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  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 7, Iss. 1 — Jan. 4, 2012
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Point-spread function engineering to reduce the impact of spherical aberration on 3D computational fluorescence microscopy imaging

Shuai Yuan and Chrysanthe Preza  »View Author Affiliations


Optics Express, Vol. 19, Issue 23, pp. 23298-23314 (2011)
http://dx.doi.org/10.1364/OE.19.023298


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Abstract

Wavefront encoding (WFE) with different cubic phase mask designs was investigated in engineering 3D point-spread functions (PSF) to reduce their sensitivity to depth-induced spherical aberration (SA) which affects computational complexity in 3D microscopy imaging. The sensitivity of WFE-PSFs to defocus and to SA was evaluated as a function of phase mask parameters using mean-square-error metrics to facilitate the selection of mask designs for extended-depth-of-field (EDOF) microscopy and for computational optical sectioning microscopy (COSM). Further studies on pupil phase contribution and simulated WFE-microscope images evaluated the engineered PSFs and demonstrated SA insensitivity over sample depths of 30 μm. Despite its low sensitivity to SA, the successful WFE design for COSM maintains a high sensitivity to defocus as it is desired for optical sectioning.

© 2011 OSA

1. Introduction

Point-spread function (PSF) engineering, achieved by placing a phase mask at the pupil plane of the imaging lens to encode the wavefront emerging from an imaging system, has been implemented successfully to enhance optical system properties [1

1. E. R. Dowski Jr and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34(11), 1859–1866 (1995). [CrossRef] [PubMed]

,2

2. S. R. P. Pavani and R. Piestun, “Three dimensional tracking of fluorescent microparticles using a photon-limited double-helix response system,” Opt. Express 16(26), 22048–22057 (2008). [CrossRef] [PubMed]

]. In extended depth-of-field (EDOF) microscopy, wavefront encoding (WFE) with a cubic phase mask (CPM), designed to reduce PSF sensitivity to defocus, produces an intermediate (encoded) image which is then digitally processed to decode the desired information [3

3. S. C. Tucker and W. T. Cathey, and E. Dowski Jr., “Extended depth of field and aberration control for inexpensive digital microscope systems,” Opt. Express 4(11), 467–474 (1999). [CrossRef] [PubMed]

,4

4. M. R. Arnison, C. J. Cogswell, C. J. R. Sheppard, and P. Török, “Wavefront coding fluorescence microscopy using high aperture lenses,” in Optical imaging and microscopy: techniques and advanced systems, P. Török and F.-J. Kao, eds. (Springer-Verlag, Berlin, 2003), pp. 143–165.

]. In the final processed image, all structures distributed in a three-dimensional (3D) object are visible and well-focused in the 2D images acquired as the microscope is focused at different depths of the object (regardless of the location of the structures in the 3D volume). In this paper, we present a study that explores PSF engineering with WFE using different CPM-based designs with the goal to render the PSF less sensitive to depth-induced spherical aberration (SA).

Computational imaging plays a significant role in the advances achieved in 3D fluorescence microscopy [5

5. P. M. Carlton, J. Boulanger, C. Kervrann, J.-B. Sibarita, J. Salamero, S. Gordon-Messer, D. Bressan, J. E. Haber, S. Haase, L. Shao, L. Winoto, A. Matsuda, P. Kner, S. Uzawa, M. Gustafsson, Z. Kam, D. A. Agard, and J. W. Sedat, “Fast live simultaneous multiwavelength four-dimensional optical microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107(37), 16016–16022 (2010). [CrossRef] [PubMed]

]. Traditional wide-field microscopy has been transformed to quantitative 3D imaging by coupling digital processing to the measured data in order to reduce the impact of aberrations [6

6. J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999). [CrossRef] [PubMed]

,7

7. J.-B. Sibarita, “Deconvolution Microscopy,” in Microscopy Techniques, J. Rietdorf, ed. (Springer Berlin / Heidelberg, 2005), pp. 1288–1291.

] and improve optical sectioning [8

8. D. A. Agard, “Optical sectioning microscopy: cellular architecture in three dimensions,” Annu. Rev. Biophys. Bioeng. 13(1), 191–219 (1984). [CrossRef] [PubMed]

,9

9. J.-A. Conchello and J. W. Lichtman, “Optical sectioning microscopy,” Nat. Methods 2(12), 920–931 (2005). [CrossRef] [PubMed]

]. The widespread use of computational optical sectioning microscopy (COSM) [8

8. D. A. Agard, “Optical sectioning microscopy: cellular architecture in three dimensions,” Annu. Rev. Biophys. Bioeng. 13(1), 191–219 (1984). [CrossRef] [PubMed]

] has motivated the development of new computational methodologies to reduce the impact of depth-induced SA, on the 3D image quality [10

10. C. Preza and J. A. Conchello, “Depth-variant maximum-likelihood restoration for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A 21, 1593–1601 (2004).

12

12. S. Yuan and C. Preza, “3D fluorescence microscopy imaging accounting for depth-varying point-spread functions predicted by a strata interpolation method and a principal component analysis method,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XVIII, (SPIE, 2011), 79040M.

] by accounting for the fact that 3D microscope imaging is inherently depth variant (i.e., as the imaging depth increases within a sample, the 3D PSF changes due in part to a refractive index (RI) mismatch between imaging layers [13

13. S. F. Gibson and F. Lanni, “Experimental Test of an analytical Model of Aberration in an Oil-Immersion Objective Lens Used in 3-dimensional Light Microscopy,” J. Opt. Soc. Am. A 9(1), 154–166 (1992). [CrossRef] [PubMed]

,14

14. P. Török, P. Varga, and G. Nemeth, “Analytical solution of the diffection integrals and interpretation of wave-front distortion when light is focused through a planar interface between materials of mismatched refractive indices,” J. Opt. Soc. Am. A 12(12), 2660–2671 (1995). [CrossRef]

]). When the depth variability is significant, the use of multiple depth-variant (DV) PSFs is necessary in data processing to reduce undesirable computation artifacts [10

10. C. Preza and J. A. Conchello, “Depth-variant maximum-likelihood restoration for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A 21, 1593–1601 (2004).

,15

15. J. G. McNally, C. Preza, J.-A. Conchello, and L. J. Thomas Jr., “Artifacts in computational optical-sectioning microscopy,” J. Opt. Soc. Am. A 11(3), 1056–1067 (1994). [CrossRef] [PubMed]

,16

16. J. W. Shaevitz and D. A. Fletcher, “Enhanced three-dimensional deconvolution microscopy using a measured depth-varying point-spread function,” J. Opt. Soc. Am. A 24(9), 2622–2627 (2007). [CrossRef] [PubMed]

]. As expected, computational complexity increases with the number of 3D DV-PSFs used in the computations [17

17. C. Preza and V. Myneni, “Quantitative depth-variant imaging for fluorescence microscopy using the COSMOS software package,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing Xvii, SPIE 7570 (SPIE, 2010), 757003.

] and thus, reducing the sensitivity of the 3D PSF to SA would reduce computational load and processing time, because fewer DV-PSFs would be used in the computations [12

12. S. Yuan and C. Preza, “3D fluorescence microscopy imaging accounting for depth-varying point-spread functions predicted by a strata interpolation method and a principal component analysis method,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XVIII, (SPIE, 2011), 79040M.

,18

18. C. Preza and J.-A. Conchello, “Depth-variant maximum-likelihood restoration for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A 21(9), 1593–1601 (2004). [CrossRef] [PubMed]

].

Three CPM-based designs [1

1. E. R. Dowski Jr and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34(11), 1859–1866 (1995). [CrossRef] [PubMed]

,22

22. H. Zhao, Y. C. Li, H. J. Feng, Z. H. Xu, and Q. Li, “Cubic sinusoidal phase mask: Another choice to extend the depth of field of incoherent imaging system,” Opt. Laser Technol. 42(4), 561–569 (2010). [CrossRef]

,23

23. G. Carles, A. Carnicer, and S. Bosch, “Phase mask selection in wavefront coding systems: A design approach,” Opt. Lasers Eng. 48(7-8), 779–785 (2010). [CrossRef]

] are investigated in this study to determine: a) a phase mask design that renders the PSF insensitive to both defocus and SA, suitable for EDOF microscopy; and b) another phase mask design that renders the PSF insensitive to SA only but not to defocus, suitable for COSM. Because COSM strives to provide the best 3D image of the underlying object intensity by removing from each optical section contributions due to out-of-focus structures, sensitivity to defocus is a desirable imaging characteristic. To our knowledge, this is the first study that investigates integrating WFE to COSM in order to reduce the impact of SA. WFE with a double-helix phase mask integrated to a widefield microscope to improve resolution has recently been presented as a proof of concept for a WFE-COSM system [26

26. S. Ghosh, G. Grover, R. Piestun, and C. Preza, “Effect of double-helix point-spread functions on 3D imaging in the presence of sphereical aberrations,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XVIII, Proceedings of SPIE, (SPIE, 2011), 790447.

].

The paper is organized as follows: Section 2 reviews image formation theory for a WFE microscope in the presence of SA. Methods used in the evaluation and selection of a suitable phase mask design for EDOF microscopy and WFE-COSM are presented in Section 3. Results from our investigation of WFE-PSF variability as a function of defocus and SA are presented in Section 4, and they are further discussed in Section 5. Simulated WFE-microscope images are also presented in Section 4 to demonstrate the effectiveness of the engineered WFE-PSFs.

2. Theory

2.1 Wavefront encoded PSF (WFE-PSF)

In a WFE microscope system a phase mask with phaseϕ(fx,fy), inserted at the back focal plane of the imaging lens, modifies the phase of the generalized pupil function or defocused amplitude transfer function (ATF) and thereby changes the properties of the system’s PSF. A system suffering defocus and SA is characterized by a PSF that varies with both the depth z=zi at which the microscope is focused and the depth z=zo at which the light point source is located. Thus, the defocused WFE-PSF can be described as:
hzi,zo(x,y)=|F1{H(fx,fy)ej(2π/λ)W(fx,fy;zi,zo)ejϕ(fx,fy)}|2,
(1)
where F1{}denotes a 2-D inverse Fourier transform,H(fx,fy)is the clear circular aperture pupil function, λ is the emission wavelength, and W(fx,fy;zi,zo)is the optical path length error due to defocus and SA as a function of the normalized spatial frequenciesfxand fy. Thus, the phase of the generalized pupil function or defocused WFE-ATF is given by
θ(fx,fy)=2πλW(fx,fy;zi,zo)+ϕ(fx,fy).
(2)
Equation (1) represents a single layer of the 3-D WFE-PSF atz=zi,
hWFE(x,y,zi,zo)=hzi,zo(x,y),
(3)
when a point source is located at z=zo.

2.2 Image formation model in the presence of SA

The intensity in the intermediate (i.e., not processed) image formed by a WFE-system characterized by DV 3D PSFs [Eq. (1)] can be represented by the superposition integral:
g(xi)=OhWFE(xixo,yiy0,zi,zo)s(xo)dxo,
(4)
where xi=(xi,yi,zi)is a point in the image space, xo=(xo,yo,zo)is a point in the object space O ands(xo) is the intensity of the underlying object. Equation (4) can be approximated with a strata-based model that requires only a finite number of 3D DV PSFs in the computation of the image formation model [10

10. C. Preza and J. A. Conchello, “Depth-variant maximum-likelihood restoration for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A 21, 1593–1601 (2004).

].

3. Methods

3.1 Phase mask design patterns

A family of three CPM-based designs were selected from the literature for evaluation in this study: the CPM [1

1. E. R. Dowski Jr and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34(11), 1859–1866 (1995). [CrossRef] [PubMed]

]; the generalized CPM (GCPM) [23

23. G. Carles, A. Carnicer, and S. Bosch, “Phase mask selection in wavefront coding systems: A design approach,” Opt. Lasers Eng. 48(7-8), 779–785 (2010). [CrossRef]

]; and the sinusoidal CPM (SCPM) [22

22. H. Zhao, Y. C. Li, H. J. Feng, Z. H. Xu, and Q. Li, “Cubic sinusoidal phase mask: Another choice to extend the depth of field of incoherent imaging system,” Opt. Laser Technol. 42(4), 561–569 (2010). [CrossRef]

]. The CPM was chosen because it is the first phase mask designed to achieve EDOF through WFE while the recently published GCPM and the SCPM are interesting variations of the CPM, worth investigating, as evident from the mathematical function of the phase ϕ(fx,fy)for each mask: CPM:ϕ(fx,fy)=α(fx3+fy3)GCPM:ϕ(fx,fy)=α(fx3+fy3)+β(fx2fy+fxfy2)SCPM:ϕ(fx,fy)=α(fx3+fy3)+β(sin(ωfx)+sin(ωfy)), where α in radians represents the strength of the CPM [1

1. E. R. Dowski Jr and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34(11), 1859–1866 (1995). [CrossRef] [PubMed]

], and β, also in radians, weighs the contribution of the second term in GCPM and SCPM affecting the deviation of these masks from the CPM. As evident from Eq. (2), if the strength of the CPM is large enough, the phase variation induced by CPM can dominate the phase variation in the pupil plane due to defocus and SA and render the WFE system insensitive to both. In this study, we investigate how the values of the parameters α, β, and ω affect the WFE-PSF’s sensitivity to defocus and SA. Based on reported values of the phase mask parameters in the literature, we chose to investigate the sensitivity of the WFE-PSFs computed by setting the value of α equal to 10, 20, 30, 50, 70, 100, 150 and 200 in all the mask designs; the value of β equal to -α/4, -α/3, -α/2, -α, −2α, −3α and −4α in the GCPM, and to α/3, α/2, α, 2α and 3α in the SCPM; and the value of ω equal to -π/2, -π/4, π/4 and π/2 in the SCPM. Typical and selected patterns of phase mask designs from the study presented in this paper computed within a circular pupil aperture (on a 201 x 201 grid withfxandfy ranging from [-1 1] over the 201 columns/rows) are shown in Fig. 1
Fig. 1 Phase wrapped from –π to π for phase-mask designs computed within a circular pupil aperture. Phase pattern for: (a) CPM (α = 30); (b) SCPM (α = 30, β = α/2 and ω = π/2); (c) GCPM (α = 30 and β = −3α); (d) GCPM selected for EDOF microscopy (α = 150 and β = −3α); and (e) GCPM selected for COSM (α = 50 and β = -α).
.

3.2 Computation of WFE-PSFs

In this study, 3D DV complex amplitude PSFs for conventional widefield microscopy were computed using the Gibson and Lanni PSF model [13

13. S. F. Gibson and F. Lanni, “Experimental Test of an analytical Model of Aberration in an Oil-Immersion Objective Lens Used in 3-dimensional Light Microscopy,” J. Opt. Soc. Am. A 9(1), 154–166 (1992). [CrossRef] [PubMed]

] which is readily available to us via the PSF computation module of our COSM Open Software (COSMOS) package (http://cirl.memphis.edu/COSMOS). The conventional PSFs are computed over a clear circular aperture (CCA) and thus we refer to them as CCA-PSFs when we want to distinguish them from WFE-PSFs based on a phase mask. A total of 13 DV CCA-PSFs were computed on either a 512 x 512 x 300 grid or a 1024 x 1024 x 300 grid with voxels of size 0.1 x 0.1 x 0.1 μm3 assuming that: a) the light point source is located at a different depth (zo=0, 5, 10, 15, 20, 25, 30, 40,…, 90 and 100 μm) in water (RI, nwater = 1.33) below the coverslip; and b) a 60x/1.2 NA oil-immersion objective lens (RI, noil = 1.515) and an emission wavelength λemission = 633 nm are used to image the point source.

3.3 Simulated 3D images

Simulated intermediate images (i.e., images before any digital processing) from a WFE-widefield microscope (Figs. 10
Fig. 10 Simulated unprocessed EDOF images of 3 objects with different amounts of SA, computed using a model for a high-NA WFE-microscope that includes the selected GCPM-EDOF mask design. (a) XZ (top) and XY (bottom) projection views of Object 1. (b) XZ center cut views of the simulated image when the top bead is at depth 0 μm (top), 10 μm (middle) and 20 μm (bottom). (c) XY cut views of simulated images at different distances away from the best focal plane shown by the dotted lines in the XZ image of (b): zi = −5 μm (left); 0 μm (middle); and 5 μm (right).
and 11
Fig. 11 Simulated intermediate images of the 3-bead objects with different amounts of depth-induced SA computed using a model for a high-NA WFE-microscope that includes the selected GCPM-COSM mask design. (a) XZ center cut views of simulated image when the top bead in the object is at depth: 0 μm (top); 10 μm (middle); and 20 μm (bottom). (b) XY cut views of simulated images at different distances away from the best focal plane shown by the dotted lines in the top XZ image of (a): zi = −5 μm (left); 0 μm (middle); and 5 μm (right).
) were computed using a strata-based approach [10

10. C. Preza and J. A. Conchello, “Depth-variant maximum-likelihood restoration for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A 21, 1593–1601 (2004).

] that approximates Eq. (4) and 3 computer-generated test objects that consist of 3 small spheres (1 μm in diameter) centered at different depths. In Object 1, the spheres are at depths 0, 5, and 10 μm [Fig. 10(a)] on a 256 x 256 x 300 grid with voxels of size 0.1 x 0.1 x0.1 μm3. The origin of the xoyo plane is at the center of the grid, and the origin along the zo axis is placed at the 100th xoyo plane. The (xo, yo) coordinates of the three spheres are: (−5 μm, 5 μm), (0 μm, 0 μm), and (5 μm, −5 μm), from top to bottom. Objects 2 and 3 are similar to Object 1 except that the depths of all the spheres are increased by 10 μm and 20 μm, respectively, in order to allow simulation of images with larger amounts of SA. The RI of the 3 objects is assumed to be equal to 1.33, while the immersion medium of the lens is oil (i.e., RI = 1.515).

In order to simulate the image of Object 1 using the strata model [10

10. C. Preza and J. A. Conchello, “Depth-variant maximum-likelihood restoration for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A 21, 1593–1601 (2004).

], 3 DV WFE-PSFs were computed as described in Section 3.2 assuming the point source is located at depths zo= 0, 5, and 10 μm, while PSFs associated with each stratum were approximated using a linear interpolation method, and two PSFs computed at the depths that form the boundaries of a stratum. Additional DV WFE-PSFs were computed at depths zo= 15, 20, 25, and 30 μm to generate the simulated images for Objects 2 and 3.

3.4 Metrics for phase mask design evaluation and selection

An appropriate metric for the design evaluation and selection is PSF variability as a function of defocus and SA. To compare the performance of the phase mask designs and select the design that best met our imaging goals we investigated the sensitivity of WFE-PSFs (obtained for different phase mask designs by varying the mask parameters) to both defocus and SA using 2 metrics (Table 1

Table 1. Equations for Evaluation Metrics

table-icon
View This Table
). To quantify sensitivity to defocus, we proposed a normalized mean square error (NMSE) that measures the difference between the XY layers of a 3D WFE-PSF given by Eq. (5), where hZk,zo(x,y) represents a layer atzi=Zk and hZc,zo(x,y) represents the layer with the best focused 2D PSF at zi=Zc. For a 3D WFE-PSF without SA (zo=0μm) the best focused layer (atzi=0) is in the center of the 3D PSF volume. As previously established, in the presence of SA the best focus shifts along the optical axis [19

19. G. Saavedra, I. Escobar, R. Martinez-Cuenca, E. Sanchez-Ortiga, and M. Martínez-Corral, “Reduction of spherical-aberration impact in microscopy by wavefront coding,” Opt. Express 17(16), 13810–13818 (2009). [CrossRef] [PubMed]

], and thus a newzi=Zc is determined from the conventional PSF. In general, a small NMSE2D value over a long range of zi indicates that the XY layers of the PSF are similar over the range, (i.e., the PSF is insensitive to defocus) and thus good EDOF is achieved in the WFE microscope.

Similarly, sensitivity to SA was quantified with another NMSE which measures the difference between 3D DV WFE-PSFs, described by Eq. (6) in Table 1. In Eq. (6), hd(x,y,z) is the 3D WFE-PSF with SA due to a point source located at depth zo=dbelow the coverslip in the presence of a RI mismatch between imaging layers, and h0(x,y,z)is the 3D WFE-PSF without any SA (i.e., the point source is located at depth zo=0 μm below the coverslip at the interface between the sample and the lens’ immersion medium).

In all the WFE-PSF cases that resulted from varying the mask parameters as described in Section 3.1, the NMSE2D and NMSE3D were computed and plotted as a function of defocus and point source location, respectively, in order to investigate the sensitivity of the PSF to defocus and SA. The NMSE analysis was performed using WFE-PSFs on a 512 x 512 grid. PSFs computed on a larger grid were cropped to 512 x 512. Although cropping of the WFE-PSF can affect the values of NMSE2D and NMSE3D, we found that the effect was minimal and the results did not change significantly.

3.5 Merit function for phase mask parameter selection suitable for COSM

For COSM it is necessary to find a phase mask design that reduces the impact of SA without simultaneously extending the DOF of the microscope at the same time. Towards this end, a merit function that combines the two NMSE metrics in Table 1 in a way that reflects this goal was used for the design selection:
R=NMSE¯2D/NMSE¯3D,
(7)
where NMSE¯2D and NMSE¯3D are the average values of the NMSE2D and the NMSE3D, respectively, over the investigation range. In this study, NMSE¯2Dwas averaged over the range zi= −5 μm to 5 μm while NMSE¯3Dwas averaged over the range zo=0 to 30 μm. For COSM, a large R value is preferred which can be achieved by increasing the NMSE2D value, (i.e. by increasing the sensitivity to defocus) while decreasing the value of the NMSE3D, (i.e. by lowering the sensitivity to SA). For all design parameter sets investigated (Section 3.1), Eq. (7) was calculated and the parameter set that yielded the largest Rvalue was selected for COSM for each mask type.

3.6 Computation of phase due to defocus and SA from 2D CCA-ATFs

As noted earlier, if the phase variation due to the mask is large enough, it can dominate the 2D ATF phase variation due to defocus and SA [first term in Eq. (2)] and render the WFE microscope insensitive to both aberrations as desirable for EDOF microscopy. For COSM, it is desirable that the mask’s phase variation dominates only the phase variation due to SA. To confirm that the selected phase mask designs for the two microscopy modalities met these design selection goals, their phase values were compared to the phase variation (due to defocus) and SA of the 2D CCA-ATF. CCA-ATFs were obtained by computing the 2D Fourier transform of each layer (at z=zi) of the conventional complex-amplitude 3D PSFs with different amounts of SA.

The phase term due to defocus was obtained from the 2D CCA-ATFs computed from a PSF without SA. The CCA-ATF phase term due to SA was isolated from the CCA-ATF phase term due to defocus by computing the phase difference between CCA-ATFs with SA and CCA-ATFs without SA at the same defocus distance zi.

3.7 Computation of 2D Modulation Transfer Function (MTF)

4. Results

4.1 Effect of mask parameter α on the sensitivity of the CPM-PSF to defocus and SA

Results from studying the effect of parameter α on the variability of the CPM-PSF as a function of defocus and SA are summarized in Fig. 3
Fig. 3 NMSE plots for CPM-PSFs for different α values reported in panel (c). NMSE2D values plotted vs. defocus reported as the distance from the designed best focal plane (FP), for PSFs: (a) without SA (zo = 0 μm); and (b) with SA (zo = 20 μm). (c) NMSE3D computed for PSFs with increasing amount of SA, plotted vs. zo. A semi-log scale is used for plotting the NMSE2D to facilitate visualization of the small differences between the curves especially near the minimum values of the curves.
. The NMSE2D computed for CPM PSFs without SA (zo = 0 μm) and with SA (zo = 20 μm) are plotted as a function of defocus (i.e., the distance from the designed best focal plane [FP]) in Fig. 3(a) and Fig. 3(b), respectively, for different α values. Overall, the values of the NMSE2D decrease as the value of α increases indicating that better EDOF is achieved. This result is consistent with a prior reported result [1

1. E. R. Dowski Jr and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34(11), 1859–1866 (1995). [CrossRef] [PubMed]

]. Figure 3(c) plots the NMSE3D computed using PSFs with increasing amounts of SA (denoted by the increasing Z location of the point source which is equivalent to zo) for different α values. It is evident that using a larger value for α in the CPM pattern renders the PSF less sensitive to both defocus and SA.

4.2 Effect of mask parameters on the sensitivity of the GCPM-PSF to defocus and SA

The effects of the parameters α and β in the GCPM function on the WFE-PSF were investigated using 56 GCPM designs that resulted from the α-β combinations described in Section 3.1. Some of the results in this study are summarized in Fig. 4
Fig. 4 NMSE plots for GCPM-PSFs for different α values reported in panel (a). NMSE2D computed for PSFs without SA and plotted vs. defocus: (a) β = -α/3; (b) β = -α; and (c) β =  3α. NMSE3D computed for PSFs with increasing amount of SA and plotted vs. zo: (d) for β = −3α and different values of α; and (e) for α = 150 and different values of β. A semi-log scale is used for plotting the NMSE2D to facilitate visualization of the small differences between the curves.
. The NMSE2D, computed for GCPM PSFs without SA (zo = 0 μm), is plotted as a function of defocus for different α values [listed in Fig. 4(a)] and β values. As shown, when β = -α, the change in the α value has little effect on the NMSE2D [Fig. 4(b)] indicating that the resulting 3D GCPM PSFs are similar in this case. When β = −3α [Fig. 4(c)], the NMSE2D plots show greater variability than the one observed when β = -α/3 [(Fig. 4(a)], and they also show a similar trend as the one observed for the CPM design [Fig. 3(a)]. Moreover in this case, the NMSE2D is lower for most α values than in other β cases, indicating a better achieved EDOF. This finding is consistent with prior reported results [23

23. G. Carles, A. Carnicer, and S. Bosch, “Phase mask selection in wavefront coding systems: A design approach,” Opt. Lasers Eng. 48(7-8), 779–785 (2010). [CrossRef]

]. The smallest NMSE2D is achieved at α = 150 for β = −3α

among all conditions investigated in this study. Similar NMSE2D plots (not shown) were obtained for PSFs with SA (for zo = 20 μm).

Figure 4(d) plots the NMSE3D vs. zo, computed using PSFs with increasing amounts of SA obtained for different values of α when β = −3α. Similarly, Fig. 4(e) plots the computed NMSE3D for different values of β when α = 150. Among all conditions investigated in our study, the smallest NMSE3D is achieved for α = 150 when β = −3α.

4.3 Effect of mask parameters on the sensitivity of the SCPM-PSF to defocus and SA

4.4 Phase mask design selection for EDOF microscopy

For EDOF microscopy the WFE-PSF is engineered to be insensitive to both defocus and SA. Towards this end, we selected phase mask parameters for each mask type based on the studies presented in Sections 4.1-4.3. Naturally, it is desirable to choose the parameters that yield the smallest NMSE2D values over defocus as this supports the WFE-PSF’s insensitivity to defocus. In all the cases investigated for the CPM, GCPM and SCPM, these parameters also yielded the smallest NMSE3D.

Figure 6
Fig. 6 Comparison of NMSE plots for WFE-PSFs engineered with selected masks designs suitable for EDOF microscopy using a CPM (α = 200), a GCPM (α = 150 and β = −3α) and a SCPM (α = 150, β = α/3 and ω = -π/2). NMSE2D computed from PSFs without SA (a) and with SA when zo = 20 μm (b), plotted using a semi-log scale vs. defocus. (c) NMSE3D computed for PSFs with increasing amount of SA plotted vs. zo.
compares the sensitivity to defocus and SA of WFE-PSFs engineered with the 3 selected CPM-based masks. The NMSE2D computed from PSFs without SA (zo = 0 μm) and with SA (zo = 20 μm) for the 3 selected CPM-based designs is plotted as a function of defocus in Fig. 6(a) and Fig. 6(b), respectively. Figure 6(c) plots NMSE3D computed using PSFs with increasing amounts of SA vs. zo for the 3 CPM-based designs. Among the 3 designs, the selected SCPM design (with α = 150, β = α/3 and ω = -π/2) achieves the best NMSE2D performance, while the selected GCPM design (α = 150 and β = −3α) achieves the best NMSE3D performance and an acceptable NMSE2D performance. Thus, from this comparison study, the GCPM design is selected for EDOF microscopy in the presence of SA.

4.5 Phase mask design selection for COSM

Unlike EDOF microscopy, in COSM there is a conflict between optical sectioning and the EDOF characteristic achieved with CPM-based WFE. For COSM the WFE-PSF is engineered to be insensitive to SA without losing its optical sectioning ability (i.e. sensitivity to defocus). Towards this end, we used the merit function R [Eq. (7)] to select the best phase mask designparameters from the studies presented in Sections 4.1-4.3. For comparison purposes, the sensitivity of the CCA-PSF to defocus and to SA is compared to results obtained for the WFE-PSFs that yielded the largest R value among all the designs investigated (Fig. 7
Fig. 7 Comparison of NMSE plots for WFE-PSFs engineered with selected masks designs suitable for COSM using a CPM (α = 50), a GCPM (α = 50 and β = -α) and a SCPM (α = 30, β = α/2 and ω = -π/4). NMSE2D computed from PSFs without SA (a) and with SA when zo = 20 μm (b), plotted vs. defocus. (c) NMSE3D computed for PSFs with increasing amount of SA plotted vs. zo. (d) R plotted as a function of α. The straight dashed line marks the CCA R value.
). The NMSE2D computed from PSFs without SA (zo = 0 μm) and with SA (zo = 20 μm) is plotted as a function of defocus in Fig. 7(a) and Fig. 7(b), respectively. Figure 7(c) plots NMSE3D computed using PSFs with increasing amounts of SA vs. zo for the 3 CPM-based designs. As it is evident, the CCA-PSF has the largest sensitivity to both defocus and SA among the compared PSFs [Figs. 7(a-c)]. This is also confirmed by comparing the resulting peak R values in Fig. 7(d), whereR is plotted as a function of α. The R value (=1.11) for the CCA-PSF is much less than all three peak R values for the selected designs:R= 2.03, 1.70, and 1.67 for the GCPM, the CPM, and the SCPM designs, correspondingly [Fig. 7(d)].

It is interesting to note that for some parameters, the CPM and SCPM designs yield an R value smaller than the value for CCA [Fig. 7(d)]. This demonstrates that the phase mask must be properly designed to ensure that the use of WFE in COSM is beneficial. The results in Fig. 7 show that the selected GCPM design (α = 50 and β = -α) achieves the best performance for both NMSE2D and NMSE3D, and has the largest Rvalue among the 3 CPM-based designs investigated in this study. Therefore, the selected GCPM design was chosen for COSM in the presence of SA.

4.6 Comparison of selected GCPM to the CCA-ATF phase

To further investigate the degree to which selected GCPM designs for the two imaging modalities met the design selection goals, we compared their phases to the phase terms due to defocus and SA of the conventional defocused 2D ATF (or CCA-ATF), computed at different imaging conditions as discussed in Section 3.6. Figure 8
Fig. 8 Comparison of the GCPM phase to CCA-ATF phase changes due to defocus and SA. Phase profile vs. normalized frequency,fx, for fy=0 of the CCA-ATF phase due to: (a) defocus only, for different defocus values; (b) SA only, for different amounts of SA. (c) Comparison of the phase change due to: SA (zi = 0 μm, zo = 30 μm); defocus (zi = −15 μm, zo = 0 μm), the selected GCPM-EDOF phase mask (α = 150, β = −3α), and the selected GCPM-COSM phase mask (α = 50, β = -α). (d) Derivative with respect to fx of the phase functions shown in (c).
summarizes the CCA-ATF phase analysis results. Figure 8(a) plots profiles of the ATF phase due to defocus only, as a function of the normalized spatial frequency in the back focal plane of the objective lens for different amounts of defocus. Figure 8(b) compares profiles from the ATF phase due to SA only, for different amounts of SA, demonstrating the increase in the phase variation over the back focal plane and the investigated depth range. The phase due to SA was found to be approximately linear with depth.

Figure 8(c) compares profiles of the CCA-ATF phase due to different terms: defocus (zi=15μm,zo=0μm), SA (zi=0μm,zo=30μm), the selected GCPM-EDOF phase mask (α = 150, β = −3α), and the selected GCPM-COSM phase mask (α = 50, β = -α). As evident, the selected GCPM-COSM mask has a phase variation which dominates the phase variations due to the SA term only for investigated depths not exceeding 30 μm [Figs. 8(b) and (c)] but it does not dominate the phase due to the defocus term. The latter is important for COSM since optical sectioning requires sensitivity to defocus. On the other hand, the GCPM-EDOF has absolute phase values that dominate all the other phase terms. In addition and as is evident in Fig. 8(d), where the derivatives with respect tofx of the phase functions shown inFig. 8(c) are compared, the phase variation due to the selected GCPM-EDOF mask dominates those from both the defocus and SA phase terms as is desirable for this imaging modality.

4.7 WFE-PSFs with selected designs

WFE-PSFs with different amounts of SA computed with the selected phase mask designs for EDOF and COSM, respectively, are shown in the different rows of Fig. 9
Fig. 9 XY and XZ cut-view images through the center of 3D WFE-PSFs for a 60x/1.2NA oil-immersion lens computed without SA for zo = 0 μm (top row) and with SA for zo = 10 μm (second row), and for zo = 20 μm (third row) using different types of mask designs: (a) CCA, i.e. a conventional PSF suitable for COSM; (b) GCPM (α = 50, β = -α) suitable for COSM; and (c) GCPM (α = 150, β = −3α) suitable for EDOF microscopy.
(the SA term is equal to zero in the first row and it increases in rows 2 and 3 of the figure). Figure 9(a) shows XZ cut-view images of the conventional widefield PSF which is equivalent to a WFE-PSF with CCA. XY (left) and XZ (right) cut-view images of the GCPM-PSF selected for COSM [Fig. 9(b)] and of the GCPM-PSF selected for EDOF microscopy [Fig. 9(c)], respectively. As depicted in Fig. 9, both engineered GCPM-PSFs show reasonable invariance to SA and features with respect to defocus sensitivity consistent with our design selection goals. Images of the GCPM-PSF engineered for COSM show desirable changes with defocus, while the corresponding images of the GCPM-PSF engineered for EDOF show an EDOF over the observed 30-μm z range.

4.8 Simulated 3D intermediate images from EDOF microscopy

To evaluate the effect of the selected GCPM on the WFE microscope, simulated intermediate images from EDOF microscopy were generated using the GCPM-PSF in Fig. 9(c) and the three-sphere test objects described in Section 3.3 [Fig. 10(a)]. XZ views of the simulated images for different amounts of SA, achieved by placing the spheres at deeper depths, demonstrate qualitatively the EDOF and SA invariance characteristics achieved with the selected GCPM design and WFE [Fig. 10(b)]. This is also demonstrated by the XY cut-view images from different Z planes (shown by the dotted lines in [Fig. 10(b)] of the 3D image with different amounts of SA [Fig. 10(c)]. This result suggests that even in the presence of SA, with the use of WFE and our selected GCPM, the single 2D-PSF deconvolution approach used in EDOF microscopy [4

4. M. R. Arnison, C. J. Cogswell, C. J. R. Sheppard, and P. Török, “Wavefront coding fluorescence microscopy using high aperture lenses,” in Optical imaging and microscopy: techniques and advanced systems, P. Török and F.-J. Kao, eds. (Springer-Verlag, Berlin, 2003), pp. 143–165.

] may still be adequate for processing these images in order to obtain the final EDOF microscopy images without any processing artifacts.

4.9 Simulated 3D intermediate images from WFE-COSM

To evaluate the effect of the GCPM selected for COSM on the WFE microscope, simulated images from widefield microscopy were generated using the GCPM-PSF in Fig. 9(b) and the three-sphere test objects described in Section 3.3. XZ views from 3 simulated images with different amounts of SA shown in Fig. 11(a) appear very similar, confirming that SA-invariance is achieved with the selected GCPM design and WFE. This is also demonstrated by the XY cut-view images from different planes of the 3D image with different amounts of SA shown in each column of Fig. 11(b). In addition, the resulting WFE-microscope images appear to change with defocus as evidenced both by the intensity distribution of the XZ images and by the change in the appearance of the three beads in the XY images in each row of Fig. 11(b).

Overall results in Fig. 11 show that the use of the selected GCPM renders the WFE microscope less sensitive to SA without making it insensitive to defocus. This result suggests that using a DV stratum-based approach with a single stratum and two 3D WFE-PSFs (one atzo=0μmand one at the largest depth within the sample) may be adequate for processing these intermediate 3D images to obtain the final 3D COSM images without processing artifacts.

5. Discussion

Our results show that both parameters α and β have a large impact on the achieved EDOF in the GCPM and SCPM designs. We found that the interaction between α and β can at some times neutralize their impact on the achieved EDOF [Figs. 4(a-c)]. In the GCPM case, we showed that when β = -α, changing the value of α did not affect the achieved EDOF (Fig. 4b). Our study confirms that increasing the value of the parameter α in the CPM-based designs improves the achieved EDOF [Figs. 3(a), 4(c), and 5(a)] of the imaging systems as previously reported for the CPM [1

1. E. R. Dowski Jr and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34(11), 1859–1866 (1995). [CrossRef] [PubMed]

]. Our results also suggest that a large α value should be chosen for EDOF applications.

However, large α values can cause an increase in sampling and manufacturing error. This is because, with a large α value, the mask’s wrapped phase values increase rapidly between 0 and 2π [Fig. 1(d)] and more samples are required in discretizing the phase function in order to represent the phase values accurately in this interval. In simulated experiments with a fixed grid size, it was evident that increasing the α value reduced the number of pixels available to represent one phase wrapping which could result in phase inaccuracies due to sampling error. Additionally, if the error in manufacturing the mask is proportional to the original unwrapped CPM phase term, then it is possible that the error would increase with an increasing α value.

Furthermore, the intensity of the CPM-based WFE-PSFs spreads over a larger area in the xy plane [Figs. 2(b) and 9(c)] compared to the CCA-PSF [Fig. 2(a)]. The large extend of the WFE-PSF in the xy plane imposes new constraints on image dimensions used for the finite-discrete domain implementation of the forward and inverse imaging problems (due to the crosstalk between multiple fluorescence points in the underlying specimen), which increase computational complexity. The tradeoff between achieved EDOF and computational complexity can be controlled by the chosen α value.

Two well-known issues of CPM-based EDOF microscopy are: (1) image shifting; and (2) low SNR in the final image. We investigated both these issues for several selected designs proposed in this study. As previously established, a lateral shift of the CPM-PSF peak at out-of-focus planes (evident more clearly in Fig. 12(a)
Fig. 12 The effect of α on the CPM-PSF (a & b), on the GCPM-PSF (c & d) and on frequency content (e). PZ cut views (where P is the diagonal line along x = y) of the WFE-PSF using: (a) CPM with α = 30; (b) CPM with α = 150; (c) GCPM with α = 30 and β = −3α and (d) GCPM with α = 150, and β = −3α. We note that the CPM-PSF image in (a) looks qualitatively similar to the image in Fig. 4.8(a) of [21]. (e) Normalized MTF intensity profiles vs.fp(the frequency along the diagonal line fx=fy). MTFs shown are without SA (zo = 0 μm) and with SA (zo = 20 μm) using different masks: (i) CCA; (ii) GCPM with α = 150 and β = −3α; and (iii) GCPM with α = 30 and β = −3α.
than it is in Fig. 2(b) due to the PZ plane view used instead of the XZ plane), introduces artifacts in the final reconstructed image where specimen features away from best focus appear laterally shifted. Our results summarized in Fig. 12 support that these artifacts could be greatly reduced or even eliminated using different mask parameters or designs. For example, a larger α reduces the lateral shift in the CPM-PSF [Fig. 12(b)] while in general, GCPM PSFs do not exhibit a lateral shift within their EDOF range [Figs. 12(c) and (d)].

The study presented here is a proof of concept and it provides a methodology for engineering PSFs with reduced variability due to SA using computed theoretical PSFs. Although the CCA-PSF model used in this investigation includes apodisation effects, it does not include vectorial effects [27

27. 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(1-3), 55–63 (2003). [CrossRef]

] which can affect the appearance of the WFE-PSF and consequently the EDOF range for a high NA system predicted by our methodology [4

4. M. R. Arnison, C. J. Cogswell, C. J. R. Sheppard, and P. Török, “Wavefront coding fluorescence microscopy using high aperture lenses,” in Optical imaging and microscopy: techniques and advanced systems, P. Török and F.-J. Kao, eds. (Springer-Verlag, Berlin, 2003), pp. 143–165.

]. Similarly, the predicted depth within the sample over which the WFE-PSF shows reduced sensitivity to SA could be affected by inaccuracies in the PSF model. Based on comparison studies between experimental and theoretical CPM-PSFs [21

21. M. R. Arnison, “Phase control and measurement in digital microscopy,” Ph.D. dissertation (University of Sydney, Sydney, 2004).

], it is expected that the presented methodology could provide more accurate predictions if it is applied to WFE-PSFs computed using a model that includes vectorial effects [4

4. M. R. Arnison, C. J. Cogswell, C. J. R. Sheppard, and P. Török, “Wavefront coding fluorescence microscopy using high aperture lenses,” in Optical imaging and microscopy: techniques and advanced systems, P. Török and F.-J. Kao, eds. (Springer-Verlag, Berlin, 2003), pp. 143–165.

,27

27. 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(1-3), 55–63 (2003). [CrossRef]

] instead of the CCA-PSF model used in this study.

6. Conclusion

In this study, we evaluated WFE with CPM-based designs with a goal to engineer 3D PSFs that are less sensitive to depth-induced SA (due to a RI mismatch in the imaging layers) than the conventional widefield microscopy PSF for a high-NA lens. Three existing CPM-based designs were investigated using two MSE metrics that quantify the PSF’s sensitivity to defocus and SA. Phase mask designs were selected using these metrics and evaluated for two microscopy applications. Our results show that among the evaluated phase mask designs, the generalized cubic phase mask (GCPM) provided more suitable designs that met our selection goals for EDOF microscopy and for COSM. Achieved EDOF and SA invariance were also evaluated with a comparison of the ATF phase variation (due to SA and defocus) in the pupil plane to the phase variation due to the GCPMs selected for each of the two microscopy applications. The desired imaging characteristics were also confirmed by comparing simulated intermediate images with different amounts of SA and defocus from the resulting WFE-microscope in each case, and were computed using several simple test objects. Our results show that both GCPM designs selected by this study render the WFE-microscope less susceptible to SA for sample depths of 30 μm (when the RI mismatch is between water and oil) and that the GCPM selected for COSM does not render the microscope insensitive to defocus as desired for optical sectioning. Additional studies are currently underway to investigate image restoration of the final 3D images from the intermediate WFE images using the WFE-PSFs. These studies will further confirm the benefits of the engineered PSFs proposed by this study.

Acknowledgement

This work was supported by the National Science Foundation (NSF CAREER award DBI-0844682 and NSF IDBR award DBI-0852847, PI: C. Preza) and the University of Memphis.

References and Links

1.

E. R. Dowski Jr and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34(11), 1859–1866 (1995). [CrossRef] [PubMed]

2.

S. R. P. Pavani and R. Piestun, “Three dimensional tracking of fluorescent microparticles using a photon-limited double-helix response system,” Opt. Express 16(26), 22048–22057 (2008). [CrossRef] [PubMed]

3.

S. C. Tucker and W. T. Cathey, and E. Dowski Jr., “Extended depth of field and aberration control for inexpensive digital microscope systems,” Opt. Express 4(11), 467–474 (1999). [CrossRef] [PubMed]

4.

M. R. Arnison, C. J. Cogswell, C. J. R. Sheppard, and P. Török, “Wavefront coding fluorescence microscopy using high aperture lenses,” in Optical imaging and microscopy: techniques and advanced systems, P. Török and F.-J. Kao, eds. (Springer-Verlag, Berlin, 2003), pp. 143–165.

5.

P. M. Carlton, J. Boulanger, C. Kervrann, J.-B. Sibarita, J. Salamero, S. Gordon-Messer, D. Bressan, J. E. Haber, S. Haase, L. Shao, L. Winoto, A. Matsuda, P. Kner, S. Uzawa, M. Gustafsson, Z. Kam, D. A. Agard, and J. W. Sedat, “Fast live simultaneous multiwavelength four-dimensional optical microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107(37), 16016–16022 (2010). [CrossRef] [PubMed]

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J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999). [CrossRef] [PubMed]

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J.-B. Sibarita, “Deconvolution Microscopy,” in Microscopy Techniques, J. Rietdorf, ed. (Springer Berlin / Heidelberg, 2005), pp. 1288–1291.

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D. A. Agard, “Optical sectioning microscopy: cellular architecture in three dimensions,” Annu. Rev. Biophys. Bioeng. 13(1), 191–219 (1984). [CrossRef] [PubMed]

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J.-A. Conchello and J. W. Lichtman, “Optical sectioning microscopy,” Nat. Methods 2(12), 920–931 (2005). [CrossRef] [PubMed]

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C. Preza and J. A. Conchello, “Depth-variant maximum-likelihood restoration for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A 21, 1593–1601 (2004).

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Z. Kam, P. Kner, D. A. Agard, and J. W. Sedat, “Modelling the application of adaptive optics to wide-field microscope live imaging,” J. Microsc. 226(1), 33–42 (2007). [CrossRef] [PubMed]

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S. Yuan and C. Preza, “3D fluorescence microscopy imaging accounting for depth-varying point-spread functions predicted by a strata interpolation method and a principal component analysis method,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XVIII, (SPIE, 2011), 79040M.

13.

S. F. Gibson and F. Lanni, “Experimental Test of an analytical Model of Aberration in an Oil-Immersion Objective Lens Used in 3-dimensional Light Microscopy,” J. Opt. Soc. Am. A 9(1), 154–166 (1992). [CrossRef] [PubMed]

14.

P. Török, P. Varga, and G. Nemeth, “Analytical solution of the diffection integrals and interpretation of wave-front distortion when light is focused through a planar interface between materials of mismatched refractive indices,” J. Opt. Soc. Am. A 12(12), 2660–2671 (1995). [CrossRef]

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J. G. McNally, C. Preza, J.-A. Conchello, and L. J. Thomas Jr., “Artifacts in computational optical-sectioning microscopy,” J. Opt. Soc. Am. A 11(3), 1056–1067 (1994). [CrossRef] [PubMed]

16.

J. W. Shaevitz and D. A. Fletcher, “Enhanced three-dimensional deconvolution microscopy using a measured depth-varying point-spread function,” J. Opt. Soc. Am. A 24(9), 2622–2627 (2007). [CrossRef] [PubMed]

17.

C. Preza and V. Myneni, “Quantitative depth-variant imaging for fluorescence microscopy using the COSMOS software package,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing Xvii, SPIE 7570 (SPIE, 2010), 757003.

18.

C. Preza and J.-A. Conchello, “Depth-variant maximum-likelihood restoration for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A 21(9), 1593–1601 (2004). [CrossRef] [PubMed]

19.

G. Saavedra, I. Escobar, R. Martinez-Cuenca, E. Sanchez-Ortiga, and M. Martínez-Corral, “Reduction of spherical-aberration impact in microscopy by wavefront coding,” Opt. Express 17(16), 13810–13818 (2009). [CrossRef] [PubMed]

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S. Mezouari and A. R. Harvey, “Phase pupil functions for reduction of defocus and spherical aberrations,” Opt. Lett. 28(10), 771–773 (2003). [CrossRef] [PubMed]

21.

M. R. Arnison, “Phase control and measurement in digital microscopy,” Ph.D. dissertation (University of Sydney, Sydney, 2004).

22.

H. Zhao, Y. C. Li, H. J. Feng, Z. H. Xu, and Q. Li, “Cubic sinusoidal phase mask: Another choice to extend the depth of field of incoherent imaging system,” Opt. Laser Technol. 42(4), 561–569 (2010). [CrossRef]

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S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “High-resolution imaging using integrated optical systems,” Int. J. Imaging Syst. Technol. 14(2), 67–74 (2004). [CrossRef]

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S. S. Sherif, W. T. Cathey, and E. R. Dowski, “Phase plate to extend the depth of field of incoherent hybrid imaging systems,” Appl. Opt. 43(13), 2709–2721 (2004). [CrossRef] [PubMed]

26.

S. Ghosh, G. Grover, R. Piestun, and C. Preza, “Effect of double-helix point-spread functions on 3D imaging in the presence of sphereical aberrations,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XVIII, Proceedings of SPIE, (SPIE, 2011), 790447.

27.

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(1-3), 55–63 (2003). [CrossRef]

OCIS Codes
(180.6900) Microscopy : Three-dimensional microscopy
(350.4600) Other areas of optics : Optical engineering
(110.7348) Imaging systems : Wavefront encoding

ToC Category:
Microscopy

History
Original Manuscript: August 15, 2011
Revised Manuscript: October 4, 2011
Manuscript Accepted: October 13, 2011
Published: November 1, 2011

Virtual Issues
Vol. 7, Iss. 1 Virtual Journal for Biomedical Optics

Citation
Shuai Yuan and Chrysanthe Preza, "Point-spread function engineering to reduce the impact of spherical aberration on 3D computational fluorescence microscopy imaging," Opt. Express 19, 23298-23314 (2011)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-19-23-23298


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References

  1. E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt.34(11), 1859–1866 (1995). [CrossRef] [PubMed]
  2. S. R. P. Pavani and R. Piestun, “Three dimensional tracking of fluorescent microparticles using a photon-limited double-helix response system,” Opt. Express16(26), 22048–22057 (2008). [CrossRef] [PubMed]
  3. S. C. Tucker and W. T. Cathey, and E. Dowski., “Extended depth of field and aberration control for inexpensive digital microscope systems,” Opt. Express4(11), 467–474 (1999). [CrossRef] [PubMed]
  4. M. R. Arnison, C. J. Cogswell, C. J. R. Sheppard, and P. Török, “Wavefront coding fluorescence microscopy using high aperture lenses,” in Optical imaging and microscopy: techniques and advanced systems, P. Török and F.-J. Kao, eds. (Springer-Verlag, Berlin, 2003), pp. 143–165.
  5. P. M. Carlton, J. Boulanger, C. Kervrann, J.-B. Sibarita, J. Salamero, S. Gordon-Messer, D. Bressan, J. E. Haber, S. Haase, L. Shao, L. Winoto, A. Matsuda, P. Kner, S. Uzawa, M. Gustafsson, Z. Kam, D. A. Agard, and J. W. Sedat, “Fast live simultaneous multiwavelength four-dimensional optical microscopy,” Proc. Natl. Acad. Sci. U.S.A.107(37), 16016–16022 (2010). [CrossRef] [PubMed]
  6. J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods19(3), 373–385 (1999). [CrossRef] [PubMed]
  7. J.-B. Sibarita, “Deconvolution Microscopy,” in Microscopy Techniques, J. Rietdorf, ed. (Springer Berlin / Heidelberg, 2005), pp. 1288–1291.
  8. D. A. Agard, “Optical sectioning microscopy: cellular architecture in three dimensions,” Annu. Rev. Biophys. Bioeng.13(1), 191–219 (1984). [CrossRef] [PubMed]
  9. J.-A. Conchello and J. W. Lichtman, “Optical sectioning microscopy,” Nat. Methods2(12), 920–931 (2005). [CrossRef] [PubMed]
  10. C. Preza and J. A. Conchello, “Depth-variant maximum-likelihood restoration for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A21, 1593–1601 (2004).
  11. Z. Kam, P. Kner, D. A. Agard, and J. W. Sedat, “Modelling the application of adaptive optics to wide-field microscope live imaging,” J. Microsc.226(1), 33–42 (2007). [CrossRef] [PubMed]
  12. S. Yuan and C. Preza, “3D fluorescence microscopy imaging accounting for depth-varying point-spread functions predicted by a strata interpolation method and a principal component analysis method,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XVIII, (SPIE, 2011), 79040M.
  13. S. F. Gibson and F. Lanni, “Experimental Test of an analytical Model of Aberration in an Oil-Immersion Objective Lens Used in 3-dimensional Light Microscopy,” J. Opt. Soc. Am. A9(1), 154–166 (1992). [CrossRef] [PubMed]
  14. P. Török, P. Varga, and G. Nemeth, “Analytical solution of the diffection integrals and interpretation of wave-front distortion when light is focused through a planar interface between materials of mismatched refractive indices,” J. Opt. Soc. Am. A12(12), 2660–2671 (1995). [CrossRef]
  15. J. G. McNally, C. Preza, J.-A. Conchello, and L. J. Thomas., “Artifacts in computational optical-sectioning microscopy,” J. Opt. Soc. Am. A11(3), 1056–1067 (1994). [CrossRef] [PubMed]
  16. J. W. Shaevitz and D. A. Fletcher, “Enhanced three-dimensional deconvolution microscopy using a measured depth-varying point-spread function,” J. Opt. Soc. Am. A24(9), 2622–2627 (2007). [CrossRef] [PubMed]
  17. C. Preza and V. Myneni, “Quantitative depth-variant imaging for fluorescence microscopy using the COSMOS software package,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing Xvii, SPIE 7570 (SPIE, 2010), 757003.
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