Hagai Kirshner

Quantitative evaluation of software packages for single-molecule localization microscopy.

The quality of super-resolution images obtained by single-molecule localization microscopy (SMLM) depends largely on the software used to detect and accurately localize point sources. In this work, we focus on the computational aspects of super-resolution microscopy and present a comprehensive evaluation of localization software packages. Our philosophy is to evaluate each package as a whole, thus maintaining the integrity of the software. We prepared synthetic data that represent three-dimensional structures modeled after biological components, taking excitation parameters, noise sources, point-spread functions and pixelation into account. We then asked developers to run their software on our data; most responded favorably, allowing us to present a broad picture of the methods available. We evaluated their results using quantitative and user-interpretable criteria: detection rate, accuracy, quality of image reconstruction, resolution, software usability and computational resources. These metrics reflect the various tradeoffs of SMLM software packages and help users to choose the software that fits their needs.

3-D PSF fitting for fluorescence microscopy: implementation and localization application

Localization microscopy relies on computationally efficient Gaussian approximations of the point spread function for the calculation of fluorophore positions. Theoretical predictions show that under specific experimental conditions, localization accuracy is significantly improved when the localization is performed using a more realistic model. Here, we show how this can be achieved by considering three‐dimensional (3‐D) point spread function models for the wide field microscope. We introduce a least‐squares point spread function fitting framework that utilizes the Gibson and Lanni model and propose a computationally efficient way for evaluating its derivative functions. We demonstrate the usefulness of the proposed approach with algorithms for particle localization and defocus estimation, both implemented as plugins for ImageJ.

FALCON: fast and unbiased reconstruction of high-density super-resolution microscopy data

Super resolution microscopy such as STORM and (F)PALM is now a well known method for biological studies at the nanometer scale. However, conventional imaging schemes based on sparse activation of photo-switchable fluorescent probes have inherently slow temporal resolution which is a serious limitation when investigating live-cell dynamics. Here, we present an algorithm for high-density super-resolution microscopy which combines a sparsity-promoting formulation with a Taylor series approximation of the PSF. Our algorithm is designed to provide unbiased localization on continuous space and high recall rates for high-density imaging, and to have orders-of-magnitude shorter run times compared to previous high-density algorithms. We validated our algorithm on both simulated and experimental data, and demonstrated live-cell imaging with temporal resolution of 2.5 seconds by recovering fast ER dynamics.

A Unified Formulation of Gaussian Versus Sparse Stochastic Processes—Part II: Discrete-Domain Theory

This paper is devoted to the characterization of an extended family of continuous-time autoregressive moving average (CARMA) processes that are solutions of stochastic differential equations driven by white Lévy innovations. These are completely specified by: 1) a set of poles and zeros that fixes their correlation structure and 2) a canonical infinitely divisible probability distribution that controls their degree of sparsity (with the Gaussian model corresponding to the least sparse scenario). The generalized CARMA processes are either stationary or nonstationary, depending on the location of the poles in the complex plane. The most basic nonstationary representatives (with a single pole at the origin) are the Lévy processes, which are the non-Gaussian counterparts of Brownian motion. We focus on the general analog-to-discrete conversion problem and introduce a novel spline-based formalism that greatly simplifies the derivation of the correlation properties and joint probability distributions of the discrete versions of these processes. We also rely on the concept of generalized increment process, which suppresses all long range dependencies, to specify an equivalent discrete-domain innovation model. A crucial ingredient is the existence of a minimally supported function associated with the whitening operator L; this B-spline, which is fundamental to our formulation, appears in most of our formulas, both at the level of the correlation and the characteristic function. We make use of these discrete-domain results to numerically generate illustrative examples of sparse signals that are consistent with the continuous-domain model.

On the Approximation of

Most signal processing applications are based on discrete-time signals although the origin of many sources of information is analog. In this paper, we consider the task of signal representation by a set of functions. Focusing on the representation coefficients of the original continuous-time signal, the question considered herein is to what extent the sampling process keeps algebraic relations, such as inner product, intact. By interpreting the sampling process as a bounded operator, a vector-like interpretation for this approximation problem has been derived, giving rise to an optimal discrete approximation scheme different from the Riemann-type sum often used. The objective of this optimal scheme is in the min-max sense and no bandlimitedness constraints are imposed. Tight upper bounds on this optimal and the Riemann-type sum approximation schemes are then derived. We further consider the case of a finite number of samples and formulate a closed-form solution for such a case. The results of this work provide a tool for finding the optimal scheme for approximating an L2 inner product, and to determine the maximum potential representation error induced by the sampling process. The maximum representation error can also be determined for the Riemann-type sum approximation scheme. Examples of practical applications are given and discussed

L_{2}

Many sources of information are of analog or continuous-time nature. However, digital signal processing applications rely on discrete data. We consider the problem of approximating L2 inner products, i.e., representation coefficients of a continuous-time signal, from its generalized samples. Adopting a robust approach, we process these generalized samples in a minimax optimal sense. Specifically, we minimize the worst approximation error of the desired representation coefficients by proper processing of the given sample sequence. We then extend our results to criteria which incorporate smoothness constraints on the unknown function. Finally, we compare our methods with the piecewise-constant approximation technique, commonly used for this problem, and discuss the possible improvements by the suggested schemes.

Inner Products From Sampled Data

In this work, we investigate the relationship between continuous-time autoregressive (AR) models and their sampled version. We consider uniform sampling and derive criteria for uniquely determining the continuous-time parameters from sampled data; the model order is assumed to be known. We achieve this by removing a set of measure zero from the collection of all AR models and by investigating the asymptotic behavior of the remaining set of autocorrelation functions. We provide necessary and sufficient conditions for uniqueness of general AR models, and we demonstrate the usefulness of this result by considering particular examples. We further exploit our theory and introduce an estimation algorithm that recovers continuous-time AR parameters from sampled data, regardless of the sampling interval. We demonstrate the usefulness of our algorithm for various Gaussian and non-Gaussian AR processes.

Minimax Approximation of Representation Coefficients From Generalized Samples

The problem of estimating continuous-domain autoregressive moving-average processes from sampled data is considered. The proposed approach incorporates the sampling process into the problem formulation while introducing exponential models for both the continuous and the sampled processes. We derive an exact evaluation of the discrete-domain power-spectrum using exponential B-splines and further suggest an estimation approach that is based on digitally filtering the available data. The proposed functional, which is related to Whittles likelihood function, exhibits several local minima that originate from aliasing. The global minimum, however, corresponds to a maximum-likelihood estimator, regardless of the sampling step. Experimental results indicate that the proposed approach closely follows the Cramér-Rao bound for various aliasing configurations.

On the Unique Identification of Continuous-Time Autoregressive Models From Sampled Data

A Sobolev reproducing-kernel Hilbert space approach to image interpolation is introduced. The underlying kernels are exponential functions and are related to stochastic autoregressive image modeling. The corresponding image interpolants can be implemented effectively using compactly-supported exponential B-splines. A tight l2 upper-bound on the interpolation error is then derived, suggesting that the proposed exponential functions are optimal in this regard. Experimental results indicate that the proposed interpolation approach with properly-tuned, signal-dependent weights outperforms currently available polynomial B-spline models of comparable order. Furthermore, a unified approach to image interpolation by ideal and nonideal sampling procedures is derived, suggesting that the proposed exponential kernels may have a significant role in image modeling as well. Our conclusion is that the proposed Sobolev-based approach could be instrumental and a preferred alternative in many interpolation tasks.

A Sampling Theory Approach for Continuous ARMA Identification

We consider the problem of sampling continuous-time auto-regressive processes on a uniform grid. We investigate whether a given sampled process originates from a single continuous-time model, and address this uniqueness problem by introducing an alternative description of poles in the complex plane. We then utilize Kroneckers approximation theorem and prove that the set of non-unique continuous-time AR(2) models has Lebesgue measure zero in this plane. This is a key aspect in current estimation algorithms that use sampled data, as it allows one to remove the sampling rate constraint that is imposed currently.