Abstract

One of the ongoing challenges in single particle fluorescence microscopy resides in estimating the axial position of particles with sub-resolution precision. Due to the complexity of the diffraction patterns generated by such particles, the standard fitting methods used to estimate a particle’s lateral position are not applicable. A new approach for axial localization is proposed: it consists of a maximum-likelihood estimator based on a theoretical image formation model that incorporates noise. The fundamental theoretical limits on localization are studied, using Cramér-Rao bounds. These indicate that the proposed approach can be used to localize particles with nanometer-scale precision. Using phantom data generated according to the image formation model, it is then shown that the precision of the proposed estimator reaches the fundamental limits. Moreover, the approach is tested on experimental data, and sub-resolution localization at the 10 nm scale is demonstrated.

© 2005 Optical Society of America

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References

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Applied Optics

P. Török and R. Varga, "Electromagnetic diffraction of light focused through a stratified medium," Applied Optics 36(11), 2305-2312 (1997).
[CrossRef] [PubMed]

Biophys J.

V. Levi, Q. Ruan, and E. Gratton, "3-D Particle Tracking in a Two-Photon Microscope: Application to the study of molecular dynamics in cells," Biophys J. 88, 2919-2928 (2005).
[CrossRef] [PubMed]

M. K. Cheezum,W. F.Walker, andW. H. Guilford, "Quantitative Comparison of Algorithms for Tracking Single Fluorescent Particles," Biophys J. 81, 2378-2388 (2001).
[CrossRef] [PubMed]

R. E. Thompson, D. R. Larson, and W. W. Webb, "Precise Nanometer Localization Analysis for Individual Fluorescent Probes," Biophys J. 82, 2775-2783 (2002).
[CrossRef] [PubMed]

R. J. Ober, S. Ram, and S.Ward, "Localization Accuracy in Single-Molecule Microscopy," Biophys J. 86, 1185-1200 (2004).
[CrossRef] [PubMed]

H. P. Kao and A. S. Verkman, "Tracking of Single Fluorescent Particles in Three Dimensions: Use of Cylindrical Optics to Encode Particle Position," Biophys J. 67, 1291-1300 (1994).
[CrossRef] [PubMed]

Chem. Phys. Lett.

A. Van Oijen, J. K¨ohler, J. Schmidt, M.M¨uller, and G. Brakenhoff, "3-Dimensional super-resolution by spectrally selective imaging," Chem. Phys. Lett. 292, 183-187 (1998).
[CrossRef]

ISBI 2004

N. Subotic, D. Van De Ville, and M. Unser, "On the Feasibility of Axial Tracking of a Fluorescent Nano-Particle Using a Defocusing Model," in Proceedings of the Second IEEE International Symposium on Biomedical Imaging: From Nano to Macro (ISBI’04), pp. 1231-1234 (Arlington VA, USA, 2004).

J. Microsc.

A. Egner and S. W. Hell, "Equivalence of the Huygens-Fresnel and Debye approach for the calculation of high aperture point-spread functions in the presence of refractive index mismatch," J. Microsc. 193, 244-249 (1999).
[CrossRef]

S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, "Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index," J. Microsc. 169, 391-405 (1993).
[CrossRef]

M. G. L. Gustafsson, D. A. Agard, and J.W. Sedat, "I5M: 3D widefield light microscopy with better than 100nm axial resolution," J. Microsc. 195, 10-16 (1999).
[CrossRef] [PubMed]

J. Opt. Soc. Am

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 8(10), 1601-1613 (1991).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

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 (2002).
[CrossRef]

S. Hell and E. H. K. Stelzer, "Fundamental improvement of resolution with a 4Pi-confocal fluorescence microscope using two-photon excitation," Opt. Commun. 93, 277-282 (1992).
[CrossRef]

Opt. Lett.

Rev. Sci. Instrum.

N. Bobroff, "Position measurement with a resolution and noise-limited instrument," Rev. Sci. Instrum. 57, 1152-1157 (1986).
[CrossRef]

Single Molecules

U. Kubitscheck, "Single Protein Molecules Visualized and Tracked in the interior of Eukaryotic Cells," Single Molecules 3, 267-274 (2002).
[CrossRef]

Other

In practice, the number of possible optical sections is constrained by the exposure time, the dynamics of the biological process under study, and photobleaching of the fluorescent labels.

M. Gu, Advanced Optical Imaging Theory (Springer, 2000).

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Figures (11)

Fig. 1.
Fig. 1.

Coordinate system and notational conventions used in this paper.

Fig. 2.
Fig. 2.

(a) xz-section (top) and axial intensity profile (bottom) for a source located at the interface between the coverslip and specimen layer, in which case the PSF is symmetric. (b) Corresponding CRB for different values of the quantization factor c (where c is given in units of 1/|A|2). The decrease of the bound is proportional to this factor. At a defocus distance of 0.5 μm, these values of c correspond to the following SNRs: 7.4, 18.0, 28.4, and 38.8 dB.

Fig. 3.
Fig. 3.

(a) xz-sections of the theoretical PSF corresponding to point sources located at different depths zp of the specimen. (b) CRBs corresponding to the PSFs shown in (a), where c = 3000 (in units of 1/|A|2).

Fig. 4.
Fig. 4.

Result of the ML estimation from simulated acquisitions. For every point, the estimation was performed 50 times with a single acquisition (using different realizations of the noise). The standard deviation of the estimates matches the CRB well, showing that our ML estimator is optimal. The singularity around 0.25 μm is due to the mathematical properties of the first derivative of the PSF, which is close to zero when the focus is near to the particle’s position.

Fig. 5.
Fig. 5.

(a) xz-section of a z-stack of a bead located at zp = 22.1 μm. (b) xz-section of the PSF model corresponding to the parameters from (a).

Fig. 6.
Fig. 6.

Comparison of acquisitions of a bead located at zp = 22.1 μm with their counterparts generated from the theoretical model. The distances indicate the amount by which the acquisitions are defocused.

Fig. 7.
Fig. 7.

Localization results for three different beads. The values plotted are the deviation Δz = p - z ref, where z ref is the reference position estimated using all acquisitions. The respective reference values are, from top to bottom: 22.050 μm, 22.073 μm, and 22.081 μm, with the corresponging averages of the estimations: 22.046 μm, 22.069 μm, and 22.085 μm.

Fig. 8.
Fig. 8.

Standard deviation of the localization results with respect to the CRB, displayed over the range in which the estimations were performed.

Fig. 9.
Fig. 9.

Optimal focal positions for a variety of acquisition settings. For a single acquisition, (a) and (b) clearly show the influence of the particle’s depth on the optimal position; this is notably due to the focal shift that occurs as a particle moves deeper into the specimen (here a 4 μm thick section is considered). The optimal position is indicated by the vertical bars. (c), (d) Optimal focal positions when two acquisitions are used for two different sections of the sample. (e), (f) Scenario with three acquisitions. The optimal acquisition settings are considerably different from the uniform ones, and their effect on the CRB is substantial.

Fig. 10.
Fig. 10.

Comparison of the Cramér-Rao bounds for the scalar and vectorial formulations of the PSF. The xz-sections of the PSF and the dotted line in the CRB plots were obtained using the vectorial model. As the source moves deeper into the specimen, the difference between the two models becomes increasingly negligible.

Fig. 11.
Fig. 11.

Schematic representation of the optical path difference relative to two sources: the first in the design configuration (PQRS, with zp = 0), and the second in an arbitrary position zp (path ABCD).

Equations (32)

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PSF ( x x p , y y p , z z p ) = A 0 1 e iW ( ρ , z z p ) J 0 ( k ( x x p ) 2 + ( y y p ) 2 NA ρ ) ρ d ρ 2 ,
PSF ( r , z z p ) = A 0 1 e iW ( ρ , z z p ) J 0 ( kr NA ρ ) ρ d ρ 2 .
q ̄ ( x , y , z n z p ) = c PSF ( x , y , z n z p ) ,
P ( q ( x , y , z n z p ) ) = e q ̅ ( x , y , z n z p ) q ̄ ( x , y , z n z p ) q ̄ ( x , y , z n z p ) q ( x , y , z n z p ) ! ,
n = 1 N x , y 𝓢 P ( q ( x , y , z n z p ) ) ,
Var ( z ̂ p 1 ) E [ 2 z p 2 ln n = 1 N x , y 𝓢 P ( q ̄ ( x , y , z n z p ) ) ] ,
Var ( z ̂ p ) 1 n = 1 N x , y 𝓢 q ̄ ( x , y , z n z p ) 1 ( z p q ̄ ( x , y , z n z p ) ) 2 ,
z p ln n = 1 N x , y 𝓢 P ( q ) = n = 1 N x , y 𝓢 q ̄ z p ( q q ̄ 1 ) 0 .
z p ln n = 1 N x , y 𝓢 P ( q ) n = 1 N x , y 𝓢 q ̄ z p ( q q ̄ 1 )
+ n = 1 N x , y 𝓢 ( 2 q ̄ z p 2 ( q q ̄ 1 ) ( q ̄ z p ) 2 q q ̄ 2 ) ( z p z ̂ p ) 0 ,
z ̂ p ( m + 1 ) = z ̂ p ( m ) n = 1 N x , y 𝓢 ( q ̄ z p ( q q ̄ 1 ) ) n = 1 N x , y 𝓢 ( 2 q ̄ z p 2 ( q q ̄ 1 ) ( q ̄ z p ) 2 q q ̄ 2 ) ,
z ̂ p ( 0 ) = arg max z p n = 1 N x , y 𝓢 ( q μ q ) ( q ̄ μ q ̄ ) n = 1 N x , y 𝓢 ( q μ q ) 2 n = 1 N x , y 𝓢 ( q ̄ μ q ̄ ) 2
P ( q ) = e ( q ̄ + σ b 2 ) ( q ̄ + σ b 2 ) q q ! .
Var ( z ̂ p ) 1 n = 1 N x , y 𝓢 ( z p q ̄ ( x , y , z n z p ) ) 2 q ̄ ( x , y , z n z p ) + σ b 2 ,
z ̂ p ( m + 1 ) = z ̂ p ( m ) n = 1 N x , y 𝓢 ( q ̄ z p ( q q ̄ + σ b 2 1 ) ) n = 1 N x , y 𝓢 ( 2 q ̄ z p 2 ( q q ̄ + σ b 2 1 ) ( q ̄ z p ) 2 q ( q ̄ + σ b 2 ) 2 ) .
arg min z 1 , , z N a b ( 1 n = 1 N x , y , 𝓢 1 q ̄ ( q ̄ z p ) 2 ) d z p ,
1 k ( W ( ρ , z n z p ) ) = OPD ( z p z n ) n i 2 NA 2 ρ 2
+ z p { n s 2 NA 2 ρ 2 n i n s n i 2 NA 2 ρ 2 }
+ t g { n g 2 NA 2 ρ 2 n i n g n i 2 NA 2 ρ 2 }
t g * { n g * 2 NA 2 ρ 2 n i n g * n i 2 NA 2 ρ 2 }
t i * { n i * 2 NA 2 ρ 2 n i n i * n i 2 NA 2 ρ 2 } .
q ̄ ( x , y , z n z p ) z p = 2 k A 2 0 1 sin ( W ( ρ , z n z p ) ) J 0 ( kr NA ρ ) ρ d ρ
· 0 1 cos ( W ( ρ , z n z p ) ) J 0 ( kr NA ρ ) g ( ρ ) ρ d ρ
2 k A 2 0 1 cos ( W ( ρ , z n z p ) ) J 0 ( kr NA ρ ) ρ d ρ ,
· 0 1 sin ( W ( ρ , z n z p ) ) J 0 ( kr NA ρ ) g ( ρ ) ρ d ρ ,
q ̄ ( x , y , z n z p ) z p 2 = 2 k 2 A 2 ( 0 1 cos ( W ( ρ , z n z p ) ) J 0 ( kr NA ρ ) g ( ρ ) ρ d ρ ) 2
2 k 2 A 2 0 1 sin ( W ( ρ , z n z p ) ) J 0 ( kr NA ρ ) ρ d ρ
· 0 1 sin ( W ( ρ , z n z p ) ) J 0 ( kr NA ρ ) g ( ρ ) 2 ρ
+ 2 k 2 A 2 ( 0 1 cos ( W ( ρ , z n z p ) ) J 0 ( kr NA ρ ) g ( ρ ) ρ d ρ ) 2
2 k 2 A 2 0 1 cos ( W ( ρ , z n z p ) ) J 0 ( kr NA ρ ) ρ d ρ
· 0 1 cos ( W ( ρ , z n z p ) ) J 0 ( kr NA ρ ) g ( ρ ) 2 ρ d ρ ,
g ( ρ ) = ( ( 1 n i n s ) n i 2 NA 2 ρ 2 + n s 2 NA 2 ρ 2 ) .

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