Faster STORM using compressed sensing
TL;DR: A sparse-signal recovery technique using compressed sensing to analyze images with highly overlapping fluorescent spots that allows an activated fluorophore density an order of magnitude higher than what conventional single-molecule fitting methods can handle.
Abstract: In super-resolution microscopy methods based on single-molecule switching, the rate of accumulating single-molecule activation events often limits the time resolution. Here we developed a sparse-signal recovery technique using compressed sensing to analyze images with highly overlapping fluorescent spots. This method allows an activated fluorophore density an order of magnitude higher than what conventional single-molecule fitting methods can handle. Using this method, we demonstrated imaging microtubule dynamics in living cells with a time resolution of 3 s.
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TL;DR: This report reports multiplexed error-robust FISH (MERFISH), a single-molecule imaging method that allows thousands of RNA species to be imaged in single cells by using combinatorial FISH labeling with encoding schemes capable of detecting and/or correcting errors.
Abstract: INTRODUCTION: The copy number and in- tracellular localization of RNA are important regulators of gene expression. Measurement of these properties at the transcriptome scale in single cells will give answers to many ques- tions related to gene expression and regulation. Single-molecule RNA imaging approaches, such as single-molecule fluorescence in situ hybrid- ization(smFISH), are powerful toolsforcount- ing and mappingRNA; however, the number of RNA species that can be simultaneously im- aged in individual cells has been limited. This makes it challenging to perform transcriptomic analysis of single cells in a spatially resolved manner. Here, we report multiplexed error- robust FISH (MERFISH), a single-molecule im- aging method that allows thousands of RNA species to be imaged in single cells by using combinatorial FISH labeling with encoding schemes capable of detecting and/or correct- ing errors. RATIONALE: We labeled each cellular RNA with a set of encoding probes, which contain targeting sequences that bind the RNA and readout sequences that bind fluorescently la- beled readout probes. Each RNA species is encodedwithaparticular combinationofread- out sequences. We used successive rounds of hybridization and imaging, each with a differ- ent readout probe, to identify the readout se- quences bound to each RNA and to decode the RNA. In principle, combinatorial labeling al- lows the number of detectable RNA species to
1,576 citations
Cites methods from "Faster STORM using compressed sensi..."
...By using more advanced image analysis algorithms to better resolve overlapping images of individual molecules, such as compressed sensing (39, 40), it would be possible to extend the resolvable density by approximately fourfold and thus allow nearly the entire transcriptome, except for the top 2% most expressed genes, to be imaged all together....
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TL;DR: In this Review,luorescence nanoscopy uniquely combines minimally invasive optical access to the internal nanoscale structure and dynamics of cells and tissues with molecular detection specificity and the labelling of individual molecules to enable their visualization has emerged as a central challenge.
Abstract: Fluorescence nanoscopy uniquely combines minimally invasive optical access to the internal nanoscale structure and dynamics of cells and tissues with molecular detection specificity. While the basic physical principles of 'super-resolution' imaging were discovered in the 1990s, with initial experimental demonstrations following in 2000, the broad application of super-resolution imaging to address cell-biological questions has only more recently emerged. Nanoscopy approaches have begun to facilitate discoveries in cell biology and to add new knowledge. One current direction for method improvement is the ambition to quantitatively account for each molecule under investigation and assess true molecular colocalization patterns via multi-colour analyses. In pursuing this goal, the labelling of individual molecules to enable their visualization has emerged as a central challenge. Extending nanoscale imaging into (sliced) tissue and whole-animal contexts is a further goal. In this Review we describe the successes to date and discuss current obstacles and possibilities for further development.
726 citations
TL;DR: This work introduces a measure based on Fourier ring correlation (FRC) that can be computed directly from an image and demonstrates its validity and benefits on two-dimensional (2D) and 3D localization microscopy images of tubulin and actin filaments.
Abstract: Resolution in optical nanoscopy (or super-resolution microscopy) depends on the localization uncertainty and density of single fluorescent labels and on the sample's spatial structure. Currently there is no integral, practical resolution measure that accounts for all factors. We introduce a measure based on Fourier ring correlation (FRC) that can be computed directly from an image. We demonstrate its validity and benefits on two-dimensional (2D) and 3D localization microscopy images of tubulin and actin filaments. Our FRC resolution method makes it possible to compare achieved resolutions in images taken with different nanoscopy methods, to optimize and rank different emitter localization and labeling strategies, to define a stopping criterion for data acquisition, to describe image anisotropy and heterogeneity, and even to estimate the average number of localizations per emitter. Our findings challenge the current focus on obtaining the best localization precision, showing instead how the best image resolution can be achieved as fast as possible.
649 citations
Lawrence Livermore National Laboratory1, Harvard University2, Massachusetts Institute of Technology3, University of the Basque Country4, University of Liège5, Forschungszentrum Jülich6, Queen's University Belfast7, University of Coimbra8, University of Zaragoza9, Martin Luther University of Halle-Wittenberg10, Max Planck Society11
TL;DR: This article discusses how the real-space approach has allowed for the recent development of new ideas for the simulation of electronic systems, and the exact solution of the Schrödinger equation for low-dimensionality systems.
Abstract: Real-space grids are a powerful alternative for the simulation of electronic systems. One of the main advantages of the approach is the flexibility and simplicity of working directly in real space where the different fields are discretized on a grid, combined with competitive numerical performance and great potential for parallelization. These properties constitute a great advantage at the time of implementing and testing new physical models. Based on our experience with the Octopus code, in this article we discuss how the real-space approach has allowed for the recent development of new ideas for the simulation of electronic systems. Among these applications are approaches to calculate response properties, modeling of photoemission, optimal control of quantum systems, simulation of plasmonic systems, and the exact solution of the Schrodinger equation for low-dimensionality systems.
436 citations
TL;DR: The foundations of SPT are described together with novel optical implementations that nowadays allow the investigation of single molecule dynamic events with increasingly high spatiotemporal resolution using molecular densities closer to physiological expression levels.
Abstract: Optical microscopy has for centuries been a key tool to study living cells with minimum invasiveness The advent of single molecule techniques over the past two decades has revolutionized the field of cell biology by providing a more quantitative picture of the complex and highly dynamic organization of living systems Amongst these techniques, single particle tracking (SPT) has emerged as a powerful approach to study a variety of dynamic processes in life sciences SPT provides access to single molecule behavior in the natural context of living cells, thereby allowing a complete statistical characterization of the system under study In this review we describe the foundations of SPT together with novel optical implementations that nowadays allow the investigation of single molecule dynamic events with increasingly high spatiotemporal resolution using molecular densities closer to physiological expression levels We outline some of the algorithms for the faithful reconstruction of SPT trajectories as well as data analysis, and highlight biological examples where the technique has provided novel insights into the role of diffusion regulating cellular function The last part of the review concentrates on different theoretical models that describe anomalous transport behavior and ergodicity breaking observed from SPT studies in living cells
426 citations
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TL;DR: In this paper, the authors considered the model problem of reconstructing an object from incomplete frequency samples and showed that with probability at least 1-O(N/sup -M/), f can be reconstructed exactly as the solution to the lscr/sub 1/ minimization problem.
Abstract: This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal f/spl isin/C/sup N/ and a randomly chosen set of frequencies /spl Omega/. Is it possible to reconstruct f from the partial knowledge of its Fourier coefficients on the set /spl Omega/? A typical result of this paper is as follows. Suppose that f is a superposition of |T| spikes f(t)=/spl sigma//sub /spl tau//spl isin/T/f(/spl tau/)/spl delta/(t-/spl tau/) obeying |T|/spl les/C/sub M//spl middot/(log N)/sup -1/ /spl middot/ |/spl Omega/| for some constant C/sub M/>0. We do not know the locations of the spikes nor their amplitudes. Then with probability at least 1-O(N/sup -M/), f can be reconstructed exactly as the solution to the /spl lscr//sub 1/ minimization problem. In short, exact recovery may be obtained by solving a convex optimization problem. We give numerical values for C/sub M/ which depend on the desired probability of success. Our result may be interpreted as a novel kind of nonlinear sampling theorem. In effect, it says that any signal made out of |T| spikes may be recovered by convex programming from almost every set of frequencies of size O(|T|/spl middot/logN). Moreover, this is nearly optimal in the sense that any method succeeding with probability 1-O(N/sup -M/) would in general require a number of frequency samples at least proportional to |T|/spl middot/logN. The methodology extends to a variety of other situations and higher dimensions. For example, we show how one can reconstruct a piecewise constant (one- or two-dimensional) object from incomplete frequency samples - provided that the number of jumps (discontinuities) obeys the condition above - by minimizing other convex functionals such as the total variation of f.
14,587 citations
TL;DR: This work introduced a method for optically imaging intracellular proteins at nanometer spatial resolution and used this method to image specific target proteins in thin sections of lysosomes and mitochondria and in fixed whole cells to image retroviral protein Gag at the plasma membrane.
Abstract: We introduce a method for optically imaging intracellular proteins at nanometer spatial resolution. Numerous sparse subsets of photoactivatable fluorescent protein molecules were activated, localized (to approximately 2 to 25 nanometers), and then bleached. The aggregate position information from all subsets was then assembled into a superresolution image. We used this method--termed photoactivated localization microscopy--to image specific target proteins in thin sections of lysosomes and mitochondria; in fixed whole cells, we imaged vinculin at focal adhesions, actin within a lamellipodium, and the distribution of the retroviral protein Gag at the plasma membrane.
7,924 citations
TL;DR: A high-resolution fluorescence microscopy method based on high-accuracy localization of photoswitchable fluorophores that can, in principle, reach molecular-scale resolution is developed.
Abstract: We have developed a high-resolution fluorescence microscopy method based on high-accuracy localization of photoswitchable fluorophores. In each imaging cycle, only a fraction of the fluorophores were turned on, allowing their positions to be determined with nanometer accuracy. The fluorophore positions obtained from a series of imaging cycles were used to reconstruct the overall image. We demonstrated an imaging resolution of 20 nm. This technique can, in principle, reach molecular-scale resolution.
7,213 citations
TL;DR: In this paper, the authors considered the problem of recovering a vector x ∈ R^m from incomplete and contaminated observations y = Ax ∈ e + e, where e is an error term.
Abstract: Suppose we wish to recover a vector x_0 Є R^m (e.g., a digital signal or image) from incomplete and contaminated observations y = Ax_0 + e; A is an n by m matrix with far fewer rows than columns (n « m) and e is an error term. Is it possible to recover x_0 accurately based on the data y?
To recover x_0, we consider the solution x^# to the l_(1-)regularization problem min ‖x‖l_1 subject to ‖Ax - y‖l(2) ≤ Є, where Є is the size of the error term e. We show that if A obeys a uniform uncertainty principle (with unit-normed columns) and if the vector x_0 is sufficiently sparse, then the solution is within the noise level ‖x^# - x_0‖l_2 ≤ C Є. As a first example, suppose that A is a Gaussian random matrix; then stable recovery occurs for almost all such A's provided that the number of nonzeros of x_0 is of about the same order as the number of observations. As a second instance, suppose one observes few Fourier samples of x_0; then stable recovery occurs for almost any set of n coefficients provided that the number of nonzeros is of the order of n/[log m]^6. In the case where the error term vanishes, the recovery is of course exact, and this work actually provides novel insights into the exact recovery phenomenon discussed in earlier papers. The methodology also explains why one can also very nearly recover approximately sparse signals.
6,727 citations
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TL;DR: It is shown that it is possible to recover x0 accurately based on the data y from incomplete and contaminated observations.
Abstract: Suppose we wish to recover an n-dimensional real-valued vector x_0 (e.g. a digital signal or image) from incomplete and contaminated observations y = A x_0 + e; A is a n by m matrix with far fewer rows than columns (n << m) and e is an error term. Is it possible to recover x_0 accurately based on the data y?
To recover x_0, we consider the solution x* to the l1-regularization problem min \|x\|_1 subject to \|Ax-y\|_2 <= epsilon, where epsilon is the size of the error term e. We show that if A obeys a uniform uncertainty principle (with unit-normed columns) and if the vector x_0 is sufficiently sparse, then the solution is within the noise level \|x* - x_0\|_2 \le C epsilon.
As a first example, suppose that A is a Gaussian random matrix, then stable recovery occurs for almost all such A's provided that the number of nonzeros of x_0 is of about the same order as the number of observations. Second, suppose one observes few Fourier samples of x_0, then stable recovery occurs for almost any set of p coefficients provided that the number of nonzeros is of the order of n/[\log m]^6. In the case where the error term vanishes, the recovery is of course exact, and this work actually provides novel insights on the exact recovery phenomenon discussed in earlier papers. The methodology also explains why one can also very nearly recover approximately sparse signals.
6,226 citations