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Super‐resolution structured‐illumination microscopy (SIM) is a powerful technique that allows one to surpass the diffraction limit by up to a factor two. Yet, its practical use is hampered by its sensitivity to imaging conditions which makes it prone to reconstruction artefacts. In this work, we present FlexSIM, a flexible SIM reconstruction method capable to handle highly challenging data. Specifically, we demonstrate the ability of FlexSIM to deal with the distortion of patterns, the high level of noise encountered in live imaging, as well as out‐of‐focus fluorescence. Moreover, we show that FlexSIM achieves state‐of‐the‐art performance over a variety of open SIM datasets. [ABSTRACT FROM AUTHOR]

On-grid based direction-of-arrival (DOA) estimation methods rely on the resolution of a difficult group-sparse optimization problem that involves the ℓ2,0 pseudo-norm. In this work, we show that an exact relaxation of this problem can be obtained by replacing the ℓ2,0 term with a group minimax concave penalty with suitable parameters. This relaxation is more amenable to non-convex optimization algorithms as it is continuous and admits less local (not global) minimizers than the initial ℓ2,0-regularized criteria. We then show on numerical simulations that the minimization of the proposed relaxation with an iteratively reweighted ℓ2,1 algorithm leads to an improved performance over traditional approaches. [ABSTRACT FROM AUTHOR]

Single-particle cryo-electron microscopy (cryo-EM) reconstructs the three-dimensional (3D) structure of bio-molecules from a large set of 2D projection images with random and unknown orientations. A crucial step in the single-particle cryo-EM pipeline is 3D refinement, which resolves a high-resolution 3D structure from an initial approximate volume by refining the estimation of the orientation of each projection. In this work, we propose a new approach that refines the projection angles on the continuum. We formulate the optimization problem over the density map and the orientations jointly. The density map is updated using the efficient alternating-direction method of multipliers, while the orientations are updated through a semi-coordinate-wise gradient descent for which we provide an explicit derivation of the gradient. Our method eliminates the requirement for a fine discretization of the orientation space and does away with the classical but computationally expensive template-matching step. Numerical results demonstrate the feasibility and performance of our approach compared to several baselines. [ABSTRACT FROM AUTHOR]

This paper is devoted to the analysis of necessary (not sufficient) optimality conditions for the ℓ 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 algorithms only ensure the convergence to suboptimal points that verify some necessary optimality conditions. The degree of restrictiveness of these conditions is thus directly related to the performance of the algorithms. Within this context, our first goal is to provide a comprehensive review of commonly used necessary optimality conditions as well as known relationships between them. Then, we complete this hierarchy of conditions by proving new inclusion properties between the sets of candidate solutions associated with them. Moreover, we go one step further by providing a quantitative analysis of these sets. Finally, we report the results of a numerical experiment dedicated to the comparison of several algorithms with different optimality guaranties. In particular, this illustrates the fact that the performance of an algorithm is related to the restrictiveness of the optimality condition verified by the point it converges to. [ABSTRACT FROM AUTHOR]

Numerous nonconvex continuous penalties have been proposed to approach the ℓ0 pseudonorm for optimization purpose. Apart from the theoretical results for convex ℓ1 relaxation under restrictive hypotheses, only few works have been devoted to analyze the consistency, in terms of minimizers, between the ℓ0-regularized least squares functional and relaxed ones using continuous approximations. In this context, two questions are of fundamental importance: do relaxed functionals preserve global minimizers of the initial one? Do these approximations introduce unwanted new (local) minimizers? In this paper we answer these questions by deriving necessary and sufficient conditions on such ℓ0 continuous approximations in order that each (local and global) minimizer of the underlying relaxation is also a minimizer of the ℓ2-ℓ0 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 ℓ0-norm within the ℓ2-ℓ0 minimization framework. As the inferior limit of this class of penalties, we get the recently proposed continuous exact ℓ0 penalty. Finally, state of the art penalties, such as minimax concave penalty, smoothly clipped absolute deviation, or capped-ℓ1, are analyzed according to the proposed class of exact continuous penalties. [ABSTRACT FROM AUTHOR]

High resolution imaging of molecules at the cell-substrate interface is required for understanding key biological processes. Here we propose a complete pipeline for multi-angle total internal reflection fluorescence microscopy (MA-TIRF) going from instrument design and calibration procedures to numerical reconstruction. Our custom setup is endowed with a homogeneous field illumination and precise excitation beam angle. Given a set of MA-TIRF acquisitions, we deploy an efficient joint deconvolution/reconstruction algorithm based on a variational formulation of the inverse problem. This algorithm offers the possibility of using various regularizations and can run on graphics processing unit (GPU) for rapid reconstruction. Moreover, it can be easily used with other MA-TIRF devices and we provide it as an open-source software. This ensemble has enabled us to visualize and measure with unprecedented nanometric resolution, the depth of molecular components of the fibronectin assembly machinery at the basal surface of endothelial cells. [ABSTRACT FROM AUTHOR]