Improvement of two-dimensional structuredillumination microscopy with an incoherentillumination pattern of tunable frequencyHASTI SHABANI,1,† ANA DOBLAS,1,† GENARO SAAVEDRA,2 EMILIO SANCHEZ-ORTIGA,2 AND

CHRYSANTHE PREZA1,*1Department of Electrical and Computer Engineering, The University of Memphis, Memphis, Tennessee 38152, USA23D Imaging and Display Laboratory, Department of Optics, University of Valencia, Burjassot, E-46199, Spain*Corresponding author: cpreza@memphis.edu

Received 3 October 2017; revised 22 November 2017; accepted 13 December 2017; posted 14 December 2017 (Doc. ID 308353);published 19 January 2018

In two-dimensional structured illumination microscopy (2D-SIM), high-resolution images with optimal opticalsectioning (OS) cannot be obtained simultaneously. This tradeoff can be overcome by using a tunable-frequency2D-SIM system and a proper reconstruction method. The goal of this work is twofold. First, we present a computa-tional approach to reconstruct optical-sectioned images with super-resolution enhancement (OS-SR) by usinga tunable SIM system. Second, we propose an incoherent tunable-frequency 2D-SIM system based on a Fresnelbiprism implementation. Integration of the proposed computational method with this tunable structured illumi-nation (SI) system results in a new 2D-SIM system that is advantageous compared to other 2D-SIM systems withcomparable complexity, because it provides high-resolution OS images independent of the objective lens used, with-out the presence of coherent noise and without reducing the contrast of the structured pattern, as in other incoherentimplementations. Evaluation of our proposed system is demonstrated with comparative studies of simulated andexperimental reconstructed images to validate our theoretical findings. Our experimental results show a simulta-neous improvement of the lateral resolution by a factor of 1.8× with the desired OS capability achieved in theresulting OS-SR combination image. Our experimental results also verify that our system can provide betterOS capability than the commercial Zeiss ApoTome-SIM system in the investigated study. © 2018 Optical

Society of America

OCIS codes: (180.6900) Three-dimensional microscopy; (110.2945) Illumination design; (110.1758) Computational imaging;

(100.6640) Superresolution; (100.3020) Image reconstruction-restoration.

https://doi.org/10.1364/AO.57.000B92

1. INTRODUCTION

Structured illumination microscopy (SIM) is a techniqueknown for its potential to enhance both the axial and lateralresolution of conventional widefield fluorescence microscopybeyond the classical Abbe diffraction limit. In SIM, the under-lying specimen is excited using a spatially varying structuredillumination (SI) pattern, which enables high-resolutionthree-dimensional (3D) images to be retrieved using the propercomputational reconstruction process. Among all types of SIMsystems, there are two SIM approaches based on which tradi-tional SIM systems have been developed and successfully com-mercialized. The first SIM approach, proposed by Neil andWilson [1,2], is based on the generation of the SI patternvia the incoherent projection of a grating onto the sampleplane. The main advantages of this implementation are:

(1) the image of the grating is axially localized in the samplevolume, which provides good optical-sectioning (OS) capabil-ity; (2) it does not present coherent noise; and (3) it has beensuccessfully implemented in a commercial microscope as anadd-on device (Zeiss ApoTome.2). Nonetheless, this systemis not aimed as a super-resolution technique because improve-ment in the lateral resolution is not significant due to the use oflow modulation frequency structured patterns. This is becausehigh-frequency information gets severely attenuated twice bythe incoherent optical transfer function (OTF), due to itswell-known shape, of both the illumination and detectionimaging systems.

The second SIM approach was developed to enhance thelateral resolution. Super-resolution (SR) performance wasachieved by Gustafsson [3] by illuminating the sample using

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a sinusoidal pattern created by the interference of two lightbeams emerging from a diffraction grating (2D-SIM). Withthis implementation, there is a tradeoff between OS capability(by filling the frequency information in the OTF’s missingcone) and doubling the lateral resolution [4,5]. To overcomethis tradeoff, Gustafsson et al. [4] proposed illuminating thefluorescent sample with three mutually coherent beams froma diffraction grating and therefore the structured pattern variedin both the lateral and axial directions (3D-SIM). In commer-cial SIM systems based on Ref. [4], the SI pattern is generatedusing coherent illumination from a laser, which is known toresult in images with coherent noise. As with the ApoTome-based SIM system, the most important limitation of commer-cial SIM systems (such as Zeiss Elyra, Nikon N-SIM, andDeltaVision OMX and OMX SR from GE Healthcare LifeSciences) is that their performance is objective lens dependent.Because only a discrete number of gratings is available, the OSand SR improvement is limited to only a particular set of ob-jective lenses. This problem can be solved by the use of electro-optical devices (e.g., reflective and transmissive spatial lightmodulators [SLMs]) [6–17], which allow rapid generation ofmultiple SI patterns in a controlled way using either coherentor incoherent illumination. However, these devices are costly.In this study, we present an alternative cost-effective approachfor a compact 2D-SIM system with a tunable modulation-frequency SI pattern implemented using incoherent illumina-tion. In our proposed setup, a Fresnel biprism [18,19] isilluminated by a wavefront emerging from one spatially inco-herent linear source, and it produces a SI pattern in which themodulation frequency changes with the location of the Fresnelbiprism along the optical axis. The use of a biprism in a coher-ent-based tunable-frequency 2D-SIM was previously demon-strated [20]. An additional advantage of this system is thatthe modulation contrast of the SI pattern is not attenuatedby the illumination incoherent OTF (as in the case of theApoTome-based SIM), and thus its modulation frequencycan always be set as high as the cutoff frequency of the objectivelens used independently of the wavelength. Thus, our system issuitable for multi-color SR SIM imaging.

Additionally, to take advantage of the tunability of ourproposed SIM system, we also present a computational methodthat provides OS images with SR. Our computational approachcan be applied to any tunable-frequency 2D-SIM by recordingphase-shifted images from two independent modulationfrequencies. The lower spatial modulation frequency providesOS capability by filling the missing cone, and the other oneproduces SR performance by almost doubling the cutofffrequency of the conventional widefield system. These two da-tasets are properly combined to provide simultaneous OS andSR in the reconstructed image. Although, in the last decade,other computational methods have been proposed to enhancethe resolution in widefield imaging based in part on a 2D-SIMapproach [21–26], to our knowledge, this is the first time thattwo data sets from a tunable-frequency 2D-SIM system areused to enhance both OS and SR capabilities. Preliminary re-sults of this work have been presented in [27,28]. In this paper,we investigate experimentally the approach proposed in [27]based on the optical system provided in [28] in order to show

a simultaneous improvement of the lateral resolution by a factorof 1.8× with the desired OS capability achieved by combiningthe OS and SR information. Here, we compare the OS capabil-ity of our system to the one provided by the commercial ZeissApoTome-SIM. To complete the study of [27], we have inves-tigated different filters to combine the two different datasets.Moreover, since the structured pattern created by theFresnel biprism cannot be described by a pure sinusoid, wehave proposed a computational approach to correct theenvelope distorted by the SI pattern.

The paper is organized as follows. Section 2 reviews the basictheory of 2D-SIM and presents our computational approach tocombine OS and SR information. The performance of our ap-proach is validated numerically and compared to the 3D-SIM[4] performance in simulation. In Section 3, we describe ourincoherent-based tunable-frequency SIM system. Section 4shows experimental implementation of our system and someexperimental results obtained with our approach. Experi-mental comparison of results from our system and the ZeissApoTome.2 system is also shown in Section 4. The mainachievements of this study are summarized in Section 5.

2. TRADEOFFS IN 2D-SIM PERFORMANCE

A. BackgroundIn a 2D-SIM system, the specimen under investigation isilluminated by a SI pattern, usually generated by interferingtwo coherent beams [3]. Therefore, the fluorescent sample isilluminated by an axially extended sinusoidal pattern,I�x; y� � 1� cos�2πumx � ϕ�, with modulation frequencyum and initial phase ϕ. Since the imaging formation is a linearshift-invariant process, the 3D recorded image, D�·�, is givenby the 3D convolution of a uniformly scaled copy of the 3Ddistribution emitted from the objectO�·�, and the 3D intensitypoint spread function (PSF) of the imaging system, [3,5]:

D�x; z� � 1

M 2

��1� cos

�2π

umM

x � ϕ

��O�xM

;zM 2

��

⊗3 jh�x; z�j2; (1)

where x � �x; y� are the transverse coordinates, h�x; z� is theconventional widefield amplitude PSF, ⊗3 stands for the3D convolution operator, and M is the lateral magnificationof the imaging system. The high-resolution information, whichis encoded inside the compact support of OTF, needs to beretrieved by a computational method. This task can be per-formed by capturing three modulated images in which theSI pattern has been laterally shifted. After straightforward math-ematical manipulation, the 3D irradiance distribution of thesynthetic SIM forward image, DSIM�·�, can be expressed asthe 3D convolution of the fluorescence intensity distributionemitted from the object, O�·�, and a synthetic 3D PSF,hSIM�·�, provided by the SI pattern [5]:

DSIM�x; z� �1

M 2 O�xM

;zM 2

�⊗3 hSIM�x; z�; (2)

where

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hSIM�x; z� � jh�x; z�j2 cos2�2π

2umM

x�: (3)

Similar to the conventional widefield system, it is moreillustrative to analyze the SIM imaging process in the spatialfrequency domain. The 2D-SIM system is characterized bya synthetic OTF function, H SIM�·�, obtained by taking theFourier transform of Eq. (3):

H SIM�u; w� � H �u; w� � 1

2H�u −

umM

; v; w�

� 1

2H�u� um

M; v; w

�; (4)

where u � �u; v� and w are the transverse and the axial fre-quency coordinates, respectively. The restored image, O�·�,can be computed by solving a set of linear equations, processingwith a generalized Wiener filter, and finally shifting and prop-erly combining the retrieved frequency components, a processdetailed in Ref. [4]. Without loss of generality, we assumeM � 1. Eq. (4) shows that the synthetic OTF is obtainedas the sum of three-shifted replicas of the conventional wide-field OTF, H �·�. Consequently, the synthetic OTF of 2D-SIMis enlarged along the u axis compared to the conventional wide-field OTF. As it is already known, this enlargement providesresolution improvement along the x direction, which is the di-rection in which the SI pattern changes. In fact, the perfor-mance of 2D-SIM depends on the modulation frequency ofthe illumination pattern [4,5]. Fig. 1 shows the meridional sec-tion in the uw plane of the synthetic OTF for two differentmodulation frequencies (um � 0.5uc and um � uc , where ucis the cutoff frequency of the imaging system). From this figure,it is clear that there is a tradeoff between the OS and SRcapabilities of the SIM system depending on the modulationfrequency [4,5]. The best OS image [Fig. 1(a)] does not havedouble lateral resolution, and the highest improvement oflateral resolution [Fig. 1(b)] is not accompanied by any im-provement of the OS capability. Therefore, the conventional2D-SIM systems are not able to provide simultaneously thehighest improvement in lateral resolution and OS.

B. Proposed Computational ApproachTo overcome the described issue of 2D-SIM and to provideboth SR and OS, simultaneously, an approach to combine datafrom two modulation frequencies is proposed and investigatedhere. This approach requires that two SIM datasets (of threeor nine images total: three phase shifts at one or three orienta-tion angles of the pattern fringe direction, respectively) withdifferent um should be captured: one that supports OS and

the other SR. The same reconstruction method described aboveand detailed in Ref. [4] is first applied to the two 2D-SIM data-sets to yield two restored images: OOS�·� and OSR�·� with OSand SR content, respectively. In this work, we propose a com-putational approach to combine the information from the OSand SR restored images as follows:

OOS−SR�x; z� � OOS�x; z� � hHP�x; z� ⊗3 OSR�x; z�; (5)

where hHP�·� is a spherically symmetric high-pass filter with aradial profile in frequency domain defined by

hHP�ρ; n� �8<:

�ρ∕uOS�n ρ ≤ uOS

1 uOS < ρ ≤ uSR0 otherwise

; (6)

where uOS and uSR are the modulation frequencies for OS andSR datasets, respectively, ρ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 � v2 � w2

p, and n is the

order of the high-pass filter, which is n � 0, 1, 2, 3, indicatingno filter, linear, parabolic, and cubic filter, respectively. The goalof our approach is to retrieve high-frequency information fromthe SR dataset while keeping the OS frequency informationunaltered. Therefore, the filter should attenuate the low spatialfrequencies of the SR part, since we are interested in retainingthe information at these frequencies from the OS part. Basedon qualitative and quantitative comparisons (not shown), theperformance of the parabolic and the cubic filters was found tobe similar. This is consistent with the goal of our approach.Since the parabolic filter performs slightly better in resolvingthe fine structure of the object, we used it in the OS-SR com-bination results shown in the paper. The design of a filteroptimal in some sense may yield better results; however it isbeyond the scope of this paper.

To investigate the restoration performance of the proposedapproach, a synthetic 3D test object was simulated on a 256 ×256 × 256 grid, as shown in Fig. 2(a). An outer sphericalshell and inner spherical beads to mimic a cell’s structure inbiological studies compose this object. While the diameterand the thickness of the spherical shell is 3 μm and200 nm, respectively, the diameter of each bead is 125 nm,and the closest distance between two neighboring beads is equalto 100 nm. The latter is below the resolution limit of the con-ventional widefield microscope based on the parameters usedfor the microscope in this simulation study: emission wave-length at λ � 515 nm, and the imaging lens’ numerical aper-ture (NA) is equal to 1.4. Therefore, the cutoff frequency is5.4 μm−1�uc � 2 NA∕λ�. The phase ϕ of the SI pattern ischanged by 2π∕3 for the three phase images along three ori-entation angles: θ � 0 deg, 60 deg, and 120 deg.

The lateral and axial views of the restored images of thetest object (Fig. 2) show that the OS-SR combination result[Fig. 2(e)] incorporates the benefits of both the OS result[Fig. 2(c)] and SR result [Fig. 2(d)] images. To evaluatehow well simultaneous OS and SR is achieved by our proposedmethod, we compare this result with one we obtained using3D-SIM [4], which is the benchmark for 3D OS images withdouble lateral resolution [Fig. 2(f )]. Fig. 3 shows a quantitativecomparison of these results [Figs. 2(a), 2(e) and 2(f )]. As ex-pected, the performance of the proposed method is worse than3D-SIM in the achieved OS as evident by the axial views

Fig. 1. Frequency support of the synthetic 2D-SIM OTF dependson the modulation frequency (um) of the SI pattern. uw-section of the3D synthetic OTF obtained from Eq. (4) withM � 1 and a SI patternwith: (a) um � 0.5uc and (b) um � uc . Note that either the missingcone is completely filled providing OS (a) or the effective cutoff fre-quency is doubled providing SR (b).

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[middle row in Figs. 2(e) and 2(f ), respectively] because moreaxial frequencies are filled in the missing cone in 3D-SIM thanin our approach. Theoretically, both methods should providethe same lateral increase of the synthetic OTF, since the modu-lation frequency of the pattern we simulated for the 3D-SIMimage and the second image of our approach is the same(um � 0.9uc � 4.8 μm−1). Our simulated results show thatthe contrast of the lateral performance in our approach isslightly better [top and bottom rows in Figs. 2(e) and 2(f ), re-spectively]. This result might be related to the use of the apod-ization filter [Eq. (6)] in our restoration method [Eq. (5)], inwhich the SR information is weighted by the filter in order toenhance the high-frequency content, and thus the contrast ofthe restored images can be increased. However, because such afilter is not used in the 3D-SIM approach, we did not apply itto the 3D-SIM result [Fig. 2(f )] shown here, and this couldbe the reason for the contrast discrepancy. Overall, these noise-less simulation results verify the simultaneous OS and SR

performance achieved by our approach, which is expectedcompared to 3D-SIM.

The current implementation of our proposed method usessix phase-shifted images (three images per each modulationfrequency) for each orientation angle of the pattern, as opposedto the five phase-shifted images required in 3D-SIM. Thisresults in a 20% data-acquisition increase over 3D-SIM.However, it is worth noting that the two 2D-SIM datasets usedin our approach have the same widefield component, and thusin fact only five phase-shifted images are needed to restore theOS-SR image. We plan to take advantage of this fact in ourfuture implementation, thereby avoiding the current data-acquisition increase.

3. INCOHERENT TUNABLE-FREQUENCY2D-SIM BASED ON FRESNEL BIPRISM

As mentioned above, integration of our proposed method to atunable frequency SIM system is advantageous. This sectiondescribes our incoherent tunable-frequency 2D-SIM systemshown in Fig. 4. The frequency tunability is achieved by theuse of a Fresnel biprism, which is illuminated by the diffractedlight emerging from a spatially incoherent linear source [29].The main advantage of the incoherent illumination is that:(i) no coherence noise distorts the recorded SIM images[28] and (ii) the contrast of the fringes is not penalized bythe shape of the illumination OTF as in ApoTome-basedSIM systems.

The use of a Fresnel biprism to generate interference pat-terns is a well-known strategy. It is based on the fact that thiselement generates the virtual splitting of a point source locatedat a distance η from it. These two virtual sources are located atthe same plane of the original source and are symmetrically

Fig. 2. Qualitative evaluation of the proposed method. Lateral view (top row), axial view (middle row), and zoomed view of the cropped squaremarked with a dashed pink line in the top left image (bottom row) of the 3D: (a) true object; (b) deconvolved widefield image using a Wiener filter;(c) reconstructed 2D-SIM image for um � 0.5uc � 2.7 μm−1; (d) reconstructed 2D-SIM image for um � 0.9uc � 4.8 μm−1; (e) combined2D-SIM reconstructed image with proposed method [Eqs. (5) and (6)]; and (f ) reconstructed image from 3D-SIM raw data using the methodin Ref. [4]. Scale bars: 1 μm (top and middle row) and 0.3 μm (bottom row). The cutoff frequency (uc) is 5.4 μm−1 (the emission wavelength atλ � 515 nm and the imaging lens’ NA � 1.4).

Fig. 3. Quantitative comparison of the proposed method and 3Winterference SIM using the results shown in Fig. 2. Normalized inten-sity profiles from the: (a) axial view; and (b) lateral view of croppedimage along x axis.

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distributed from its original position along a line perpendicularto the edge of the biprism (Fig. 4). The distance between thesesources is given by a�η� � 2η�n − 1� tan�δ�, where n and δ arethe refractive index (in general, the dispersion of Fresnelbiprism material is known to be less than 1.3% in the visiblerange) and the refringence angle of the biprism, respectively[18,19]. Because these twin sources come from the same realsource, they are mutually coherent, and a high-contrast inter-ference pattern is generated at any transverse observation planebeyond the biprism. Note that any lateral displacement (alongthe x and y directions) of the real source generates a correspond-ing shift of the interference pattern. However, if the displace-ment occurs along the biprism’s edge direction (y direction),this shifting is not distinguishable due to the invariance ofthe fringe structure along this direction [29,30].

The invariance along the y direction of the interference pat-terns is, in fact, a key point for our incoherent SI proposal. Let usconsider that the Fresnel biprism is now illuminated using anincoherent linear source (e.g., a slit). This incoherent-illuminatedslit is aligned with the biprism’s edge and placed at a distance ηfrom it. Following the same reasoning as before, each pointsource of the slit generates the same high-contrast interferencepattern at any observation plane behind the biprism. Thus, innoiseless conditions, since each incoherent point produces thesame intensity distribution, there is just a reinforcement ofthe intensity signal compared with the case of the single-pointsource. However, in a realistic experimental situation, the result isslightly different. Note that in the presence of noise, the vertical(y) displacement of the interference pattern for each point of theslit, which is unobservable in the signal response, can be detect-able in the coherent noise pattern. Nonetheless, when consider-ing the whole set of points of the slit, an averaging effect appears,since each point source is incoherent with respect to its neighbor.This results in an incoherent superposition of all the interferencepatterns created by the point sources, washing out speckle noiseand improving signal-to-noise ratio (SNR).

It is worth to highlight that our incoherent proposed SImethod provides high-contrast interference patterns becauseit is based on incoherent superposition of coherent interference

patterns generated by each point of an incoherent slit. In thisway, when these coherent fringes are projected to the samplespace through an illumination imaging system (L1 and L2lenses in Fig. 4), each coherent interference pattern is transmit-ted without loss of visibility. This is a unique feature of usingcoherent imaging systems, and it represents a clear advantage toincoherent grating-projection SIM techniques.

To finalize the description of our SI system, we discuss itstunability capability. One can realize that the separation be-tween the virtual point sources, a�η�, can be tuned by changingthe axial position, η, of the biprism with respect to the sourceplane. Since both virtual point sources are placed at the frontfocal plane of a converging lens (L1 lens), the interference pat-tern at any observation plane (z) beyond L1 lens has a periodp�η� � λf L1∕a�η� [29], which is constant along z. Particularly,the higher the separation between the source and the biprism,the lower the value of the period is, and, as a result, a higherspatial frequency is obtained. It is important to realize that whenthis setup is used as the illumination system in SIM, the imageof the virtual sources is placed at the pupil plane of the objectivelens [Fig. 4(b)]. Therefore, the higher the separation betweenthe real source and the Fresnel biprism, the higher the separationof the virtual sources at the pupil plane, a 0�η� � f L2a�η�∕f L1,and the higher the spatial transverse frequency of the structuredpattern, um�η� � a 0�η�∕λf obj. Note that the SR performance isachieved when the separation between the virtual sources at thepupil plane occupies the whole pupil diameter, in which case themodulation frequency of the pattern coincides with the cutofffrequency of the objective lens [Fig. 4(b), bottom panel]. Asnoted earlier, the dispersion of the biprism material over the vis-ible range is insignificant, and therefore the distance a 0�η� is alsoinvariant to the wavelength used in this range. Since both themodulation frequency and the cutoff frequency of the objectivelens are affected by the wavelength in the same manner, the SRperformance of our system can always be satisfied for differentwavelengths without any change in the illumination setup.Thus, our system has the same multi-color advantage as inco-herent grid-projection SIM systems.

Taking advantage of our tunable 2D-SIM system allows usto change the modulation frequency of the SI pattern with min-imal changes; we can record the two SIM datasets with differentum (OS and SR datasets) and apply the reconstruction methodproposed in Section 2 in order to obtain a synthetic OS-SR SIMimage. It is worth to highlight that the generated SI pattern pro-vided by the Fresnel biprism is not completely a pure sinusoidalpattern, because the waves emerging from each virtual incoher-ent source are abruptly cut off by the biprism’s edge [18,19]. Toobtain a reliable evaluation of the SI pattern created by the bipr-ism, we model its transmittance by Eq. (17) of Ref. [30] andcalculate the amplitude distribution behind the Fresnel biprism.After some mathematical operations, the irradiance distribution,computed as the absolute square value of Eq. (2) of Ref. [29],the SI patterns created beyond L1 lens are expressed as

I�x; z; η� � env�x; z; η� � ϑC �x; z; η� cos�2π

a�η�λf L1

x�

� ϑS�x; z; η� sin�2π

a�η�λf L1

x�; (7)

Fig. 4. (a) Illustration of the tunable-frequency 2D-SIM using aFresnel biprism, which is illuminated by the diffracted light emergingfrom an incoherent linear source. (b) Tunable capability of our 2DSIM system. The lateral modulation frequency of the structured pat-tern (um) is changed by the axial displacement of the Fresnel biprism(η). Illustration for two modulation frequencies: half (top) or equal to(bottom) the cutoff frequency of the objective lens.

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where

env�x; z; η� � 1� C �α� � S�α� − C �β� − S�β�� C2�α� � S2�α� � C2�β� � S2�β�; (8)

ϑC �x; z; η� � 1� C �α� � S�α� − C �β� − S�β�� 2C �α�C �β� � 2S�α�S�β�; (9)

ϑS�x; z; η� � −C �α� � S�α� − C �β� � S�β�� 2C �α�S�β� − 2C �β�S�α�: (10)

C �·� and S�·� are the Fresnel integrals defined as the criterion ofAbramowitz and Stegun [31], and α and β are given by

α �ffiffiffiffiffi2η

λ

rf L1x � �n − 1�δ�f 2

L1 � ηz − ηf L1�f L1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 2L1 � ηz − ηf L1

p ;

β �ffiffiffiffiffi2η

λ

rf L1x − �n − 1�δ�f 2

L1 � ηz − ηf L1�f L1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 2L1 � ηz − ηf L1

p : (11)

From Eq. (7), it is clear that the SI pattern created by thebiprism cannot be described by a pure sinusoidal function,as shown in Fig. 5(a) where the pattern is distorted by anupper and lower envelope due to the biprism, described byEqs. (8)–(11). To investigate this issue, we simulated the effectof these envelopes by comparing the forward and restored im-ages of a 6-μm spherical bead with an ideal (a pure sinusoidalfunction) and the real [Eq. (7)] SI patterns. It is observable fromFig. 5(c) that there is a residual pattern in the restored imagefrom the data with the real pattern, as expected. The proposedapproach to reduce the effect of the envelope consists of twoparts. First, a low-pass filter is applied to the background of theraw forward images or the calibration data (such as, the imageof a flat sheet of fluorescence acquired with the same imagingconditions) to extract the frequency information of the enve-lopes in the real pattern. Then the effect of the envelopes in theforward images is reduced by point-wise dividing each raw im-age by the computed envelopes, where division by zero isavoided using a small additive constant in the denominator.Here, we assumed that the sinusoidal term in Eq. (7) is neg-ligible, which is a valid assumption based on a comparison (notshown) between the terms defined by Eqs. (9) and (10).Second, a notch filter, which filters the residual frequencyaround the modulation frequency’s vicinity, is applied to therestored image. As shown in Fig. 5(d), the proposed approachis able to reduce the residual pattern. Fig. 5(e) shows the nor-malized intensity profile along the center of the restored beadimages in Figs. 5(b)–5(d) for a quantitative comparison andverification of the envelope correction method.

4. EXPERIMENTAL VERIFICATION

This section is divided in two subsections. First, the implemen-tation of our incoherent tunable-frequency SIM system isdescribed and how experimental SIM data are acquired.After this brief description, experimental results are presentedand discussed to show: a) improvement of the resolutionachieved by using our system; b) successful integration

of our tunable SI system with our proposed method(Section 2.2) to obtain OS high-resolution images; and c) com-parison of the OS capability of our system with a commerciallyavailable OS technique (Zeiss ApoTome SIM).

A. Experimental SetupThe optical configuration of Fig. 4(a) has been implemented inan open setup. In this setup, a variable-width linear source isilluminated incoherently by the light emerging from a 488-nmlaser (Oxxius, RPMC Lasers, Inc., O’Fallon, Missouri) andpassing through a rotating diffuser that removes spatial coher-ence (while other methods could also be used). In our setup, therotating diffuser is imaged onto the linear source using a con-verging lens, and the linear source is set at the front focal planeof a converging lens (L1) whose focal length is 100 mm.A Fresnel biprism with 2.5° angle and refractive index of1.5194 at wavelength 532 nm (NewLight Photonics, Inc.,Toronto, Canada) is used. To set the two virtual images of thelinear source at the pupil plane of the objective lens, a converg-ing lens (L2) of focal length 250 mm is inserted behind the L1lens. The pupil plane of the objective lens is set at the back focalplane of the L2 lens, so that the objective lens (20X PlanNeoFluar 0.5 NA, Zeiss) recollimates the beams and createsthe interference between two plane waves in the objective lens’object focal plane, which is in the object space. This illumina-tion irradiance pattern can be thought of as a demagnified

Fig. 5. Qualitative evaluation of the proposed envelope correctionmethod to address the real SI pattern in tunable 2D-SIM based on theFresnel biprism. (a) Real pattern generated by the biprism andthe corresponding lower and upper envelope. Lateral view from the3D: (b) forward image (top row) and reconstructed image (bottomrow) with ideal sinusoidal pattern; (c) forward image with real pattern(top row) and its reconstruction with ideal sinusoidal pattern (bottomrow); and (d) reconstructed image after envelope correction applied to(c). (e) Quantitative comparison of images restored from ideal and realpattern SIM data assuming an ideal sinusoidal pattern, with andwithout envelope correction.

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replica of the irradiance pattern obtained after the L1 lens[Eq. (7)] whose lateral magnification is M ill � f obj∕f L2 �0.032. Fluorescence light emitted by the sample is recollectedagain by the same objective lens and transmitted through a di-chroic mirror of transmission cutoff wavelength λc � 506 nm.An emission filter (wavelength bandpass of 530� 50 nm ) isinserted after the dichroic mirror to block the scattered excita-tion light. The emitted fluorescence distribution is imaged ontoa high-sensitivity CMOS camera (Orca-Flash4.0 C11440-22CU, Hamamatsu) using an imaging system of lateral mag-nification M det � f TL∕f obj � 37.5. The camera has a 6.5 ×6.5 μm2 cell, which produces a 0.173 μm × 0.173 μm pixelin the lateral object space.

3D datasets are acquired by axially scanning the sample vol-ume using a motorized linear translation stage and 1-μm inter-vals, which is the smallest distance possible with this stage. Foreach transverse plane, we record three phase-shifted SIM im-ages in which the phase of the illumination pattern is shiftedby laterally translating the Fresnel biprism. For this purpose,the Fresnel biprism is mounted onto a motorized translationstage. All these steps are implemented in an acquisition softwaredeveloped in LabVIEW. With this software, we have control ofthe exposure time of the camera, the axial spacing betweentransverse planes, and the lateral displacement of the Fresnelbiprism. For all experimental data shown here, we used a4-μm lateral displacement of the biprism.

To modify the modulation frequency of our structuredpattern, the Fresnel biprism is mounted onto an optical railthat is parallel to the optical axis of our experimental system.Therefore, the tunability of the illumination pattern is obtainedby axially displacing the Fresnel biprism. Fig. 6 shows images ofa fluorescent USAF test (Ready Optics, Calabasas, California)acquired for four different axial positions of the biprism. Theseimages show a variation by a factor of 7.5 in the modulationfrequency of the pattern. Although this figure does not showhigh modulation frequencies, as we will show later, we are ableto tune the modulation frequency until the cutoff frequency ofthe objective lens, where the experimental cutoff frequency isuc � 1.9 μm−1. In general, this statement is valid for anyobjective lens used, however, in this current implementation,the focal length of the L1 and L2 lenses may have to be changed

in order to achieve the highest modulation frequency for a dif-ferent objective lens.

For ideal implementation, the linear source should beilluminated by the beam emerging from an ultra-brightlight-emitting diode (LED), which is already a spatially inco-herent source. However, because the modulation frequency ofthe illumination pattern is wavelength dependent, the higherthe spectral bandwidth of the LED, the narrower the fieldof view (FOV) of the pattern. This results in using onlyLEDs with spectral bandwidth <10 nm if one wants to havea similar FOV as the one provided by the laser-diffuser imple-mentation. Observing the experimental results of Fig. 2(c) ofRef. [28], one can realize the reduction of the FOV in terms ofthe bandwidth of the LED source.

B. Experimental ResultsFirst, we are interested in verifying the resolution performanceof our incoherent tunable-frequency SIM system. For this task,the fluorescent USAF target was imaged with our systemusing two different modulation frequencies (um � 0.5uc �0.95 μm−1 and um � 0.9uc � 1.7 μm−1) and compared theachieved resolution to the resolution of the conventional wide-field system. Fig. 7 shows the 2D Wiener-filtered image fromthe conventional raw data [Fig. 7(a)] and the reconstructedSIM images for both modulation frequencies [Figs. 7(b) and7(c)]. It is worth to mention that these images were obtainedafter applying first the envelope function compensation(Section 3) using calibration data (e.g., the SIM image froma thin fluorescent layer) and followed by the proposedreconstruction SIM method (Section 2).

From Fig. 7 one can observe the lateral resolution achievedin each case. Note that for each case, the smallest resolvedelement of the group 10 is highlighted by a dashed pink rec-tangle. Normalized intensity profiles along the vertical direction(marked by the two white arrows in Fig. 7) of group 10 are alsocompared [Fig. 7(g)] to indicate the resolution limit in eachcase. From the results in Figs. 7(d)–7(f ), we determined thehighest detectable spatial frequency for each case to be:1024 lp/mm (element 2 of group 10), 1448 lp/mm (element4 of group 10), and 1825 lp/mm (element 6 of group 10),respectively. Therefore, the spatial frequency is increased bya factor of 1.4 for um � 0.5uc and a factor 1.8 for um �0.9uc in comparison with the conventional imaging. As it isexpected from the SIM technique, the use of a modulation fre-quency closer to the cutoff frequency of the imaging systemalmost doubles the resolution limit of the conventional system.Since the variation between the theoretical and experimentalfactor is less than 10%, one could claim that the improvementof the experimental resolution is predicted by the theorydiscussed in Section 2 within the error range.

For the second experiment, we used again the USAF targetas a sample to verify our computational approach described inSection 2. To have a proof of the OS improvement, the samplewas tilted approximately 80 deg with respect to the optical axis.In Fig. 8, we show the OS reconstructed image correspondingto the SIM image for um � 0.5uc [Fig. 8(a)], the SR recon-structed image corresponding to the SIM image for um �0.9uc [Fig. 8(b)], and the OS-SR reconstructed imagecorresponding to the image after applying our approach

Fig. 6. Experimental verification of the frequency tunabilitycapability of our SIM system. Fluorescent USAF target images ac-quired at four different axial positions of the Fresnel biprism inour open setup implementation. The cutoff frequency (uc ) is1.9 μm−1 (the emission wavelength at λ � 515 nm and the imaginglens’ NA � 0.5).

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[Fig. 8(c)]. For better visualization of the enhancement of theresolution, cropped and zoomed views of groups 10 and 11 areshown in the second row of Fig. 8 for each case. As before, thedashed pink rectangles mark the smallest resolved element foreach image. Based on these images, it is clear that the OS-SRimage preserves the high spatial frequencies of the SR image,since both of them present the same resolution limit (the high-est resolved spatial frequency is 1825 lp/mm).

These experimental results coincide with those modeled inSection 2 and predicted by simulations in Figs. 2 and 3.Regarding assessment of the achieved OS, we compared theareas inside the white vertical rectangles located on the left sideof each panel in the first row of Fig. 8. From these areas, onecan observe how the out-of-focus information is better rejectedin the OS and OS-SR images than in the SR image. These re-sults demonstrate that the depth discrimination of the OS-SRimage is kept comparable to the OS case, while it has anincrease in the lateral resolution by a factor of 1.8 over the

conventional resolution, thereby providing a final image withan overall improvement in quality.

In our last experiment, we compared the performance of ourincoherent tunable-frequency 2D-SIM with the ZeissApoTome.2 SIM module on a Zeiss AxioImager.Z2 uprightmicroscope. For this experiment, as a sample we used a clusterof 6-μm spherical shells. The sample was a priori prepared bymounting spherical shells of diameter 6 μm and a shell thick-ness of 1 μm (Invitrogen, Molecular Probes, FocalCheck mi-crospheres, 6-μm fluorescent green ring stain/blue throughout)in ProLong Diamond (refractive index equal to 1.47). To fixthe shells inside the mounting medium, the sample was curedat room temperature for 24 h. The top panel of Fig. 9(a) showsa transverse section of the sample using widefield microscopy inwhich several spherical shells are clustered at different randomlocations. This cluster was captured using the ApoTome-SIMsystem [first row of Fig. 9(b)] and our incoherent tunable SIMsystem with two desired modulation frequencies [first row ofFig. 9(c)] in order to create the OS-SR image. It is importantto realize that although we have used the highest modulationfrequency possible for the ApoTome implementation, its valueis much smaller compared to those in our implementation.Note that the modulation frequency for the ApoTome systemis only 4% of the cutoff frequency of the objective lens used,while our system allows us to tune it up to 90% of the cutofffrequency.

All raw images were properly processed to obtain the re-stored images shown in Fig. 9. It is important to mention thatthe images in Figs. 9(a) and 9(b) were restored using deconvo-lution methods implemented in the Zeiss ZEN.2 software [32]in which we have taken into account an aberrated PSF at a30-μm depth, which is the determined pre-aberration of ourexperimental microscope [33], and the refractive index ofthe mounting medium. In both cases, the regularization param-eter of the deconvolution method has been properly set to getthe best image quality. The corresponding reconstructed images

Fig. 7. Achieved SR verified in experimental images of the fluores-cent USAF target: (a) and (d) widefield restoration; (b) and (e) recon-structed SIM image for um � 0.5uc � 0.95 μm−1 (OS); (c) and(f ) reconstructed SIM image for um � 0.9uc � 1.7 μm−1 (SR); and(g) intensity profile through the horizontal elements of group 10 alongthe direction marked by the arrows in panel (d). Images in (d)–(f ) arezoomed versions of the areas indicated by the yellow squares inimages (a)–(c), correspondingly. The groups marked with the pinkdashed line in (d)–(e) indicate the resolution achieved in each case.

Fig. 8. Simultaneous OS and SR achieved and verified with exper-imental images of a tilted fluorescent USAF target: (a) reconstructedSIM image for um � 0.5uc � 0.95 μm−1 (OS); (b) reconstructedSIM image for um � 0.9uc � 1.7 μm−1 (SR); and (c) recon-structed 2D-SIM image with proposed method (OS-SR combination).

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are shown in the second row of Fig. 9. Clearly, from these ex-perimental results, one can realize that our system provides betterOS capability than the commercial ApoTome-SIM system. Notethat the intensity of the out-of-focus spherical shells (marked bythe colored arrows) is less in the OS-SR image than in the othercases. Even though the contribution of the SR image is not vis-ible in the OS-SR image, since the size of the spherical shells ismuch higher than the resolution limit of our system, the OS-SRcombination shows a 5% improvement in the SNR, which wasfound to be equal to 26.3 dB, compared to the OS image, whichhas SNR � 25.1 dB. The SNR is computed by taking the ratioof the average intensity of the signal over the spherical shell withuniform intensity and the standard deviation of the backgroundintensity. Moreover, the contrasts for the OS and OS-SR imagesare 0.47 and 0.69, respectively. Therefore, in the OS-SR image,both the SNR and the contrast have been improved. It is worthnoting that the spherical shell of fluorescence is not evident in therestored images as expected. We believe that this is due to oc-clusion of the out-of-focus light within the cluster and perhapsthe presence of an air bubble near the cluster. It is clear that thereare residual fringes in Fig. 9(c), and this is because the envelopecorrection (Section 3) could not be applied to these restorations,as calibration data about the pattern’s envelope are not availablefor this dataset. In the USAF target, data shown in (Figs. 7 and 8)the envelope information could be extracted from the back-ground intensity. However, in the case of the 6-μm cluster,the object does not emit fluorescent light across the FOV,and thus calibration data (such as the image of a flat sheet offluorescence acquired with the same imaging conditions) areneeded for the envelope correction.

5. CONCLUSIONS

We have shown the performance of an incoherent-based2D-SIM system with tunable modulation frequency. In oursystem, the structured illumination pattern is generated by

splitting an incoherent linear source using a Fresnel biprism.A key feature of this setup is that the structured pattern is freefrom coherence noise, and its modulation contrast is not re-duced through the illumination system. Additionally, our sys-tem is cost effective and produces a continuous variation of thepattern’s frequency, up to the cutoff frequency of any commer-cially available objective lens independently of the wavelengthused. Based on the latter, our system is suitable for multi-spectral imaging in fluorescence microscopy. In order to takeadvantage of our tunability capability, we have also proposeda computational OS-SR algorithm that reconstructs OS imageswith double lateral resolution. Although our OS-SR method isbased on a 2D-SIM system, it provides superior performance oversingle-modulation-frequency 2D-SIM by providing simultaneousOS and SR information, which is the hallmark of 3D-SIM. Weare currently investigating the OS-SR combination with five im-ages, and we will report our studies in a future publication.Experimentally, we have applied the OS-SR algorithm to the im-age of a titled extreme USAF test. From the reconstructed OS-SRimage, we show that the resolution limit is increased by a factor of1.8×, and the OS capability of the OS-SR is also enhanced.Finally, the performance in terms of the OS capability of our tun-able 2D-SIM system was compared with the one provided by theZeiss ApoTome.2 by imaging 6-μm spherical fluorescent shells ina test sample. Our results verified that our system provides betterOS capability than the commercial ApoTome-SIM system in thecase of this test sample.

Funding. National Science Foundation (NSF) (DBI1353904); the University of Memphis; Ministerio deEconomía y Competitividad (MINECO) (DPI2015-66458-C2-1-R); Generalitat Valenciana (PROMETEOII/2014/072).

Acknowledgment. We thank our collaborators Dr.Martinez-Corral (Univ. of Valencia, Spain), Dr. S. V. King

Fig. 9. Experimental validation of improved optical sectioning achieved with our approach. Experimental images of a 6-μm spherical shells clusterfrom different systems: (a) Zeiss AxioImager.Z2 Widefield; (b) Zeiss ApoTome.2; and (c) our tunable 2D-SIM. The reconstructed images of panels(a) and (b) are obtained using the Zeiss ZEN.2 software. The arrows illustrate that our system has better optical-sectioning capability overApoTome.2. Lens: 20×∕0.5 NA; Emission wavelength: 515 nm.

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(St. Jude Children’s Research Hospital, Memphis, TN) forscientific and technical discussion and N. Patwary (PhDcandidate, the University of Memphis) for assistance withApoTome data acquisition and Zeiss ZEN processing.

†These authors contributed equally to this work.

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