reducing depth induced spherical aberration in 3d ... · reducing depth induced spherical...

Reducing depth induced spherical aberration in 3D widefield fluorescence microscopy by wavefront coding using the SQUBIC phase

mask

Nurmohammed Patwary1, Ana Doblas2, Sharon V. King1, and Chrysanthe Preza1♦, 1Dept. of Electrical and Computer Engineering, The Univ. of Memphis, Memphis, TN, USA

2Dept. of Optics, Univ. of Valencia, E-46100, Burjassot, Spain

ABSTRACT

Imaging thick biological samples introduces spherical aberration (SA) due to refractive index (RI) mismatch between specimen and imaging lens immersion medium. SA increases with the increase of either depth or RI mismatch. Therefore, it is difficult to find a static compensator for SA1. Different wavefront coding methods2,3 have been studied to find an optimal way of static wavefront correction to reduce depth-induced SA. Inspired by a recent design of a radially symmetric squared cubic (SQUBIC) phase mask that was tested for scanning confocal microscopy1 we have modified the pupil using the SQUBIC mask to engineer the point spread function (PSF) of a wide field fluorescence microscope. In this study, simulated images of a thick test object were generated using a wavefront encoded engineered PSF (WFE-PSF) and were restored using space-invariant (SI) and depth-variant (DV) expectation maximization (EM) algorithms implemented in the COSMOS software4. Quantitative comparisons between restorations obtained with both the conventional and WFE PSFs are presented. Simulations show that, in the presence of SA, the use of the SIEM algorithm and a single SQUBIC encoded WFE-PSF can yield adequate image restoration. In addition, in the presence of a large amount of SA, it is possible to get adequate results using the DVEM with fewer DV-PSFs than would typically be required for processing images acquired with a clear circular aperture (CCA) PSF. This result implies that modification of a widefield system with the SQUBIC mask renders the system less sensitive to depth-induced SA and suitable for imaging samples at larger optical depths. Keywords: Three-dimensional microscopy, wavefront encoding, point-spread function engineering, image restoration, phase mask.

1. INTRODUCTION The wavefront encoding technique has been implemented successfully in point-spread function (PSF) engineering3 by placing a phase mask at the back focal plane of the microscope objective. The PSF is aberrated due to the refractive index (RI) mismatch between the immersion medium of the microscope objective lens and the object mounting medium. Spherical aberration (SA) increases with increased focal depth and due to this depth dependence, spherical aberration is a dynamic process. Spherical aberration reduces image resolution and increases the complexity of computational optical sectioning microscopy (COSM). Different depth variant restoration methods have been developed to address the depth variability of the PSF that add computational burden. To address this problem we test a wavefront modification designed to decrease the depth dependency of PSFs. Pupil modification by a phase mask has been studied in extended depth-of-field (EDOF) microscopy3 where a cubic phase mask (CPM) was designed to gain EDOF. In EDOF imaging an intermediate encoded image is taken using a modified pupil and a computational process decodes this image. EDOF with a generalized cubic phase mask (GCPM) has been successfully studied in 3D computational microscopy2. In addition, adaptive optics and depth-variant deconvolution algorithms6 have also been used to counteract spherical aberration7-9. A strata based model in the restoration process5 represents the volume using multiple PSFs computed at specific depths to mitigate the depth variability of the PSFs. Performance of this approach depends on the thickness of the object. The number of PSFs required increases with the increase of thickness and so does the computational burden and memory requirement. Principal component analysis (PCA) is another approach to solve the depth-variant imaging problem10, 11

♦[email protected]; phone 1 901 678-4369; fax 1 901 678-5469; cirl.memphis.edu

Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XXI, edited by Thomas G. Brown, Carol J. Cogswell, Tony Wilson, Proc. of SPIE Vol. 8949, 894911

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Proc. of SPIE Vol. 8949 894911-1

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which minimizes the computational burden in restoration process but at the same time requires a lot of memory for an accurate PCA component calculation. These approaches work well in that they produce results in which artifacts due to SA are minimized, but they require a high performance computation facility. Hence, a simpler solution would be to reduce the PSF depth variability of the PSFs.

Recently, a simple but effective phase mask (SQUBIC) has been reported1 to reduce the SA impact over a large range of depth. This phase mask works effectively in reducing wavefront distortion, and hence makes the PSF invariable over a large range of depth. This depth invariability makes object restoration from the blurred image using a single PSF over a wide range feasible. In this paper, we show the performance of the SQUBIC phase mask in 3D fluorescence microscopy in restoring a 3D object. Both qualitative and quantitative comparison is presented to show the improvement gained by using SQUBIC phase mask in the back focal plane in lieu of the conventional CCA system. This paper is organized as follows. Section 2 describes the background and theory behind SQUBIC WFE-PSF. Section 3 and 4 include the simulation methods and results, respectively. Summary of conclusions and future possible improvements are discussed in Section 5.

2. BACKGROUND 2.1 SQUBIC encoded WFE-PSF

Depth induced spherical aberration comes from the wavefront distortion due to the propagation of light within layers with RI mismatch, which depends on light wavelength, object RI, and numerical aperture (NA) of the objective lens. All these factors introduce different degrees of freedom and make SA a dynamic process. The idea of the SQUBIC phase mask is to apply a fixed dominant wavefront distortion via a phase mask 2 ( , )Aφ π ϕ ρ α= 1 at the back focal plane of the objective lens. To address different imaging conditions and design parameters, variation of the value of A is considered in order to vary the amount of wavefront distortion applied. Efficiency of the SQUBIC phase mask increases with the increase of A. In practice, the microscope objective’s back focal plane is projected on a spatial light modulator (SLM) through a 4F optical system and the SLM resolution limits the value of A16. The phase function ( , )ϕ ρ α is defined as1 :

32 21 sin ( ) 1 1( , )

1 cos( ) 2ρ α

ϕ ρ αα

⎡ ⎤− −= +⎢ ⎥

−⎢ ⎥⎣ ⎦ (1)

where ρ is a normalized pupil radius, ( )1sin NAnα −= and n is the refractive index of the lens’ immersion medium.

2.2 3D DV WFE-PSF formation

In optical sectioning microscopy, a 3D volume is acquired by optically slicing the sample through the axis of light propagation (Z-axis) and capturing 2D images at multiple depths. When the specimen emitted wavefront passes through the objective lens, its Fourier transform is generated at the back focal plane12 . To calculate WFE-PSFs from CCA PSFs, a generalized pupil function is calculated by taking the Fourier transform of the CCA PSF slices and adding the SQUBIC phase function ( , )x yf fφ to the CCA pupil phase of each slice. This process is described by2 :

(2)

where, { }1F − • is 2-D inverse Fourier transform, ( , )x yH f f is clear circular aperture pupil function , λ is the emission

wavelength, and ( , ; , )x y i oW f f z z is the optical path length error due to defocus and SA as a function of the normalized

spatial frequencies xf and yf .

{ } 2(2 / ) ( , ; , ) ( , )1, ( , ) ( , ) x y i o x y

i o

j W f f z z j f fz z x yh x y F H f f e eπ λ φ−=



2.3 Forward model formation

The forward image is simulated by computing 3D DV-PSFs at each layer using the superposition integral2 :

( ) ( , , , ) ( )WFE i o i o i o oO

g h x x y y z z s d= − −∫∫∫i 0x x x (3)

where, ( , , )i i ix y z=ix is a point in the image space, ( , , )o o o ox y z=x is a point in the object space O and ( )s ox is the object of interest. 2.4 Image restoration

Intermediate images, simulated using the forward model, are restored using the space invariant expectation maximization (SIEM)13 and the depth variant expectation maximization (DVEM)5 algorithms. Both methods are iterative. In SIEM process, only one single PSF is used to restore the whole volume where in the DVEM method multiple DV-PSFs are used to restore the total volume. In the restoration process the region of interest (ROI) is divided in the axial direction into multiple strata and a PSF is computed at the edge of each stratum. Hence, the number of PSFs required in DVEM is equal to the number of strata plus one. The number of 3D deblurring computations needed is equal to the number of strata. As 3D deblurring is a computationally costly process, complexity increases with the increase in number of strata. A higher number of strata is required to achieve a more accurate restoration result. Both these algorithms are implemented in CosmEstimation module COSMOS4 .

3. SIMULATION METHODS 3.1 Depth-variant (DV) PSF

Two hundred CCA DV-PSFs were calculated in MATLAB using the Gibson-Lanni optical path distance (OPD) model14 and vectorial field approximation15 at an interval of 0.3 µm to cover 60 µm depth. PSFs were calculated on a 128×128×1024 grid with a voxel size 0.1 µm × 0.1 µm × 0.1 µm with the following parameters: (a) the light point source is located at varying depth (starting from 0 µm to 60 µm with 0.3 µm spacing) in water (RI, nwater = 1.33) below the coverslip (b) a 20X/0.8 NA objective lens and (c) the emission wavelength λemmision = 515 µm. The voxel size is chosen to satisfy the Nyquist criteria. To compute WFE-PSFs, CCA PSF data were oversampled and over-ranged i.e., the number of pixels in the x-y plane was increased by the oversampling factor (we note that if over-ranging and oversampling is done by the same factor, the frequency domain sampling remains the same). WFE-PSFs were calculated from the CCA PSFs using the methods described in section 2.2. We used A = 88 to compute the SQUBIC phase mask.

3.2 Test object and forward image formation

To demonstrate the performance of the engineered PSF with the SQUBIC phase mask, three different types of object were studied. Test object 1 (Fig. 3a) consists of three small spheres (3 μm in diameter). Coordinates of the bead centers are (16.8 µm, 12.5 µm, 20 µm), (16.8 µm, 16.8 µm, 30 µm), (16.8 µm, 21.1 µm, 40 µm) respectively. The RI of the mounting medium is assumed to be 1.33, while the immersion medium of the lens is air (i.e., RI = 1.00). A second synthetic object (Fig. 3e) has only one 3-µm in diameter bead and depth position of that bead was varied from 20 µm to 40 µm. Test object 3 has five spherical beads of 3-µm diameter (Fig. 2a). The beads are centered at different (x,y,z) coordinates: (16.8 µm, 8.3 µm, 10 µm), (16.8 µm, 12.5 µm, 20 µm), (16.8 µm, 16.8 µm, 30 µm), (16.8 µm, 21.1 µm, 40 µm) and (16.8 µm, 25.3 µm, 50 µm) respectively. All synthetic objects are simulated on a 336×336×600 grid where each voxel size is 0.1 µm × 0.1 µm × 0.1 µm. The object is set within a larger grid (336×336×1250) to allow enough empty space to completely capture the intensity of the blurred images. A 7×7×7 Gaussian kernel with standard deviation 2.5 is used to smooth the object. This test object is convolved with all 200 PSFs using the variant tab of the CosmTools4 to compute the simulated image, referred to here as the forward model image, (see Fig. 2b & c) and restored using CosmEstimation module of COSMOS software.



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may become difficult to implement the mask practically and fine sampling will be needed in computation which will increase the computational burden2 . In this study, our concentration was confined to low numerical aperture objective lens. Further study is being performed using a high numerical aperture oil-immersed objective lens.

6. ACKNOWLEDGEMENT This work is supported by the National Science Foundation (NSF CAREER award DBI-0844682, PI: C. Preza; NSF IDBR award DBI-0852847, PI: C. Preza) and The University of Memphis (through a Herff fellowship to N. Patwary).

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