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Super-resolution localization microscopy has revolutionized the observation of living structures at the cellular scale, by achieving a spatial resolution that is improved by more than an order of magnitude compared to the diffraction limit. These methods localize single events from isolated sources in repeated cycles in order to achieve super-resolution. The requirement for sparse distribution of simultaneously activated sources in the field of view dictates the acquisition of thousands of frames in order to construct the full super-resolution image. As a result, these methods have slow temporal resolution which is a major limitation when investigating live-cell dynamics. In this paper we present the use of a phase stretch transform for high-density super-resolution localization microscopy. This is a nonlinear frequency dependent transform that emulates the propagation of light through a physical medium with a specific warped diffractive property and applies a 2D phase function to the image in the frequency domain. By choosing properly the transform parameters and the phase kernel profile, the point spread function of each emitter can be sharpened and narrowed. This enables the localization of overlapping emitters, thus allowing a higher density of activated emitters as well as shorter data collection acquisition rates. The method is validated by numerical simulations and by experimental data obtained using a microtubule sample.
© 2016 Optical Society of America
1. Introduction
Localization microscopy super-resolution (SR) techniques [1–3] have led to a striking increase in resolution by more than an order of magnitude with compare to conventional fluorescence microscopy, which is limited by diffraction [4] to length scales set by the Rayleigh criterion of approximately half the wavelength of light [5,6]. Examples for these techniques are photoactivated localization microscopy (PALM) and stochastic optical reconstruction microscopy (STORM) [1,7]. The difference between these techniques is in the fluorophores used for the labeling of the sample. In PALM, photo-switchable fluorescent proteins are used. STORM uses a pair of antibodies tagged with organic fluorophores. One molecule of the pair (called activator), when excited near its absorption maximum, serves to reactivate the other molecule (called reporter) to the fluorescent state. This requires two illumination beams. Direct STORM (dSTORM) is a variation on the basic concept of STORM, but instead of paired photoswitchers, the technique is able to use conventional standalone carbocyanine dyes for organic fluorophores [8]. This means that a single illumination source is required. The post possessing and super resolution image generation for all of these localization microscopy techniques is identical.
Localization microscopy achieves SR imaging in three steps. The first step is to image light-activated fluorescent molecules that act as tiny point sources of light. Since these objects are smaller than the microscope resolution, they will appear as a diffraction-limited spots with a point spread function (PSF) given by an Airy function. The use of low light intensities and the fact that the molecules' activation is inherently random ensures that only a sparse subset is turned on at each time. Thus, these point-like light sources are well separated, so the PSF image of each one does not overlap with that of its neighbors.
The second step is localization in which the exact position of each point-like source is determined by finding the center of the PSF. This is possible for well-separated sources, because the shape of the PSFs is known in advance. Each imaged PSF is fitted to an ideal PSF using algorithms such as non-linear least squares [9,10], and maximum likelihood [11]. It is accomplished by establishing an intensity threshold value that distinguishes background from signal and determines whether a certain PSF exceeds this value. If so, the data is fitted to a model of a Gaussian profile with a single peak. If two PSFs are in close proximity such that the saddle between them is higher than the threshold, they will be indistinguishable and be considered as one, resulting in an increased localization error. The final step is to repeat the illumination and localization steps for thousands of times. A different set of separated point-like sources is detected in each time, until a sufficient density of source points has been obtained. By summing all the positions of the detected PSFs, an SR image can be built up [12]. The spatial resolution in this image can exceed the diffraction limit, because it is determined by the accuracy with which the position of each source can be estimated. The theoretically achievable localization precision for a single fluorophore is limited by the statistical shot noise that scales with the inverse square root of the number of photons collected from each fluorophore and also by the background noise [13,14]. The achievable spatial resolution reaches approximately 20 nm, an order of magnitude improvement with comparison to the diffraction limit [15–18].
The computational load associated with the analysis of the acquired data and the localization process is very high. The localization of overlapping emitters is a difficult task that can be time consuming and therefore conventional approaches localize only single emitter events that are spatially separated by a distance greater than several times the width of their PSFs [16]. This requirement dictates experimental conditions with low activation density across the field of view, thus increases the acquisition time, to order of minutes [17]. This time scale sets a serious limitation for live-cell dynamics investigation purposes.
To address this problem of overlapping emitters’ localization, several methods were recently reported. One category of methods includes statistical deconvolution techniques that iterates through the observed PSF with a guess-work of overlapping PSFs combined with different methods of compressed sensing [19] and Taylor series approximation [20]. All of these approaches are slow and requires ~10 times more computation time/frame than the other methods of single-emitter fitting [21]. Other methods use algorithms designed for graphics processing unit (GPU) analysis, and as such are relatively fast. They use the maximum likelihood technique with increasing numbers of point sources within the recorded PSF in the localization algorithm. The processing time is on the order of minutes, however it is still not fast enough [22–24].
Here we present the use of the phase stretch transform (PST) that emulates diffraction by using an all-pass phase filter with specific frequency dispersion dependencies. The output phase profile in spatial domain reveals variations in image intensity that corresponds to the PSFs locations, with a sharper profile. This transform has its roots in the Photonic Time Stretch, a temporal signal processing technique that employs temporal dispersion to slow down, capture, and digitally process fast waveforms in real time which led to the discovery of optical rogue waves and detection of cancer cells in blood with record sensitivity [25,26]. In addition it was successfully employed in analog to digital conversion [27], data compression [28] and edge detection [29]. The generalized time stretch transform encompasses linear and nonlinear dispersion as well as near and far field propagation regimes. Impact of such transformations on a signal can be understood in terms of the symmetry of the dispersion profile with even symmetry, such as the quadratic profile, producing edge detection [30].
This paper employs this transform to high density SR localization microscopy, with the aim of achieving the best possible spatial resolution for a given number of frames. The use of spatial domain filters for SR localization microscopy was previously done with a wavelet filter [31]. The wavelet segmentation algorithm for particle detection and centroid localization yields an order of magnitude improvement in the image processing time with compare to a Gaussian fitting. The localization algorithm is based on the segmentation and calculation of the centroid of each region. It is a fast technique, however the obtainable localization precision neither for an isolated emitter case nor for the overlapping emitters one, reach that of a Gaussian fitting. The PST algorithm yields the same performance as a Gaussian fitting for an isolated emitter, and for the case of overlapping emitters the PST enables the PSFs localization at a precision similar to an isolated emitter.
The application of the PST method on each frame prior to localization, effectively sharpens the PSF peak and narrows it, such that overlapping emitters can be localized as isolated point sources. The ability to localize overlapping PSFs will allow a higher density of activated emitters and shorter acquisition times. In addition, in cases where the captured image contains non-uniform intensity PSFs, the method will also highlight low intensity PSFs that can otherwise be discarded during the thresholding process in a conventional localization microscopy routine.
The PST is applied to raw image data prior to performing the emitter localization. This processing is performed in the Fourier domain and as such it is very fast, requiring only microseconds of processing per frame, therefore suitable for live cell imaging purposes.
The paper is organized as follows: First, the theoretical background for imaging of single molecules, and the PST method are presented. Next, numerical simulations for a variety of parameters are presented and fully discussed. Third, the proposed technique is applied on experimentally acquired dSTORM images to validate the approach.
2. Theoretical background
The ideal PSF of an aberration-free defocused imaging system with a finite lens aperture is given by an Airy function [32]. A mathematically simpler model is a Gaussian function [33]. The standard deviation of the Gaussian, σ, is given by setting the e−1 point of the intensity model to be equal to the Rayleigh radius, and is equal to
Where N.A. is the numerical aperture of the objective and λ is the wavelength of the emitted light.The captured intensity image is degraded by added photon shot-noise, which is a Poisson process, ηshot, with an expected value that corresponds to the noiseless pixel values and a standard deviation that equals the square root of the value of each pixel. Another additive noise is the background noise which is described by a Poisson distributed random variable, ηB, with variance Nb (assumed constant across the field of view) [16].
The signal to noise ratio (SNR) is determined by the ratio between the pure data and the additive noises (ηshot and ηb), and typical values for localization microscopy are between 5 and 100 [34].
PST emulates to propagation of electromagnetic waves through a diffractive medium with an engineered dielectric function. The result is a physics-inspired digital signal processing technique that achieves multidimensional warped Fourier domain sampling function. The magnitude of the transformed data leads to non-uniform Fourier transformation. Localization microscopy images contain isolated PSFs with a Gaussian shape. In the frequency domain it remains a Gaussian, with low frequency information, therefore the designed phase kernel of the transform has a pronounced effect on these frequencies, whereas higher frequencies that contain the wide-spread noise will be less affected. The result is a narrower PSF shape.
The PST in frequency domain can be described as follows:
where IPST(x,y) and I(x,y) are the output and input images respectively, ∡⟨⋅⟩ is the angle operator, FFT2 is the two-dimensional Fast Fourier Transform, IFFT2 is the two-dimensional Inverse Fast Fourier Transform. K͂ is the frequency response of the warped phase kernel that is described by a nonlinear frequency dependent phase:Recently, the PST method was used for edge detection [29], where the engineered phase was designed for the emphasis and detection of high frequencies in an image. Here, we were inspired by the same concept, however the need was to emphasis and detect the low frequencies associated with the PSFs and for that purpose we modified the phase kernel profile to suit this case. The PST kernel phase profile can be arbitrary but the one used here has circular symmetry with respect to frequency variables, thus it is independent in θ. The phase kernel is given by:where r = √(u2 + v2) and normalized to be between 0 and 1. S and W are real-valued numbers related to the strength (S) and wrap (W) of the phase profile. S and W are the parameters that control the narrowed shape and noise reduction of the PSF. For small wrap value (W«1), Eq. (4) can be approximated to a quadratic phase profile. This profile can narrow the full width half maximum (FWHM) of each PSF by a factor of 2 in each direction, allowing an increase by a factor of 4 in total emitter's density.The major advantage of the proposed technique over other techniques that preform a direct intensity boost on raw images like the square of the image intensity is that they also increase the image noise by the same amount. Here, the PST effects only the frequencies that contain the image information, therefore the desired data is enhanced, whereas the image noise will be filtered. Applying a low pass filter to the image will filter the image noise, however the image data will remain the same and overlapping emitters will not be influenced.
In a previous paper we suggested the use of the modified K-factor algorithm for the same purpose [35]. The PST has two major advantages in respect to the K-factor algorithm: The first is a larger narrowing of the PSF, by a factor of 2 with compared to the K-factor, resulting in the ability to detect even closer PSFs. In addition, the modified K-factor algorithm is based on the detection of contrast levels in an image and therefore is sensitive to non-uniform intensity of the PSFs across the field of view. Weaker PSFs might not be detected, as they will be erased by the algorithm. The PST highlights all the PSFs, thus enables their detection. Figure 1 illustrates the influence of the PST on a noiseless image containing two PSFs separated by a distance of 1.5σ~300nm. It is compared to the modified K-factor algorithm with the same parameters as described in Ref [36]. The K-factor reduces the saddle between the two closely spaced PSFs by a factor of about 2, whereas the PST yields a further improvement with compare to the K-factor, by another factor of 2. It should be mentioned that in order to apply both of these techniques, there must be a certain saddle, even minimal, between the peaks of the two PSFs. In cases where the overlap is such that the blurred spot appears as a single wide Gaussian, both of the techniques will not be applicable.
Fig. 1 Image of two PSFs separated by a distance of 1.5σ~300nm. Original image (blue line). Modified K-factor algorithm applied to the original image (black), and the PST method applied to the original image (red).
The PST technique is an added step during the conventional localization microscopy routine. The sample is imaged using repeated cycles of activation of small percentage of fluorophores in the sample, followed by bleaching to minimize the presence of already visualized particles in posterior acquired images. Each frame is passed on to a computer program realized in software package like MATLAB (MathWorks, Natick, MA, USA), where it is processed with the PST. Since the PST reshaped the PSF, the processed image cannot be fitted into a Gaussian, however it can be fitted into a PST processed Gaussian, therefore each PST processed frame undergoes localization of the activated PSFs using methods of single emitter fitting, where it is being fitted to a Gaussian shape that was PST processed. Reconstruction and creation of the super-resolved image is done by super-imposing all the localized positions into a meta-SR image.
3. Simulation results
To evaluate the performance of the PST method and its impact on overlapping emitters and localization precision, 1000 Monte-Carlo (MC) simulations were used to generate mock data sets with randomly positioned emitters in each set and with additive shot and background noise. The simulated model parameters matched the experimental parameters that are described in the experimental results section below. The value for σ in the model used for the PSF was calculated from Eq. (1) to be 186nm for the given imaging parameters.
The performance of the PST was compared to both a simple Gaussian fitting applied on the raw simulated data, as well as to the K-factor filtered data. The PST processed image was fitted to a PSF shape that underwent PST processing. Each image contained randomly positioned fluorophores, which were fitted using the non-linear least-squares minimization routine lsqnonlin in MATLAB. The algorithm initially detected the position of each particle as the pixel with the highest intensity in its region, and preformed a fit by taking 9 × 9 pixels around this initial location [37]. The fit produces the best estimate of the position of the emitter, and the process is repeated for 1000 MC iterations yielding a set of localization positions (x̂i,ŷi) that can be compared with the known positions (xi,yi). The root mean square (RMS) localization error is computed for the raw simulated data, the K-factor filtered simulated data and the PST processed data:
The position of each fluorophore, the distance between fluorophores, the number of arriving photons N, the background noise parameter Nb, and the parameters S, and W of the PST phase kernel profile, were all allowed to vary. All analysis was performed in MATLB.The parameters S and W control the shape and width of the phase kernel, which in turn determine the narrowing of the PSF. On one hand one wishes to narrow the PSF and on the other to reduce the noise. The parameter S controls the amplitude of the added phase. A larger S results in better noise performance but at the expense of lower spatial resolution. W controls the shape of the added phase, a larger W results in a narrower PSF but it also might produce false alarm detection of noise as PSFs since small readout values will increase the noise. In order to optimize the choice of these parameters for localization microscopy purposes, the tested scenario consisted of two closely spaced Gaussian shaped PSFs. The S and W parameters were allowed to vary, together with the SNR. The search was designed to maximize the height of the maxima of the PSFs, while minimizing the saddle between them, therefore maximizing the PSF peak to minimum saddle ratio. The best choice parameters were found to be W = 50 and S = 5 for an SNR>5. The assumption is that localization microscopy imaging is performed in conditions of good SNR, otherwise the localization accuracy will deteriorate rapidly. For cases of a lower SNR, the value of W should increase. This will further narrow the PST kernel, and a narrower band of frequencies will be used for the processing. Since the imaging conditions are of a high additive noise, the less frequencies taken for the processing, the less additive noise will be used. However, this value should be chosen such that the narrowing will not remove frequencies associated with the image data, otherwise it will cause a reduction in localization precision. Hence, for poor SNR conditions, the PST parameter values should be chosen a priori by simulating the imaging conditions.
Simulation results of multiple emitters with added shot noise (N = 5000) and Nb = 5 are presented in Fig. 2. Figure 2(a) is the original image containing non-uniform intensity PSFs. Figure 2(b) is the K-factor algorithm result and Fig. 2(c) is presents applying the PST with W = 50, S = 5. Figures 2(d)-2(f) is the magnification of the yellow area marked by '1' in Figs. 2(a)-2(c) respectively. In the original image, this area appeared as a large spot. The K-factor algorithm improved it and allowed the detection of three different emitters. The PST further narrowed the PSFs and enabled their localization with a higher precision as isolated point sources. Another advantage the PST has is its ability to highlight low intensity PSFs that are marked by the yellow areas '2' and '3′. In Fig. 2(a) they are barely visible. The K-factor algorithm, which is based on contrast levels, has erased them completely, whereas the PST method highlighted them, making them clearly visible. The PST phase kernel profile, for these parameters is illustrated in Fig. 2(g). The method sharpens the PSFs, making overlapping emitters become distinguishable. As a result, in the frequency domain, each PSF broadens. This broadening can be seen in the OTF of a single PSF Fig. 2(h). The cutoff frequency using the PST technique (red line), is higher than the cutoff frequency of the raw data (black line).
Fig. 2 Simulation results. (a) original image with additive shot noise (N = 5000) and background noise (Nb = 5). (b) The K-factor algorithm result. (c) PST result. (d)-(f) is the magnification of the yellow regions marked by 1 in (a)-(c) respectively. (g) is the PST phase kernel profile with parameters of W = 50, S = 5. (h) OTF of a single PSF for the raw data (black line) and for the PST processed image (red line).
In order to test the performance of the proposed method, two different aspects were considered. The first was to make sure that the PST processing does not affect the localization precision for an isolated emitter. Since the PST method is a post processing technique, the obtained localization precision is limited by the Cramer-Rao bound and cannot reach a lower resolution than that of the regular localization using a Gaussian performed on the original data. The major advantage of the technique is its ability to localize overlapping emitters with resolution of an isolated PSF.
Figure 3(a) compares the RMS localization error of the fitting process for an isolated emitter as a function of SNR, with 1000 Monte-Carlo iterations. The added noise includes both shot noise and background noise. The shot noise is determined by the total number of photons detected, as it is a Poisson random process with a rate that is proportional to √N (N being the number of photons) and therefore is reduced with higher N [38]. The background noise is set by the parameter Nb. The SNR was calculated by changing both the number of detected photons and also the background noise parameter, such that 0<Nb<20 to mimic low and high background noise conditions. An increase in the SNR reduces the RMS error (Fig. 3(a)). Gaussian fitting applied to raw data is presented in the blue line. The red line presents the results for PST processed Gaussian fitting applied to the PST processed raw data. The results indicate that the PST method combined with fitting using a PST processed Gaussian, yields similar results to that of a simple Gaussian fitting performed on the raw data. Therefore, the method does not affect the localization precision of an isolated emitter.
Fig. 3 (a). Error in localization for an isolated emitter as a function of the SNR. (b). The minimal resolvable distance between two point sources as a function of SNR. Both simulations contain 1000 Monte-Carlo iterations. Gaussian fitting applied to the raw image is presented in the blue line. PST applied to the raw data and fitted using a PST processed Gaussian is presented in the red line.
The second aspect was to test the PST method's ability to localize overlapping emitters with precision of an isolated emitter. Figure 3(b) simulates a scenario of two closely spaced PSFs as a function of SNR. Each of these PSFs is localized, and the minimal distance between them is varied until the obtained localization precision, for a given SNR, is identical to the isolated emitter case presented in Fig. 3(a). Gaussian fitting applied to raw data is presented in the blue line. The red line presents the results for PST processed Gaussian fitting applied on the PST processed raw data. The minimal distance at which overlapping emitters can be localized with precision of a single emitter is significantly reduces with the PST method, with minimal separation distance of 1.2σ for SNR of 100.
4. Experimental results
Single molecule imaging experiments were performed in an epi-fluorescence microscope setup consisting of an inverted microscope (Zeiss Elyra P.1, Carl Zeiss Microscopy Inc.), 1.46 N.A. total internal reflection (TIRF) objective, 642nm diode laser, and an electron multiplying CCD camera Ixon (897, Andor Technologies PLC.) with EM gain set to ≈200. The epi-fluorescence filter setup consisted of a dichroic mirror (650nm, Semrock) and an emission filter (692/40, Semrock). The sample chamber was mounted in a 3D piezostage (P-737 PIFOC Specimen-Focusing Z Stage, Physik Instrumente). 10,000 images were taken in a TIRF configuration at 40frames/second. Frames were 512 × 512 pixels with an effective pixel size of 99.8nm.
The sample contained microtubules from BSC-1 African green monkey kidney epithelial cells (American Type Culture Collection-ATCC) fixed with a mixture of paraformaldehyde and glutaraldehyle, and then blocked with 5-percent normal goat serum before being treated with Abcam rat antibody to tubulin, ab6160 as the primary antibody directed to alpha-tubulin. After washing, the specimen was treated with Invitrogen Alexa Fluor® 647 Goat Anti-Rat, A-21247 as the secondary antibody. The Cell culture methods were standard cell culture techniques for primary/secondary antibody labeling methods. The peak emission wavelength of Alexa 647 is 671nm. Raw data pre-processing included noise reduction by convolving it with a Gaussian kernel. Next, the PST method was applied to each frame using MATLAB, followed with localization and reconstruction of the SR image.
The fluorescence data included frames with simultaneously activated fluorophores such that it contained overlapping molecules. A single frame from the set is presented in Fig. 4(a). The PST was applied to each individual frame (Fig. 4(b)), where blurred spots that contained overlapping emitters (Figs. 4(c), 4(e) and 4(g)), became distinguishable and can be localized with high precision (Figs. 4(d), 4(f) and 4(h)). Figure 5(i) compares the OTF of an isolated PSF taken from the raw image in Fig. 4(a) (black line), and the PST processed frame taken from Fig. 4(b) (red line). With good agreement to the simulation model, the PST OTF is broader that the raw data's and contains more spatial frequencies, thus enables the localization of overlapping emitters with a higher precision.
Fig. 4 Individual dSTORM frame without processing (a). and after PST processing (b). Marked regions are examples where the difference can be clearly seen. (c), (e) and (g) are the magnification of the marked areas in (a).(d), (f) and (h) are is the magnification of the marked areas in (b). (i) compares the OTF of an isolated PSF taken from the raw image in (a) (black line), and the PST processed frame taken from (b) (red line).
Fig. 5 SR dSTORM reconstruction of imaging data from Alexa647 labeled microtubules sample without processing and using the PST method. Images in the upper row represent reconstructed image using 10,000 frames and those in the lower row represent reconstructed image using 2,500 frames. (a) and (d) Conventional dSTORM analysis. (b) and (e) PST processing applied on raw data coming from dSTORM regular analysis. The black and red lines in (c) presents the cross sections of the white dashed lines in (a) and (b) respectively.
Localization of all the detected PSFs in each frame for the entire set of 10,000 frames was conducted both on the raw data and on the PST processed set. The final reconstructed SR image super-imposed all the detected positions for each of the sets. Figure 5(a) is the raw data SR dSTORM image using 10,000 frames. Figure 5(b) is the PST processed SR image using 10,000 frames. The PST enabled the usage of overlapping emitters data, that is discarded in the regular dSTORM routine and thus it was able to extract more information from each frame. It was calculated that the PST SR image is denser, in terms of detected emitters, by a factor of 4 compared to Fig. 5(a) that is a less detailed image with gaps in estimated locations of the emitters where there was a high density of overlapping emitters. The black and red lines in Fig. 5(c) presents the cross sections of the white dashed lines in Figs. 5(a) and 5(b) respectively. The results indicate that both images have the same thickness, but the raw data image contains gapes associated with lower density of detected emitters. The measured width of the microtubule is ~56nm, which is in agreement with the reported value in literature [39].
For a given amount of frames, the PST processed image contains 4 times more detected emitters than the raw data. As such, a less amount of PST processed images are required in order to achieve the same super-resolution image of the raw data images. In order to determine this reduction in the number of frames, the initial correlation between the two reconstructed SR images with 10,000 frames for both the PST and the raw data was calculated. This yielded a value of 84%. Next, the amount of PST processed images that generated the SR image was reduced, while the SR image generated with the raw data remained constant with 10,000 frames. The correlation between the two images was calculated again and increased. This process was repeated until a predetermined threshold for the correlation coefficient was reached. Arbitrary, this threshold was chosen as 99%. 2500 PST processed frames yielded a 99.4% correlation coefficient. For imaging applications that requires high process speed, such as live cell imaging, one may choose this threshold differently.
Figure 6 presents the number of PST frames required to generate the SR image as a function of correlation percentage with the 10,000 raw data images reconstruction. The number of frames used to achieve this maximum correlation with the PST method was 2,500 and the reconstructed image is presented in Fig. 5(e). The same amount of frames used for reconstruction with the raw data produced a significantly less detailed image (Fig. 5(d)).
Fig. 6 The number of PST frames required to generate the SR image as a function of correlation percentage with the 10,000 raw data images reconstruction.
The penalty is the increase in processing time, which is 2μs per 512x512 pixels frame, with pixel size of 99.8nm in a simple PC (HP Compaq Elite 8300 Microtower PC with Windows 7 professional 64 bit operation system, Intel® Core i5-3470 processor, 3.20 GHz, 12 GB RAM). The time penalty for a set of 10,000 frames is 30ms
5. Conclusions
Implementation of the PST method on images acquired by SR localization microscopy techniques, like PALM and STORM, prior to the fitting process, enables the analysis of images with a higher density of activated emitters per frame, thereby the method can be a useful tool for fast live cell SR microscopy techniques operating at high density, allowing for an increase in activated fluorophore density by a factor of 4. In addition, it highlights low intensity PSFs, improving their SNR, which yields a higher localization precision.
Our validation studies performed by using simulated and experimental data showed that the PST can extract more data from each frame, as it can localize overlapping emitters with minimal distance of only 1.2σ for SNR of 100. Therefore, it can efficiently reduce the SR image acquisition time. Moreover, our method is efficient in terms of performance, which reduces computation time by orders of magnitude in comparison with the previously introduced high density algorithms.
Acknowledgments
We would also like to thank Carl .G. Ebeling for his help with the experiment and data collection.
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