Non-heuristic automatic techniques for overcoming low signal-to-noise-ratio bias of localization microscopy and multiple signal classification algorithm

Localization microscopy and multiple signal classification algorithm use temporal stack of image frames of sparse emissions from fluorophores to provide super-resolution images. Localization microscopy localizes emissions in each image independently and later collates the localizations in all the frames, giving same weight to each frame irrespective of its signal-to-noise ratio. This results in a bias towards frames with low signal-to-noise ratio and causes cluttered background in the super-resolved image. User-defined heuristic computational filters are employed to remove a set of localizations in an attempt to overcome this bias. Multiple signal classification performs eigen-decomposition of the entire stack, irrespective of the relative signal-to-noise ratios of the frames, and uses a threshold to classify eigenimages into signal and null subspaces. This results in under-representation of frames with low signal-to-noise ratio in the signal space and over-representation in the null space. Thus, multiple signal classification algorithms is biased against frames with low signal-to-noise ratio resulting into suppression of the corresponding fluorophores. This paper presents techniques to automatically debias localization microscopy and multiple signal classification algorithm of these biases without compromising their resolution and without employing heuristics, user-defined criteria. The effect of debiasing is demonstrated through five datasets of invitro and fixed cell samples.


Supplementary Note 1. Temporal patterns in fluorescence intensity in response to natural or experimentally introduced changes in photokinetics of fluorophores
It is commonly assumed that the fluorescence characteristics of the emitters do not change within the timescale of the acquisition of images for localization microscopy. The overall decrease in intensity is often attributed to the decrease in emitter density due to irreversible loss of some of them through bleaching. This is easily verified in Supplementary Figure  We illustrate other examples of changes in intensity over time due to chemical changes and external intervention using datasets of microtubules in cells. Here, due to the absence of ground truth, the number of localizations or photons correspond to the estimated numbers. Further, the quantity plotted as estimated number of photons is actually linearly proportional to the estimated number of photons, the unknown constant of proportionality being the characteristic of the measurement system and computation algorithm. We use the estimations of both rainSTORM and NSTORM for a more conclusive inference. Beside photobleaching, another reason of patterned decrease in fluorescence intensity may be the variations in the fluorescence emission efficiencies (mostly through differences in the fraction of time they spend in the on-state) potentially due to change in concentration of chemicals influencing photokinetic phenomena [S2]. This appears to be the case with the dataset of microtubules in cell 2. The image intensity shown in Supplementary Figure 2 Sometimes, in order to improve blinking or fluorescence intensity, manual intervention may be done midway in the form of adding certain chemicals, increasing the excitation power, or switching on an additional activation source. In datasets of microtubules in cells 1 and 3, an activation laser was switched on mid-way when the incidents of blinking or their intensity was heuristically concluded as quite diminished. Upon further incident of diminished blinking or intensity, the power of the activation laser was increased. This results in sudden increase in the fluorescence intensity, as observed in Supplementary  Figure 3

Supplementary Note 2. Detection of foreground in a frame
Consider an image frame k I which contains intensities at all the pixels in the k th frame (see Supplementary Figure 5 (a)). The histogram of its image intensity (HoI) is computed as () k hn, where n is a discrete intensity value and () k hn is the number of pixels with intensity value n in the image k I . In images of sparsely emitting emitters, the foreground pixels are so few that the histogram distribution essentially appears unimodal and corresponds to background (see Supplementary Figure 5 (b)). In fact, the number of foreground pixels is typically orders of magnitudes smaller than the number of background pixels. Thus, we consider logarithm of HoI (LoHoI), ( ) log( ( )) kk g n k n  . We empirically found that even () k gn is not amenable for identifying and separating the bimodal distributions (see Supplementary   Figure 5 (c)). However, the values of LoHoI have a bimodal distribution. The threshold value which separates these two distributions is computed using the popular Otsu's method of thresholding [S1]. The thresholded LoHoI is used to find 0 n after which the thresholded LoHoI remains 0 (see Supplementary

Technique 2 for debiasing LM: Weighing contributions from the frame with its average intensity.
In this technique, ( , ) k b x y is defined as: where r denotes an image pixel in the measured image stack and r N  is the total number of pixels. In this technique, we penalize all the localizations in a frame equally, irrespective of the local spatial variations in SNR.

Technique 3 for debiasing LM: Weighing contributions from the frame with its average foreground intensity.
In this technique, ( , ) k b x y is defined as: where F is the foreground. Similar to technique 2, all localizations in a frame are penalized equally.

Technique 4 for debiasing LM: Weighing with inverse of localization accuracy.
In this technique, ( , ) k b x y is defined as: is the localization accuracy of the localization at ( , ) xy in the k th frame. Here, we use the Thomson formula for computing ( , ) k xy  [S3]. Similar to technique 1, each localization in a frame is treated individually and incorporates the effect of the local signal to background ratio on the quality of localization.

Comparison of the debiasing techniques of LM:
The comparison of debiasing techniques 1-3 of LM is given in Supplementary Table 1 and Supplementary Figure 6. It is seen that the technique 1 is very effective for both the examples. We note that the techniques 1 and 4 give almost the same result. The SSIM values between the results of techniques 1 and 4 are equal to 99.4% for both the examples. The correlation between them is also very high, 97.4% for in-vitro actin filaments and 99.4% for in-vitro microtubules. Thus, it is evident that the number of estimated photons is highly correlated to inverse of localization precision. This is because the localization precision and estimated number of photons are both related to the signal to background ratio. where r denotes an image pixel in the measured image stack and r N  is the total number of pixels.  Figure 6. It is seen that techniques 1 and 2 are comparable to each other. Technique 3 provides better value of SSIM for in-vitro actin filaments data but poorer value of SSIM for in-vitro microtubules. Thus, an inference regarding it cannot be made. Fig. 10 and 13 of the main paper.

Supplementary Note 4. Histograms in
In Fig. 10 (a), histogram of intensities of original LM image in Fig. 9(a) is given. The intensity of original LM image at a pixel is denoted as ( , ) s x y . Since the original image counts the number of localizations in a pixel in the LM image, the values of ( , ) s x y are integers. Each of these integers is significantly less than the number of the frames because of the spatio-temporal sparsity of emissions. Fig. 10(a) shows the number of pixels in the original LM image at which an integer intensity value 1 e is observed. Since the number of background pixels are too many and they contribute to the first bin only, we skip the first bin. , where n is the bin number. We count the number of pixels in debiased LM image whose intensity lies in these bins. We plot this count as the function of the outer edge of the bin. Since the number of background pixels are too many and they contribute to the first bin only, we skip the first bin.