Quantitative sectioning and noise
analysis for structured illumination
microscopy
Nathan Hagen,1,2 Liang Gao,1 and Tomasz S. Tkaczyk1,∗
1 Department of Bioengineering, Rice University, Houston, Texas 77005, USA
2 Email: nhagen@optics.arizona.edu∗ ttkaczyk@rice.edu

Abstract:
Structured illumination (SI) has long been regarded as anonquantitative technique for obtaining sectioned microscopic images.
Its lack of quantitative results has restricted the use of SI sectioning toqualitative imaging experiments, and has also limited researchers’ ability
to compare SI against competing sectioning methods such as confocal
microscopy. We show how to modify the standard SI sectioning algorithm
to make the technique quantitative, and provide formulas for calculating thenoise in the sectioned images. The results indicate that, for an illumination
source providing the same spatially-integrated photon flux at the object
plane, and for the same effective slice thicknesses, SI sectioning can provide
higher SNR images than confocal microscopy for an equivalent setup whenthe modulation contrast exceeds about 0.09.
© 2011 Optical Society of America
OCIS codes: (180.0180) Microscopy; (110.4280) Noise in imaging systems; (100.3010) Image
reconstruction techniques; (180.6900) Three-dimensional microscopy; (100.6640) Superresolution.

References and links
1. D. Karadaglić and T. Wilson, “Image formation in structured illumination wide-field fluorescence microscopy,”
Micron 39, 808–818 (2008).
2. M. A. A. Neil, R. Juškaitis, and T. Wilson, “Method of obtaining optical sectioning by using structured light in aconventional microscope,” Opt. Lett. 22, 1905–1907 (1997).
3. L. Schermelleh, P. M. Carlton, S. Haase, L. Shao, L. Winoto, P. Kner, B. Burke, M. C. Cardoso, D. A. Agard,
M. G. L. Gustafsson, H. Leonhardt, and J. W. Sedat, “Subdiffraction multicolor imaging of the nuclear peripherywith 3D structured illumination microscopy,” Science 320, 1332–1336 (2008).
4. V. C. Cogger, G. P. McNerney, T. Nyunt, L. D. DeLeve, P. McCourt, B. Smedsrød, D. G. L. Couteur, and T. R.
Huser, “Three-dimensional structured illumination microscopy of liver sinusoidal endothelial cell fenestrations,”
J. Struct. Biol. 171, 382–388 (2010).
5. P. Kner, B. B. Chhun, E. R. Griffis, L. Winoto, and M. G. L. Gustafsson, “Super-resolution video microscopy oflive cells by structured illumination,” Nat. Met. 6, 339–344 (2009).
6. G. Best, R. Amberger, D. Baddeley, T. Ach, S. Dithmar, R. Heintzmann, and C. Cremer, “Structured illumination
microscopy of autofluorescent aggregations in human tissue,” Micron 42, 330–335 (2011).
7. P. J. Keller, A. D. Schmidt, A. Santella, K. Khairy, Z. Bao, J. Wittbrodt, and E. H. K. Stelzer, “Fast, high-contrast
imaging of animal development with scanned light sheet-based structured-illumination microscopy,” Nat. Meth.
7, 637–645 (2010).
8. N. Bozinovic, C. Ventalon, T. Ford, and J. Mertz, “Fluorescence endomicroscopy with structured illumination,”Opt. Express 16, 8016–8026 (2008).
9. R. Heintzmann and C. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using
a diffraction grating,” in EUROPTO Conference on Optical Microscopy, Proc. SPIE 3568, 185–196 (1999).

#154806 - $15.00 USD(C) 2012 OSA

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2 January 2012 / Vol. 20, No. 1 / OPTICS EXPRESS 403

10. M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination
microscopy,” J. Microsc. 198, 82–87 (2000).
11. L. M. Hirvonen, K. Wicker, O. Mandula, and R. Heintzmann, “Structured illumination microscopy of a living
cell,” Eur. Biophys. J. 38, 807–813 (2009).
12. J. M. Murray, P. L. Appleton, J. R. Swedlow, and J. C. Waters, “Evaluating performance in three-dimensional
fluorescence microscopy,” J. Microsc. 228, 390–405 (2007).
13. L. Gao, N. Bedard, N. Hagen, R. T. Kester, and T. S. Tkaczyk, “Depth-resolved image mapping spectrometer(IMS) with structured illumination,” Opt. Express 19, 17439–17452 (2011).
14. M. G. Somekh, K. Hsu, and M. C. Pitter, “Resolution in structured illumination microscopy: a probabilistic
approach,” J. Opt. Soc. Am. A 25, 1319–1329 (2008).
15. M. G. Somekh, K. Hsu, and M. C. Pitter, “Stochastic transfer function for structured illumination microscopy,”
J. Opt. Soc. Am. A 26, 1630–1637 (2009).
16. K. M. Kedziora, J. H. M. Prehn, J. Dobruck, and T. Bernas, “Method of calibration of a fluorescence microscopefor quantitative studies,” J. Microsc. 244, 101–111 (2011).
17. J. C. Waters, “Accuracy and precision in quantitative fluorescence microscopy,” J. Cell Biol. 1858, 1135–1148(2009).
18. A. Esposito, S. Schlachter, G. S. Schierle, A. D. Elder, A. Diaspro, F. S. Wouters, C. F. Kaminski, and A. I. Iliev,“Quantitative fluorescence microscopy techniques,” Methods Mol. Biol. 586, 117–142 (2009).
19. R. Heintzmann and P. A. Benedetti, “High-resolution image reconstruction in fluorescence microscopy withpatterned excitation,” Appl. Opt. 45, 5037–5045 (2006).
20. M. A. A. Neil, T. Wilson, and R. Juškaitis, “A light efficient optical sectioning microscope,” J. Microsc. 189,
114–117 (1998).
21. M. J. Cole, J. Siegel, S. E. D. Webb, R. Jones, K. Dowling, M. J. Dayel, D. Parsons-Karavassis, P. M. W.
French, M. J. Lever, L. O. D. Sucharov, M. A. A. Neil, R. Juvskaitis, and T. Wilson, “Time-domain whole-field
fluorescence lifetime imaging with optical sectioning,” J. Microsc. 203, 246–257 (2001).
22. L. H. Schaefer, D. Schuster, and J. Schaffer, “Structured illumination microscopy: artefact analysis and reduction
utilizing a parameter optimization approach,” J. Microsc. 216, 165–174 (2004).
23. L. G. Krzewina and M. K. Kim, “Single-exposure optical sectioning by color structured illumination microscopy,” Opt. Lett. 31, 477–479 (2006).
24. A. L. Barlow and C. J. Guerin, “Quantization of widefield fluorescence images using structured illumination andimage analysis software,” Microsc. Res. Tech. 70, 76–84 (2007).
25. F. Chasles, B. Dubertret, and A. C. Boccara, “Optimization and characterization of a structured illumination
microscope,” Opt. Express 15, 16130–16141 (2007).
26. S. D. Konecky, A. Mazhar, D. Cuccia, A. J. Durkin, J. C. Schotland, and B. J. Tromberg, “Quantitative optical
tomography of sub-surface heterogeneities using spatially modulated structured light,” Opt. Express 17, 14780–
14790 (2009).
27. M. F. Langhorst, J. Schaffer, and B. Goetze, “Structure brings clarity: structured illumination microscopy in cellbiology,” Biotechnol. J. 4, 858–865 (2009).
28. T. A. Erickson, A. Mazhar, D. Cuccia, A. J. Durkin, and J. W. Tunnell, “Lookup-table method for imaging opticalproperties with structured illumination beyond the diffusion theory regime,” J. Biomed. Opt. 15, 036013 (2010).
29. K. Wicker and R. Heintzmann, “Single-shot optical sectioning using polarization-coded structured illumination,”
J. Opt. 12, 084010 (2010).
30. T. Wilson, “Optical sectioning in fluorescence microscopy,” J. Microsc. 242, 111–116 (2010).
31. S. Gruppetta and S. Chetty, “Theoretical study of multispectral structured illumination for depth resolved imaging
of non-stationary objects: focus on retinal imaging,” Biomed. Opt. Express 2, 255–263 (2011).
32. M. A. A. Neil, R. Juškaitis, and T. Wilson, “Real time 3D fluorescence microscopy by two beam interference
illumination,” Opt. Commun. 153, 1–4 (1998).
33. V. Poher, H. X. Zhang, G. T. Kennedy, C. Griffin, S. Oddos, E. Gu, D. S. Elson, M. Girkin, P. M. W. French,
M. D. Dawson, and M. A. A. Neil, “Optical sectioning microscopes with no moving parts using a micro-stripe
array light emitting diode,” Opt. Express 15 (2007).
34. L. Mortara and A. Fowler, “Evaluations of charge-coupled device (CCD) performance for astronomical use,” inSolid state imagers for astronomy, Proc. SPIE 290, 28–33 (1981).

1.

Introduction

Structured illumination (SI) is an optical sectioning technique compatible with widefield imaging microscopy, and has been shown to provide a depth resolution comparable to confocal
microscopy [1]. Since its invention [2], SI microscopy has been widely used as a sectioning
tool in bioimaging research, both at the cellular level — such as in 3D imaging of cellular nuclear periphery [3], cellular fenestrations [4], tubulin and kinesin dynamics [5] and at the tissue#154806 - $15.00 USD(C) 2012 OSA

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level — such as in imaging of autofluorescence aggregation in human eye [6], zebrafish development [7], rat colonic mucosa [8]. In addition, SI has also been used as a super-resolution
technique to break the diffraction limit [5, 9–11]. SI thus has the ability to maintain the highlight collection capability of widefield imaging [12, 13] while also removing out-of-plane light.
It has, however, been criticized as being a non-quantitative technique, and for producing noisy
data in comparison to the sectioned images derived from confocal microscopes. The analysis
below shows that SI can easily be made quantitative by properly scaling the standard sectioning algorithm, and we also provide analytical expressions for the resulting noise in SI-sectioned
images. Although Somekh et al. [14, 15] provide numerical simulations of the noise properties
of SI-sectioned images, they use a non-quantitative algorithm and do not include the effects ofout-of-focus light on the noise.
Quantitative sectioned images allow one to perform photon counting as if the regions aboveand below the sectioned layer were not present. While the resulting photon number estimate
will be noisier than would be the case for imaging the slice without out-of-focus layers present,
the mean value of the correctly scaled algorithm will equal to the mean photon count one
would obtain with a standard widefield microscope. This permits researchers to use standard
methods [16] of correcting for the objective lens’ numerical aperture, optical transmission, anddetector quantum efficiency to determine photon counts at the sectioned plane relative to thenumber of photoelectrons detected at the sensor plane. Allowing quantitative data to be obtainedwith SI sectioning thus gives researchers the ability to perform measurements of radiance, absolute reflectance, fluorophore quantum yield, and absolute fluorophore concentration withinvolumetric media [17, 18].
Finally, the analytical formulas for noise also enable us to roughly define an operational range
for the modulation contrast at the section plane, such that for any contrast above this value one
can expect SI to provide higher SNR data than confocal microscopy. For any contrast below
this value, confocal imaging will out-perform SI.
2.

Sectioning algorithm

The general approach in SI is to illuminate the object with a sinusoidal illumination pattern of
the form [2](1)
si (x, y) = 12 1 + m cos(ν x + φi ) ,at each of three spatial phases φ1 = 0, φ2 = 2π /3, and φ3 = 4π /3. Although other structured
forms are possible [19], this one is particularly easy to implement. The quantity m is the modulation contrast (a number varying from 0 to 1), and ν is the modulation spatial frequency. The
factor of 1/2 placed in front, not present in previous work, is used here in order to represent thefact that half of the illumination light is absorbed or reflected by the grid placed in the illumination path. If we take the limit m → 0, we obtain standard widefield illumination of half theintensity that one would obtain without the grid in place. Note that s represents a normalized
illumination amplitude, ranging from 0 to 1.
Ignoring the effects of optical blurring, the resulting modulated images gi (x, y) are given bygi (x, y) = 12 d(x, y) + si (x, y) f (x, y)(2)

for a planar object distribution f (x, y) and out-of-focus light d(x, y), both scaled to what one
would obtain with standard widefield imaging. For fluorescence imaging, the absolute brightness f of the object contains the illumination irradiance I, the fluorophore quantum yield q, anda factor Ω resulting from integrating the angular distribution of fluorescence emission over thenumerical aperture of the imaging optics: ffluor = IqΩ. For brightfield imaging, f is simply theillumination I multiplied by the object reflectance R: fbright = IR. Thus, the expression for gi#154806 - $15.00 USD(C) 2012 OSA

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is valid for the cases of both fluorescence imaging and brightfield imaging, but with a subtle
difference in what f means for each case.
In order to obtain an optically sectioned image i(x, y) at the focal plane, the most common
algorithm used is [1, 2, 8, 20–31]
(3)istd (x, y) = (g1 − g2 )2 + (g1 − g3 )3 + (g2 − g3 )2 ,based on square law detection. Alternative illumination patterns allow for different processing
approaches [19]. SI for superresolution, for example, relies on the Moire effect to detect lightemitted outside the conventional bandwidth limit.
Since the algorithm (3) operates on each pixel independently, we have dropped the spatialarguments (x, y) as unnecessary. (These can be added back into each equation at any point.)
Inserting Eqs. (1) into (2) and applying trigonometric identities, we obtain the result
3m
istd = √ f ,
2 2

(4)

Thus, the sectioned
image is a copy of the object distribution, as we expect, but scaled by the√factor 3m/(2 2). The quantitatively-scaled algorithm is obtained by multiplying Eq. (3) by the
inverse of this factor:√ 
2 2i(x, y) =(g1 − g2 )2 + (g1 − g3 )3 + (g2 − g3 )2 .
(5)3m
Note that the algorithm assumes that the out-of-focus light d(x, y) does not change with a shift
in the illumination pattern. A consequence of this result is that in order to obtain quantitative
results for the sectioned image, one must estimate the modulation contrast m. A further assumption required is that of linearity, which in fluorescence imaging is limited to weakly fluorescentstructures [1].
The scale factor in front of the square root differs from that given in previous studies. The
factor of 2 in the numerator appears as a result of the 1/2 scaling introduced into our definitionof si (x, y) and thus is new. All previous authors have also assumed ideal modulation (m = 1).
This is an assumption which introduces a large error into the quantitative result. Moreover,since the modulation m(x, y) is in general spatially varying, the error introduced is generally nota simple scalar factor for the whole image. As a whole, the literature shows wide disagreement
over√the appropriate scale factor to place in front of the square root. Refs [20, 22, 26, 28, 32, 33]
use 2/3, which is appropriate when m = 1 and the factor of 1/2 in s(x, y) is not used. Refs [2,
23–25,27,29] use a scale factor of 1, which is the most appropriate choice√√ for a non-quantitative2) [8], 1/ 2 [1, 31], orapproach,
while
other
authorsuse
alternative
factors
such
as
1/(3
√
3 2 [21] without explanation.
In practice, one finds that even for ideal samples m cannot achieve the maximum value of
1. The modulation contrast, however, remains excellent (m > 0.5) in thin samples in which thesectioned plane is taken near the surface, but poor (m < 0.1) in dense tissue samples (in which
multiple scattering is present) and in deeper layers of thinly scattering media.
3.

Widefield image algorithm

In addition to the sectioned image algorithm (5), it is well known that one can form a widefield
image representation iw (x, y) from the modulated images by
iw = 23 (g1 + g2 + g3 ) .

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(Our scale factor differs from that of previous authors due to the factor of 1/2 used in thedefinition of s(x, y).) Once again inserting the formulas for the illumination and modulated
images, Eqs. (1) and (2), we obtain(7)
iw = d + f .
The widefield image is a simple sum of the sectioned plane and the out-of-focus contribution.
For quantitative work, we also want to estimate the variance of the widefield image, for which 2   2
we insert Eqs. (1), (2), and (6) into the standard variance formula var(iw ) = iw − iw andsolve. Here the angle brackets · represent an expectation value. The first term in the varianceformula is 2  4
iw = 9 (g1 + g2 + g3 )2
= 49 g21 + g22 + g33 + g1 g2 + g1 g3 + g1 g2 + g2 g3 + g1 g3 + g2 g3
= 49 94 d 2 + f 2 (s21 + s22 + s23 ) + 32 d f (s1 + s2 + s3 ) + 2 f 2 (s1 s2 + s1 s3 + s2 s3 )   
 (8)= d 2 + 49 (s21 + s22 + s23 ) f 2 + 43 (s1 + s2 + s3 ) d f
 2
8+ 9 (s1 s2 + s1 s3 + s2 s3 ) f .
Here
we

 have  assumed
  that d and fare independent of one another, so that terms such as
d + f = d + f and d f = d f . Because the si represent a normalized illumination
distribution, the stochastic properties of the system are present only within the out-of-focus
light d and the slice’s lightf , and not in the illumination s.
 distribution
The second moment i2w thus separates into four terms, each of which can be consideredseparately. Using trigonometric identities, we obtain for the modulation factor in each term
s1 + s2 + s3 = 32 ,(9)

s21 + s22 + s23 = 34 + 38 m2 ,3 2
m ,s1 s2 + s1 s3 + s2 s3 = 34 − 16

(10)
(11)

giving the result 2   2
    
iw = d + var(d) + 2 d f + f 2 ,   2
where we have used d 2 = d + var(d).
The second term in the variance formula is easily obtained from Eq. (7) as    2
    2 2
iw = ( d + f )2 = d + 2 d f + f .

Putting the two results together produces
var(iw ) = var(d) + var( f ) .

(12)

Just as the mean of the widefield image is a simple sum of the planar slice and the out-of  value
focus light, iw = d + f , the variance of the widefield image is also a simple sum of thecomponent variances.
4.

Variance and SNR of sectioned images

Next we can try to follow the same procedure for the sectioning algorithm (5) to obtain the   2variance of the sectioned image, var(i) = i2 − i . The second term in the variance formula 2  2
is easily obtained from Eqs. (4) and (5) as i = f . The first term can be obtained byinserting Eq. (6) to give 2  8(13)
i = 9m2 [(g1 − g2 )2 + (g1 − g3 )2 + (g2 − g3 )2 ] .
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Received 20 Sep 2011; revised 9 Dec 2011; accepted 13 Dec 2011; published 21 Dec 2011
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The next step is to substitute Eqs. (1) and (2) into this formula, but here one must be careful.
di andEach modulated image gi provides different samples
 fi of the stochastic out-of-focus
and slice light distribution, such that terms like d12 − d1 d2 will be nonzero. That is, since d1      2is independent of d2 , we can write d1 d2 = d1 d2 , whereas d12 = d1 + var(d  1 ). Once
all cross-terms are eliminated from inside an expectation value, one can then take di → dand var(di ) = var(d), and likewise for the fi as well. Thus, the first quadratic term inside theexpectation value of Eq. (13) is (g1 − g2 )2 = ( 12 d1 + s1 f1 − 12 d2 − s2 f2 )2           
= 14 d12 + 14 d22 − 12 d1 d2 + s21 f12 s22 f22 + s1 d1 f1 − s2 d1 f2
      
− s1 d2 f1 + s2 d2 f2 − 2s1 s2 f1 f2 .
      2
Letting all di → d and di2 → d + var(d), we have



 2
 
(g1 − g2 )2 = 12 var(d) + (s21 + s22 ) f 2 − 2s1 s2 f .

Doing this also for the second and third squared terms in Eq. (13) and combining gives
 2
 2
 2 
8 32
22
var(d)
−
2(s
i =
s
+
s
s
+
s
s
)
f
+
2(s
+
s
+
s
)
f
1
2
1
3
2
3
1
2
3
9m2 2


 2
 
3
3
3 3
8 3
var(d) − 2 − m2 f + 2 + m2 f 2
=
2
9m 2
4 16
4 8
 2
4
4
2
+
var( f ) ,var(d) + f +
=
3m2
3 3m2
where we have also used the results of Eqs. (10) and (11). Incorporating this result into thevariance formula obtains
var(i) =4
24+ 2 var( f ) .
var(d) +
2
3m
3 3m

(14)

The variance of the sectioned image is thus dependent on the out-of-focus contribution in addition to the variance of f . (Recall that f represents the light obtained from a single standard
widefield image of just the planar slice itself.) Both terms contain a dependence on the modulation contrast, so that as the modulation approaches zero (m → 0), the variance in the sectioned
image increases without bound, as we should expect.
The theoretical expression for the sectioned image variance Eq. (14) indicates that when
the illumination produces ideal modulation contrast at the√slice plane (m = 1), the sectioning
algorithm amplifies noise in the slice image by a factor of 2 (in standard deviation) relative tothe stochastic noise in f . For most cases, a large out-of-focus contribution is present, and thisnot only reduces the modulation contrast but adds to the noise as well. Then m becomes small,and in this regime the variance approximates tovar(i) ≈
4 
var(d) + var( f ) ,2
3m

so that the signal-to-noise
ratio of the sectioned image in the weak modulation regime is reduced√
by a factor 2/(m 3) relative to that of a standard widefield image,

√  
i
m 3 f
SNR(i) ≈
1/2 = 2 σ (i ) .
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Received 20 Sep 2011; revised 9 Dec 2011; accepted 13 Dec 2011; published 21 Dec 2011
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5.

Estimating the modulation contrast

The quantitative sectioning algorithm (5) requires knowing the modulation contrast m in orderto properly scale the result. This value is generally not known a priori and so must be estimated,but one can use the modulated images themselves to provide the estimated value m̂. For eachof the three modulated images, we obtain a “modulation map” by normalizing the modulated
images using the widefield algorithm asgi
gi (x, y)= 2.
iw (x, y)
3 (g1 + g2 + g3 )Since μi can also be written as μi = 12 + m2 cos ν x − φi , subtracting 1/2 from μi produces a
result proportional to m, so that

μi (x, y) =m2  2cos (ν x) + cos2 ν x + 23π + cos2 ν x + 43π
4
3 2
= m ,
8

(μ1 − 12 )2 + (μ2 − 12 )2 + (μ3 − 12 )2 =and an estimate of m is thus given by1/2
8(μ1 − 12 )2 + (μ2 − 12 )2 + (μ3 − 12 )2.
(15)m̂(x, y) =3This suggests that, for high SNR data, one can calculate the sectioned image pixel-by-pixel by
combining Eqs. (5) and (15) to give
1/2
1 (g1 − g2 )2 + (g1 − g3 )2 + (g2 − g3 )2(16)i(x, y) = √ g1 1

g3
g2
1
1 1/2
3
+
+
−
−
−
iw
2
iw
2
iw
2
As an example, we measured the three modulated images g1 , g2 , and g3 on a fluorescentbead sample and use the algorithm in Eq. (15) to obtain the estimated modulation m̂(x, y) at
every pixel in the image. Our experimental setup (see Sec. 6.1) achieves approximately uniform
modulation across the image, and so if we first threshold the image iw to prevent noisy pixels
from skewing the estimate, we obtain a histogram of m̂ across the image, as shown in Fig. 1. The
histogram suggests that the pixel-by-pixel estimate of m will be quite noisy, so that a much moreaccurate estimate can be achieved by averaging m̂ across the image. Or, if there is a significant
contribution of outliers, one can use the histogram median as a more robust estimate. For Fig. 1,
the mean and median values are m̂ = 0.48 and 0.49 respectively. We can note, however, thattaking the mean or median are only valid for homogeneous samples.
From a theoretical standpoint, the roughly Gaussian shape to the histogram is expected, but
the long tail at the lower values of m̂ is not, and may be the result of some beads aggregating
together to create a thicker layer at some locations within the image. If the thickness exceeds
the sectioning depth, then the modulation contrast will drop.
While Fig. 1 shows that the median modulation contrast can be accurately estimated from asingle image, the use of Eq. (15) to estimate m(x, y) requires exceedingly high signal-to-noise
ratio images in order to achieve a reasonable accuracy for every pixel in the image. Its practicaluse, therefore, requires collecting a large number of photons (perhaps by summing a sequenceof static images) or some kind of spatial processing in order to reduce the effects of noise.
6.

Experimental results

In order to test our theoretical results, we conducted several experiments on a Zeiss Axio Imager
Z1 microscope equipped with an Apotome module, a Zeiss AxioCam MRm monochromatic#154806 - $15.00 USD(C) 2012 OSA

Received 20 Sep 2011; revised 9 Dec 2011; accepted 13 Dec 2011; published 21 Dec 2011
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Number of elements

250

mean = 0.48
median = 0.49

200

150

100

50

0

0

0.1

0.2

0.3

^
m

0.4

0.5

0.6

0.7

Fig. 1. A histogram of the estimated modulation contrast, m̂, obtained from (15).

camera (1388 × 1040 pixels), and an HB-100 mercury lamp illumination source. The objective
lens used for all of the experiments is a Zeiss Plan-Apochromatic 20× objective (NA = 0.8).
In order compare measurements against theory, we use the ratio r of measured noise to theestimated shot noise. Whereas the measured noise is obtained by taking the standard deviationof a sequence of 1000 measurements, the photon shot noise standard deviation σp is estimatedby taking the square root of the mean number of photons collected. For a standard widefieldmeasurement, this number should be close to 1, but for SI-sectioned images the noise is larger
than one would expect from the shot noise alone, so that the theory predicts r > 1.
The first step of the experiment involves measuring the camera gain in order to scale digitalcounts to detected photelectrons. This involves imaging a uniformly illuminated field (createdby Köhler illumination) at the microscope sample stage with different illumination intensities.
To remove the effects of pixel response nonuniformity, we implemented the following procedures: [34]1. At each illumination intensity, two wide field images I1 and I2 are acquired.
2. The standard deviation σc is calculated for a 200 × 200 pixel area in the difference image
I1 − I2 .
3. The signal variance is calculated by var( fc ) = σc2 /2. The scaling factor of 2 accounts forthe increased noise due to the image subtraction operation in Step 2.
The quantities fc and σc here indicate the measured intensity and standard deviation in unitsof counts and not photons. The resulting measured signal vs. variance at different illumination
intensities is shown in Fig. 2, and gives an estimated gain g = 4.1 photons/count.
In order to provide a baseline reference for later SI noise measurements, we imaged a microscope slide containing a sparse layer of fluorescent beads (Molecular Probes Fluosphere F8853,
peak emission at 515 nm, 2 μ m diameter) in wide-field mode (i.e. without the structured gridplaced in the illumination path). To prepare a uniformly distributed sample, the fluorescentbeads were suspended by vortex mixing and sonicated. The suspensions was then dropped onto
a microscope slide and sealed with a cover slip. A total of 1000 images of the sample  wereacquired in a time sequenced experiment. The measured mean fluorescent intensity f of thefluorescent beads is 4.33 × 104 counts (obtained by summing all pixels at the bead location),and the standard deviation σn of fluorescent intensity is 112 counts. Thus the ratio of measured#154806 - $15.00 USD(C) 2012 OSA

Received 20 Sep 2011; revised 9 Dec 2011; accepted 13 Dec 2011; published 21 Dec 2011
2 January 2012 / Vol. 20, No. 1 / OPTICS EXPRESS 410

3500

Signal (counts)

3000
2500
2000
1500
1000
500
0
0

200

400

Variance

600

800

Fig. 2. Signal vs. variance measurement, showing experimental data (dots) and the fitted
line. The line slope of 4.1 gives an estimate for the gain of g = 4.1 photons/count.

noise to the photon shot noise is:r=

112 counts × 4.1 photons/countσm== 1.09 .
σp(4.33 × 104 counts × 4.1 photons/count)1/2

Since r is very close to 1, we can say that the imaging system is shot-noise limited.
6.1.

Sectioned imaging of 2 μ m fluorescent beads without out-of-focus light

We first measured the axial PSF of the SI-sectioned measurements by imaging sub-resolution
green fluorescent nanoparticles (175 nm diameter spheres, from Invitrogen), using the softwarerecommended VL grid (17.5 lines/mm) on the Apotome. The FWHM of resulting measured
axial PSF is 3.6 μ m, indicating that our fluorescent bead sample (peak emission at 515 nm,
2 μ m diameter) is sufficiently thin that no out-of-focus light will be present. For the quantitativealgorithm, we estimate the modulation contrast for this setup using the data shown in Fig. 1,
giving m̂ = 0.49 from the median of the distribution. A total of 1000
sectioned images wereacquired in a time sequenced experiment. The mean intensity f and its standard deviation
σm are calculated for 5 different beads, with results shown in Table 1. The ratio of measured
noise to the photon noise is calculated as:r=

σm (photons)σm (photons)
σm (counts) × 4.1 photons/count=  
1/2 =  
1/2 . (17)σp (photons)f (photons)f (counts) × 4.1 photons/count)

On the other hand, the theoretical value of the ratio r, assuming Poisson noise, is obtained by
 2
1/2
4
var(i)
4 1/2
σm
2
3 + 3m2 var( f )
√
+ 2
=
=
r̂ =
= √
= 2.49 .
σp
3 3m
f
f
The theoretical value of 2.49 thus closely corresponds to the measured mean value of 2.41
given in Table 1. That is, use of the sectioning algorithm with the three modulated images onthe (planar) sample has reduced the SNR by a factor of 2.41 from that of a single (unmodulated)widefield image.
The calculated  widefield image iw of the same sample is obtained by algorithm (6). The mean
intensity iw , standard deviation σm and ratio r are calculated for the same five beads, withresults shown in Table 1. The measured value of the noise ratio r = 1.04 closely corresponds tothe theoretical result of r = 1 obtained from Eq. (12).
#154806 - $15.00 USD(C) 2012 OSA

Received 20 Sep 2011; revised 9 Dec 2011; accepted 13 Dec 2011; published 21 Dec 2011
2 January 2012 / Vol. 20, No. 1 / OPTICS EXPRESS 411

Table 1. Measurements on a sample of five different fluorescent  beads without out-offocus light. These are the measured mean widefield
intensityf and its standard deviation σm ( f ), the calculated widefield intensity iw and its standard deviation σm (iw ), and thecorresponding
ratios r obtained with Eq. (17). The modulation contrast m used to calculate 
f is estimated using Eq. (15).

Bead
1
2
3
4
5

6.2.

Sectioned image 
f (cts)σm ( f ) (cts)5.03 × 104

6.29 × 104
4.61 × 104
5.04 × 104
6.17 × 104r

Calculated widefield image 
iw (cts) σm (iw ) (cts)
r

279
292
259
277
276

2.53
2.35
2.44
2.50
2.24

4.99 × 104
5.51 × 104
4.59 × 104
4.93 × 104
5.60 × 104

mean:

2.41

mean:

114
122
115
110
119

1.03
1.06
1.08
1.00
1.02
1.04

Sectioned imaging of 6 μ m fluorescent beads containing out-of-focus light

In order to measure noise amplification in SI in the case when out-of-focus light is present,
we imaged a sample of 6 μ m diameter green fluorescent beads. Since the sectioning thicknessof the Apotome in this setup is 3.6 μ m, there will be some out-of-focus light present. A totalof 1000 sectioned images were acquired in a time sequenced experiment. The mean intensity
f and its standard deviation σm are calculated for five different beads, with results shown in
Table 2. The theoretical value of the ratio r, which we write as r̂, is given by 1/2
var(i)
1
4 
m2
√
var(
f
)
=√
var(d)
+
1
+
2
f
f 3m2
   
d + f
 1/2
1
2
4 
√
√
.
var(d)
+
var(
f
)
≈√
=
f 3m2
f
m 3
      
If we use the relation iw = d + f , let m̂ = 0.49, and substitute the mean iw with its
measured value iw , we have√2iwr̂ = √ √ = 2.36 iw / f .
m 3 f
r̂ =

σm
=
σp

The resulting theoretical r̂ values for the five beads are shown in Table 2, and indicate a close
correspondence with the experimentally measured value for r. Note that the variation in r̂
among the five beads selected may be due to a variation in the relative amount of fluorescenceemitted by the bead from within the sectioned plane to that emitted from outside the sectioned
plane.

#154806 - $15.00 USD(C) 2012 OSA

Received 20 Sep 2011; revised 9 Dec 2011; accepted 13 Dec 2011; published 21 Dec 2011
2 January 2012 / Vol. 20, No. 1 / OPTICS EXPRESS 412

 
Table 2. The mean intensity of wide-field images, iw , and standard deviation σm of fivedifferent fluorescent beads, obtained from experiment and theory.

Bead
1
2
3
4
5

7.

Experiment 
f (cts) σm (cts)1.70 × 105
1.20 × 105
10.29 × 104
1.89 × 105
9.05 × 104

Theoryriw (cts)

r̂

836
669
511
750
526

4.09
3.90
3.24
3.50
3.54

5.01 × 105
4.13 × 105
2.32 × 105
5.04 × 105
2.80 × 105

4.11
4.38
3.56
3.85
4.15

mean:

3.66

mean:

4.01

Conclusion

Although it has often been argued that structure illumination sectioning microscopy is a nonquantitative technique, we have shown that a quantitative version of the algorithm can be obtained by adding a proper scaling factor. Quantitative scaling does require that one estimate themodulation contrast m, and this adds an extra step of complexity, but Eq. (15) provides a simple
means of obtaining such an estimate. A consequence of ignoring this scaling factor, as the sectioning algorithms have up to this point, is that in z-stack volumetric images (x, y, z) the deeper
layers will appear artificially darkened. A result of the SI sectioning approach, however, is thatsince it removes out-of-focus light from the sectioned image after detection, it suffers fromthe shot noise of both the section image and all out-of-focus planes. While this has long been
known, little has been known about the quantitative correlation between the noise amplification
in SI microscopy and the out-of-focus light or other imaging parameters.
The theoretical analysis given above has made no assumptions about the properties of thenoise other than to assume that the variance is the primary quantity of interest, and thus theresults remain valid across all noise regimes (read noise limited, shot noise limited, etc.).
The noise amplification indicated by the variance result may be taken as an argument that SIsectioning is a poor substitute for confocal sectioning due to the loss in SNR. But this is not thewhole story. The compatibility of SI with widefield imaging also allows orders of magnitude
greater light throughput than that achievable by confocal microscopy, such that one can use
lower-intensity light sources and still obtain 100–200× increases in photon collection abovethat of scanning laser illumination [12,13]. In this case, taking 150 as a representative value forincreased light collection, SI sectioning can provide better SNR than confocal sectioning when
4
< 150
3m2

→

m > 0.094 .

An additional advantage SI microscopy has is the ability to reject any residual DC light, suchas that generated by stray light or reflections within the optical system, though this comes at
an SNR penalty. Whether SI-sectioning or confocal sectioning produces better SNR images isdependent on the microscope setup and the object under analysis, but our theoretical results
provide support for the common empirical observation that SI-sectioning gives lower quality
results when imaging deep within tissue.
Acknowledgments
This work was supported in part by the National Institute of Health under grants R01CA124319 and R21-EB009186.
#154806 - $15.00 USD(C) 2012 OSA

Received 20 Sep 2011; revised 9 Dec 2011; accepted 13 Dec 2011; published 21 Dec 2011
2 January 2012 / Vol. 20, No. 1 / OPTICS EXPRESS 413

Quantitative sectioning and noise
analysis for structured illumination
microscopy: errata
Nathan Hagen,1,2 Liang Gao,1 and Tomasz S. Tkaczyk1,∗
1 Department of Bioengineering, Rice University, Houston, Texas 77005, USA
2 nhagen@optics.arizona.edu∗ ttkaczyk@rice.edu

Abstract: We correct errors in two equations of our original manuscript.
© 2012 Optical Society of America
OCIS codes: (180.0180) Microscopy; (110.4280) Noise in imaging systems; (100.3010) Image
reconstruction techniques; (180.6900) Three-dimensional microscopy; (100.6640) Superresolution.

References and links
1. N. Hagen, L. Gao, and T. S. Tkaczyk, “Quantitative sectioning and noise analysis for structured illumination
microscopy,” Opt. Express 20, 403–413 (2012).

Equations (3) and (5) of our original manuscript [1] contain errors in the power of the (g1 − g3 )image difference term, used to calculate the sectioned image. The corrected equation for the
standard (non-quantitative) sectioned image algorithm is(3)istd (x, y) = (g1 − g2 )2 + (g1 − g3 )2 + (g2 − g3 )2 ,and for the quantitative version√ 
2 2(g1 − g2 )2 + (g1 − g3 )2 + (g2 − g3 )2 .
i(x, y) =3m(5)

The corrections have no effect on the remainder of the manuscript.

#163029 - $15.00 USD(C) 2012 OSA

Received 14 Feb 2012; published 17 Feb 201227 February 2012 / Vol. 20, No. 5 / OPTICS EXPRESS 5343

Quantitative sectioning and noise
analysis for structured illumination
microscopy: erratum
Ting Ai Chen,1,5 Nathan Hagen,2,6 Liang Gao,3 and Tomasz S.
Tkaczyk4,7
1 Research Laboratory for Biomedical Optics and Molecular Imaging, Institute of Biomedicaland Health Engineering, Shenzhen Institutes of Advanced Technology, Chinese Academy of
Sciences, 1068 Xueyuan Boulevard, Nanshan, Shenzhen 518055, China
2 College of Optical Sciences, University of Arizona, 1630 E. University Boulevard, Tucson
Arizona 85721, USA
3 Department of Biomedical Engineering, Washington University, 1 Brookings Drive, St. Louis
Missouri 63130, USA
4 Department of Bioengineering, Rice University, 6100 Main Street, Houston Texas 77005,
USA
5 ta.chen@siat.ac.cn
6 nhagen@optics.arizona.edu
7 ttkaczyk@rice.edu

Abstract:
We correct an error in the original manuscript, where an
unrecognized assumption was made about the relationship between theout-of-focus light and the in-focus light. We summarize the condition underwhich the assumption may still hold, and mention alternative methods
researchers can use to obtain accurate quantitative sectioning.
© 2015 Optical Society of America
OCIS codes: (180.0180) Microscopy; (110.4280) Noise in imaging systems; (100.3010) Image
reconstruction techniques; (180.6900) Three-dimensional microscopy; (100.6640) Superresolution.

References and links
1. N. Hagen, L. Gao, and T. S. Tkaczyk, “Quantitative sectioning and noise analysis for structured illumination
microscopy,” Opt. Express 20(1), 403–413 (2012).

Section 5 in the original manuscript provides two expressions for the normalized modulated
images µi : µi = 12 + m2 cos(νx − φi ) and µi = gi /iw . By equating the two expressions, one canobtain an estimate for the modulation factor m at each pixel (Eq. (15) in the original paper).
However, if one uses the second expression and substitutes the equation for the ith measured
image, gi = 21 d + si f , together with iw = d + f , one obtains the result
µi =

1 m
f
+ cos(νx − φi )
.
2 2
d+ f

That is, the two expressions can be equated only when d  f — when the out-of-focus light issmall in comparison to the in-focus light. This is an implicit assumption that went unrecognizedin the original manuscript and will cause serious quantitative error.
The problem is that in the expression for the measurement images,
gi =#235068© 2015 OSA

d+ f mf+cos(νx − φi ) ,2
2

Received 8 Oct 2015; published 12 Oct 201519 Oct 2015 | Vol. 23, No. 21 | DOI:10.1364/OE.23.027633 | OPTICS EXPRESS 27633

the amplitude of the modulation m and the brightness of the in-focus signal f are coupled andcannot be separated from one another using the measurement images alone. As a result, a weakmodulation factor such as that produced by a blurred illumination pattern will produce the same
effect on the measurement as a weak signal from the in-focus plane.
Quantitative results are still possible, but require additional steps. For example, in order toseparate the effects of m from f , one can create phantoms of known f such that the two factors
m and f can be separated from one another. Care must be taken to ensure that the phantom will
have similar optical properties to the sample of interest, and that the various phantoms have
known signal f at several different depths within the tissue, so that one can measure the change
in modulation m as a function of depth. Another method would be to apply a Monte Carlosimulation of the tissue properties in order to model the modulation depth at the section plane
as a function of the modulation depth incident at the surface.

#235068© 2015 OSA

Received 8 Oct 2015; published 12 Oct 201519 Oct 2015 | Vol. 23, No. 21 | DOI:10.1364/OE.23.027633 | OPTICS EXPRESS 27634