# Sparsity-based background removal for STORM super-resolution images

Patris Valera  1,2,* Josué Page Vizcaíno  1,2 and Tobias Lasser1,2  1Munich
Institute of Biomedical Engineering, Technical University of Munich, Germany
2TUM School of Computation, Information and Technology, Technical University
of Munich, Germany<EMAIL_ADDRESS>
††journal: opticajournal††articletype: Research Article

Single-molecule localization microscopy techniques, like stochastic optical
reconstruction microscopy (STORM), visualize biological specimens by
stochastically exciting sparse blinking emitters. The raw images suffer from
unwanted background fluorescence, which must be removed to achieve super-
resolution. We introduce a sparsity-based background removal method by
adapting a neural network (SLNet) from a different microscopy domain. The
SLNet computes a low-rank representation of the images, and then, by
subtracting it from the raw images, the sparse component is computed,
representing the frames without the background. We compared our approach with
widely used background removal methods, such as the median background removal
or the rolling ball algorithm, on two commonly used STORM datasets, one glial
cell, and one microtubule dataset. The SLNet delivers STORM frames with less
background, leading to higher emitters’ localization precision and higher-
resolution reconstructed images than commonly used methods. Notably, the SLNet
is lightweight and easily trainable (<5 min). Since it is trained in an
unsupervised manner, no prior information is required and can be applied to
any STORM dataset. We uploaded a pre-trained SLNet to the Bioimage model zoo,
easily accessible through ImageJ. Our results show that our sparse
decomposition method could be an essential and efficient STORM pre-processing
tool.

##  1 Introduction

Super-resolution microscopy (SRM) assembles a group of optical microscopy
techniques that allow fluorescent imaging at (sub-)cellular level and at
unprecedented resolutions. Unlike the traditional light microscopy techniques,
these methods exceed the Abbe diffraction resolution limit, resulting in a
10-nanometer distance limit between two light point sources and a lateral
resolution of 20 nanometers [1, 2]. One type of SRM is single-molecule
localization microscopy (SMLM) [1, 2, 3, 4, 5], which uses the activation of
fluorophores to visualize biological structures and is categorized into
(fluorescence) photo-activation localization microscopy ((f)PALM) [6, 7] and
(direct) stochastic optical reconstruction microscopy ((d)STORM) [8, 9]. SMLM
methods stochastically activate a sparse set of fluorophores at different time
steps and capture these in multiple image acquisitions. In a post-processing
step, the centroids of the emitters are estimated and accumulated into a high-
resolution reconstructed image with nanometer-level precision [1].

One drawback of SMLM is its sensibility to background signals often present in
fluorescence microscopy (e.g., autofluorescence and non-bleached out-of-focus
molecules [9, 10]). These signals affect the ability to accurately localize
the emitters’ centroids within the sample, known as localization precision
[11, 12], ultimately leading to decreased image resolution and artifacts in
the reconstruction. Therefore, a background removal pre-processing (see Fig.
1) is crucial for precise and high-resolution image reconstruction in STORM.

Figure 1: Left: The three main steps conducted in STORM, where the background
removal methods are applied in the pre-processing stage. Right: The sparse
decomposition process and the architecture of the neural network SLNet. The
network approximates the low-rank representation stack from the input stack,
and the sparse images are calculated as a subtraction of the two. The
parameter _k_ controls the size of the input stack (here, $k=3$).

There have been several attempts to remove the background, such as median
background removal [13, 14], "rolling ball" algorithm [15, 16], or image
filtering [17, 18], but most are either too general or too computationally
expensive. The background in fluorescence microscopy is heterogeneous and non-
uniform, but most methods assume otherwise [17]. For example, a standard way
is to compute the median intensity of all frames and subtract it from them to
get images without a prevalent background. Although it may be simple, this
method generally removes an atypical uniform background that leaves
unnecessary fluorescence in the images or can also wrongly remove
fluorophores. On the other hand, the "rolling ball" algorithm removes the
background by "rolling" a ball of a certain radius under the pixel intensity
histogram of each frame and then subtracts the area below it. This algorithm
is an intuitive method to remove uniform background intensity levels, but as
such, it has limitations; for example, at times, it tends to include a
disproportionately large amount of weak spots or generate more artifacts in
the background [15]. Another widely used technique is image filtering, which
includes temporal and band-pass filters applied to correct the background.
However, they have proven insufficient in cases with structured and non-
uniform backgrounds [17, 19].

As a novel and refined solution to the aforementioned problems, we propose to
use the SLNet [20], previously used for background removal in zebrafish light
field 3D microscopy. The SLNet is a fast, computationally inexpensive, sparse-
low-rank decomposition-based method to compute and remove the background of
fluorescent static images (see Fig. 1). It is trained in an unsupervised
fashion by minimizing a custom-crafted loss function that aims to approximate
the raw images with a low-rank representation using three sample images at
different time points. Then, the sparse components, the images without the
background, are computed by subtracting the low-rank components from the raw
images.

This work compares the SLNet and different background removal methods as a
pre-processing step, aided by ThunderSTORM [21], an open-source plug-in for
ImageJ [22], which performs localization and reconstruction. A hyper-parameter
optimization of the SLNet was also performed, including parameters like the
number of images, learning rate, the required training epochs, etc.

In subsection 2.1, we first present the sparse-low-rank decomposition on which
our method is based. Then, in subsection 2.2, we describe the SLNet neural
network adapted from [20] and explain the unsupervised training of the SLNet
in subsubsection 2.2.1. The datasets we used are defined in subsection 2.3,
and the reconstruction method of the super-resolution images is presented in
subsection 2.4. In the experimental results in section 3, we describe the
methods we used to compare our findings and present our quantitative
evaluation in subsection 3.1 with 2 FWHM analyses in subsubsection 3.1.1 and a
SQUIRREL analysis in subsubsection 3.1.2. In the hyperparameter search results
in subsection 3.2, we describe the effect of 2 crucial parameters on the SLNet
training and the sparsity output level.

##  2 Methods

In short, our method uses the sparse decomposition implemented with the help
of the neural network SLNet by assuming that the blinking emitters’ raw frames
used for the STORM reconstruction consist of low-rank components representing
the background and sparse components representing the specimen’s essential
signals.

###  2.1 Sparse-low-rank Decomposition

The sparse-low-rank decomposition (SD) method, also known as robust principal
component analysis (PCA) [23, 24], computes the low-rank and sparse components
of a matrix $M$ with the constraint that the sum of the components equals the
matrix $M$. In our case, the data is a set of images with size $m\times n$ and
recorded at $k$ time points, giving us $M_{k,m,n}\in\mathbb{R}_{\geq
0}^{k\times m\times n}$. We can use sparse decomposition to decompose temporal
stacks of the frames into sparse and low-rank components by rearranging
$M_{k,m,n}$ as $M_{k,mn}\in\mathbb{R}_{\geq 0}^{k\times mn}$ and minimizing
the convex optimization problem:

$\begin{split}\min_{L,S}&\quad{\lVert L\rVert}_{*}+\lambda{\lVert
S\rVert}_{1}\\\ \text{subject \ to}&\quad L+S=M_{k,mn}\end{split}$ (1)

where

  * •

$L\in\mathbb{R}_{\geq 0}^{k\times mn}$ is the low-rank component

  * •

${\lVert L\rVert}_{*}$ denotes the nuclear norm (the sum of all singular
values) of the low-rank component

  * •

$S\in\mathbb{R}_{\geq 0}^{k\times mn}$ is the sparse component

  * •

${\lVert S\rVert}_{1}$ denotes the $L_{1}$-norm (component-wise sum of
absolute values of all entries) of the sparse component

  * •

$\lambda>0$ is the weighting parameter that controls the ratio between the
low-rank and sparse components and thus the level of sparseness.

This convex optimization problem is NP-hard, as it requires minimizing both
$L_{1}$-norm and nuclear norm at the same time. Different solutions have been
found to solve this problem, such as the iterative thresholding approach [25,
26], the accelerated proximal gradient approach [27], the dual approach [28],
and methods of augmented Lagrange multipliers [23], which are primarily used
due to faster convergence [23, 24]. However, we solve it with the procedure
proven by [29] and used by [20], where the low-rank matrix $L$ is computed
with a neural network. Then, the sparse matrix $S$ is calculated as the
subtraction of the original image with the low-rank component, equaling
$(M-L)_{\geq 0}$.

###  2.2 Neural Network SLNet

The SLNet comes into action for approximating the low-rank component of the
input:

$\mathcal{N}_{\Theta}^{SL}(M_{k,m,n})\approx L_{k,mn}$ (2)

It takes as an input the set of images $M_{k,m,n}$ and generates the low-rank
component $L_{k,mn}$ of the stack of images, parametrized by $\Theta$. After
finding the low-rank component, the sparse component S (the resulting images)
is calculated as

$S=(M_{k,mn}-\mathcal{N}_{\Theta}^{SL}(M_{k,m,n}))_{\geq 0}$ (3)

SLNet is a simple and small neural network consisting of two convolutional
layers and one rectified linear unit (ReLU) activation function (see Fig. 1).
This small architecture brings the benefits that the network takes less space
and performs an efficient computation. The network takes a stack of raw images
$M_{k}$ converted to a tensor with the shape $(batch\\_size,k,m,n)$ as input
and outputs a tensor with the same size, which represents the low-rank
components of the input images. The weight initialization of SLNet is
essential for the output $L$. If the weights are too large, the output $L$ has
higher entries than the input, and the resulting sparse component $S$ from
their subtraction would have only $0$s as its elements, and thus the learning
would stop. Therefore, the weights $\Theta$ are first initialized with the
Kaiming method [30], which considers the non-linearity of the ReLU activation
function and prevents the exponential growth or shrinking of the input. For
further details on the architecture and training methodology, we refer the
reader to the publicly available source code [31].

####  2.2.1 Network Training

The network is trained in an unsupervised manner by minimizing the augmented
Lagrangian loss function:

$\min_{\Theta}\quad{\lVert
M-\Gamma_{\mu}(\mathcal{N}_{\Theta}^{SL}(M_{k,m,n}))\rVert}_{1}+\alpha\cdot\overline{\lvert
S\rvert}+\overline{\lvert R\rvert}$ (4)

with $M$ the raw images, $\alpha\cdot\overline{\lvert S\rvert}$ the scaled
average of the sparse component $S$ weighted by an $\alpha$ parameter, and
$\overline{\lvert R\rvert}$ the average reconstruction error. The operator
$\Gamma_{\mu}(\cdot)$ performs singular value shrinkage [32, 33] and enforces
its output to have a low rank by setting the eigenvalues of the rectangular
matrix of the input’s SVD to 0 if they are smaller than a threshold $\mu$,
meaning $\Sigma_{<\mu}=0$ and decreasing the remaining eigenvalues.
Mathematically, the operator is expressed as:

$\begin{split}\Gamma_{\mu}(X)=\ &U\
[\mathrm{sign}(\Sigma)\cdot\max(|\Sigma|-\mu,0)]\ V^{*}\\\ &\text{where
}X=U\Sigma V^{*}\end{split}$ (5)

The threshold $\mu$ and the parameter $\alpha$ are critical user-defined
parameters that control the degree of the low rank of the SLNet output $L$ and
the level of sparseness of the result $S$. The network training was configured
with a sparseness threshold $\mu$ set to 0.01 and the alpha parameter set to
12. This configuration was selected based on the sparsity of the output frames
from differently trained neural networks and the reconstruction quality they
provided. The effects of these parameters are further explained in subsection
3.2.

We trained the network for 100 epochs because it produced satisfactory sparse
results and reconstructions. The training took 2 minutes and 37 seconds on a
workstation with an AMD Ryzen 9 3950X 16-core Processor, and using an NVIDIA
GeForce RTX 2080 Ti GPU. The training and testing took about 8 minutes. For
the sparse decomposition, we use only three frames at a time, chosen based on
a time index $t\geq 0$ that shifts them. The frames are $M_{t}$, $M_{t+50}$
and $M_{t+100}$ (derived from the original SLNet [20]). Consequently, the
trained SLNet can be used in parallel after the microscope records 100
specimen images.

###  2.3 Dataset descriptions

We used two datasets with distinct samples to train and test the neural
network and generate other methods’ results. The first experimental data we
used is the dataset obtained from [34], which includes actin-labeled glial
cells of a rat’s hippocampal neurons. We trained the network with the first
100 images of the BIN10 frames and tested on the remaining 5890 frames. The
second is a microtubule dataset obtained from [35], consisting of microtubules
immunolabeled with Alexa Fluor 647 phalloidin. We used the first 7000 out of
60000 frames for testing.

###  2.4 STORM super-resolution reconstruction

After acquiring the images and pre-processing them to get high-quality raw
images, the most critical step is the STORM high-resolution image
reconstruction performed using the ImageJ [22] plugin ThunderSTORM [21]. To
compare the different results and find out the most suitable combination of
image filters [36] with background removal methods, the image pre-processing
was done with all the supported filters on ThunderSTORM:

  * •

averaging filter (kernel size 3 pixels),

  * •

Gaussian filter (sigma 1.6 pixels),

  * •

lowered Gaussian filter (sigma 1.6 pixels),

  * •

difference-of-Gaussians filter (sigma1 1.0 pixels, sigma2 1.6 pixels),

  * •

difference of averaging filters (first kernel size 3 pixels, second kernel
size 5 pixels),

  * •

wavelet filter (B-spine, order 3, scale 2),

  * •

median filter (box pattern with kernel size 3 pixels).

Additionally, the first testing dataset (the glial cells) contains a high
density of emitters, and as a fluorescence live cell imaging, the fluorophores
of the neurons are on the move. Consequently, the centroids of the fluorescent
molecules are distant, and we needed to post-process the reconstructed images
with lateral drift correction with cross-correlation (five bins at 5 x
magnification).

##  3 Experiments and Results

To assess our method, we compared the following pre-processing methods:

  * •

The raw non-preprocessed images

  * •

Median background removal (implemented in Python by subtracting the median
value of all pixels in all frames to the raw data)

  * •

Rolling ball background subtraction (provided as the plug-in "Rolling Ball
Background Subtraction" [37, 38] in ImageJ. The glial cell and microtubules
datasets were processed without smoothing and a radius of 3 and 10 pixels,
chosen heuristically.)

  * •

The SLNet with $\alpha=12$ and $\mu=0.01$.

These methods are compared on the STORM datasets (described in sec. 2.3)
qualitatively in figures 2 and 3, and quantitatively using the metrics
described in sec. 3.1, presented in figures 4 and 5.

The reconstruction of the raw images was challenging due to high
autofluorescence caused by the densely packed fluorophores. Figures 2 and 3
show the best reconstruction of each method and three magnified areas with
densely packed structures to illustrate the improvement of our method. The
SLNet outputs have individual emitters sufficiently separated, leading to a
better approximation of the coordinates of centroids and resulting in sharper
and more defined images compared to other methods.

Figure 2: Super-resolution reconstruction of glial cells using ThunderSTORM
and three background removal methods: median, "rolling ball" with radius 3
pixels and our SLNet background removal method. (1), (2), (3) Magnified views
of three selected regions with scale bar $1\mu m$. Figure 3: Super-resolution
reconstruction of microtubules using ThunderSTORM and three background removal
methods: median, "rolling ball" with radius 10 pixels and our SLNet background
removal method. (1), (2), (3) Magnified views of three selected regions with
scale bar $1\mu m$.

###  3.1 Quantitative comparison

As super-resolution methods achieve sub-Abbe-limit resolutions, no methodology
exists to generate or measure ground-truth data. Hence, data-driven approaches
are used to compare different pre-processing and reconstruction methods, as
described in this section.

####  3.1.1 FWHM evaluation

To evaluate the resolving power of the STORM reconstructions, we analyzed the
full width at half maximum (FWHM) of:

  * •

The single emitters after pre-processing, where we extracted the position of
emitters (38 for the glial cells and 45 for the microtubules) on a single
thresholded frame of the raw data using the "Analysing Particles…" [39]
command of ImageJ.

  * •

Single microtubules/filament extensions on the reconstructed images from both
datasets. We used 20 microtubules to measure their transversal FWHM and
evaluate the statistical distribution. We manually selected microtubules that
were visually small and close to each other, thus challenging to reconstruct
due to the proximity of the fluorophores in those areas.

Fig. 4 shows the FWHM analysis on both datasets for the emitter and the
reconstructed images.

The SLNet FWHM measurements show a better central tendency for both datasets
since the median is lower than the other methods in all cases. As a result,
our method appears better at improving spatial resolution. We observed that
the other methods had some extreme outliers, whereas in the SLNet results, the
measurements were more uniform. In addition, our method has shown symmetry in
all measurements (especially in the FWHM analysis of the reconstructed glial
cells), generally leading to a bias towards lower FWHM values. From these
results, we concluded that the sparse decomposition tends to produce lower
FWHM values, has less variability, and is less prone to outliers.

Figure 4: The measurements were taken for the raw frames and the three methods
(SLNet, Median Background Removal, and "Rolling Ball" algorithm) on the glial
cells and microtubules dataset [34, 35]. Left: Box plot presenting
quantitative results of around 45 measured emitters’ FWHM on the STORM emitter
images. Right: Box plot presenting quantitative results of around 20 measured
tubules’ FWHM of the reconstructed super-resolution images. Figure 5: Wide-
field images of both datasets and SQUIRREL analysis comparing the four
respective STORM reconstructions. SLNet delivers better linearity in
reconstruction concerning the wide-field image of datasets.

####  3.1.2 SQUIRREL Analysis

As a second step of the evaluation, we performed a SQUIRREL analysis (see Fig.
5) with the help of NanoJ- SQUIRREL [40], an ImageJ plug-in. SQUIRREL stands
for Super-resolution QUantitative Image Rating and Reporting of Error
Locations, because it assesses and maps errors and artifacts of super-
resolution images concerning their dataset wide-field image. A SQUIRREL
analysis calculates the scaled Pearson’s coefficient (RSP) and the smallest
resolution scaled error (RSE). A higher RSP and a lower RSE indicate a better
agreement and linearity of the reconstruction with the wide-field image. Our
conclusions based on the FWHM analysis were confirmed by analyzing each
method’s best-reconstructed images with NanoJ-SQUIRREL. It proved that the
SLNet super-resolution STORM image had the highest RSP values and the smallest
RSE for both datasets, hence having the highest agreement with the wide-field
image from all other STORM frames. The "rolling ball" images achieved the
second-best agreement.

###  3.2 SLNet hyperparameter search: Sparsity threshold and alpha

The threshold $\mu$ from the SLNet singular value shrinking operator (Equation
5) and the $\alpha$ parameter from the loss function (Equation 4) control the
low rank of the SLNet predicted dense components and, consequently, also the
sparsity of the output sparse frames. The $\mu$ parameter indicates how the
eigenvalues of the input frames should be shrunk to lower the input frames’
rank and to produce the sparse component at the end. To demonstrate this
behavior affected by the $\mu$ threshold, we trained the SLNet for 100 epochs
on 100 images with different $\mu$ parameters, i.e., with $\mu=\\{0,\ 0.2,\
0.5,\ 1,\ 3,\ 10,\ 20,\ 30\\}$. On the other hand, the $\alpha$ parameter
influences the loss function by weighting the sparse component. We trained the
network with $\alpha=\\{0,\ 0.2,\ 0.5,\ 1,\ 3,\ 7,\ 12,\ 20,\ 30\\}$ and
tested with the remaining raw frames. Both parameters are user-defined and
influence our background removal method directly.

Figure 6: Comparison of the sparsity level of glial cells [34] frames from
SLNets trained with various $\mu$ and $\alpha$ values with the median
background removal and the "rolling ball" algorithm images.

The degree of sparseness (sparsity) is the percentage of zero elements out of
all the elements as in:

$sparsity=\frac{\\#\ zero\ elements}{\\#\ total\ elements}\times 100$ (6)

Fig. 6 shows how these parameters influence the level of sparseness compared
with the other methods. The results verify the idea behind the sparse
decomposition and our unsupervised training of the network with the help of
the loss function because the sparsity of the results increases as the
threshold $\mu$ and the $\alpha$ parameter increase. As seen, even when both
parameters are set to 0, the network produces a low-rank starting point, and
the decomposition outputs a sparser result. The $\alpha$ parameter influences
the sparsity faster than the $\mu$ threshold since, as seen from the graphs,
for $\alpha$ = 3, we get 77% sparsity, whereas for $\mu$ = 3, 34%.

Additionally, all SLNet results have a higher sparseness level than the other
method’s outcomes, constant at either 5% or 18%. As such, our method implies
numerous benefits, such as significantly reducing the computational time, the
duration of the STORM reconstruction, and the flexibility to alter the
background levels based on the concrete data. By analyzing these results and
considering the varying background levels in each dataset, we concluded that
both parameters are crucial, and the user should choose them based on the
dataset and the desired sparseness level. Nevertheless, the user should
consider that training the network for too long or choosing too high values
for both parameters, the output images would have very high sparsity and be
counterproductive for the STORM reconstruction.

##  4 Discussion

On the grounds of implementing a more performant approach to deal with the
problem of constant or autofluorescence background that occurs in high-
resolution fluorescence microscopy, we introduce and adapt a sparse
decomposition-based background removal method using the neural network SLNet
(fig. 1) in working with STORM raw images. Subsequently, SLNet proved to be an
efficient option as a convolutional neural network to produce the low-rank
components of the input images and perform the sparse decomposition. This
decomposition computes the sparse images without the unnecessary background,
ready to be used for the STORM super-resolution reconstruction.

The sparsity-based background removal method presented is computationally
efficient due to the simple architecture of the SLNet and the fast training
and evaluation time. Furthermore, we demonstrated how the parameters $\mu$ and
$\alpha$ control the sparseness level of the output frames due to their
influence on the loss function (fig. 6). Both these parameters are very user-
friendly because they are easily reconfigured for the model training. They are
also crucial in STORM since the studied samples and used microscopic systems
differ in the level of fluorescence they produce. Hence, the user can control
the sparseness based on the present background and obtain satisfactory
decomposition results.

To evaluate our method, we compared the reconstruction of other commonly used
methods, such as the median background removal and the "rolling ball"
algorithm (fig. 2, 3). The reconstruction was executed with the software
ThunderSTORM, and we encountered that the most satisfactory results were
derived from a combination of image filters (part of the ThunderSTORM
workflow) and the SLNet. Additionally, the reconstruction of our outcomes was
faster (up to 80% faster for a reconstruction without any filters) than the
other methods due to the sparsity of the frames. Measuring the FWHM of the
emitters in the STORM frames and small areas like microtubules or thin
filament extensions in the reconstructed images (fig. 4) delivered lower FWHM
values and proved that our sparse decomposition method achieved the highest
resolution. A SQUIRREL analysis (fig. 5) proved the better agreement and
linearity of the SLNet’s result out of all other super-resolution
reconstructed images compared with the respective wide-field frame.

Overall, our method accurately processed sparse frames with less background,
leading to a high localization precision of the emitters’ centroids.
Consequently, the fluorophores were adequately distant and visible, and the
specimen’s structure was defined and shaper. In contrast, the other
reconstructed images were blurry in highly dense areas, and the tubules
appeared fused with each other. However, we saw (Fig. 3 bottom right tubules)
that the sparse decomposition is not beneficial when parts of the specimen
occur in distant image regions since it approximates those molecules as
background and removes them.

##  5 Conclusion

In this paper, we presented a method to remove the background of STORM images
using a sparse decomposition that assumes the background represents the low-
rank components and the specimen represents the sparse ones. It was
implemented by adapting the neural network SLNet to work on STORM, and we
assessed that it outperformed some of the most used approaches by scientists.
The best case to use SLNet is when studying specimens with a detailed and
overlapping structure. Our method could be a beneficial tool in the STORM
microscopy environment that is specific to its input and delivers high-
resolution images due to its fast computational time and the flexibility
offered in the image sparseness levels.

Funding Content in the funding section will be generated entirely from details
submitted to Prism. Authors may add placeholder text in the manuscript to
assess length, but any text added to this section will be replaced during
production and will display official funder names along with any grant numbers
provided. If additional details about a funder are required, they may be added
to the Acknowledgments, even if this duplicates information in the funding
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please include funder names and grant numbers in the manuscript.

Disclosures The authors declare no conflicts of interest.

Data Availability Statement Data underlying the results presented in this
paper are available in [34, 35]. The code is available in [31].

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