## A comparative study of high resolution microscopy imaging modalities using a three-dimensional resolution measure

Optics Express, Vol. 17, Issue 26, pp. 24377-24402 (2009)

http://dx.doi.org/10.1364/OE.17.024377

Acrobat PDF (2987 KB)

### Abstract

From an acquired image, single molecule microscopy makes possible the determination of the distance separating two closely spaced biomolecules in three-dimensional (3D) space. Such distance information can be an important indicator of the nature of the biomolecular interaction. Distance determination, however, is especially difficult when, for example, the imaged point sources are very close to each other or are located near the focal plane of the imaging setup. In the context of such challenges, we compare the limits of the distance estimation accuracy for several high resolution 3D imaging modalities. The comparisons are made using a Cramer-Rao lower bound-based 3D resolution measure which predicts the best possible accuracy with which a given distance can be estimated. Modalities which separate the detection of individual point sources (e.g., using photoactivatable fluorophores) are shown to provide the best accuracy limits when the two point sources are very close to each other and/or are oriented near parallel to the optical axis. Meanwhile, modalities which implement the simultaneous imaging of the point sources from multiple focal planes perform best when given a near-focus point source pair. We also demonstrate that the maximum likelihood estimator is capable of attaining the limit of the accuracy predicted for each modality.

© 2009 Optical Society of America

## 1. Introduction

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4. M. P. Gordon, T. Ha, and P. R. Selvin, “Single-molecule high-resolution imaging with photobleaching,” Proc. Natl. Acad. Sci. USA **101**, 6462–6465 (2004). [CrossRef] [PubMed]

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6. S. Ram, E. S. Ward, and R. J. Ober, “Beyond Rayleigh’s criterion: A resolution measure with application to single-molecule microscopy,” Proc. Natl. Acad. Sci. USA **103**, 4457–4462 (2006). [CrossRef] [PubMed]

7. E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science **313**, 1642–1645 (2006). [CrossRef] [PubMed]

8. S. T. Hess, T. P. K. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. **91**, 4258–4272 (2006). [CrossRef] [PubMed]

9. M. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods **3**, 793–795 (2006). [CrossRef] [PubMed]

10. A. Sharonov and R. M. Hochstrasser, “Wide-field subdiffraction imaging by accumulated binding of diffusing probes,” Proc. Natl. Acad. Sci. USA **103**, 18911–18916 (2006). [CrossRef] [PubMed]

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14. S. Ram, P. Prabhat, J. Chao, E. S. Ward, and R. J. Ober, “High accuracy 3D quantum dot tracking with multifocal plane microscopy for the study of fast intracellular dynamics in live cells,” Biophys. J. **95**, 6025–6043 (2008). [CrossRef] [PubMed]

15. S. R. P. Pavani and R. Piestun, “Three dimensional tracking of fluorescent microparticles using a photon-limited double-helix response system,” Opt. Express **16**, 22048–22057 (2008). [CrossRef] [PubMed]

16. S. Weiss, “Fluorescence spectroscopy of single biomolecules,” Science **283**, 1676–1683 (1999). [CrossRef] [PubMed]

6. S. Ram, E. S. Ward, and R. J. Ober, “Beyond Rayleigh’s criterion: A resolution measure with application to single-molecule microscopy,” Proc. Natl. Acad. Sci. USA **103**, 4457–4462 (2006). [CrossRef] [PubMed]

21. J. Chao, S. Ram, A. V. Abraham, E. S. Ward, and R. J. Ober, “A resolution measure for three-dimensional microscopy,” Opt. Commun. **282**, 1751–1761 (2009). [CrossRef]

6. S. Ram, E. S. Ward, and R. J. Ober, “Beyond Rayleigh’s criterion: A resolution measure with application to single-molecule microscopy,” Proc. Natl. Acad. Sci. USA **103**, 4457–4462 (2006). [CrossRef] [PubMed]

21. J. Chao, S. Ram, A. V. Abraham, E. S. Ward, and R. J. Ober, “A resolution measure for three-dimensional microscopy,” Opt. Commun. **282**, 1751–1761 (2009). [CrossRef]

22. T. D. Lacoste, X. Michalet, F. Pinaud, D. S. Chemla, A. P. Alivisatos, and S. Weiss, “Ultrahigh-resolution multi-color colocalization of single fluorescent probes,” Proc. Natl. Acad. Sci. USA **97**, 9461–9466 (2000). [CrossRef] [PubMed]

23. A. Agrawal, R. Deo, G. D. Wang, M. D. Wang, and S. Nie, “Nanometer-scale mapping and single-molecule detection with color-coded nanoparticle probes,” Proc. Natl. Acad. Sci. USA **105**, 3298–3303 (2008). [CrossRef] [PubMed]

3. X. Qu, D. Wu, L. Mets, and N. F. Scherer, “Nanometer-localized multiple single-molecule fluorescence microscopy,” Proc. Natl. Acad. Sci. USA **101**, 11298–11303 (2004). [CrossRef] [PubMed]

4. M. P. Gordon, T. Ha, and P. R. Selvin, “Single-molecule high-resolution imaging with photobleaching,” Proc. Natl. Acad. Sci. USA **101**, 6462–6465 (2004). [CrossRef] [PubMed]

**103**, 4457–4462 (2006). [CrossRef] [PubMed]

5. K. A. Lidke, B. Rieger, T. M. Jovin, and R. Heintzmann, “Superresolution by localization of quantum dots using blinking statistics,” Opt. Express **13**, 7052–7062 (2005). [CrossRef] [PubMed]

7. E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science **313**, 1642–1645 (2006). [CrossRef] [PubMed]

8. S. T. Hess, T. P. K. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. **91**, 4258–4272 (2006). [CrossRef] [PubMed]

9. M. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods **3**, 793–795 (2006). [CrossRef] [PubMed]

10. A. Sharonov and R. M. Hochstrasser, “Wide-field subdiffraction imaging by accumulated binding of diffusing probes,” Proc. Natl. Acad. Sci. USA **103**, 18911–18916 (2006). [CrossRef] [PubMed]

*z*) position [24

24. S. Ram, E. S. Ward, and R. J. Ober, “How accurately can a single molecule be localized in three dimensions using a fluorescence microscope?” Proc. SPIE **5699**, 426–435 (2005). [CrossRef] [PubMed]

*z*-localization problem has direct implications on the distance problem. In [21

21. J. Chao, S. Ram, A. V. Abraham, E. S. Ward, and R. J. Ober, “A resolution measure for three-dimensional microscopy,” Opt. Commun. **282**, 1751–1761 (2009). [CrossRef]

25. P. Prabhat, S. Ram, E. S. Ward, and R. J. Ober, “Simultaneous imaging of different focal planes in fluorescence microscopy for the study of cellular dynamics in three dimensions,” IEEE Trans. Nanobiosci. **3**, 237–242 (2004). [CrossRef]

*z*-)axis, MUM can be used to visualize and study cellular processes over a large depth range (e.g., [14

14. S. Ram, P. Prabhat, J. Chao, E. S. Ward, and R. J. Ober, “High accuracy 3D quantum dot tracking with multifocal plane microscopy for the study of fast intracellular dynamics in live cells,” Biophys. J. **95**, 6025–6043 (2008). [CrossRef] [PubMed]

26. P. Prabhat, Z. Gan, J. Chao, S. Ram, C. Vaccaro, S. Gibbons, R. J. Ober, and E. S. Ward, “Elucidation of intracellular recycling pathways leading to exocytosis of the Fc receptor, FcRn, by using multifocal plane microscopy,” Proc. Natl. Acad. Sci. USA **104**, 5889–5894 (2007). [CrossRef] [PubMed]

*z*position can be determined [14

14. S. Ram, P. Prabhat, J. Chao, E. S. Ward, and R. J. Ober, “High accuracy 3D quantum dot tracking with multifocal plane microscopy for the study of fast intracellular dynamics in live cells,” Biophys. J. **95**, 6025–6043 (2008). [CrossRef] [PubMed]

28. M. J. Mlodzianoski, M. F. Juette, G. L. Beane, and J. Bewersdorf, “Experimental characterization of 3D localization techniques for particle-tracking and super-resolution microscopy,” Opt. Express **17**, 8264–8277 (2009). [CrossRef] [PubMed]

**282**, 1751–1761 (2009). [CrossRef]

*z*-axis. As we showed in [21

**282**, 1751–1761 (2009). [CrossRef]

*z*direction.

## 2. The four imaging modalities

**SIM-SNG (simultaneous detection, single focal plane) modality**represents the conventional fluorescence imaging setup where a single image is acquired of a pair of point sources and is used to estimate the distance of separation. A simulated image for this modality is shown in Fig. 1(a), where a relatively large 500 nm distance of separation was specified to clearly illustrate the presence of two point sources. For much smaller distances of separation, the two spots would overlap significantly and be difficult to visually distinguish as two.

**SEP-SNG (separate detection, single focal plane) modality**separates the detection of two closely spaced point sources either spectrally or temporally, and images each point source from one and the same focal plane. For ease of presentation, however, and without loss of generality, we will assume for this modality the model of temporal separation that relies on the use of photoactivatable or photoswitchable fluorophores (e.g., [7

7. E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science **313**, 1642–1645 (2006). [CrossRef] [PubMed]

8. S. T. Hess, T. P. K. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. **91**, 4258–4272 (2006). [CrossRef] [PubMed]

9. M. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods **3**, 793–795 (2006). [CrossRef] [PubMed]

**SIM-MUM (simultaneous detection, multifocal plane) modality**(e.g., [14

**95**, 6025–6043 (2008). [CrossRef] [PubMed]

26. P. Prabhat, Z. Gan, J. Chao, S. Ram, C. Vaccaro, S. Gibbons, R. J. Ober, and E. S. Ward, “Elucidation of intracellular recycling pathways leading to exocytosis of the Fc receptor, FcRn, by using multifocal plane microscopy,” Proc. Natl. Acad. Sci. USA **104**, 5889–5894 (2007). [CrossRef] [PubMed]

**SEP-MUM (separate detection, multifocal plane) modality**(e.g., [29

29. M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, “Three-dimensional sub-100 nm resolution fluorescence microscopy of thick samples,” Nat. Methods **5**, 527–529 (2008). [CrossRef] [PubMed]

## 3. The 3D resolution measures for the four imaging modalities

30. S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidimens. Syst. Sig. Process. **17**, 27–57 (2006). [CrossRef]

**282**, 1751–1761 (2009). [CrossRef]

**282**, 1751–1761 (2009). [CrossRef]

### 3.1. The SIM-SNG modality

*θ*=(

*d*,

*ϕ*,

*ω*,

*s*,

_{x}*s*,

_{y}*s*),

_{z}*θ*∈Θ, from a single image of two point sources (Fig. 1(a)). The symbol Θ denotes the parameter space which is an open subset of R

^{6}. As shown in Fig. 2, the six parameters of

*θ*collectively describe the geometry of a pair of point sources

*P*

_{1}and

*P*

_{2}in 3D space. Parameter

*d*denotes the distance that separates

*P*

_{1}and

*P*

_{2}, parameter

*ϕ*denotes the angle which the

*xy*-plane projection of the line segment

*P*

_{1}

*P*

_{2}forms with the positive

*x*-axis, parameter

*ω*denotes the angle which

*P*

_{1}

*P*

_{2}forms with the positive

*z*-axis, and parameters

*s*, and

_{x}, s_{y}*s*denote the coordinates of the midpoint between

_{z}*P*

_{1}and

*P*

_{2}.

*θ*(which includes the distance of separation

*d*) from an image of the point sources

*P*

_{1}and

*P*

_{2}, a mathematical model that accurately describes the image is required. To account for the intrinsically stochastic nature of the photon emission (and hence the photon detection) process, this image, which we assume to be acquired during the time interval [

*t*

_{0},

*t*], is modeled as a spatio-temporal random process [30

30. S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidimens. Syst. Sig. Process. **17**, 27–57 (2006). [CrossRef]

*P*

_{1}and

*P*

_{2}as an inhomogeneous Poisson process with intensity function Λ

*(*

_{θ}*τ*),

*τ*≥

*t*

_{0}. The spatial portion models the image space coordinates at which the photons are detected as a sequence of independent random variables with probability density functions

*(*

_{θ}*τ*) is simply the sum of the rates at which photons are detected from

*P*

_{1}and

*P*

_{2}(Eq. (3)), and each density function

*f*is simply a weighted sum of the images (i.e., the point spread functions) of the two point sources (Eq. (4)). Note that in general, both Λ

_{θ,τ}*(*

_{θ}*τ*) and

*θ*(i.e., the parameters to be estimated).

*(*

_{θ}*τ*) and

**I**(

*θ*), this matrix provides a measure of the amount of information the image carries about the parameter vector

*θ*to be estimated. Since

*θ*is a six-parameter vector,

**I**(

*θ*) is a 6-by-6 matrix.

**I**(

*θ*) is a lower bound on the covariance matrix of any unbiased estimator

*θ*̂ of the parameter vector

*θ*(Eq. (8)). It then follows that element (1,1) of

**I**

^{-1}(

*θ*), which corresponds to the distance parameter

*d*(i.e., the first parameter in

*θ*), is a lower bound on the variance of the distance estimates of any unbiased estimator. The 3D resolution measure for the SIM-SNG modality is accordingly defined as the quantity

### 3.2. The SEP-SNG, the SIM-MUM, and the SEP-MUM modalities

*θ*. Then, by applying the definition given in Section 3.1, the resolution measure follows easily as the square root of the first main diagonal element of the inverse matrix.

*(*

_{θ}*τ*) and the probability density functions

*(*

_{θ}*τ*) will simply be the photon detection rate of only point source

*P*

_{1}or

*P*

_{2}, and not a sum of their detection rates. Likewise, each spatial density function

*f*will entail only the point spread function of

_{θ,τ}*P*

_{1}or

*P*

_{2}, and not a sum of their point spread functions. For images acquired from different focal planes by the SIM-MUM or the SEP-MUM modality, the spatial density functions

**I**(

*θ*), the 3D resolution measure for the modality follows easily as the quantity

## 4. Comparison of modalities using the 3D resolution measure

*z*-axis, we look at how their resolution measures depend on the parameters

*d*,

*sz*, and

*ω*(Fig. 2), respectively. More specifically, we plot the resolution measures as functions of two of the three parameters at a time, and we do so for all three possible pairings of the parameters.

*P*

_{1}and

*P*

_{2}that emit photons of wavelength

*λ*=655 nm. The image of each point source is assumed to be described by the classical 3D point spread function of Born and Wolf [17]. That is, the point spread functions of

*P*

_{1}and

*P*

_{2}are each of the form

*z*

_{0}is the axial position of the point source in the object space,

*n*is the numerical aperture of the objective lens, and

_{a}*n*is the refractive index of the object space medium. For our comparisons, the objective lens is assumed to have a numerical aperture of

*n*=1.4 and a magnification of

_{a}*M*=100. The refractive index of the object space medium is assumed to be

*n*=1.515.

### 4.1. Small distances of separation

*d*and axial position

*s*. For a given axial position, a deterioration of the limit of the distance estimation accuracy (i.e., an increase in the resolution measure) is observed for both modalities as the distance of separation is decreased from 200 nm to 0 nm. This deterioration is nonlinear, progressing relatively slowly at larger values of

_{z}*d*, but becoming significantly sharper when the value of d is in the low tens of nanometers. (Note that at d=0 nm, the resolution measure is infinity due to the Fisher information matrix becoming singular. This special scenario corresponds to the degenerate case where the images of the point sources completely coincide. Also, note that for some

*s*values close to 0 nm in Fig. 3(a) where the resolution measure is very large, the described pattern of deterioration does not hold. Specifically, the pattern is interrupted by a sharp deterioration of the resolution measure over values of

_{z}*d*that put one of the point sources near the focal plane at

*s*=0 nm. This problem of especially poor depth discrimination near focus is discussed in Section 4.2.)

_{z}*d*and

*s*. For a given value of

_{z}*s*, the resolution measures for both modalities stay relatively small and constant regardless of the distance of separation. (Note the exception at

_{z}*d*=0 nm, where the resolution measure is infinity. In addition, the exception to the rule is again present for some near-focus

*s*values in Fig. 3(b) where the resolution measure is very large.)

_{z}*s*=367.2 nm from each of the four 3D plots of Fig. 3. At this axial position, a distance of

_{z}*d*=200 nm can be estimated with best possible accuracies of ±22.95 nm and ±18.51 nm when the SIM-SNG and the SIM-MUM modalities, respectively, are used. These numbers correspond to approximately ±10% of the 200 nm distance. When the SEP-SNG and the SEP-MUM modalities are used, however, the best possible accuracies that can be expected are ±9.92 nm and ±12.70 nm, respectively, or approximately just ±5% of the 200 nm distance. Though a nontrivial, factor of two improvement in the limit of the accuracy can already be seen at

*d*=200 nm when the SEP-SNG and the SEP-MUM modalities are used, the advantage of using separate detection is even more significant at smaller distances.

*d*=100 nm, for example, best possible accuracies of ±45.34 nm and ±33.26 nm can be expected from the SIM-SNG and the SIM-MUM modalities, respectively. These numbers are no better than ±30% of the 100 nm distance. When the SEP-SNG and the SEP-MUM modalities are used, however, the resolution measures are ±9.58 nm and ±12.60 nm, respectively, corresponding to just approximately ±10% of the 100 nm distance. If the distance is halved again to

*d*=50 nm, then the limits of the accuracy for the SIM-SNG and the SIM-MUM modalities deteriorate further to ±88.48 nm and ±64.60 nm, respectively. These numbers are greater than the 50 nm distance, and are clearly unacceptable. In sharp contrast, by using the SEP-SNG and the SEP-MUM modalities instead, resolution measures of ±9.47 nm and ±12.54 nm, respectively, can still be expected. These limits of the accuracy correspond to a perhaps still acceptable ±20% and ±25% of the 50 nm distance, respectively.

**282**, 1751–1761 (2009). [CrossRef]

*d*=50 nm from Fig. 4(a) as a function of the expected number of detected photons. For each modality, the improvement (i.e., decrease) in the resolution measure can roughly be described by an inverse square root dependence on the increase in photon count. By doubling the expected photon count to 10000 per point source, for example, the best possible accuracies with which the 50 nm distance can be estimated are improved to ±51.37 nm and ±41.01 nm for the SIM-SNG and the SIM-MUM modalities, respectively, and ±5.96 nm and ±7.72 nm for the SEP-SNG and the SEP-MUM modalities, respectively. A further tenfold increase to 100000 photons per point source would improve the resolution measures to less than ±24% of the 50 nm distance for the SIM-SNG (±11.68 nm) and the SIM-MUM (±10.73 nm) modalities, and to less than ±4% for the SEP-SNG (±1.51 nm) and the SEP-MUM (±1.81 nm) modalities.

*d*and the orientation angle

*ω*. On the one hand, for a given value of

*ω*, the nonlinear deterioration of the resolution measure with decreasing distance of separation is observed for the SIM-SNG (Fig. 5(a)) and the SIM-MUM (Fig. 5(c)) modalities. On the other hand, for a given angle, the resolution measure remains low and essentially constant over all distances (except

*d*=0 nm) for the SEP-SNG (Fig. 5(b)) and the SEP-MUM (Fig. 5(d)) modalities. (Note that there is an exception at

*ω*=0°, where the resolution measure is infinity for all modalities regardless of the distance

*d*. This is due to the Fisher information matrix being singular when one point source is situated exactly in front of the other.)

### 4.2. Near-focus depth discrimination

*z*position of a point source can be determined is especially poor when it is located near the focal plane [24

24. S. Ram, E. S. Ward, and R. J. Ober, “How accurately can a single molecule be localized in three dimensions using a fluorescence microscope?” Proc. SPIE **5699**, 426–435 (2005). [CrossRef] [PubMed]

*x*

_{01},

*y*

_{01},

*z*

_{01}) and (

*x*

_{02},

*y*

_{02},

*z*

_{02}) (see Fig. 2) of the two point sources, it follows that especially poor accuracy for determining the distance of separation can also be expected when either of the point sources is near-focus. In more technical terms, the unknown parameter vector

*θ*can alternatively be defined as (

*x*

_{01},

*y*

_{01},

*z*

_{01},

*x*

_{02},

*y*

_{02},

*z*

_{02}), and estimates of the distance of separation

*d*can be obtained indirectly by simply computing the Euclidean distance using the estimated coordinates. Therefore, a poor accuracy in the estimation of

*z*

_{01}or

*z*

_{02}can be expected to translate to a poor accuracy in the determination of

*d*.

**282**, 1751–1761 (2009). [CrossRef]

*d*, as the two sharp increases of the resolution measure near the focal plane

*s*=0 nm. (Note that the presence of two sharp increases can be seen more clearly in the 2D plot of Fig. 6(b). Additionally, we note the exception at

_{z}*d*=0 nm where the resolution measure is infinity regardless of the axial position.) The sharp deterioration below the focal plane corresponds to point source

*P*

_{1}coming very close to being in focus (i.e., value of

*z*

_{01}approaching 0 nm), and the sharp deterioration above the focal plane corresponds to point source

*P*

_{2}coming very close to being in focus (i.e., value of

*z*

_{02}approaching 0 nm).

*z*-localization accuracy translates to poor (good) distance estimation accuracy, we plot in Fig. 6(a) the 2D slice

*d*=200 nm from Fig. 3(b), together with the corresponding limits of the localization accuracy of the axial coordinates

*z*

_{01}and

*z*

_{02}of the two point sources at each axial (

*s*) position of the point source pair. This limit of the

_{z}*z*-localization accuracy is based on the same theoretical framework [30

30. S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidimens. Syst. Sig. Process. **17**, 27–57 (2006). [CrossRef]

**95**, 6025–6043 (2008). [CrossRef] [PubMed]

*z*-localization accuracy of the two point sources. Above a defocus of roughly |

*s*|=500 nm, both the resolution measure and the limits of the

_{z}*z*-localization accuracy exhibit a deteriorating trend as the point source pair is positioned farther away, in either direction, from the focal plane at

*s*=0 nm. Below a defocus of roughly |

_{z}*s*|=500 nm, the resolution measure exhibits, as explained above, the two sharp increases about the focal plane. The sharp deterioration below the focal plane, which corresponds to

_{z}*P*

_{1}coming close to being in focus, coincides accordingly with the sharp deterioration of the

*z*-localization accuracy of

*P*

_{1}, wherein

*z*

_{01}approaches 0 nm. Similarly, the sharp deterioration above the focal plane corresponds to

*P*

_{2}coming close to being in focus, and coincides with the sharp deterioration of the

*z*-localization accuracy of

*P*, wherein

_{2}*z*

_{02}approaches 0 nm.

*s*=0 nm are absent. Instead, for a given value of

_{z}*d*(except

*d*=0 nm), the resolution measure remains flat in the near-focus region, indicating that good distance estimation accuracy can be expected from the SIM-MUM modality even when one of the point sources is near-focus. This is due to the fact that while a point source may be near-focus from the perspective of the focal plane at

*s*=0 nm, it is a good distance away from the second focal plane at

_{z}*s*=500 nm. The image acquired from the second focal plane therefore contains enough information to compensate for the lack thereof in the image acquired from the first focal plane. Thus, by using both images in the estimation of the distance of separation, the combined information is sufficient to yield a good accuracy.

_{z}*d*not equal to 0 nm. Importantly, the SEP-MUM modality is the only modality out of the four that is able to overcome both the small distance of separation problem and the near-focus depth discrimination problem. By combining the principles of separate detection and MUM, Fig. 3(d) shows that it is able to achieve a comparatively low and flat resolution measure over all distances of separation 200 nm and below (except

*d*=0 nm), and over the four-micron axial range centered at the focal plane.

*z*-axis. This outward deterioration can be explained intuitively by the fact that the farther away two closely spaced point sources are from focus, the more they will appear to be a single point source in the acquired image, and hence the more difficult the distance estimation.

*d*=200 nm from each of the four 3D plots of Fig. 3. To give an example of the important advantage gained with MUM when a point source is near-focus, consider the point source pair centered at

*s*=64.60 nm, which corresponds to

_{z}*z*

_{02}=-6.11 nm and hence places point source

*P*

_{2}near the focal plane. With the SIM-SNG and the SEP-SNG modalities, extremely poor resolution measures of ±525.60 nm and ±277.07 nm (out of the plot’s range), respectively, can be expected in estimating the 200 nm distance of separation. In significant contrast, resolution measures of no worse than ±10% of the 200 nm distance can be expected from the SIM-MUM (±19.23 nm) and the SEP-MUM (±12.74 nm) modalities.

*s*and the orientation angle

_{z}*ω*. For a given value of

*ω*, all four modalities lose accuracy as a non-near-focus point source pair is moved away from the focal plane(s). In the near-focus region, however, the SIM-SNG (Fig. 7(a)) and the SEP-SNG (Fig. 7(b)) modalities exhibit the pair of sharp deteriorations about the focal plane, whereas the SIM-MUM (Fig. 7(c)) and the SEP-MUM (Fig. 7(d)) modalities do not. (Note that when

*ω*=90° such that the two point sources have the same

*z*position (i.e.,

*z*

_{01}=

*z*

_{02}), only a single point deterioration is observed exactly at the focal plane

*s*=0 nm in Figs. 7(a) and 7(b). In this special scenario, the resolution measure actually remains small for axial positions immediately about

_{z}*s*=0 nm. Also, as noted in Section 4.1, the resolution measure is infinity when

_{z}*ω*=0°. This is the case for all modalities for any

*s*.)

_{z}### 4.3. Near-parallel orientations with respect to the z-axis

*z*-axis (i.e., the angle

*ω*; see Fig. 2) is an important determinant of the limit of the distance estimation accuracy [21

**282**, 1751–1761 (2009). [CrossRef]

*ω*=90°; perpendicular to the

*z*-axis). This is because this side-by-side orientation can be expected to produce the least overlap between the point spread functions of the two point sources in the acquired image. However, as this point source pair is rotated towards parallel alignment with the

*z*-axis (

*ω*=0°), such that one point source starts to go in front of, and hence obscure the other, the distance estimation problem becomes tougher as more point spread function overlap can be expected.

*d*(except

*d*=0 nm) in these plots, it can be seen that the 3D resolution measure increases as the point source pair is rotated from the side-by-side orientation of

*ω*=90° towards the front-and-back orientation of

*ω*=0°. This deterioration of the resolution measure progresses relatively slowly when going from

*ω*=90° to roughly ω=45°, but continues at a significantly faster rate as

*ω*decreases further before leveling off when approaching 0° (at which point the resolution measure is infinity as mentioned previously).

*ω*) can be significantly lessened by the same principle. This can be seen in Figs. 5(b) and 5(d), where for a given value of

*d*(except

*d*=0 nm), the resolution measures for the SEP-SNG and the SEP-MUM modalities, respectively, stay low and almost flat over the entire range of values for

*ω*(except at

*ω*=0° where they are infinity). The advantage conferred by these two modalities here can again be attributed to the removal of the overlap of the point spread functions of the two point sources, which in this case is caused by small values of

*ω*. It is important to note at this point that, by also being able to overcome the deteriorative effect associated with near-parallel orientations, the SEP-MUM modality is the only modality considered here that is capable of addressing all three challenges associated with distance, depth discrimination, and orientation.

*d*=200 nm from each of the four 3D plots of Fig. 5. At values of ω larger than approximately 45°, resolution measures of around, or better than, ±20 nm (i.e., ±10% of the 200 nm distance) can be expected for all four modalities. At

*ω*=75°, for example, the resolution measures for the SIM-SNG, the SEP-SNG, the SIM-MUM, and the SEP-MUM modalities are ±8.10 nm, ±5.82 nm, ±6.62 nm, and ±5.96 nm, respectively. At values of

*ω*smaller than approximately 45°, however, substantial differences can be observed between the modalities. Whereas the SIM-SNG and the SIM-MUM modalities both lose a significant amount of accuracy, the SEP-SNG and the SEP-MUM modalities experience only a small deterioration. At

*ω*=7.5°, for example, poor limits of the accuracy of ±96.23 nm and ±51.42 nm can be expected, respectively, from the SIM-SNG and the SIM-MUM modalities. These numbers correspond, respectively, to about ±50% and ±25% of the 200 nm distance. In contrast, for the same small angle, much better resolution measures of ±12.77 nm and ±17.37 nm can be expected, respectively, from the SEP-SNG and the SEP-MUM modalities. In fact, across the entire range of values for

*ω*(except ω=0°), these two modalities are able to maintain accuracies of better than ±10% of the 200 nm distance.

*ω*is decreased from 90° to 0°. For the SEP-SNG (Fig. 7(b)) and the SEP-MUM (Fig. 7(d)) modalities, the resolution measure remains, for a given

*s*, relatively low and flat throughout the entire range of angles (except

_{z}*ω*=0°). (Note that due to the near-focus depth discrimination problem (Section 4.2), the patterns described are not observed in Figs. 7(a) and 7(b) for some

*s*values in the region occupied by the pair of sharp deteriorations about the focal plane. In these cases, the patterns are interrupted by a sharp deterioration over values of ω which put one of the point sources near the focal plane.)

_{z}## 5. Maximum likelihood estimation with simulated images

**282**, 1751–1761 (2009). [CrossRef]

### 5.1. Methods

*N*pixels produced by the SIM-SNG modality, the log-likelihood function is then given by

_{p}*p*(

_{θ,k}*z*),

_{k}*k*=1, …,

*N*, is the probability that

_{p}*z*photons are detected at the kth pixel, and is given by Eq. (7). The parameter vector

_{k}*θ*that is estimated is as given in Section 3.1, consisting of the six parameters (including the distance of separation

*d*) that together describe the 3D geometry of a point source pair (Fig. 2).

*N*pixels produced by the SEP-SNG, the SIM-MUM, or the SEP-MUM modality, the log-likelihood function is also as given by Eq. (2), but with one important caveat. Depending on whether the image is of point source

_{p}*P*

_{1}or

*P*

_{2}or both, and of focal plane 1 or 2,

*p*(

_{θ,k}*z*) of Eq. (7) is evaluated using the appropriate redefinitions of the intensity function Λθ and the spatial density functions

_{k}*d*, the overall log-likelihood function to be maximized is just the sum of the log-likelihood functions for the individual images.

*µ*m by 13

*µ*m pixels. In addition to the imaging modality, the various data sets differ in terms of the point source pair’s distance of separation

*d*, axial position

*s*, and orientation angle

_{z}*ω*.

*e*

^{-}and a standard deviation of 8

*e*

^{-}.

*and*

_{θ}*P*

_{1}or

*P*

_{2}or both, and of focal plane 1 or 2. Also, since a 50:50 splitting of the collected fluorescence between the two focal planes is assumed for the SIM-MUM and the SEP-MUM modalities, the expected photon count per point source is halved to 2500, and the mean of the background noise at each pixel is halved to 40 photons for these modalities.

32. “EstimationTool,” http://www4.utsouthwestern.edu/wardlab/estimationtool.

33. “FandPLimitTool,” http://www4.utsouthwestern.edu/wardlab/fandplimittool.

### 5.2. Results

*d*=200 nm, an axial position of

*s*=400 nm that places both point sources well away from the focal plane (

_{z}*z*

_{01}=450 nm,

*z*

_{02}=350 nm), and an orientation of

*ω*=60° that is far from parallel alignment with the

*z*-axis. Given this relatively easy scenario, resolution measures of no worse than ±7% of the 200 nm distance are predicted and attained for the SIM-SNG (±13.71 nm) and the SIM-MUM (±12.09 nm) modalities. Even better accuracy limits of no worse than ±5% of 200 nm are predicted and attained for the SEP-SNG (±7.82 nm) and the SEP-MUM (±9.59 nm) modalities.

*d*=100 nm and an orientation of

*ω*=30° that is closer to parallel alignment with the z-axis. These two data sets demonstrate that, given the more challenging separation distance and orientation, good accuracy limits of ±11.12 nm and ±15.47 nm are still predicted and attained, respectively, for the SEP-SNG and the SEP-MUM modalities. (Poor accuracy limits of ±73.22 nm and ±56.78 nm are predicted for the SIM-SNG and the SIM-MUM modalities, respectively.) Data sets 7 and 8 entail a point source pair with the same distance of separation and orientation as that of data sets 1 through 4, but with an axial position of

*s*=75 nm that places point source

_{z}*P*

_{2}only 25 nm away from the focal plane. Given this point source that is close to the focal plane, these data sets show that excellent accuracy limits of ±12.29 nm and ±9.58 nm are still predicted and attained, respectively, for the SIM-MUM and the SEP-MUM modalities. (Poor accuracy limits of ±85.27 nm and ±49.23 nm are predicted for the SIM-

*d*=100 nm,

*ω*=30°, and

*s*=75 nm, a good accuracy limit of ±15.54 nm is still predicted and attained for the SEP-MUM modality. (Poor accuracy limits of ±488.43 nm, ±66.84 nm, and ±57.33 nm are predicted for the SIM-SNG, the SEP-SNG, and the SIM-MUM modalities, respectively.)

_{z}*θ*is asymptotically Gaussian distributed with asymptotic mean

*θ*and covariance

**I**

^{-1}(

*θ*) (e.g., [34]), where the latter is the Cramer-Rao lower bound (Eq. (8)) on which the resolution measure is based. This means that our estimator is asymptotically unbiased and asymptotically efficient. However, in the case of finite data sets such as those shown in Table 1, it is difficult to analytically determine the accuracy of our complex estimator and whether or not it is biased. We have therefore relied on simulation studies instead. Having examined many data sets such as those shown in Table 1, we can draw the conclusion that if there is bias, then it is a very small one provided that the resolution measure is small compared to the estimated distance. For example, for every data set in Table 1, the mean differs from the true distance by less than 1% of the true distance. These data sets entail relatively small resolution measures that are no worse than ±16% of the corresponding true distances. Importantly, this shows that under the practically desirable condition where the resolution measure predicts a good accuracy limit, the maximum likelihood estimator is able to recover the true distance with very little bias if it in fact exists. Moreover, under this condition, we can conclude based on the consistency with which it does so in our simulation studies, that the maximum likelihood estimator is able to attain the resolution measure.

## 6. Conclusions

*z*-axis. We have shown that the conventional fluorescence imaging setup generally performs poorly under these conditions. However, modalities that implement the MUM technique are able to overcome the near-focus depth discrimination problem, and modalities that employ the separate detection of fluorophores are able to deal with point source pairs with small distances of separation and/or near-parallel orientations. Moreover, though it does not always yield the best limit of the distance estimation accuracy numerically, the modality that combines the principles of separate detection and multifocal plane imaging is the only modality that is able to provide generally good accuracy limits when all three problems are present.

35. S. F. Gibson and F. Lanni, “Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy,” J. Opt. Soc. Am. A **9**, 154–166 (1992). [CrossRef] [PubMed]

36. P. Török and P. Varga, “Electromagnetic diffraction of light focused through a stratified medium,” Appl. Opt. **36**, 2305–2312 (1997). [CrossRef] [PubMed]

37. O. Haeberlé, “Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: Conventional microscopy,” Opt. Commun. **216**, 55–63 (2003). [CrossRef]

*n*=1.2, an immersion medium refractive index of

_{a}*n*=1.33, and a magnification of

*M*=63. We see that while the plots of Fig. 9 do not show the same numerical values as in Figs. 4(a), 6(b), and 8, they demonstrate the same general behavior of the resolution measures as functions of a point source pair’s distance of separation, axial position, and orientation.

## Appendix

## A.1 The 3D resolution measure for the SIM-SNG modality

*P*

_{1}and

*P*

_{2}(Fig. 1(a)) that is acquired during the time interval [

*t*

_{0},

*t*], and from which the unknown parameter vector

*θ*=(

*d,ϕ,ω, s*) is to be estimated (see Fig. 2). Assuming at first the absence of any extraneous noise sources (e.g., sample autofluorescence, detector dark current, detector readout), such an image is formed strictly from the photons detected from the point source pair. Since photon emission (and hence photon detection) is stochastic by nature, this image is modeled as a spatio-temporal random process [30

_{x}, s_{y}, s_{z}**17**, 27–57 (2006). [CrossRef]

_{1}and Λ

_{2}are the photon detection rates of

*P*

_{1}and

*P*

_{2}, respectively. The spatial aspect models the image space coordinates of the detected photons as a sequence of independent random variables with probability density functions

*f*is given by

_{θ,τ}*x,y*) ∈ ℝ

^{2},

*ε*(

_{i}*τ*)=Λ

*(*

_{i}*τ*)/(Λ

_{1}(

*τ*)+Λ

_{2}(

*τ*)), i=1, 2, τ≥

*t*

_{0},

*M*denotes the lateral magnification of the microscope, (

*x*

_{01},

*y*

_{01},

*z*

_{01}) and (

*x*

_{02},

*y*

_{02},

*z*

_{02}) denote, respectively, the 3D coordinates of the positions of

*P*

_{1}and

*P*

_{2}in the object space (see Fig. 2), and

*P*

_{1}and

*P*

_{2}. Note that the position coordinates of

*P*

_{1}and

*P*

_{2}are functions of the unknown parameter vector

*θ*, and can be written explicitly as

*x*

_{01}=

*s*+

_{x}*d*sin

*ω*cos

*ϕ*/2,

*y*

_{01}=

*s*+

_{y}*d*sin

*ω*sin

*ϕ*/2,

*z*

_{01}=

*s*+

_{z}*d*cos

*ω*/2,

*x*

_{02}=

*s*-

_{x}*d*sinω cos

*ϕ*/2,

*y*

_{02}=

*s*-

_{y}*d*sin

*ω*sin

*ϕ*/2, and

*z*

_{02}=

*s*-

_{z}*d*cos

*ω*/2. Also, the point spread functions

**17**, 27–57 (2006). [CrossRef]

*z*

_{0}),

*z*

_{0}∈ℝ, in the object space.

*N*pixels, however, the collected data is a sequence of independent random variables {

_{p}*S*

_{θ,1}, …,

*S*}, where

_{θ},N_{p}*S*is the number of photons detected at the

_{θ,k}*k*pixel. In [30

^{th}**17**, 27–57 (2006). [CrossRef]

*S*is Poisson-distributed with mean

_{θ,k}*t*

_{0},

*t*] is again the image acquisition time interval,

*C*is the region in the detector plane that is occupied by the

_{k}*k*pixel, and the functions Λ

^{th}*(*

_{θ}*τ*) and

*f*(

_{θ,τ}*x,y*) are as defined above.

*N*pixels is modeled as a sequence of independent random variables {

_{p}*𝓣*

_{θ,1}, …,

*𝓣*

_{θ,Np}}, where

*𝓣*

_{θ,k}is the total photon count at the

*k*pixel. At each pixel

^{th}*k*, the total photon count

*𝓣*

_{θ,k}is modeled as the sum of the three mutually independent random variables

*S*,

_{θ,k}*B*, and

_{k}*W*. The random variable

_{k}*S*represents again the number of photons from the point source pair which are detected at the

_{θ,k}*k*pixel. It is dependent on the unknown parameter vector

^{th}*θ*, and is Poisson-distributed with mean

*µ*(

_{θ}*k*) given by Eq. (5). The random variable

*B*represents the number of spurious photons at the

_{k}*k*pixel which arise from extraneous noise sources such as sample autofluorescence and the detector dark current. It is assumed to be Poisson-distributed with mean

^{th}*β*(

*k*). The random variable

*W*represents the number of photons at the

_{k}*k*pixel which are due to measurement noise sources such as the detector readout process. It is assumed to be Gaussian-distributed with mean

^{th}*η*and standard deviation

_{k}*σ*. Note that neither

_{k}*B*nor

_{k}*W*is dependent on

_{k}*θ*, which contains parameters pertaining only to the point source pair.

**I**(

*θ*) corresponding to an image of

*N*pixels is then given by [30

_{p}**17**, 27–57 (2006). [CrossRef]

*k*=1, …,

*N*,

_{p}*ν*(

_{θ}*k*)=

*µ*(

_{θ}*k*)+

*β*(

*k*), and

*θ*̂ is any unbiased estimator of

*θ*, the 3D resolution measure is defined as the quantity

*d*.

## A.2 The 3D resolution measure for the SEP-SNG modality

*P*

_{1}only, and the other of point source

*P*

_{2}only (Fig. 1(b)). We first consider an image of point source

*P*

_{1}that is acquired during the interval [

*t*

_{0},

*t*

_{1}]. Since

*P*

_{1}is the only point source that is detected, the intensity function of the Poisson process is simply given by the photon detection rate of

*P*

_{1}, i.e., Λ

*(*

_{θ}*τ*)=Λ

_{1}(

*τ*),

*t*

_{0}≤

*τ*≤

*t*

_{1}. Likewise, the spatial density function will involve only the point spread function of

*P*

_{1}, and is given by

*τ*has been dropped from

*f*because unlike the density function of Eq. (4), the

_{θ}*f*here does not depend on time. Substituting these redefined functions into Eq. (5), the number of photons

_{θ}*Sθ,*detected at the

_{k}*k*pixel of this image is then Poisson-distributed with mean

^{th}*µ*(

_{θ}*k*) is superscripted with (1) to denote image of

*P*

_{1}. Finally, by evaluating Eq. (6) with

*µ*

^{(1)}

*, we obtain the Fisher information matrix*

_{θ}**I**

^{(1)}(

*θ*) which corresponds to the image of

*P*

_{1}.

*P*

_{2}acquired during the time interval [

*t*

_{2},

*t*

_{3}] that is disjoint from [

*t*

_{0},

*t*

_{1}], we get the mean photon count

*µ*(

_{θ}*k*) denotes image of

*P*

_{2}. Substituting it into Eq. (6), we obtain the Fisher information matrix

**I**

^{(2)}(

*θ*) which corresponds to the image of

*P*

_{2}.

## A.3 The 3D resolution measure for the SIM-MUM modality

*t*

_{0},

*t*]. Since each image acquired during this time is that of both point sources

*P*

_{1}and

*P*

_{2}, the intensity function of the temporal portion of its respective spatio-temporal random process will remain the same as that for the SIM-SNG modality. That is, it is still the sum of the photon detection rates of

*P*

_{1}and

*P*

_{2}, given by Eq. (3).

*z*coordinates of point sources

*P*

_{1}and

*P*

_{2}(i.e.,

*z*

_{01}and

*z*

_{02}) be given with respect to focal plane 1. Furthermore, let

*M*denote the lateral magnification associated with focal plane 1. Then, for the image corresponding to focal plane 1, each density function

*f*is exactly as given by Eq. (4). Accordingly, the Fisher information matrix

_{θ,τ}**I**

^{(1)}(

*θ*) corresponding to the image from focal plane 1 is readily given by Eq. (6).

*z*above focal plane 1. Then, with respect to focal plane 2, the

_{f}*z*coordinates

*z*

_{01}and

*z*

_{02}of the same point sources become

*z*

_{01}-Δ

*z*

*f*and

*z*

_{02}-Δ

*z*, respectively. Moreover, the lateral magnification

_{f}*M*′ that is associated with focal plane 2 can be determined using the geometrical optics-based relationship [38

38. L. Tao and C. Nicholson, “The three-dimensional point spread functions of a microscope objective in image and object space,” J. Microsc. **178**, 267–271 (1995). [CrossRef] [PubMed]

*n*is the refractive index of the object space medium, and

*L*is the tube length of the microscope. By substituting the modified axial positions and magnification into Eq. (4), we obtain the density functions

*f*will be given by

_{θ,τ}*ε*

_{1}and

*ε*

_{2}are as defined for Eq. (4). Finally, by using Eq. (13) in Eq. (5) and substituting the result into Eq. (6), we obtain the Fisher information matrix

**I**

^{(2)}(

*θ*) corresponding to the image from focal plane 2.

**I**(

*θ*) for the SIM-MUM modality is then given by Eq. (11), and the 3D resolution measure is just the quantity

## A.4 The 3D resolution measure for the SEP-MUM modality

*P*

_{1}only, is acquired simultaneously by two cameras during time interval [

*t*

_{0},

*t*

_{1}], but from two distinct focal planes. The other pair, each of point source

*P*

_{2}only, is acquired simultaneously by the same two cameras during time interval [

*t*

_{2},

*t*

_{3}] that is disjoint from [

*t*

_{0},

*t*

_{1}], but from the same two distinct focal planes. Each of the four images is hence characterized by a unique combination of time and focal plane, and can be assumed to be formed independently of the other three images.

*P*

_{1}(

*P*

_{2}) acquired during [

*t*

_{0},

*t*

_{1}] ([

*t*

_{2},

*t*

_{3}]).

*M*′ is again the lateral magnification of focal plane 2 given by Eq. (12), and Δ

*z*is again the spacing between the two focal planes. Use of Eq. (14) (Eq. (15)) gives the matrix for the image of

_{f}*P*

_{1}(

*P*

_{2}) acquired during [

*t*

_{0},

*t*

_{1}] ([

*t*

_{2},

*t*3]).

## Acknowledgments

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**OCIS Codes**

(000.5490) General : Probability theory, stochastic processes, and statistics

(030.5290) Coherence and statistical optics : Photon statistics

(100.6640) Image processing : Superresolution

(180.2520) Microscopy : Fluorescence microscopy

(180.6900) Microscopy : Three-dimensional microscopy

(110.3055) Imaging systems : Information theoretical analysis

**ToC Category:**

Microscopy

**History**

Original Manuscript: November 12, 2009

Revised Manuscript: December 14, 2009

Manuscript Accepted: December 17, 2009

Published: December 18, 2009

**Virtual Issues**

Vol. 5, Iss. 1 *Virtual Journal for Biomedical Optics*

**Citation**

Jerry Chao, Sripad Ram, E. Sally Ward, and Raimund J. Ober, "A comparative study of high resolution microscopy imaging modalities using a three-dimensional resolution measure," Opt. Express **17**, 24377-24402 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-26-24377

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