Point-spread function engineering to reduce the impact of spherical aberration on 3D computational fluorescence microscopy imaging

doi:10.1364/OE.19.023298

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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.

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

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]

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

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]

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

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

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]

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

] and improve optical sectioning [8

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

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

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

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

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

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]

]). 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

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

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]

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

,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]

In this study, we investigate the use of WFE with different CPM-based designs as another approach to deal with 3D imaging in the presence of SA. In a previous study, one type of CPM-based design was assessed on its ability to render WFE confocal microscopy less sensitive to SA using simulation [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]

]. Radial phase masks (quartic and logarithmic) have also been derived to extend DOF and to alleviate SA effects when the system is well focused [20

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]

]. In another study, the engineered CPM-PSF has been shown to be susceptible to SA, albeit less than the conventional PSF [21

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

]. The goal of this investigation is to reduce the impact of SA on the 3D CPM-PSF and other WFE-PSF obtained from alternative CPM-based designs (proposed for EDOF purposes in previous studies [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

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]

–25

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]

]).

Three CPM-based designs [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

,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

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.

Previous studies on deriving and optimizing CPM-based designs for EDOF focused on improving the information (or frequency) transfer performance of the WFE system. Consequently, metrics based on the Fisher information [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

], the optical transfer function and the modulation transfer function [4

,22

,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]

] have been used in CPM design investigation and/or optimization. In order to enable PSF engineering that addresses the impact of both defocus and SA for COSM and EDOF microscopy, we propose a metric for the design selection that is directly associated with PSF variability as a function of defocus and SA. We use merit functions based on the mean square error computed between the layers of the WFE-PSF to quantify defocus sensitivity as well as between two 3D DV WFE-PSFs (with different amounts of SA) to quantify SA sensitivity of the WFE-PSF. These merit functions are used to guide the selection of the mask design for both COSM and EDOF microscopy.

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 phase

ϕ (f_{x}, f_{y})

, 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 = z i

at which the microscope is focused and the depth

z = z o

at which the light point source is located. Thus, the defocused WFE-PSF can be described as:

h_{z_{i}, z_{o}} (x, y) = {| F^{- 1} {H (f_{x}, f_{y}) e^{j (2 π / λ) W (f_{x}, f_{y}; z_{i,} z_{o})} e^{j ϕ (f_{x}, f_{y})}} |}^{2},

(1)

where

F^{- 1} {•}

denotes a 2-D inverse Fourier transform,

H (f_{x}, f_{y})

is the clear circular aperture pupil function, λ is the emission wavelength, and

W (f_{x}, f_{y}; z_{i}, z_{o})

is the optical path length error due to defocus and SA as a function of the normalized spatial frequencies

f_{x}

and

f_{y}

. Thus, the phase of the generalized pupil function or defocused WFE-ATF is given by

θ (f_{x}, f_{y}) = \frac{2 π}{λ} W (f_{x}, f_{y}; z_{i,} z_{o}) + ϕ (f_{x}, f_{y}) .

(2)

Equation (1) represents a single layer of the 3-D WFE-PSF at

z = z_{i}

h_{W F E} (x, y, z_{i}, z_{o}) = h_{z_{i},}_{z_{o}} (x, y),

(3)

when a point source is located at

z = z o

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 (x_{i}) = \underset{O}{∭} h_{W F E} (x_{i} - x_{o}, y_{i} - y_{0}, z_{i}, z_{o}) s (x_{o}) d x_{o},

(4)

where

x_{i} = (x_{i}, y_{i}, z_{i})

is a point in the image space,

x_{o} = (x_{o}, y_{o}, z_{o})

is a point in the object space O and

s (x_{o})

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

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

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

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

]. 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 (f_{x}, f_{y})

for each mask:

\begin{matrix} CPM ​ : ϕ (f_{x}, f_{y}) = α (f_{x}^{3} + f_{y}^{3}) \\ GCPM: ϕ (f_{x}, f_{y}) = α (f_{x}^{3} + f_{y}^{3}) + β (f_{x}^{2} f_{y} + f_{x} f_{y}^{2}) \\ SCPM: ϕ (f_{x}, f_{y}) = α (f_{x}^{3} + f_{y}^{3}) + β (\sin (ω f_{x}) + \sin (ω f_{y})), \end{matrix}

where α in radians represents the strength of the CPM [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 with

f_{x}

and

f_{y}

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

] 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 (

z_{o} =

0, 5, 10, 15, 20, 25, 30, 40,…, 90 and 100 μm) in water (RI, n_water = 1.33) below the coverslip; and b) a 60x/1.2 NA oil-immersion objective lens (RI, n_oil = 1.515) and an emission wavelength λ_emission = 633 nm are used to image the point source.

The DV WFE-PSFs were calculated for all the phase mask designs using Eq. (1) by first computing the 2D Fourier transform of the conventional complex-amplitude PSFs (computed with COSMOS as described above) using Matlab (Mathworks, Natick, MA), thereby obtaining the conventional ATF. The phase patterns (Fig. 1) were resized to a 194 x 194 grid (or 388 x 388) and then padded to a 512 x 512 (or 1024 x 1024) grid so that the circular apertures in the patterns match the circular pupil aperture of the objective lens. A 1024 x 1024 grid was used in the computation of some of the DV WFE-PSFs in order to obtain PSF images without artifacts. Figure 2 shows XY and XZ cut views of WFE-PSFs obtained with different masks without SA [Figs. 2(a-d)] and with SA [Figs. 2(e-h)]. The images of the CPM-PSF with SA [Fig. 2(f)] are consistent with images in a previous study [21

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

]. As evident in Fig. 2, the appearance of the WFE-PSFs changes with SA. This demonstrates the need to investigate and select mask parameters that could reduce the SA impact on WFE-PSFs.

Fig. 2 XY and XZ view of WFE-PSFs for a 60x/1.2NA oil-immersion lens using different types of mask designs with and without SA. PSFs without SA, i.e., z_o = 0 μm (top row) computed using a: (a) CCA; (b) CPM (α = 30); (c) GCPM (α = 30 and β = −3α); and (d) SCPM (α = 30, β = α/2 and ω = π/2). PSFs with SA for z_o = 20 μm (bottom row) computed with a: (e) clear circular aperture; (f) CPM (α = 30); (g) GCPM (α = 30 and β = −3α); and (h) SCPM (α = 30, β = α/2 and ω = π/2).

3.3 Simulated 3D images

Simulated intermediate images (i.e., images before any digital processing) from a WFE-widefield microscope (Figs. 10 and 11 ) were computed using a strata-based approach [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 x_oy_o plane is at the center of the grid, and the origin along the z_o axis is placed at the 100th x_oy_o plane. The (x_o, y_o) 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).

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): z_i = −5 μm (left); 0 μm (middle); and 5 μm (right).

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): z_i = −5 μm (left); 0 μm (middle); and 5 μm (right).

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

z_{o} =

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

z_{o} =

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 ). 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

h_{Z_{k}, z_{o}} (x, y)

represents a layer at

z_{i} =

Z_k and

h_{Z_{c}, z_{o}} (x, y)

represents the layer with the best focused 2D PSF at

z_{i} =

Z_c. For a 3D WFE-PSF without SA (

z_{o} = 0

μm) the best focused layer (at

z_{i} = 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

], and thus a new

z_{i} =

Z_c is determined from the conventional PSF. In general, a small NMSE_2D value over a long range of z_i 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.

Table 1 Equations for Evaluation Metrics

NMSE between a PSF’s XY layers	NMSE between two 3D PSFs
${NMSE}_{2D} = \frac{{‖ h_{Z_{k}, z_{o}} (x, y) - h_{Z_{c}, z_{o}} (x, y) ‖}^{2}}{{‖ h_{Z_{c}, z_{o}} (x, y) ‖}^{2}}$ (5)	${NMSE}_{3D} = \frac{{‖ h_{d} (x, y, z) - h_{0} (x, y, z) ‖}^{2}}{{‖ h_{0} (x, y, z) ‖}^{2}}$ (6)

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),

h_{d} (x, y, z)

is the 3D WFE-PSF with SA due to a point source located at depth

z_{o} = d

below the coverslip in the presence of a RI mismatch between imaging layers, and

h_{0} (x, y, z)

is the 3D WFE-PSF without any SA (i.e., the point source is located at depth

z_{o} = 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 NMSE_2D and NMSE_3D 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 NMSE_2D and NMSE_3D, 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 = {\bar{NMSE}}_{2D} / {\bar{NMSE}}_{3D},

(7)

where

{\bar{NMSE}}_{2D}

and

{\bar{NMSE}}_{3D}

are the average values of the NMSE_2D and the NMSE_3D, respectively, over the investigation range. In this study,

{\bar{NMSE}}_{2D}

was averaged over the range

z_{i} =

−5 μm to 5 μm while

{\bar{NMSE}}_{3D}

was averaged over the range

z_{o} =

0 to 30 μm. For COSM, a large

R

value is preferred which can be achieved by increasing the NMSE_2D value, (i.e. by increasing the sensitivity to defocus) while decreasing the value of the NMSE_3D, (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

R

value 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 = z_{i}

) 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 z_i.

3.7 Computation of 2D Modulation Transfer Function (MTF)

A comparison between 2D defocused MTFs of the conventional (CCA) and CPM-WFE systems has been used in a previous study to investigate and quantify the anticipated SNR reduction in EDOF [4

]. Towards this end, 2D MTFs of the CCA and GCPM-WFE systems were computed from PSFs with and without SA to investigate the impact of SA on the anticipated SNR in the final EDOF and WFE-COSM images. In order to study only the effect of SA, 2D XY layers corresponding to the best evident focus were selected (i.e., with zero or minimal defocus due to the PSF discretization). For the comparison, 3 MTFs for each system were generated by computing the 2D Fourier transform of Eq. (3) with: a)

z_{i} = 0

μm and

z_{o} = 0

μm; b)

z_{i} = - 1.1

μm and

z_{o} = 10

μm; and c)

z_{i} = 1.5

μm and

z_{o} = 20

μm. To improve accuracy in the computed MTF, the 2D CCA-PSFs were computed using a 4096 x 4096 grid and the CPM-based phase patterns (Sect. 3.1) were computed on a 2001 x 2001 grid. MTF profiles were compared along the normalized spatial frequency

f_{p}

, which is along the diagonal line

f_{x} = f_{y}

in the frequency plane.

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 . The NMSE_2D computed for CPM PSFs without SA (

z_{o}

= 0 μm) and with SA (

z_{o}

= 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 NMSE_2D decrease as the value of α increases indicating that better EDOF is achieved. This result is consistent with a prior reported result [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 NMSE_3D computed using PSFs with increasing amounts of SA (denoted by the increasing Z location of the point source which is equivalent to

z_{o}

) 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.

Fig. 3 NMSE plots for CPM-PSFs for different α values reported in panel (c). NMSE_2D values plotted vs. defocus reported as the distance from the designed best focal plane (FP), for PSFs: (a) without SA (z_o = 0 μm); and (b) with SA (z_o = 20 μm). (c) NMSE_3D computed for PSFs with increasing amount of SA, plotted vs. z_o. A semi-log scale is used for plotting the NMSE_2D to facilitate visualization of the small differences between the curves especially near the minimum values of the curves.

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 . The NMSE_2D, computed for GCPM PSFs without SA (

z_{o}

= 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 NMSE_2D [Fig. 4(b)] indicating that the resulting 3D GCPM PSFs are similar in this case. When β = −3α [Fig. 4(c)], the NMSE_2D 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 NMSE_2D 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

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 NMSE_2D is achieved at α = 150 for β = −3α

Fig. 4 NMSE plots for GCPM-PSFs for different α values reported in panel (a). NMSE_2D computed for PSFs without SA and plotted vs. defocus: (a) β = -α/3; (b) β = -α; and (c) β = 3α. NMSE_3D computed for PSFs with increasing amount of SA and plotted vs. z_o: (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 NMSE_2D to facilitate visualization of the small differences between the curves.

among all conditions investigated in this study. Similar NMSE_2D plots (not shown) were obtained for PSFs with SA (for

z_{o}

= 20 μm).

Figure 4(d) plots the NMSE_3D vs. z_o, computed using PSFs with increasing amounts of SA obtained for different values of α when β = −3α. Similarly, Fig. 4(e) plots the computed NMSE_3D for different values of β when α = 150. Among all conditions investigated in our study, the smallest NMSE_3D 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

The effects of the parameters α, β and ω, in the SCPM function, on the WFE-PSF were evaluated using 160 SCPM designs that resulted from the α-β-ω parameter sets described inSection 3.1. Among the investigated parameter sets, the set α = 150, β = α/3 and ω = -π/2 resulted in the smallest computed NMSE_2D and NMSE_3D. Some of the results from this study are summarized in Fig. 5 . The NMSE_2D, computed from SCPM PSFs without SA, is plotted as a function of defocus for different α values when β = α/3 and ω = -π/2 [Fig. 5(a)]. Similar NMSE_2D plots (not shown) were obtained for PSFs with SA (

z_{o}

= 20 μm). Figure 5(b) plots the NMSE_3D vs. z_o, computed using PSFs with increasing amounts of SA obtained with the same parameter sets as in the case of the NMSE_2D [Fig. 5(a)]. Overall, the trends observed in these results (Fig. 5) are similar with the ones for the GCPM case (Fig. 4).

Fig. 5 NMSE plots for SCPM-PSFs for different α values when β = α/3 and ω = -π/2. (a) NMSE_2D computed for PSFs without SA and plotted vs. defocus using a semi-log scale. (b) NMSE_3D computed for PSFs with increasing amount of SA and plotted vs. z_o.

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 NMSE_2D 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 NMSE_3D.

Figure 6 compares the sensitivity to defocus and SA of WFE-PSFs engineered with the 3 selected CPM-based masks. The NMSE_2D computed from PSFs without SA (z_o = 0 μm) and with SA (z_o = 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 NMSE_3D computed using PSFs with increasing amounts of SA vs. z_o 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 NMSE_2D performance, while the selected GCPM design (α = 150 and β = −3α) achieves the best NMSE_3D performance and an acceptable NMSE_2D performance. Thus, from this comparison study, the GCPM design is selected for EDOF microscopy in the presence of SA.

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). NMSE_2D computed from PSFs without SA (a) and with SA when z_o = 20 μm (b), plotted using a semi-log scale vs. defocus. (c) NMSE_3D computed for PSFs with increasing amount of SA plotted vs. z_o.

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 ). The NMSE_2D computed from PSFs without SA (z_o = 0 μm) and with SA (z_o = 20 μm) is plotted as a function of defocus in Fig. 7(a) and Fig. 7(b), respectively. Figure 7(c) plots NMSE_3D computed using PSFs with increasing amounts of SA vs. z_o 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), where

R

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)].

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). NMSE_2D computed from PSFs without SA (a) and with SA when z_o = 20 μm (b), plotted vs. defocus. (c) NMSE_3D computed for PSFs with increasing amount of SA plotted vs. z_o. (d) R plotted as a function of α. The straight dashed line marks the CCA R value.

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 NMSE_2D and NMSE_3D, and has the largest

R

value 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 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.

Fig. 8 Comparison of the GCPM phase to CCA-ATF phase changes due to defocus and SA. Phase profile vs. normalized frequency,

f_{x}

, for

f_{y} = 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 (z_i = 0 μm, z_o = 30 μm); defocus (z_i = −15 μm, z_o = 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

f_{x}

of the phase functions shown in (c).

Figure 8(c) compares profiles of the CCA-ATF phase due to different terms: defocus (

z_{i} = - 15 μm, z_{o} = 0 μm

), SA (

z_{i} = 0 μm, z_{o} = 30 μm

), 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 to

f_{x}

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 (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.

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 z_o = 0 μm (top row) and with SA for z_o = 10 μm (second row), and for z_o = 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.

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

] 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 at

z_{o} = 0 μm

and 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

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) 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)].

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

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

]. (e) Normalized MTF intensity profiles vs.

f_{p}

(the frequency along the diagonal line

f_{x} = f_{y}

). MTFs shown are without SA (z_o = 0 μm) and with SA (z_o = 20 μm) using different masks: (i) CCA; (ii) GCPM with α = 150 and β = −3α; and (iii) GCPM with α = 30 and β = −3α.

In previous studies [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]

], a reduction in the CPM-MTF intensity compared to the CCA-MTF intensity was shown which supports, as expected, the reduction of SNR in the final reconstructed images. Our results shown in Fig. 12(e) predicted the same result from the GCPM MTF. Moreover, increasing α from 30 to 150 in the GCPM design caused the corresponding MTF value to drop 36% ± 6% for frequencies in the range of 0.05 to 0.8 along f_p (Fig. 12e). Thus, it is anticipated that the EDOF-GCPM design will yield final reconstructed images with lower SNR than GCPM designs with a smaller α value. Another finding from Fig. 12(e) is that for the GCPM designs investigated in this study, a limited amount of SA had very little impact on its MTF characteristics, while the α value had some impact on the MTF characteristics as discussed above. For applications with large noise or applications requiring high SNR, the α value must be carefully selected. Furthermore, in some cases, EDOF and SA insensitivity, associated with a high α value, might have to be traded off to reach a required SNR level.

For the COSM application, the merit function R [Eq. (7)] was introduced to facilitate the selection of the mask parameters. Increasing the contribution of NMSE_2D in Eq. (7) ensures good optical sectioning while increasing the contribution of NMSE_3D ensures that better SA-insensitivity is achieved in the WFE-PSF. In most cases, the mask parameters associated with a small NMSE_2D also yield a small NMSE_3D. Clearly there is a tradeoff in selecting the mask parameters for COSM requiring that some insensitivity to SA must be sacrificed in order for the system to have adequate sensitivity to defocus. In this study, sacrificing insensitivity to SA by reducing the contribution of the NMSE_3D as in Eq. (7) yielded a WFE-PSF with desirable characteristics. Other choices to weigh the contributions of the 2 NMSE metrics are possible and could be investigated. The selection of the phase mask parameters for COSM could also be guided by evaluating the mask’s impact on the restored (final) images. The MSE metrics used in this study are directly related to the PSF engineering goals and facilitated the selection of the designs so that the goals could be met. However, determination of optimal phase mask designs requires the development of appropriate optimization methods.

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

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

]. 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

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

,27

] 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.

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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. 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]
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.	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]
20.	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]
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]
24.	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]
25.	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

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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]
J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods19(3), 373–385 (1999). [CrossRef] [PubMed]
J.-B. Sibarita, “Deconvolution Microscopy,” in Microscopy Techniques, J. Rietdorf, ed. (Springer Berlin / Heidelberg, 2005), pp. 1288–1291.
D. A. Agard, “Optical sectioning microscopy: cellular architecture in three dimensions,” Annu. Rev. Biophys. Bioeng.13(1), 191–219 (1984). [CrossRef] [PubMed]
J.-A. Conchello and J. W. Lichtman, “Optical sectioning microscopy,” Nat. Methods2(12), 920–931 (2005). [CrossRef] [PubMed]
C. Preza and J. A. Conchello, “Depth-variant maximum-likelihood restoration for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A21, 1593–1601 (2004).
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]
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.
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]
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]
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]
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]
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.
C. Preza and J.-A. Conchello, “Depth-variant maximum-likelihood restoration for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A21(9), 1593–1601 (2004). [CrossRef] [PubMed]
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. Express17(16), 13810–13818 (2009). [CrossRef] [PubMed]
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]
M. R. Arnison, “Phase control and measurement in digital microscopy,” Ph.D. dissertation (University of Sydney, Sydney, 2004).
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]
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]
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]
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]
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.
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]

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