OSA's Digital Library

Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editor: Gregory W. Faris
  • Vol. 4, Iss. 13 — Dec. 2, 2009
« Show journal navigation

Automatic deconvolution of 4Pi-microscopy data with arbitrary phase

Giuseppe Vicidomini, Stefan W. Hell, and Andreas Schönle  »View Author Affiliations


Optics Letters, Vol. 34, Issue 22, pp. 3583-3585 (2009)
http://dx.doi.org/10.1364/OL.34.003583


View Full Text Article

Acrobat PDF (246 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We propose a maximum a posteriori-based method that solves an important practical problem in the deconvolution of 4Pi images by simultaneously delivering an estimate of both the object and the unknown phase. The method was tested in simulations and on data from both test samples and biological specimen. It generates object estimates that are free from interference artifacts and reliably recovers arbitrary relative phases. Based on vectorial focusing theory, our theoretical analysis allowed for a simple and efficient implementation of the algorithm. Taking several 4Pi images at different relative phases of the interfering beams is shown to improve the robustness of the approach.

© 2009 Optical Society of America

By using two opposing objective lenses, the family of 4Pi microscopes [1

1. S. Hell and E. H. K. Stelzer, J. Opt. Soc. Am. A 9, 2159 (1992). [CrossRef]

] fundamentally improves the axial resolution of fluorescence microscopes. In the so-called 4Pi type A arrangement, the spherical wavefronts created by focusing the excitation laser beam through both lenses are coherently combined in the focus while emission is collected incoherently through at least one of the lenses and usually focused through a pinhole by the tube lens. Its effective point-spread function (E-PSF) thus depends on the relative phase ϕ of the two beams at the common focus and features several narrow diffraction maxima leading to periodic artifacts that must be removed by image restoration [2

2. M. Nagorni and S. W. Hell, J. Opt. Soc. Am. A 18, 36 (2001). [CrossRef]

]. This makes precise knowledge of the E-PSF heff essential, but ϕ is often not known a priori and has to be estimated in a step preluding the actual estimation of the specimen function f. To this end, different methods were proposed in the literature, but their successful implementation relies on particular hardware arrangements [3

3. S. W. Hell, C. M. Blanca, and J. Bewersdorf, Opt. Lett. 27, 888 (2002). [CrossRef]

, 4

4. C. M. Blanca, J. Bewersdorf, and S. W. Hell, Opt. Commun. 206, 281 (2002). [CrossRef]

] or special abilities of the operator [5

5. D. Baddeley, C. Carl, and C. Cremer, Appl. Opt. 45, 7056 (2006). [CrossRef] [PubMed]

]. Here, we propose to eliminate this extra step of phase estimation from the restoration algorithm by employing a fully automated parametric blind deconvolution (PBD) [6

6. J. Markham and J.-A. Conchello, J. Opt. Soc. Am. A 16, 2377 (1999). [CrossRef]

] method based on a maximum a posteriori (MAP) approach that simultaneously estimates both the phase difference and the specimen function.

The optical transfer function (OTF) of a 4Pi microscope features regions of weak transmission. Hell et al. [3

3. S. W. Hell, C. M. Blanca, and J. Bewersdorf, Opt. Lett. 27, 888 (2002). [CrossRef]

] showed that these critical frequencies are transmitted much less efficiently for destructive than for constructive interference, rendering the deconvolution process more robust against noise in the latter case. It may therefore be advantageous to take a number of different images gk at different phase values φk=ϕ+δk by imposing a known phase shift δk, with k=1,,Nδ and δ1=0. This so called phase-diversity (PD) approach ensures that at least one of the exposures is close to the constructive case; however, by splitting the photons between the individual images, the signal-to-noise ratio (SNR) is reduced. To explore this possibility we use a general multi-image approach for the formulation of the problem in Bayesian terms that thus aims at maximizing the a posteriori probability of observing the specimen f and the phase ϕ, given a series of images g={gk},
P(f,ϕ|g)=P(g|f,ϕ)P(f)P(ϕ)P(g).
(1)
P(g|f,ϕ) is the probability of observing g, while P(ϕ), P(g), and P(f) denote the a priori probability distributions for the phase, for the series of images, and for the specimen, respectively. We assume that, for any voxel n and any image k, the value gk(n) is a realization of an independent Poisson random variable with mean value (Hkf)(n), and therefore
P(g|f,ϕ)=k,npoi[gk(n)|(Hkf)(n)],
(2)
where poi(g|λ)=λgexp(λ)g!, and we introduced the notation (Hkf)(n) for the discretization of the convolution heff(r,φk)f(r). Importantly, Hkf, can be easily calculated for given ϕk by means of fast Fourier transforms (FFTs) with complexity O(NlogN), N being the number of voxels. However, the writing Hk as a convolution operator implies a space-invariant E-PSF and thus is constant phase over the whole image. While this condition is not always met in life-cell imaging, it is a reasonable assumption in many practical cases, in particular when imaging fixed cells. In this case, constant phase can be achieved experimentally, e.g., by using the water-miscible TDE (2,2’-Thidodiethanol) mounting medium [7

7. M. Lang, T. Müller, J. Engelhardt, and S. W. Hell, Opt. Express 15, 2459 (2007). [CrossRef] [PubMed]

] or optical phase compensation.

Next, we deconvolved experimental data acquired with a 4Pi type A module [2

2. M. Nagorni and S. W. Hell, J. Opt. Soc. Am. A 18, 36 (2001). [CrossRef]

] using two opposing objective lenses (HCS PL APO 1001.46 OIL CORR, Leica Microsystem, Germany). Figure 2 shows recovery of the phase for fluorescent beads (100nm Crimson beads, Molecular Probes, Ore., USA). The phase ϕ was varied by linearly changing the optical path length through one of the arms of the interferometric setup. Not surprisingly, our method recovers the phase reliably on this simplest of all possible samples. More importantly we therefore assessed the performance of PBD on images of biological relevance. We therefore applied it to data acquired during an investigation of the Golgi apparatus in Vero cells [7

7. M. Lang, T. Müller, J. Engelhardt, and S. W. Hell, Opt. Express 15, 2459 (2007). [CrossRef] [PubMed]

]. The images show GM 130 protein located in the cis-Golgi network immuno-labeled using Cy3 and embedded in TDE (Fig. 3 ). In this particular example the operator had tried to adjust to constructive interference. Deconvolving with this value leads to artifacts, while application of the PDB method results in significantly improved deconvolved data and estimates the actual phase at ϕ=5.88. We thus demonstrated that the MAP-PBD approach is a suitable tool to estimate both the object function and the relative phase in 4Pi microscopy.

We thank Roman Schmidt for help with the acquisition of the bead data and Marion Lang for providing the Golgi data. We also thank Alexander Egner and Johann Engelhardt for useful discussions. This work has been supported by the German Federal Ministry of Education and Research through the project INVERS.

Fig. 1 (a) Test phantom and (b) example of a simulated image with ϕ=7π8, NA 1.46 (oil), λexc=635nm, λem=680nm, pinhole 0.5AU. (c) Deconvolution with known phase and (d) PBD (phase estimated, ϕE=2.66). (e) DKL(f,fE) and (f) ϕE as functions of ϕ for SI and PD mode. Both graphs are obtained as the average of five different simulated images. For all simulations we fixed the total number of photons k,ngk(n), resulting in a maximum of expected photons of 50 (SNR7:1) and 25 (SNR5:1) for SI and PD images, respectively.
Fig. 2 Bead images (λexc=635nm, λem=680nm, SNR6:1) and object estimate in the case of constructive [(a), (b)] and destructive [(c), (d)] phase. (e) Robust estimation of arbitrary phases ϕ using the SI or PD (Nϕ=2,δ2=π).
Fig. 3 (a) Axial (xz) slice of a 3D stack of a Golgi apparatus (λexc=568nm, λem=605nm, SNR7:1). (b) Deconvolved data without phase estimation (ϕ is set to zero) and (c) with simultaneous phase estimation (estimated phase ϕE=5.88). Intensity profiles at site indicated by arrows (d). The residual sidelobes in the deconvolved data without phase estimation are removed by PBD. Scale bar, 0.5μm.
1.

S. Hell and E. H. K. Stelzer, J. Opt. Soc. Am. A 9, 2159 (1992). [CrossRef]

2.

M. Nagorni and S. W. Hell, J. Opt. Soc. Am. A 18, 36 (2001). [CrossRef]

3.

S. W. Hell, C. M. Blanca, and J. Bewersdorf, Opt. Lett. 27, 888 (2002). [CrossRef]

4.

C. M. Blanca, J. Bewersdorf, and S. W. Hell, Opt. Commun. 206, 281 (2002). [CrossRef]

5.

D. Baddeley, C. Carl, and C. Cremer, Appl. Opt. 45, 7056 (2006). [CrossRef] [PubMed]

6.

J. Markham and J.-A. Conchello, J. Opt. Soc. Am. A 16, 2377 (1999). [CrossRef]

7.

M. Lang, T. Müller, J. Engelhardt, and S. W. Hell, Opt. Express 15, 2459 (2007). [CrossRef] [PubMed]

8.

J. C. Bezdek, R. J. Hathaway, R. E. Howard, C. A. Wilson, and M. P. Windham, J. Optim. Theory Appl. 54, 471 (1987). [CrossRef]

9.

G. Vicidomini, P. Boccacci, A. Diaspro, and M. Bertero, J. Microsc. 234, 47 (2009). [CrossRef] [PubMed]

10.

C. Kelley, Iterative Method for Optimization (SIAM, 1999), Vol. 18.

11.

I. Csiszár, Ann. Stat. 19, 2032 (1991). [CrossRef]

OCIS Codes
(110.0180) Imaging systems : Microscopy
(180.2520) Microscopy : Fluorescence microscopy
(180.6900) Microscopy : Three-dimensional microscopy
(100.1455) Image processing : Blind deconvolution

ToC Category:
Microscopy

History
Original Manuscript: July 2, 2009
Revised Manuscript: October 6, 2009
Manuscript Accepted: October 9, 2009
Published: November 13, 2009

Virtual Issues
Vol. 4, Iss. 13 Virtual Journal for Biomedical Optics

Citation
Giuseppe Vicidomini, Stefan W. Hell, and Andreas Schönle, "Automatic deconvolution of 4Pi-microscopy data with arbitrary phase," Opt. Lett. 34, 3583-3585 (2009)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=ol-34-22-3583


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. S. Hell and E. H. K. Stelzer, J. Opt. Soc. Am. A 9, 2159 (1992). [CrossRef]
  2. M. Nagorni and S. W. Hell, J. Opt. Soc. Am. A 18, 36 (2001). [CrossRef]
  3. S. W. Hell, C. M. Blanca, and J. Bewersdorf, Opt. Lett. 27, 888 (2002). [CrossRef]
  4. C. M. Blanca, J. Bewersdorf, and S. W. Hell, Opt. Commun. 206, 281 (2002). [CrossRef]
  5. D. Baddeley, C. Carl, and C. Cremer, Appl. Opt. 45, 7056 (2006). [CrossRef] [PubMed]
  6. J. Markham and J.-A. Conchello, J. Opt. Soc. Am. A 16, 2377 (1999). [CrossRef]
  7. M. Lang, T. Müller, J. Engelhardt, and S. W. Hell, Opt. Express 15, 2459 (2007). [CrossRef] [PubMed]
  8. J. C. Bezdek, R. J. Hathaway, R. E. Howard, C. A. Wilson, and M. P. Windham, J. Optim. Theory Appl. 54, 471 (1987). [CrossRef]
  9. G. Vicidomini, P. Boccacci, A. Diaspro, and M. Bertero, J. Microsc. 234, 47 (2009). [CrossRef] [PubMed]
  10. C. Kelley, Iterative Method for Optimization (SIAM, 1999), Vol. 18.
  11. I. Csiszár, Ann. Stat. 19, 2032 (1991). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 3
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited