Single-molecule localization microscopy methods offer high spatial resolution, but they are not always suitable for live cell imaging due to limited temporal resolution. One strategy is to increase the density of photoactivated molecules present in each image, however suitable analysis algorithms for such data are still lacking. We present 3denseSTORM, a new algorithm for localization microscopy which is able to recover 2D or 3D super-resolution images from a sequence of diffraction limited images with high densities of photoactivated molecules. The algorithm is based on sparse support recovery and uses a Poisson noise model, which becomes critical in low-light conditions. For 3D data reconstruction we use the astigmatism and biplane imaging methods. We derive the theoretical resolution limits of the method and show examples of image reconstructions in simulations and in real 2D and 3D biological samples. The method is suitable for fast image acquisition in densely labeled samples and helps facilitate live cell studies with single molecule localization microscopy.
© 2014 Optical Society of America
Single-molecule localization microscopy (SMLM) represents a family of methods for super-resolution imaging of fluorescent samples which is capable of very high spatial resolution. SMLM methods such as photoactivated localization microscopy (PALM)  and stochastic optical reconstruction microscopy (STORM)  separate molecules into a sequence of (typically) thousands of images of sparsely distributed, stochastically photoactivated molecules so that single molecules can be localized with a precision which is not limited by diffraction. The final super-resolution image is then reconstructed from a list of the coordinates of the single molecules localized throughout the image sequence.
SMLM methods have been demonstrated in live cell imaging . However, SMLM is not always suitable for live cell applications due to limited temporal resolution. Recently, this issue has been addressed by increasing the density of photoactivated molecules in each acquired image [4–17]. This improves temporal resolution as fewer images are required for reconstruction of the final super-resolution image. However, analysis of such images poses another challenge because the images of single photoactivated molecules start to overlap as the density increases. One of the effective approaches for analyzing such data is to utilize sparse support recovery (SSR) methods . Recently, several 2D methods have been introduced that use SSR with Gaussian noise models, such as “compressed sensing STORM” (CSSTORM) , “structured sparse model and Bayesian information criterion” (SSC-BIC) , and “fast localization algorithm based on a continuous-space formulation” (FALCON) . Previously published algorithms for 3D localization in images with high molecular density include those of Babcock, et al. , (astigmatic imaging; maximum-likelihood estimation of PSF) and Gu, et al.  (biplane imaging; SSR).
It is important to recall that low-light images are subject to photon counting noise which follows Poisson statistics . This phenomenon can be a limiting factor for successful image reconstruction, especially when imaging in extreme low-light conditions with short exposure times. However, correctly handling Poisson distributed noise can lead to difficult optimization problems. This is perhaps one reason why most of the current SSR algorithms use a Gaussian noise model.
We introduce a new algorithm for localization microscopy, 3denseSTORM, which is able to reconstruct 3D super-resolution images from a sequence of diffraction limited images with high densities of photoactivated molecules. The algorithm is based on a combination of SSR and maximum-likelihood estimation and uses a Poisson noise model. For 3D imaging we use astigmatism , biplane detection , and dual objective STORM  methods, however the algorithm is flexible enough to allow 3D localization of molecules by other approaches such as the double helix point spread function method . We demonstrate that the proposed algorithm performs well in low-light conditions and with high molecular density, making it suitable for fast image acquisition in densely labeled samples. We also derive the theoretical resolution limits of the method. Finally, we experimentally validated the algorithm on simulated and real data and show that it outperforms the other current methods in the field.
2.1 Problem formulation
In a SMLM experiment, imaging can be described as a convolution of photoactivated fluorophores in the sample (point-sources of light) with a kernel described by the point spread function (PSF) of the microscope. We construct the convolution operation as a matrix multiplication of the signal x by a Toeplitz matrix H in which each column is a shifted copy of a vectorized convolution kernel given by the PSF. Because photon counting processes follow Poisson statistics , the image formed in the camera can be modeled by
To recover an unknown signal x from measurements y acquired by the camera, we need to solve the inverse problem posed by Eq. (1). Suppose that the number of photons collected by a single camera pixel follows the Poisson distribution and that all measurements are independent and identically distributed. The joint density function for all measurements can be expressed as a likelihood function of unknown parameters x and b,
This problem is ill-posed, since n m, and solving it by conventional maximum likelihood estimation methods would not give a stable solution. Following the principles of sparse support recovery , additional information that the signal is sparse, i.e., mostly zeroes, can be utilized by introducing a regularization term. This, together with the fact that the signal x is nonnegative, leads to the choice of an exponential prior
Taking the negative logarithm of the a posteriori probability term and omitting the constant terms leads to the following
This optimization problem is difficult to solve, because the Poisson log-likelihood term does not have a Lipschitz-continuous gradient, therefore the Hessian matrix, which is often used by optimization algorithms, is severely ill-conditioned and the optimization fails to converge. This problem has become a topic of an active research and recently there have been several solutions proposed, mainly based on relaxation or approximation schemes. The simplest approach is to apply a variance stabilizing transform such as the Anscombe transform . Another method uses a linear approximation of the Hessian around the point of evaluation . Other methods apply variable splitting and the augmented Lagrangian method , which together form the basis for the alternating direction of multiplier method (ADMM)  and its close relative, the split Bregman method . There are also other probability-based methods employing maximum-likelihood expectation-maximization algorithms .
2.2 Detection of molecules using sparse support recovery
Our approach is based on ADMM , mainly because of the fast convergence of this algorithm and positive results reported in the literature. Following the ADMM optimization scheme, we decomposed the complex problem in Eq. (5) into a series of simpler sub-problems described by Eqs. (6)-(10).
The optimization process is based on the following iterative scheme, where k = 0, 1, 2, … denotes the iteration. First, the sample backgroundis estimated from the imageusing an iterative low-pass filtering algorithm . Then a vector of weights for the regularization term in Eq. (5) is chosen proportional to the local uncertainty of the background . Because the data follows a Poisson distribution we use Here f(∙) is a function which interpolates the estimated low-resolution background imageto each element of the high-resolution signal x. Next, the iterative scheme alternates between regularized least-squares estimatesof the unknown signal x in Eq. (6) and maximum-likelihood estimatesof the measured image y in Eq. (7). Simultaneously, sparse and non-negative solutions are enforced by Eq. (8). The overall scheme can be written as
The algorithm is initialized with k = 0, and and terminates when there is no further significant improvement of the solution, or when the maximum number of iterations is reached. The user-specified parameters η and μ control the speed of the convergence of the algorithm and play an important role in balancing the solutions of sub-problems in Eqs. (6)-(8). The user-specified parameter β sets the minimum signal to noise ratio of detected molecules. We empirically determined the parameters β = 2.44, η = 9·10−3, and μ = 0.1. In our experience, when analyzing data with very low signal to noise ratios, it may be useful to set the value of η about ten times smaller to slow down the convergence of the algorithm thereby acquiring more detections.
The algorithm can be simply modified to account for a Gaussian noise model by settingin Eq. (7) andin Eq. (9), i.e., by iterating only over Eqs. (6), (8), and (10). This yields the same update scheme that is used in FALCON .
2.3 Extension to 3D (3denseSTORM)
Each column of the measurement matrix H is created as a shifted copy of a vectorized PSF. This can be extended for 3D imaging by creating the measurement matrix such that it contains Toeplitz blocks of vectorized sections of the 3D PSF at different axial positions. This approach is applicable for any PSF model in general, including measured PSFs. Figure 1 shows an example of construction of Toeplitz blocks in the measurement matrix H for 2D and for 3D SMLM imaging methods. In our experiments, the PSF for 2D imaging is created as a rotationally symmetric Gaussian function, while the PSF for 3D astigmatic imaging uses an elliptical Gaussian function with a defocusing curve as described in . For 3D biplane imaging there are two PSFs, one for each plane. Both are modeled as rotationally symmetric Gaussian functions, using a defocusing curve described in .
2.4 Theoretical density limits for resolving molecules
The theoretical upper bound for the density of molecules, at which an algorithm based on sparse support recovery can still correctly resolve two molecules, can be derived based on the restricted isometry property (RIP)  of the measurement matrix H. In general, a matrix is said to satisfy the RIP of order K if there exists a constantsuch that the inequality31] that signals with K non-zero elements can be perfectly recovered withrelaxation if
In practice, it is nearly impossible to compute the constantusing the RIP concept due to high combinatorial complexity. Instead, mutual coherence of the matrix H can be used as it has been shown  that if H has unit-norm columns, then H satisfies the RIP of order K withfor allThe mutual coherence of the matrix H is defined as the largest absolute inner product between any two columns
2.4.1 2D case
The theoretical limit for resolving two molecules in the 2D case can be derived using the mutual coherence defined in Eq. (12). Figure 2 shows a plot of the mutual coherence as a function of multiples of the full-width at half-maximum (FWHM) of PSF for 2D SMLM imaging. Our results indicate that the closest distance at which two molecules can be resolved using a sparse support recovery algorithm based onnorm regularization is approximately 0.78·FWHM. This corresponds to a maximum molecular density of 1.84 / FWHM2.
2.4.2 3D case
The situation in the 3D case is more complicated, because the mutual coherence of the PSF needs to be calculated for every focal plane with respect to all other focal planes. The results are displayed in Fig. 3. As expected, the diameter of the imaged PSF is larger when a molecule is further away from the focal plane and the value of the mutual coherence increases. Consequently, the reconstructed lateral resolution for 3D data is expected to be worse than in the 2D case.
3.1 3denseSTORM reconstruction algorithm
The flowchart in Fig. 4 indicates multiple steps to localize molecules performed by 3denseSTORM. First, the sparse support of the input signal is determined using the SSR scheme in a 3 × oversampled grid with a 100 nm step in the axial direction. Because theregularization term in Eq. (8) introduces a bias towards zero of the recovered molecular intensities, we perform debiasing with a fixed spatial support . This ensures good starting conditions for further processing. Next, approximate positions and intensities of molecules are extracted and continuous refinement is performed by maximum-likelihood estimation according to [15,33]. Finally, post-processing can be applied, such as removal of outliers, drift correction, etc.
The way in which the measurement matrix H is formed is important for practical use, because the full representation of the matrix leads to very large memory consumption (H scales quadratically with the size of the input image) and thus time consuming calculations. Algorithms such as CSSTORM , L1H , or 3D CS analysis , use this full representation which makes them practically impossible to use for analysis of larger images or for 3D analysis where the size of the measurement matrix grows even faster than in the 2D case. 3denseSTORM benefits from the fact that each of the minimization problems in Eqs. (6)-(8) has a closed form solution  and that the calculations involving the measurement matrix can be effectively performed in the frequency domain. Furthermore, the discrete Fourier transform diagonalizes the measurement matrix and thus only the diagonal elements are required for the calculations.
3.2 Simulated 3D SMLM data
We designed Monte-Carlo simulations to quantitatively evaluate the performance of 3denseSTORM in terms of the localization accuracy, detection rate, and recovered molecular density. We generated a series of experiments in which the density of the molecules varied from 0.1 to 20 molecules/μm2 with a step of 0.5 molecules/μm2 between each experiment (i.e., 41 independent simulations). The data set in each experiment contained 100 images 32 × 32 pixels in size. The pixel size was set to 80 nm and the FWHM of in focus molecules was 260 nm. All molecules were placed randomly inside a central region of 20 × 20 pixels within an axial range of −400 nm to + 400 nm. Each molecule was generated with an integral intensity of 2500 photons. Background offset of 70 photons was added to each image and each such image was additionally corrupted with Poisson noise. Data for astigmatic imaging were generated using ThunderSTORM , a comprehensive ImageJ plugin for PALM and STORM data analysis and simulations.
For visual examination of the localization accuracy of the detected molecules, we generated a second simulated 3D SMLM data set with molecules randomly distributed in the shape of a trefoil knot. As in the other simulations, the data consisted of 100 images of size 32 × 32 pixels. Each image contained 30 molecules, where every molecule was modeled with an integral intensity of 2500 photons. Background offset of 70 photons was added to each image and each generated image was corrupted with Poisson noise. The axial position of the set of molecules was modulated by the generating function of a trefoil knot (a sine function) ranging from −400 nm to + 400 nm.
3.3 Cellular samples
U2-OS cells were maintained in DMEM supplemented with 10 % FCS, 100 U/ml penicillin, and 100 U/ml streptomycin (all from Invitrogen, Carlsbad, CA, USA) at 37 °C, 5% CO2, and 100 % humidity. Mowiol containing 1,4-diazabicyclo[2.2.2]octane (DABCO) was from Fluka (St. Louis, Missouri). Cells were grown on high precision #1.5 coverslips (Zeiss, Jena, Germany). Before imaging, cells were first washed with PBS, then fixed with methanol for 5 min at −20 °C. For imaging of tubulin, we labeled the cells with mouse anti tubulin monoclonal antibody (T5168, Sigma, St. Louis, Missouri) for 30 min at room temperature. We then labeled the primary antibodies for 30 min at room temperature with Alexa 532-labeled goat anti mouse secondary antibody (A11002, Invitrogen). After washing with PBS, we finally mounted the coverslips in mowiol containing 100 mM mercaptoethylamine (Sigma) and sealed them onto clean slides with clear nail polish. 100 nm tetraspeck beads for measuring the microscope's PSF and establishing the calibration for 3D imaging were from Invitrogen.
We used an Olympus IX70 microscope equipped with an Olympus planapochromatic 100 × / 1.40 NA oil immersion objective and a NEO sCMOS camera (Andor, Belfast, Northern Ireland). The back-projected CCD pixel size in the sample was 65 nm. A 405 nm, 10 mW diode laser and a 532 nm, 1000 mW DPSS laser (Dragon laser, ChangChun, China) were filtered using bandpass filters and combined with dichroic mirrors (Chroma, Bellows Falls, VT, USA), diffused with a laser speckle reducer (Optotune, Dietikon, Switzerland), then coupled into a 0.39 NA, 600 μm diameter multimode optical fiber (M29L01, Thor Labs, Newton, New Jersey). The fiber output was imaged into the sample using a critical illumination setup. This configuration resulted in an evenly illuminated field. We closed the microscope’s field stop so that only a small area of the sample (~30 μm diameter) was illuminated by the full laser power. We isolated Alexa 532 fluorescence using a TIRF filter set (Chroma). For 3D imaging, we introduced a 500 mm focal length cylindrical lens in front of the sCMOS camera (LJ1144RM-A, Thor Labs). Image sequences were acquired using Andor IQ software. We typically recorded 400 frames with an exposure time of 100 ms. For the sake of simplicity we did not apply any additional corrections for pixel-dependent parameters of the sCMOS camera in any of the evaluated algorithms.
4.1 Quantitative performance evaluation
We used the 3D simulated data set described in Section 3.2 to evaluate the performance of 3denseSTORM in terms of the detection rate (F1-score, recovered density) and localization accuracy (root mean square error between ground truth and localized molecular positions in lateral and axial dimensions). The results were compared to standard single-molecule fitting performed by ThunderSTORM with the default settings  and to 3D DAOSTORM . All three methods use astigmatic imaging and refinement of localized molecules based on maximum-likelihood estimation. The main difference between these algorithms is the detection method for finding the imaged molecules. The performance of 3denseSTORM was also evaluated for biplane imaging. To demonstrate the importance of accounting for a Poisson noise model, we also show the results for 3denseSTORM with a Gaussian noise model as described in Section 2.2.
Detection of a molecule is counted as a true-positive if it is located within 200 nm lateral radius from its ground truth position. The F1-score was calculated according to  and the recovered density as the number of true positive detections per μm2. The localization accuracy was determined as the root mean square of displacements between true positive detections and their ground truth positions.
Our results indicate that single-molecule fitting is not able to recover densities higher than about 3.5 molecules/μm2, while 3D DAOSTORM saturates at about 7.5 molecules/μm2. 3denseSTORM can recover densities up to about 13.5 molecules/μm2 for astigmatic imaging and up to about 15 molecules/μm2 for biplane imaging and thus provides the best detection rate compared to other tested methods, see Fig. 5. The Gaussian version of 3denseSTORM performed slightly worse than the Poisson version. The theoretical limit of 18.6 molecules/μm2 was estimated as described in Section 2.4. It is important to recall that the density limit is directly related to the FWHM of PSF.
All of the evaluated algorithms achieved similar localization accuracy in the lateral direction, but single-molecule fitting consistently had the worst performance, and 3denseSTORM with biplane imaging had the best. In the axial direction, the results for single-molecule fitting are noticeably worse than for the other tested algorithms. 3denseSTORM with biplane imaging achieves a better detection rate and better localization accuracy than the astigmatic approach, which is in agreement with theoretical analysis based on Fisher information theory .
4.2 Visual examination
A second simulated data set with molecules distributed in the shape of a trefoil knot was used to evaluate the performance of the algorithms, see Fig. 6. As expected, the low detection rate of single-molecule fitting methods makes the shape of the reconstructed knot incomplete. This is especially noticeable near the intersections. Moreover, the recovered axial positions of many of the molecules in these areas are not correct. 3D DAOSTORM achieves much higher detection rates and the localization accuracy in the axial direction is better. 3denseSTORM provides the highest detection rate and the localized molecules preserve the shape and continuity of the 3D shape. Also the color-coded axial position of the molecules is in good agreement with the ground-truth visualization of the simulation.
However, there is one key difference between results obtained by astigmatic and biplane imaging methods, see the intersections of the knot in areas indicated by arrows in Fig. 6. Here all algorithms for astigmatic imaging fail to localize molecules correctly in the lower intersection of the knot while biplane imaging gives almost perfect results. We attribute this behavior to a coincidence of the sample structure with orientation of elliptical projections of the imaged PSF in the case of astigmatic imaging. This effect is most significant when vertically oriented ellipses lie along a horizontal structure and horizontally oriented ellipses lie along a vertical structure. This situation leads to large areas of high intensity in the raw data and the localization algorithm fails to assign the correct position to the detected molecules. Biplane imaging is less prone to this kind of problem because the PSF in each imaging plane is rotationally symmetric, making biplane imaging methods more robust in estimating the axial position. To quantify the results, we again measured the detection rate and localization error for each method, see Table 1.
4.3 Real data analysis
To evaluate the quality of 2D image reconstruction with 3denseSTORM, we used a publicly available data set from the single molecule localization challenge website . The data set, “Tubulins-high density,” contributed by Nicolas Olivier, Debora Keller and Suliana Manley, consists of 500 images of 128 × 128 pixels. We compared 3denseSTORM with 3D DAOSTORM and with single molecule fitting performed by ThunderSTORM. We used ThunderSTORM with the default settings. 3D DAOSTORM was run in a 2D mode with the threshold set for high signal-to-noise (SNR) data. The results in Fig. 7 show that both 3D DAOSTORM and 3denseSTORM reconstruct the data well, even after only 100 frames. Single molecule fitting performed by ThunderSTORM suffers from low detection rates. The reconstructed images produced by 3denseSTORM look sharper in high density areas compared to other processing methods, see the region indicated by the yellow square in Fig. 7.
To further evaluate 3denseSTORM for 3D data reconstruction, we acquired a series of images of immuno-labeled microtubules in U2-OS cells using astigmatic imaging. We analyzed two data sets labeled as ROI 1 (70 × 70 pixels) and ROI 2 (47 × 58 pixels), each 400 frames long. We used an exposure time of 100 ms, so each data set was acquired in ~40 seconds. Both data sets suffer from very low SNR due to high background caused mainly by out of focus fluorescence. We adjusted the threshold in 3D DAOSTORM for low SNR data according to recommendations in the software documentation. The results displayed in Fig. 8 show 2D intensity projections to compare how the methods reconstruct sample features. In ROI 1 the arrows indicate areas of high molecular density where 3denseSTORM clearly outperforms the other two algorithms. In ROI 2 the line segments indicate where the intensity profiles were plotted. 3denseSTORM successfully resolved two parallel microtubules while the other methods did not. We also visualized the result of 3denseSTORM as a 3D image with color-coded z-coordinates in Fig. 8.
To provide insight into computational complexity of 3denseSTORM, we measured execution times of all three tested algorithms while processing the real data sets from Figs. 7 and 8. The comparison was performed on a standard PC with Intel Core i5-3570 CPU and with NVIDIA GeForce GTX 670 GPU. The GPU was used to accelerate computation of the fast Fourier transforms used in 3denseSTORM. Data processing was performed in 4 parallel threads (one per each core of the CPU). The results are summarized in Table 2.
Previously, application of sparse support recovery (SSR) methods for processing single-molecule localization microscopy (SMLM) data was shown to outperform single-molecule or multi-emitter fitting approaches, offering better recovery of molecules at high densities and lower localization errors . This effectively enables imaging with a higher density of molecules, so that a super-resolution image can be reconstructed from far fewer images compared to single-molecule or multi-emitter fitting approaches. This in turn helps facilitate live cell imaging as data can be acquired much faster, potentially capturing dynamic, 3D movements in the sample.
Here we proposed a new algorithm, 3denseSTORM, aimed at processing SMLM data with high molecular density. The algorithm is capable of processing both 2D and 3D single-molecule data and can be extended for any kind of PSF. Unlike most other currently available methods based on SSR, 3denseSTORM is memory efficient (representation of the full measurement matrix is avoided), computationally efficient (closed-form solutions of all minimization problems are utilized), and uses a Poisson noise model in both the molecule detection and position refinement steps. Together, this allows for processing of larger 3D images acquired under the extreme low light conditions which are encountered when imaging single molecules. The algorithm is also robust to relatively high background levels and low SNR.
We verified the performance of 3denseSTORM using extensive Monte-Carlo simulations with 3D SMLM data. The results showed that 3denseSTORM can recover the true molecular density about 60% better than 3D DAOSTORM. Also the detection rate as measured by the F1-score and the localization accuracy in both lateral and axial directions were improved compared to other processing algorithms. We also derived the theoretical limits for the maximum density of molecules which can be recovered by methods based on SSR and we showed that 3denseSTORM achieved detection rates close to this limit.
This work was supported by the Czech Science Foundation (P302/12/G157, P205/12/P392, 14-15272P), by Charles University in Prague (PRVOUK P27/LF1/1 and UNCE 204022), by OPPK CZ.2.16/3.1.00/24010 and OPVK CZ.1.07/2.3.00/30.0030, and by the Biotechnology and Biomedicine Center of the Academy of Sciences and Charles University in Vestec.
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