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Super-Resolution Image Reconstruction forHigh-Density 3D Single-Molecule MicroscopyJiaqing Huang, Student Member, IEEE, Mingzhai Sun, Jianjie Ma, and Yuejie Chi, Member, IEEE

Abstract—Single-molecule localization based super-resolutionmicroscopy, by localizing a sparse subset of stochastically ac-tivated emitters in each frame, achieves sub-diffraction-limitspatial resolution. Its temporal resolution, however, is constrainedby the maximal density of activated emitters that can besuccessfully reconstructed. The state-of-the-art three-dimensional(3D) reconstruction algorithm based on compressed sensingsuffers from high computational complexity and gridding errordue to model mismatch. In this paper, we propose a novelsuper-resolution algorithm for 3D image reconstruction, dubbedTVSTORM, which promotes the sparsity of activated emitterswithout discretizing their locations. Several strategies are pursuedto improve the reconstruction quality under the Poisson noisemodel, and reduce the computational time by an order-of-magnitude. Numerical results on both simulated and cell imagingdata are provided to validate the favorable performance of theproposed algorithm.

Index Terms—Image reconstruction, super-resolution, sparseinverse problems, total-variation norm.

I. INTRODUCTION

Illuminated by light of certain wavelength, fluorophores canemit light of longer wavelength which can be captured by afluorescence microscope. Patterns of fluorescence in cells ortissues thus can be visualized after separating the emitting lightfrom the illumination light. The fluorescence microscopy hasfound numerous applications in the biological field, includingbut not limited to, discovering patterns of certain proteins,tracking the movement of molecules, studying intracellularsignaling, etc. However, due to the optical diffraction, theresolution of a conventional fluorescence microscopy is limitedto the Rayleigh limit, 0.61λ/NA [2], where λ is the wavelengthof emission light, and NA is the numerical aperture of theobjective lens.

In the past few decades, several novel imaging techniqueshave been developed to break the diffraction limit by overan order-of-magnitude both in the lateral and axial directions[3]–[6]. Among these techniques, single-molecule based super-resolution techniques, such as stochastic optical reconstruction

This work is supported in part by NSF under the grants CCF-1527456and ECCS-1650449, and by ONR under the grant N00014-15-1-2387. Part ofthis work will be presented at 2016 IEEE 13th International Symposium onBiomedical Imaging (ISBI) [1].

J. Huang and Y. Chi are with the Department of Electrical and ComputerEngineering, The Ohio State University, Columbus, OH 43210 USA (e-mail:huang.1273@osu.edu; chi.97@osu.edu).

M. Sun is with the Department of Precision Machinery and PrecisionInstrumentation, University of Science and Technology of China, Hefei, Anhui230026 China (e-mail: mingzhai@ustc.edu.cn).

J. Ma is with the Department of Surgery, Davis Heart and Lung ResearchInstitute, The Ohio State University, Columbus, OH 43210 USA (email:jianjie.ma@osumc.edu).

microscopy (STORM) [5] and photo-activated localizationmicroscopy (PALM) [6], improve the spatial resolution sig-nificantly by activating and localizing a sparse subset ofemitters within the nanometer scale in each frame. The secretbehind the resolution enhancement is the precise localizationof sparsely activated emitters within each frame, and repeatedactivation to allow different subsets of emitters turning on forlocalization across frames. The final super-resolution image isthen constructed by superimposing the reconstructed emitterlocations from all frames.

To extend STORM/PALM to 3D imaging, several tech-niques have been proposed to engineer the system’s pointspread function (PSF) in order to encode the axial location ofthe emitters, including single-camera imaging with astigma-tism [7], biplane imaging [8], multiple-camera imaging com-bined with astigmatism [9], hybrid astigmatic/biplane imaging[10], and double-helix PSF engineering [11]. In this paper, weconsider the most straightforward 3D system setup which issingle-camera imaging with astigmatism, which places a weakcylindrical lens into the optical path to modify the ellipticity ofthe emitter’s PSF along the axial direction, making it possibleto differentiate emitters at different axial locations. Extensionsto other 3D system setups are straightforward.

As the temporal resolution is limited by the number offrames needed to acquire a super-resolution image, it isdesirable to develop image reconstruction algorithms that canhandle higher emitter density per frame. However, a highdensity of activated emitters causes their PSFs to overlapspatially, which invalidates the widely used single emitterlocalization methods [5], [12], [13]. To overcome this, severalmethods [14]–[17] have been proposed, among which DAOS-TORM [17] achieves the best performance by simultaneouslyfitting overlapping PSFs, while its extension to 3D imagereconstruction, 3D-DAOSTORM [18], was shortly developedafterwards. However, it is demonstrated in [19] and [20]that, DAOSTORM is outperformed by compressed sensingbased reconstruction algorithm (CSSTORM) [8], [9], [11],[19], [21], [22], which has been shown as a robust and highperformance algorithm for high-density super-resolution imagereconstruction for both 2D and 3D imaging. For each frame,CSSTORM first imposes a fine-grained grid to model thelocations of activated emitters as a sparse signal in a discretedictionary, of which the image on the camera becomes linearmeasurements, then solves a sparse recovery problem basedon `1 minimization to invert the emitters’ locations. How-ever, this introduces an inevitable mismatch between the truecontinuous-valued location of the emitter, and its estimatedlocation on the discrete grid [23]. To reduce the artifact by

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the mismatch error, the grid needs to be fine enough, whichresults in an extremely large dictionary, making the computa-tion very expensive. Moreover, heuristic post-processing stepsare typically added to CSSTORM to enhance performance.Another algorithm called 3B analysis [24] incorporates thetemporal information of STORM/PALM images and utilizesthe emitter blinking statistics to reconstruct a high-resolutionimage using a Bayesian approach. However, it is still based ona discrete grid and suffers from long computation time evenwith parallel computing [25].

To address the artifact introduced by discretization, Minet al. proposes an algorithm called FALCON that combinessparsity-promoting formulation with a Taylor series approxi-mation of the PSF [26]. Its 3D extension for astigmatic andbiplane imaging, FALCON3D, is proposed later in [10]. BothFALCON and FALCON3D are demonstrated to reduce the dis-cretization artifact and computational cost by eliminating theneed of a fine-grained grid. However, the off-grid techniquesused in [10], [26] still suffer from approximation errors on apre-selected coarse grid.

For 2D super-resolution image reconstruction, another algo-rithm called MempSTORM has been proposed [27] based on2D spectrum estimation techniques in the signal processingliterature. MempSTORM directly estimates the continuous-valued location of the activated emitters without any discretiza-tion by transforming the image to the spectral domain. It isdemonstrated to handle the same level of emitter density asCSSTORM with much faster computational speed. A similaralgorithm, ALOHA [28], is also proposed based on low-rank Hankel structured matrix in the Fourier space with thecapability to infer the PSF. However, both MempSTORM andALOHA cannot be readily extended to 3D super-resolutionimage reconstruction. Moreover, in all the above mentionedworks, the noise is modeled as an additive bounded or Gaus-sian noise without considering its discrete characteristics ofphoton counting.

In this paper, we propose a novel super-resolution frame-work for 3D image reconstruction under the Poisson noisemodel, which is more appropriate for photon count data.Given the 3D PSF profile, the camera image is treated asan observation drawn from a Poisson distribution whoseparameters are determined by a point source signal composedof a sum of Dirac measures, each shifted and scaled by thelocations and intensities of the activated emitters. In [29], thetotal-variation norm is used as a continuous analog of the `1norm for finite-dimensional vectors to promote emitter sparsitywithout discretizing their locations via convex optimization.Recently, Rao et.al. [30] and Boyd et.al. [31] have developedfast and efficient algorithms for solving generic total-variationnorm regularized sparse inverse problems respectively, usingconditional gradient with enhancement and truncation (Co-GEnT) and alternating descent conditional gradient (ADCG)via extending the classical conditional gradient descent method[32]. In particular, it is demonstrated in [31] that the ADCGframework can be implemented for 2D super-resolution imag-ing. Built on these pioneering algorithmic advances, we aimto reconstruct the point source signal by maximizing thePoisson log-likelihood while keeping its cardinality small. We

propose a new algorithm, dubbed TVSTORM, that is closein spirit to orthogonal matching pursuit [33] but operateson an infinite-dimensional signal. Our algorithm falls intothe framework of ADCG [31] with carefully-designed mod-ifications in order to handle the Poisson likelihood as wellas the 3D PSF profile for improving performance. Specifi-cally, in each iteration, TVSTORM first selects a new pointsource and adds it to the current estimate, whose locationis determined by a first-order linearization of the Poissonlog-likelihood function over a coarse grid, and then refinesthe estimate of all the included point sources by gradientdescent using backtracking line search. Through numericalexperiments, TVSTORM demonstrates an order-of-magnitudeimprovement on the computational cost over CSSTORM dueto the elimination of optimizing over a fine-grained grid. It alsoshows significant improvement on the localization accuracyover CSSTORM in terms of detection rate, false discoveryrate and precision, without adding post-processing steps.

Preliminary results of TVSTORM were reported in [1],with the current paper significantly extending and refining thedetails of the algorithm with extensive numerical simulationsfor both synthetic and real data.

The rest of the paper is organized as follows. Section IIdescribes the problem formulation of super-resolution imagereconstruction. We present the proposed TVSTORM algorithmin Section III. Numerical experiments on both synthetic andreal data are demonstrated in Section IV. Finally, we concludein Section V.

II. SUPER-RESOLUTION IMAGE RECONSTRUCTION

We begin by introducing the imaging system of 3D single-molecule localization microscopy. In each frame, a sparse sub-set of emitters are activated. Let θ(i) = [θ

(i)x , θ

(i)y , θ

(i)z , θ

(i)I ] =

[θ(i)L , θ

(i)I ] be the parameters for the ith emitter, where θ(i)

L =

[θ(i)x , θ

(i)y , θ

(i)z ] ∈ S are the coordinates in x, y and z dimen-

sions, respectively,

S ={θL = [θx, θy, θz]

∣∣∣θx ∈ (xmin, xmax), θy ∈ (ymin, ymax),

θz ∈ (zmin, zmax)},

and θ(i)I > 0 denotes its intensity. Let Θ =

{θ(1), θ(2), ..., θ(P )} be the set of parameters, where Pis the total number of emitters. We can write the set ofactivated emitters χ = χ(x, y, z|Θ) as a sparse superpositionof point sources, given as a point source signal:

χ = χ(x, y, z|Θ) =

P∑i=1

θ(i)I δ(x−θ(i)

x , y − θ(i)y , z − θ(i)

z ), (1)

where δ(x − x0, y − y0, z − z0) is a Dirac measure locatedat (x0, y0, z0). For notational convenience, we also use χ(Θ)to denote χ(x, y, z|Θ). We denote the admissible set of χ asG = {χ = χ(Θ)|Θ = {θ(i)}Pi=1, P ∈ Z+, θ

(i)L ∈ S, θ(i)

I ≥0, 1 ≤ i ≤ P}.

Due to diffraction, the point source signal χ is low-passfiltered by the PSF of the microscopy before forming theimage, whose shape is modeled as a 2D Gaussian function

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with the ellipticity determined by the location along the zdirection:

K(x, y|x0, y0, z0) =1

2πσx(z0)σy(z0)e−(

(x−x0)2

2σx(z0)2+

(y−y0)2

2σy(z0)2

),

(2)where σx(z0) and σy(z0) are the standard deviations in the xand y directions, which are calibrated as [7]:

σx(z0) = σx,0

√√√√1 +

4∑i=2

Ax,i

(z0 − cxdx

)i,

σy(z0) = σy,0

√√√√1 +

4∑i=2

Ay,i

(z0 − cydy

)i,

(3)

where σx,0, σy,0, Ax,i, Ay,i, i = 2, 3, 4, cx, cy , dx and dy areknown parameters of the defocusing function [9].

The expected photon rate received at the (u, v)th camerapixel, denoted as µ(u, v), can be written as a convolutionbetween the PSF in (2) and the point source signal in (1),integrated over the area of a pixel:

µ(u, v|χ) =

∫ u+ 12

u− 12

∫ v+ 12

v− 12

(K ∗ χ)(x, y) dxdy, (4)

where ∗ denotes convolution, and

(K ∗ χ)(x, y) =

P∑i=1

θ(i)I K(x, y|θ(i)

x , θ(i)y , θ(i)

z ). (5)

Therefore, (4) can be rewritten as

µ(u, v|χ)

=

P∑i=1

θ(i)I

4

[Q

(u− θ(i)

x + 12√

2σx(θ(i)z )

)−Q

(u− θ(i)

x − 12√

2σx(θ(i)z )

)]

×

[Q

(v − θ(i)

y + 12√

2σy(θ(i)z )

)−Q

(v − θ(i)

y − 12√

2σy(θ(i)z )

)]

=

P∑i=1

θ(i)I φ(u, v|θ(i)

L ), (6)

where Q(x) = 2√π

∫ x0e−t

2

dt and φ(u, v|θ(i)L ) is the image

contributed by a single emitter located at θ(i)L .

The number of photons hitting the camera at the (u, v)th

pixel, denoted as y(u, v), follows an independent Poissondistribution with the parameter µ(u, v|χ), given as

Pr(y(u, v) = z|χ) =µ(u, v|χ)ze−µ(u,v|χ)

z!, z ∈ Z+. (7)

Denote the camera image as y = {y(u, v)}. The objectiveof super-resolution is to estimate the point source signal χ(Θ),given the observed image y.

III. PROPOSED APPROACH

In this section, we describe the proposed TVSTORM algo-rithm for high-density 3D super-resolution imaging under thePoisson noise model explained in Section II.

A. Poisson Loss Function

We first define the loss function, `(y|χ), as the negativePoisson log-likelihood of observing y given χ. According to(7),

`(y|χ) = −log

(∏u

∏v

Pr(y(u, v)|χ)

)=∑u

∑v

(µ(u, v|χ)− y(u, v)log(µ(u, v|χ))) + C,

where C is a constant that does not depend on χ. As suggestedin [34], for Poisson log-likelihood, it is advantageous tointroduce a small offset 0 < β � 1 inside the logarithmicterm to improve stability, so that we modify `(y|χ) as

`(y|χ) =∑u

∑v

[µ(u, v|χ)− y(u, v)log(µ(u, v|χ) + β)]

(8)after dropping the constant term. Instead of discretizing χ overa finite grid, we wish to minimize `(y|χ) while keeping thecardinality of χ small.

B. Description of TVSTORM

Since the point source signal χ is defined over an infinite-dimensional space, it is challenging to optimize. Nonethe-less, recent pioneering work in [31] has developed ADCGfor solving total-variation norm regularized sparse inverseproblems that are defined over infinite-dimensional measures,which can be regarded as a continuous analog of conjugategradient descent for solving `1-norm minimization. Inspiredby [31], we develop an iterative algorithm tailored to oursetting, denoted as TVSTORM, that is close in spirit toorthogonal matching pursuit but operates over the continuous-valued parameter space.

The description of TVSTORM is given in Algorithm 1.TVSTORM is an iterative algorithm, where in each iteration,a new point source is first selected and added to the currentestimate of χ, and then the estimate of χ is refined by gradientdescent using backtracking line search for all the parametersfixing the number of point sources. The algorithm stops whenthe intensity of the most recently added point source fallsbelow a given threshold.

Let χ(t) = χ(x, y, z|Θ(t)) be the estimate of the emitterobject at the tth iteration, where Θ(t) represents the parametersof the point sources in χ(t). Let µ(χ(t)) = {µ(u, v|χ(t))}denote the noise-free image generated from χ(t) accordingto (4). At the (t+ 1)

th iteration, the SELECT step aims toadd one point source (with parameter θ(t+1)) to χ(t), whichwe wish to find by minimizing a first-order Taylor seriesapproximation of `(y|χ). Following [31], this can be done byfirst calculating the partial derivative of `(y|χ(t)) with respectto µ(χ(t)), given as ∂`(y|χ(t))

∂µ(χ(t)), and then picking the source

location θ(t+1)L that maximizes

argminθL∈S

⟨∂`(y|χ(t))

∂µ(χ(t)), φ(θL)

⟩, (9)

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Algorithm 1 TVSTORM1: Input Parameter: threshold γ2: t← 03: Θ(0) ← ∅4: χ(0) ← χ(x, y, z|Θ(0))5: repeat6: . SELECT7: Find the emitter location θ(t+1)

L using (10);8: Find the emitter intensity θ(t+1)

I using (11);9: Update: Θ(t+1) ← Θ(t) ∪ {[θ(t+1)

L , θ(t+1)I ]}

10: . REFINE11: Θ(t+1) ← REFINE(y, Θ(t+1))12: χ(t+1) ← χ(x, y, z|Θ(t+1))13: t← t+ 114: until θ(t)

I < γ

where 〈·, ·〉 denotes the inner product. Instead of optimizingover the continuous regime S, we select the location of thenew point source as

θ(t+1)L = argmin

θL∈Scoarse

⟨∂`(y|χ(t))

∂µ(χ(t)), φ(θL)

⟩, (10)

where Scoarse is a coarse grid over S. We only require acoarse grid since the locations will be refined afterwards. Theintensity of the added emitter is chosen as the solution to

θ(t+1)I = argmin

θI>0

∥∥∥y − µ(χ(t))− θI · φ(θ(t+1)L )

∥∥∥2

2. (11)

Again we adopt the simple least-squares criteria rather thanthe likelihood since it will be refined later.

The REFINE step aims to find the maximum likelihoodestimate of χ(t+1) with the number of point sources fixed byminimizing the loss function using iterative gradient descentby using the emitter set Θ(t+1) obtained at the SELECT stepas initialization. For each parameter θ ∈ Θ(t+1), we first findthe direction that decreases the loss function by calculatingthe partial derivative of the loss function with respect to θ,whose expressions are given in Appendix A. The step sizeis then determined using backtracking line search to speedup convergence. The details of this step are described inAlgorithm 2, where PG(θj) is a projection to the constrainedparameter space. Algorithm 2 is also known as coordinatedescent.

C. Discussions

Since a coarse grid is used in the SELECT step to find thestarting locations of emitters, the main computation is com-mitted in the REFINE step, specifically, in the re-evaluationof `(y|χ(Θ)) each time Θ is altered (line 10 in Algorithm 2).Assume the image size is n × n. According to (6), the re-evaluation of `(y|χ(Θ)) requires O(|Θ| · n2) calculations.Since each time the re-evaluation only requires changing theparameters of one emitter (line 8 in Algorithm 2), efficientimplementation of this step requires only O(n2) calculationsby maintaining χ(Θ \ {θ(i)}) when refining the parameters ofthe ith emitter.

Algorithm 2 REFINE(y,Θ)

1: Input Parameters: α0, τ ∈ (0, 1), c ∈ (0, 1)2: repeat3: for every θ(i) in Θ do4: for every parameter θ(i)

j in θ(i) do5: α← α0

6: Θ← Θ7: repeat

8: θ(i)j ← PG

(θ

(i)j − α

∂`(y|χ(Θ))

∂θ(i)j

)9: α← α× τ

10: until `(y|χ(Θ)) ≤ `(y|χ(Θ))− αc∥∥∥∥∂`(y|χ(Θ))

∂θ(i)j

∥∥∥∥2

2

11: Θ← Θ12: end for13: end for14: until convergence

The stopping criteria of TVSTORM is determined bythreshold γ in Algorithm 1. Fortunately, we find the perfor-mance of TVSTORM relatively insensitive to the selection ofγ. In our implementation, we set γ to half of the expectedemitter intensity. In the REFINE step, the selection of α0

determines the maximum step size, while τ regulates theshrinking factor of the step size and c involves the Armijo-Goldstein condition [35] (line 10 in Algorithm 2). To avoidevaluating χ(Θ) at too many points, we set α0 = 103,τ = 10−1 and c = 10−6.

Our algorithm can be viewed as an instance of ADCG [31],which provides a general framework for measure optimiza-tion. TVSTORM does not assume the total-variation normof the reconstruction is bounded; rather, it pursues a greedystrategy to add new emitters and stops when the intensityof the new emitter is small enough judged by the stoppingcriteria. Moreover, the REFINE step does not prune the setof included emitters and applies line search to optimize theemitter parameters.

D. Handling background noise

Since typically the image is not homogenous in certainareas, e.g., center regions of the cell, a background termis added in our model (6) to take this inhomogeneity intoaccount, i.e.,

µ(u, v|χ)

=

P∑i=1

θ(i)I

4

[Q

(u− θ(i)

x + 12√

2σx(θ(i)z )

)−Q

(u− θ(i)

x − 12√

2σx(θ(i)z )

)]

×

[Q

(v − θ(i)

y + 12√

2σy(θ(i)z )

)−Q

(v − θ(i)

y − 12√

2σy(θ(i)z )

)]+ θb, (12)

where θb is an extra parameter to be estimated. This changeaccounts for various background across different blocks. It isstraightforward to modify the REFINE step in Algorithm 2,where we apply gradient descent using backtracking linesearch to refine θb.

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(a) Original image (b) CSSTORM before de-biasing

(c) CSSTORM after de-biasing (d) TVSTORMFig. 1. Emitter localization using CSSTORM and TVSTORM. (a) Originalimage; (b) CSSTORM before debiasing; (c) CSSTORM after debiasing; and(d) TVSTORM. We use an ellipsoid to represent the x, y and z locations ofan emitter. The center of ellipsoid is determined by the x and y positions andthe shape is determined by the z position. Red represents the results from ourcalculation and white represents the actual location from the simulation.

IV. NUMERICAL EXPERIMENTS

We conduct numerical simulations and real experimentsto demonstrate the preferable performance of TVSTORM.The images used for numerical simulations are generated byrandomly distributing certain amount of emitters in a 3D or 2Dspace. The number of emitters is determined by the requireddensity of each simulation; the volume of 3D space is 0.8µm × 0.8 µm × 0.8 µm and the volume of 2D space is0.8 µm × 0.8 µm unless otherwise stated. The pixel size ofthe generated images is 75 nm × 75 nm, which matchesour real system setting. Equations (2) and (3) are used togenerate the PSF model for simulation whose parameters arecalibrated in [9]. The simulated images are then corrupted withPoisson noise with the means of the detected photon numberdetermined by the expected number of photons of each pixel,as described in Section II.

A. Comparisons with CSSTORM on 3D Image Reconstruction

We first examine the reconstruction quality usingCSSTORM and TVSTORM on a single frame. We generatean image with four emitters that are randomly distributed ina 3D space, with intensity of 300 photons each, as shownin Fig. 1 (a). For CSSTORM, an up-sampling factor of 8in lateral direction and 9 in axial direction is used in thediscretization. The output from CSSTORM typically requirespost-processing such as de-biasing [9] in order to mitigatethe gridding error, while TVSTORM does not includepost-processing steps. Fig. 1 (b) and (c) show the imagereconstruction from CSSTORM before and after de-biasing,

where we use an ellipsoid to represent the spatial locations ofan emitter with the center representing its lateral position andthe shape representing its axial position. The reconstructionis shown in red, while the ground truth is shown in white.As seen from Fig. 1 (b), the reconstruction from CSSTORMbefore de-biasing contains many false positives. After de-biasing, nearby output emitters are clustered together butone emitter is missing, as shown in Fig. 1 (c). Contrarily,the reconstruction from TVSTORM, as shown in Fig. 1 (d),identifies all emitters correctly with high precision.

Next, to evaluate the average performance of TVSTORM,we generate a series of STORM images under differentdensities (0.75 emitter/µm3 to 11.25 emitters/µm3). Theemitters are randomly distributed in a 3D volume with theintensity set as 500. The images are then corrupted withPoisson noise. Fig. 2 compares the performance of TVSTORMwith CSSTORM in terms of identified density, false discoveryrate, precision and execution time with respect to the emitterdensity. The identified density is defined as the number ofcorrectly detected emitters per area. The false discovery rateis defined as the ratio of incorrectly detected emitters in alldetected emitters. The precision is defined as the standarddeviation of localization errors. Indeed, TVSTORM is ableto detect more emitters with an improved precision whilemaintaining a lower false discovery rate than CSSTORM.Additionally, the execution time of TVSTORM is much fasterthan that of CSSTORM due to the elimination of a fine-grainedgrid during optimization.

Density (emitters/µm3)0 1 2 3 4 5 6 7 8 9 1011

Iden

tifie

d D

ensi

ty (e

mitt

ers/µ

m3 )

0123456789

1011

(a)

CSSTORMTVSTORM

Density (emitters/µm3)0 1 2 3 4 5 6 7 8 9 1011

Fals

e D

isco

very

Rat

e

0

0.05

0.1

0.15

0.2(b)

CSSTORMTVSTORM

Density (emitters/µm3)0 1 2 3 4 5 6 7 8 9 1011

Prec

isio

n (n

m)

10

20

30

40

50

60(c)

CSSTORMTVSTORM

Density (emitters/µm3)0 1 2 3 4 5 6 7 8 9 1011

Exec

utio

n Ti

me

(s)

10-2

10-1

100

101

102(d)

CSSTORMTVSTORM

Fig. 2. 3D image reconstruction performance comparisons between TVS-TORM and CSSTORM: (a) identified density, (b) false discovery rate, (c)precision and (d) execution time with respect to the emitter density.

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B. Examining Different Photon and Background Levels

We then evaluate the performance of TVSTORM as signal-to-noise ratio (SNR) varies. We generate a series of STORMimages with emitters randomly distributed in a 3D volumeunder different SNR levels by varying the emitter photonnumbers from 100 to 1000, covering the photon levels ofnormal fluorescent proteins and fluorescent dyes. Fig. 3 showsthat the performance of TVSTORM deteriorates with thedecrease of the SNR. However, even with an emitter photonnumber as low as 100, TVSTORM is still able to detect alarge portion of emitters with small false discovery rate atan acceptable precision, as shown by the identified density,false discovery rate, and precision with respect to the emitterdensity at different photon numbers in Fig. 3 (a), (b) and (c),respectively. By explicitly considering Poisson noise model,TVSTORM is relatively insensitive to the SNR and allowsthe use of fluorophores with lower photon yield.

Density (emitters/µm3)0 1 2 3 4 5 6 7 8 9 1011Id

entif

ied

Den

sity

(em

itter

s/µ

m3 )

0123456789

1011

(a)Photons: 100Photons: 400Photons: 700Photons: 1000

Density (emitters/µm3)0 1 2 3 4 5 6 7 8 9 1011

Fals

e D

isco

very

Rat

e

0

0.05

0.1

0.15

0.2

0.25(b)

Photons: 100Photons: 400Photons: 700Photons: 1000

Density (emitters/µm3)0 1 2 3 4 5 6 7 8 9 1011

Prec

isio

n (n

m)

10

20

30

40

50

60

70

(c)Photons: 100Photons: 400Photons: 700Photons: 1000

Density (emitters/µm3)0 1 2 3 4 5 6 7 8 9 1011

Exec

utio

n Ti

me

(s)

0

0.5

1

1.5

2

2.5

3(d)

Photons: 100Photons: 400Photons: 700Photons: 1000

Fig. 3. Performance of 3D image reconstruction across different photon levels:(a) identified density, (b) false discovery rate, (c) precision and (d) executiontime with respect to the emitter density.

We also evaluate the performance of TVSTORM underdifferent background noise levels. A series of STORM imageswith background level from 0 to 400 photons are generatedand analyzed using TVSTORM. The emitter photon level isset to 1500. As Fig. 4 (a) shows, TVSTORM is still able tocapture a substantial amount of emitters when the backgroundis as high as 400 photons, with a relatively low false discoveryrate and acceptable precision, shown in Fig. 4 (b) and (c).

C. Block Width Selection

The image acquired at the camera is very large andcannot be handled as a whole. It is typically divided into

Density (emitters/µm3)0 1 2 3 4 5 6 7 8 9 10 11Id

entif

ied

Den

sity

(em

itter

s/µ

m3 )

0123456789

1011

(a)Background: 0Background: 100Background: 200Background: 300Background: 400

Density (emitters/µm3)0 1 2 3 4 5 6 7 8 9 10 11

Fals

e D

isco

very

Rat

e

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35(b)

Background: 0Background: 100Background: 200Background: 300Background: 400

Density (emitters/µm3)0 1 2 3 4 5 6 7 8 9 10 11

Prec

isio

n (n

m)

10

20

30

40

50

60

70

80

90

(c)Background: 0Background: 100Background: 200Background: 300Background: 400

Density (emitters/µm3)0 1 2 3 4 5 6 7 8 9 10 11

Exec

utio

n Ti

me

(s)

0

0.5

1

1.5

2

2.5

3

3.5

4

(d)Background: 0Background: 100Background: 200Background: 300Background: 400

Fig. 4. Performance of 3D image reconstruction across different backgroundlevels: (a) identified density, (b) false discovery rate, (c) precision and (d)execution time with respect to the emitter density.

small overlapping blocks which are analyzed independentlyto improve the computation time. To avoid compromising theperformance, blocks are overlapped and the block width is setto be larger than the PSF width. Here we study the effect ofblock width on the performance of TVSTORM by simulatingan image of size 45 µm × 45 µm and analyzing the imageunder different block widths. The photon number is set to500 and the density is 6 emitters/µm3. In our simulation, wefirst divide the total image into non-overlapping blocks of sizew×w, where w is the block width. For each block, a marginof 3 pixels in each direction is added, creating an extendedblock of size (w + 6)× (w + 6). Then we apply TVSTORMon the extended block and only emitters within the centralw × w region are kept to avoid edge effects. Fig. 5 (a), (b)and (c) show that the identified density, false discovery rateand precision maintains stable as the block width changes.However, the execution time per pixel decreases first and thenincreases with the block width, as shown in Fig. 5 (d). Thelowest execution time per pixel occurs when the block widthis 8.

D. Model Mismatch

In practice, due to the misalignment and lens imperfection,the center coordinate of the emitter’s PSF suffers from smalldrifts as the emitter moves along the axial direction [36]. Inspecific, (6) becomes

µ(u, v|χ) =

P∑i=1

θ(i)I

4

[Q

(u− θ(i)

x −∆x(θ(i)z ) + 1

2√2σx(θ

(i)z )

)

7

Block width4 8 12 16 20 24Id

entif

ied

Den

sity

(em

itter

s/µ

m3 )

0

1

2

3

4

5

6(a)

Block width4 8 12 16 20 24

Fals

e D

isco

very

Rat

e

0

0.2

0.4

0.6

0.8

1(b)

Block width4 8 12 16 20 24

Prec

isio

n (n

m)

0

20

40

60

80

100(c)

Block width4 8 12 16 20 24

Tim

e Pe

r Pix

el (s

)

0.04

0.06

0.08

0.1

0.12

0.14(d)

Fig. 5. Performance of 3D image reconstruction across different block widths:(a) identified density, (b) false discovery rate, (c) precision and (d) executiontime per pixel.

−Q

(u− θ(i)

x −∆x(θ(i)z )− 1

2√2σx(θ

(i)z )

)]

×

[Q

(v − θ(i)

y −∆y(θ(i)z ) + 1

2√2σy(θ

(i)z )

)

−Q

(v − θ(i)

y −∆y(θ(i)z )− 1

2√2σy(θ

(i)z )

)],

where ∆x(θ(i)z ) and ∆y(θ

(i)z ) are not known. To study the

effect of this model mismatch on TVSTORM, we generateimages under different mismatch level and analyze them usingTVSTORM. The deviation of the emitter lateral coordinateis generated as ∆x(z) = I sin(5z) and ∆y(z) = I cos(5z),where we vary I as different mismatch level. As shownin Fig. 6, as I increases, the performance of TVSTORMdecreases within a tolerable range until I reaches about 75nm. This indicates that calibration of the deviation alongthe axial direction is essential for accurate performance ofsuper-resolution algorithms, which can be performed usingtechniques for example in [36], and extra care should be takenwhen a large deviation is expected.

E. Performance Comparisons of 2D Image Reconstruction

Although TVSTORM is designed for 3D image reconstruc-tion, it can be easily modified for 2D setting. Here we study theperformance of TVSTORM for 2D image reconstruction. Wegenerate a series of simulated STORM images across a rangeof emitter densities (1 emitter/µm2 to 9 emitters/µm2). Theaverage number of photons for each emitter is 500. We alsoapply CSSTORM [19] and MempSTORM [27] on the same

images for comparison. An up-sampling factor of 8 is used inCSSTORM.

Fig. 7 compares the performance of TVSTORM withCSSTORM and MempSTORM in terms of identified density,false discovery rate, precision and execution time with respectto the emitter density. TVSTORM demonstrates the bestperformance in terms of identified density, false discoveryrate and precision. It is also an order-of-magnitude faster thanCSSTORM but is slower than MempSTORM, which is knownfor its extreme fast execution time.

F. Real Experiments

To demonstrate the practical applicability of TVSTORM,both 2D and 3D STORM images of microtubules stained withAlexa 647 in Hela cells are acquired with the same systemsetup and imaging condition described in [9] and analyzedwith TVSTORM. In addition, Gaussian rendering is used togenerate the final super-resolution image [26], where everyemitter location is convolved with a Gaussian kernel scaledwith its estimated intensity.

Fig. 8 (a) shows the averaged image from 10000 framesof STORM images and Fig. 8 (b) is the 3D super-resolutionimage reconstructed using TVSTORM, where the structure of3D microtubule can be well resolved with the axial coordinaterepresented in different colors. Fig. 8 (c) and (d) show the com-parison of reconstruction results of a zoom-in region betweenTVSTORM and CSSTORM. It can be seen that TVSTORMprovides a visually more continuous reconstruction of the linestructure in microtubules, because it can detect more emittersand have lower false discovery rate than CSSTORM. Similarly,the averaged image and reconstructed super-resolution imagefor 2D STORM imaging are shown in Fig. 9 (a) and (b),respectively, and demonstrate high quality reconstruction andfine details that can not be resolved in averaged low-resolutionimage.

V. CONCLUSION

In this paper, TVSTORM is proposed for 3D super-resolution image reconstruction, which aims to maximize thelikelihood under the Poisson noise using a small numberof activated emitters. TVSTORM avoids the intrinsic biasof CSSTORM due to gridding, and is computationally moreefficient, with better detection rate, false discovery rate, andprecision. Furthermore, TVSTORM can be easily adaptedto 2D super-resolution image reconstruction or other single-molecule microscopy with different PSF configurations. Ex-tensive simulation results are provided to demonstrate thesuperior performance of TVSTORM. Experimental results arealso present to show its practical applicability. In the future,we aim to incorporate the temporal information of STORMimages across neighboring frames to further improve the imagereconstruction quality.

APPENDIX

In this appendix, we derive the partial derivatives of`(y|χ(Θ)) over every parameter θ(i)

j , which is required for

8

Density (emitters/µm3)0 1 2 3 4 5 6 7 8 9 10 11

Iden

tifie

d D

ensi

ty (e

mitt

ers/µ

m3 )

0

1

2

3

4

5

6

7

8

9

10

11(a)

I = 0 nmI = 15 nmI = 30 nmI = 45 nmI = 60 nmI = 75 nm

Density (emitters/µm2)0 1 2 3 4 5 6 7 8 9 10 11

Fals

e D

isco

very

Rat

e

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1(b)

I = 0 nmI = 15 nmI = 30 nmI = 45 nmI = 60 nmI = 75 nm

Density (emitters/µm2)0 1 2 3 4 5 6 7 8 9 10 11

Prec

isio

n (n

m)

10

20

30

40

50

60

70 (c)

I = 0 nmI = 15 nmI = 30 nmI = 45 nmI = 60 nmI = 75 nm

Fig. 6. 3D image reconstruction performance comparisons across different mismatch level: (a) identified density, (b) false discovery rate, (c) precision.

Density (emitters/µm2)0 1 2 3 4 5 6 7 8 9Id

entif

ied

Den

sity

(em

itter

s/µ

m2 )

0123456789(a)

CSSTORMTVSTORMMempSTORM

Density (emitters/µm2)0 1 2 3 4 5 6 7 8 9

Fals

e D

isco

very

Rat

e

0

0.01

0.02

0.03

0.04

0.05

0.06(b)

CSSTORMTVSTORMMempSTORM

Density (emitters/µm2)0 1 2 3 4 5 6 7 8 9

Prec

isio

n (n

m)

5

10

15

20

25

30(c)

CSSTORMTVSTORMMempSTORM

Density (emitters/µm2)0 1 2 3 4 5 6 7 8 9

Exec

utio

n Ti

me

(s)

10-3

10-2

10-1

100

101(d)

CSSTORMTVSTORMMempSTORM

Fig. 7. 2D image reconstruction performance comparisons between TVS-TORM, CSSTORM and MempSTORM: (a) identified density, (b) falsediscovery rate, (c) precision and (d) execution time with respect to the emitterdensity.

the REFINE procedure as described in Algorithm 2. To beginwith, we have

∂`(y|χ(Θ))

∂θ(i)j

=∑u

∑v

(µ(u, v|Θ)− y(u, v)

µ(u, v|Θ)

∂µ(u, v|Θ)

∂θ(i)j

).

(13)

We then derive the partial derivative of µ(u, v|Θ) over θ(i)x ,

θ(i)y , θ(i)

z , and θ(i)I . Let

u(i)+ =

u− θ(i)x + 1

2√2σx(θ

(i)z )

, u(i)− =

u− θ(i)x − 1

2√2σx(θ

(i)z )

,

v(i)+ =

v − θ(i)y + 1

2√2σy(θ

(i)z )

, v(i)− =

v − θ(i)y − 1

2√2σy(θ

(i)z )

.

Then according to (6) and following simple calculus, we have

∂µ(u, v|Θ)

∂θ(i)x

=θ

(i)I

2√

2πσx(θ(i)z )

[e−(u(i)−

)2

− e−(u(i)+

)2]

×[Q(v

(i)+

)−Q

(v

(i)−

)](14)

and

∂µ(u, v|Θ)

∂θ(i)y

=θ

(i)I

2√

2πσy(θ(i)z )

[Q(u

(i)+

)−Q

(u

(i)−

)]×[e−(v(i)−

)2

− e−(v(i)+

)2]. (15)

Additionally, using the chain rule, ∂µ(u,v|Θ)

∂θ(i)z

can be writtenas:

∂µ(u, v|Θ)

∂θ(i)z

=∂µ(u, v|Θ)

∂σx(θ(i)z )

∂σx(θ(i)z )

∂θ(i)z

+∂µ(u, v|Θ)

∂σy(θ(i)z )

∂σy(θ(i)z )

∂θ(i)z

,

(16)where

∂σx(θ(i)z )

∂θ(i)z

=

σx,0

(∑4i=2 i ·Ax,i

(z0−cxdx

)i−1)

2

√1 +

∑4i=2Ax,i

(z0−cxdx

)i ,

∂σy(θ(i)z )

∂θ(i)z

=

σy,0

(∑4i=2 i ·Ay,i

(z0−cydy

)i−1)

2

√1 +

∑4i=2Ay,i

(z0−cydy

)i .

(17)

From (6), ∂µ(u,v|Θ)

∂σx(θ(i)z )

and ∂µ(u,v|Θ)

∂σy(θ(i)z )

can be calculated as:

∂µ(u, v|Θ)

∂σx(θ(i)z )

=θ

(i)I

2√πσx(θ

(i)z )

[e−(u

(i)− )2 · u(i)

− − e−(u(i)+ )2 · u(i)

+

]×[Q(v

(i)+

)−Q

(v

(i)−

)], (18)

∂µ(u, v|Θ)

∂σy(θ(i)z )

=θ

(i)I

2√πσx(θ

(i)z )

[Q(u

(i)+

)−Q

(u

(i)−

)]×[e−(v

(i)− )2 · v(i)

− − e−(v(i)+ )2 · v(i)

+

]. (19)

9

0

100

200

300

400

500

600

700

800(a)

nm

(b)

(c) (d)

Fig. 8. (a) Averaged image from 10000 frames and (b) 3D image reconstruction result of microtubules stained with Alexa 647 using TVSTORM. Comparisonof reconstruction quality between (c) TVSTORM and (d) CSSTORM. (bar: 1.5 µm)

Lastly, it is easy to derive ∂µ(u,v|Θ)

∂θ(i)I

as:

∂µ(u, v|Θ)

∂θ(i)I

=1

4

[Q(u

(i)+

)−Q

(u

(i)−

)] [Q(v

(i)+

)−Q

(v

(i)−

)].

(20)

Plugging (14), (15), (16) and (20) into (13), we obtain thepartial derivative of the loss function `(y|χ(Θ)) over θ(i)

x , θ(i)y ,

θ(i)z , and θ(i)

I as in (22), (23), (24) and (25), respectively.

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∂`(y|χ(Θ))

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∂θ(i)z

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∂σy(θ(i)z )

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]. (25)

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Jiaqing Huang (S’16) received his Ph.D. degree inElectrical Engineering from The Ohio State Univer-sity, Columbus, OH, USA in 2016, and the B.E.degree in Electrical Engineering from ShandongUniversity, Jinan, Shandong, China in 2012.

His research interests include super-resolution im-age reconstruction, signal processing, and imageprocessing.

Mingzhai Sun received the Ph.D. degree in Physicsfrom the Missouri University in 2008. From 2008-2012 he was an Associate Postdoctoral Researcherat Princeton University. From 2012-2015, he wasa Research Scientist at the Heart and Lung Re-search Institute, The Ohio State University. He iscurrently a professor at the University of Science andTechnology of China. His research interests includedevelopment of optical super-resolution microscopyfor biological and medical applications, machinelearning with applications to medical images.

Jianjie Ma received his B.S. degree in physicsfrom Wuhan University and PhD in physiology andbiophysics from Baylor College of Medicine. Heis currently the Karl P Klassen Chair of ThoracicSurgery and the Division Director of Surgical andBiomedical Sciences at The Ohio State University.His research interests are in muscle and cardio-vascular diseases, regenerative medicine and cancerbiology. He funded a university spin-off biotech-nology company (TRIM-edicine, Inc.) to translatelaboratory findings into therapeutic applications.

Yuejie Chi (S’09-M’12) received the Ph.D. degreein Electrical Engineering from Princeton Universityin 2012, and the B.E. (Hon.) degree in Electrical En-gineering from Tsinghua University, Beijing, China,in 2007. Since September 2012, she has been anassistant professor with the department of Electricaland Computer Engineering and the department ofBiomedical Informatics at The Ohio State Univer-sity. She is the recipient of several awards includingthe IEEE Signal Processing Society Young AuthorBest Paper Award, NSF Career Award, AFOSR and

ONR Young Investigator Program Awards, ORAU Ralph E. Powe JuniorFaculty Enhancement Award, and Google Faculty Research Award. She isan Elected Member of the MLSP and SPTM Technical Committees of theIEEE Signal Processing Society since January 2016. Her research interestsinclude statistical signal processing, information theory, machine learning andtheir applications in high-dimensional data analysis, network inference, activesensing and bioinformatics.