, a large size of the sample with respect to the illumination wavelength, and optical aberrations that are inherent to the system itself. In this work, we present an accurate and efficient implementation of the forward model. It relies on the exact (nonlinear) Lippmann-Schwinger equation. We address several crucial issues such as the discretization of the Green function, the computation of the far field, and the estimation of the incident field. We then deploy this model in a regularized variational-reconstruction framework and show on both simulated and real data that it leads to substantially better reconstructions than the approximate models that are traditionally used in ODT. IntroductionOptical diffraction tomography (ODT) is a noninvasive quantitative imaging modality [1,2]. This label-free technique allows one to determine a three-dimensional map of the refractive index (RI) of samples, which is of particular interest for applications that range from biology [3] to nanotechnologies [4]. The acquisition setup sequentially illuminates the sample from different angles. For each illumination, the outgoing complex wave field (i.e., the scattered field) is recorded by a digital-holography microscope [5,6]. Then, from this set of measurements, the RI of the sample can be reconstructed by solving an inverse-scattering problem. However, its resolution is very challenging due to the nonlinear nature of the interaction between the light and the sample. Related WorksTo simplify the reconstruction problem, pioneering works focused on linearized models. These include Born [1] and Rytov [7] approximations, which are valid for weakly scattering samples [8]. Although originally used to deploy direct inversion methods, these linearized models have been later combined with iterative regularization techniques to improve their robustness to noise and to alleviate the missing-cone problem [9,10].Nonlinear models that adhere more closely to the physic of the acquisition are needed to recover samples with higher variations of their refractive index. For instance, beam-propagation methods (BPM) [11][12][13][14] rely on a slice-by-slice propagation model that accounts for multiple scatterings within the direction of propagation (no reflection). Other nonlinear models include the contrast source-inversion method [15] or the recursive Born approximation [16]. Although more accurate, all these models come at the price of a large computational cost.The theory of scalar diffraction recognizes the Lippmann-Schwinger (LS) model to be the most faithful. It accounts for multiple scatterings, both in transmission and reflection. Iterative forward models that solve the LS equation have

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 square functional and relaxed ones using continuous approximations. In this context, two questions are of fundamental importance: does relaxed functionals preserve global minimizers of the initial one? Does these approximations introduce unwanted new (local) minimizers? In this paper we answer these questions by deriving necessary and sufficient conditions on such 0 continuous approximations in order that each (local and global) minimizer of the underlying relaxation is also a minimizer of the 2-0 functional and that all the global minimizers of the initial functional are preserved. Hence, a general class of penalties is provided giving a unified view of exact continuous approximations of the 0-norm within the 2-0 minimization framework. As the inferior limit of this class of penalties, we get the recently proposed CEL0 penalty. Finally, state of the art penalties, such as MCP, SCAD or Capped-1, are analyzed according to the proposed class of exact continuous penalties.

GlobalBioIm is an open-source MATLAB ® 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-algebra mechanism, GlobalBioIm allows one to easily solve inverse problems by combining elementary modules in a lego-like fashion. This user-friendly toolbox gives access to cutting-edge reconstruction algorithms, while its high modularity makes it easily extensible to new modalities and novel reconstruction methods. We expect GlobalBioIm to respond to the needs of imaging scientists looking for reliable and easy-to-use computational tools for solving their inverse problems. In this paper, we present in detail the structure and main features of the library. We also illustrate its flexibility with examples from multichannel deconvolution microscopy.

Taking benefit from recent advances in both phase retrieval and estimation of refractive indices from holographic measurements, we propose a unified framework to reconstruct them from intensity-only measurements. Our method relies on a generic and versatile formulation of the inverse problem and includes sparsity constraints. Its modularity enables the use of a variety of forward models, from simple linear ones to more sophisticated nonlinear ones, as well as various regularizers. We present reconstructions that deploy either the beam-propagation method or the iterative Lippmann-Schwinger model, combined with total-variation regularization. They suggest that our proposed (intensity-only) method can reach the same performance as reconstructions from holographic (complex) data. This is of particular interest from a practical point of view because it allows one to simplify the acquisition setup.