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Principles Of Optics Electromagnetic Theory Of Propagation Interference And Diffraction Of Light

THE theory of Fourier integrals arises out of the elegant pair of reciprocal formulae The Laplace Transform By David Vernon Widder. (Princeton Mathematical Series.) Pp. x + 406. (Princeton: Princeton University Press; London: Oxford University Press, 1941.) 36s. net.

The Laplace Transform

We present an ultrafast, precise, parameter-free method, which we term Deep-STORM, for obtaining super-resolution images from stochastically blinking emitters, such as fluorescent molecules used for localization microscopy. Deep-STORM uses a deep convolutional neural network that can be trained on simulated data or experimental measurements, both of which are demonstrated. The method achieves state-of-the-art resolution under challenging signal-to-noise conditions and high emitter densities and is significantly faster than existing approaches. Additionally, no prior information on the shape of the underlying structure is required, making the method applicable to any blinking dataset. We validate our approach by super-resolution image reconstruction of simulated and experimentally obtained data.

Deep-STORM: super-resolution single-molecule microscopy by deep learning

Imaging through scattering is an important yet challenging problem. Tremendous progress has been made by exploiting the deterministic input–output “transmission matrix” for a fixed medium. However, this “one-to-one” mapping is highly susceptible to speckle decorrelations – small perturbations to the scattering medium lead to model errors and severe degradation of the imaging performance. Our goal here is to develop a new framework that is highly scalable to both medium perturbations and measurement requirement. To do so, we propose a statistical “one-to-all” deep learning (DL) technique that encapsulates a wide range of statistical variations for the model to be resilient to speckle decorrelations. Specifically, we develop a convolutional neural network (CNN) that is able to learn the statistical information contained in the speckle intensity patterns captured on a set of diffusers having the same macroscopic parameter. We then show for the first time, to the best of our knowledge, that the trained CNN is able to generalize and make high-quality object predictions through an entirely different set of diffusers of the same class. Our work paves the way to a highly scalable DL approach for imaging through scattering media.

Deep speckle correlation: a deep learning approach toward scalable imaging through scattering media

Deep learning is becoming an increasingly important tool for image reconstruction in fluorescence microscopy. We review state-of-the-art applications such as image restoration and super-resolution imaging, and discuss how the latest deep learning research could be applied to other image reconstruction tasks. Despite its successes, deep learning also poses substantial challenges and has limits. We discuss key questions, including how to obtain training data, whether discovery of unknown structures is possible, and the danger of inferring unsubstantiated image details.

Applications, promises, and pitfalls of deep learning for fluorescence image reconstruction

Within the framework of the $\ell_0$ regularized least squares problem, we focus, in this paper, on nonconvex continuous penalties approximating the $\ell_0$-norm. Such penalties are known to better promote sparsity than the $\ell_1$ convex relaxation. Based on some results in one dimension and in the case of orthogonal matrices, we propose the continuous exact $\ell_0$ penalty (CEL0) leading to a tight continuous relaxation of the $\ell_2-\ell_0$ problem. The global minimizers of the CEL0 functional contain the global minimizers of $\ell_2 - \ell_0$, and from each global minimizer of CEL0 one can easily identify a global minimizer of $\ell_2 - \ell_0$. We also demonstrate that from each local minimizer of the CEL0 functional, a local minimizer of $\ell_2 - \ell_0$ is easy to obtain. Moreover, some strict local minimizers of the initial functional are eliminated with the proposed tight relaxation. Then solving the initial $\ell_2 - \ell_0$ problem is equivalent, in a sense, to solving it by replacing the ...

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A Continuous Exact $\ell_0$ Penalty (CEL0) for Least Squares Regularized Problem

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.

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The Sliding Frank-Wolfe Algorithm and its Application to Super-Resolution Microscopy

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.

https://hal.inria.fr/hal-01443565/document

High density molecule localization for super-resolution microscopy using CEL0 based sparse approximation

Numerous nonconvex continuous penalties have been proposed to approach the l0 pseudo-norm for optimization purpose. Apart from the theoretical results for convex l1 relaxation under restrictive hypothesis, only few works have been devoted to analyze the consistency, in terms of minimizers, between the l0-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 this approximation 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 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 CEL0 penalty. Finally, state of the art penalties, such as MCP, SCAD or Capped-l1, are analyzed according to the proposed class of exact continuous penalties.

/pdf/a-unified-view-of-exact-continuous-penalties-for-ell-2-ell-0-1b9jga09xj.pdf

A Unified View of Exact Continuous Penalties for $\ell_2$-$\ell_0$ Minimization

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.

https://iopscience.iop.org/article/10.1088/1361-6420/ab2ae9/pdf

Emmanuel Soubies

Papers

A Continuous Exact $\ell_0$ Penalty (CEL0) for Least Squares Regularized Problem

The Sliding Frank-Wolfe Algorithm and its Application to Super-Resolution Microscopy

High density molecule localization for super-resolution microscopy using CEL0 based sparse approximation

A Unified View of Exact Continuous Penalties for $\ell_2$-$\ell_0$ Minimization

Pocket guide to solve inverse problems with GlobalBioIm