2.1.

Experimental Setup

The tomographic microscope presented in this paper is based on a vertical setup of a holographic microscope for transparent objects in Mach–Zehnder configuration.³⁹ This configuration is well suited for biological samples, which are usually in a liquid environment. The crucial modification of the holographic microscope is the introduction of a reflective phase-only SLM in the beam, which illuminates the object. Moreover, the system is designed in a way that additional modifications would enable integration into a DHM in the future. The scheme of the optical system of the holographic microscope is shown in Fig. 1.

Fig. 1

Tomographic microscope setup. He-Ne: laser; SMF: single-mode fiber; CO: collimating objective; $f^{\prime }=100\mathrm{mm}$; P: linear polarizer; H: half-wave plate; BS1 and BS2: beam splitter cube; M1, M2, and M3: flat mirror; SLM: spatial light modulator; T1, T2, L1, L2, L3, and L4: lens; F: spatial filter; MO1: $100\times$ microscope objective; MO2: $40\times$ microscope objective; and CCD: charge coupled device camera.

The He-Ne laser beam ($\lambda =632.8\mathrm{nm}$) is delivered to the setup using a polarization-maintaining, single-mode fiber and is then collimated with a 100-mm focal length lens. The polarizer and half-wave plate (multiorder $\lambda /2$ plate, designed for $\lambda =633\mathrm{nm}$) are used to control the intensity and the azimuth of polarization required for the phase-only liquid crystal (LC) SLM to work with. Then, the beam is split into reference and object beams by a 50:50 beam-splitting cube. The reference beam is directed by the M2 mirror toward the CCD. It passes through a beam expander, which consists of L3 ($f^{\prime }=30\mathrm{mm}$) and L4 ($f^{\prime }=150\mathrm{mm}$). This component is useful for introducing a spatial carrier in the holograms. The object beam is directed through the M1 mirror toward the high-resolution ($1920\times 1080$ pixels, $8.0\mu \mathrm{m}$ pixel pitch, 93% fill factor) liquid crystal on silicon (LCoS) SLM, PLUTO-VIS-014 (Holoeye Photonics AG, Germany). The beam impinges on the SLM at the angle—$\alpha$ equal to 8 deg. The SLM is placed in the front focal plane (FFPL) of the T1 lens ($f^{\prime }=100\mathrm{mm}$). The T2 lens ($f^{\prime }=75\mathrm{mm}$) acts as a second lens of the (T1–T2) telescope system. The spatial filter $F$, which is used to block zero diffraction order and reflection from the SLM cover glass, is optional. The image of the SLM surface is created in the back focal plane (BFPL) of the L1 lens ($f^{\prime }=250\mathrm{mm}$). The beam is focused in the FFPL of the MO1 microscope objective (Olympus UPlanFLN $100\times /1.3$, infinity-corrected, 0.17 CS—cover slip, $\mathrm{WD}=0.2\mathrm{mm}$). If a blazed grating is displayed on the modulator, the incident plane wave is diffracted into a single diffraction order. Expanded by the telescope, the diffracted beam is focused in the FFPL of the MO1 objective and creates a tilted illumination of the sample. When a series of blazed gratings with controlled frequency and direction is displayed on the modulator, the FFPL of the MO1 is scanned (see Sec. 3). Each point in the FFPL corresponds to an inclined beam in the specimen plane. Imaging of the sample illuminated in this manner is performed using the MO2 microscope objective (Leica HCX Pl APO $40\times /1.25$ to 0.75 CS, $\mathrm{WD}=0.1\mathrm{mm}$). The plane wave, illuminating the sample, is focused in the BFPL of the MO2 and collimated using the L2 tube lens ($f^{\prime }=200\mathrm{mm}$). The CCD camera (JAI BM-500GE, $2456\times 2058$ pixels, $3.45\mu \mathrm{m}$ pixel size) is placed in the BFPL of the L2 lens. The magnification $M$ of the imaging part of the system is 38.

In this tomographic microscope, the SLM replaces two rotary mirrors, which would normally be necessary to scan the aperture of the MO1 microscope objective. Implementing SLM as an active element for scanning directions of the object-illuminating beam provides the freedom of different illumination scenarios. Below, we describe two basic types of illumination scenarios: linear and circular. In the case presented in Fig. 2(a), a spatial filter $F$ (Fig. 1) is placed in the Fourier plane of the T1 lens. Scanning the aperture of the MO1 along a circle results in illumination beams lying on a cone. In this approach, the frequency of the spatial carrier rotating in the camera plane is constant for each projection, which is beneficial for hologram processing. In the scanning method presented in Fig. 2(b), the illumination beams lie in one plane. This concept allows capturing an image for normal illumination. Here, the spatial carrier frequency is no longer constant, which might lead to issues with too low or too high fringe density.

Fig. 2

Basic illumination scenarios in tomographic measurement presented in the image plane, (a) the result of circular scanning of the back focal plane (BFPL) of the MO1 microscope objective with object beam, $k_{\mathrm{sn}}$: $n$’th sample wave vector, $n=1$ to total number of projections, $k_{\mathrm{r}}$: reference beam, ${\theta }_{\mathrm{m}}$: maximum sample beam incidence angle in the image plane, ${\varphi }_{\mathrm{n}}$: azimuth of the $n$’th projection, (b) the result linear scanning of the BFPL of the MO1 objective, ${\theta }_{\mathrm{r}}$: reference beam angle of incidence in the camera plane.

The results of data acquisition are presented in Fig. 3. The spatial carrier frequency in Figs. 3(a) and 3(b) depends on the ${\theta }_{\mathrm{m}}$ illumination angle only and for Fig. 3(c) can also be adjusted with ${\theta }_{\mathrm{r}}$ angle (with a single, mechanical tilt of the M2 mirror) to obtain optimal conditions for phase reconstruction for every projection. The spatial carrier frequency should not exceed the Nyquist limit for $\pm {\theta }_{\mathrm{m}}$.

Fig. 3

Off-axis holograms of a $23.5\mu \mathrm{m}$ PMMA microsphere: (a) circular scanning measurement (CSM) scenario, angle of incidence of the illumination beam in the specimen plane ${\theta }_{\text{in}}=53.5\mathrm{deg}$, ${\theta }_{\mathrm{r}}=0\mathrm{deg}$, ${\varphi }_{\mathrm{n}}=0\mathrm{deg}$, (b) CSM scenario, angle of incidence of the illumination beam in the specimen plane ${\theta }_{\text{in}}=53.5\mathrm{deg}$, ${\theta }_{\mathrm{r}}=0\mathrm{deg}$, ${\varphi }_{\mathrm{n}}=270\mathrm{deg}$, and (c) linear scanning measurement (LSM) scenario, ${\theta }_{\mathrm{in}}=0\mathrm{deg}$.

2.2.

Spatial Light Modulator as an Active Device in Tomographic Microscope

2.2.2.

Wavefront correction

SLMs are common tools in wavefront correction^41,42 and are also used in microscopy to improve image quality.⁴³ Phase-only, reflective modulators are also used in interferometry for wavefront shaping.⁴⁴ In this case, the correction of the object beam wavefront compensates for the shape of the SLM, which is not perfectly flat, and the aberration that is present at angles close to ${\theta }_{\max }$. Also, if a biological sample was measured, the refractive index mismatch between the immersion liquid and the specimen would result in spherical aberration. In this approach, the calibration is a convenient procedure because it is only necessary to find a sample-free region and create a look-up table (LUT) once per measurement scenario and a sample with specific refractive index.

In this paper, to display a phase image for wavefront correction, Zernike orthogonal polynomials⁴⁵ are used to fully characterize the aberrations. SLM is rectangular, therefore it is important that the polynomials are calculated over a rectangular aperture⁴⁶ instead of a unit disk. In this work, the first 15 Noll indices of the polynomials were used. Processing the reference image to calculate correction map could of course also be realized using averaging, filtering methods such as bidimensional empirical mode decomposition⁴⁷ or alternative image reconstruction technique based on orthogonal moments.⁴⁸ However, the established theory for calculating the wavefront aberrations based on the Zernike polynomials of the reference measurement appears to be the most functional approach for wavefront correction.

The pixel size of the modulator is $P_{\mathrm{m}}=8\mu \mathrm{m}$, while the camera pixel size is $P_{\mathrm{c}}=3.45\mu \mathrm{m}$. The basic condition for reconstructing the wavefront for the SLM with a camera used for imaging is to maintain the relation $P_{\mathrm{m}}>P_{\mathrm{c}}$. In the case presented here, $P_{\mathrm{m}}$ in the camera plane is $1.62\mu \mathrm{m}$, which means that the resolution of the SLM is not fully used in wavefront correction and additional errors may occur during the processing of the reference wavefront map.

The result of the calculation of the Zernike-polynomials-based wavefront map for the SLM is presented in Fig. 4. Calculation of the wavefront map does not take into account the higher frequencies in the phase. In this way, the calibration is not affected by the noise introduced by the sample-free area and only corrects aberration introduced by the optical system and the shape of the modulator.

Fig. 4

Wrapped correction phase for ${\theta }_{\mathrm{m}}$ illumination angle: (a) the reference wavefront and (b) wavefront reconstructed from calculated Zernike polynomials for coefficients 1 to 15th Noll index.

The benefit from correcting the wavefront is the fact that the quality of the reconstructed image is improved. Moreover, the sample does not need to be removed in order to place a calibration object, which is convenient. Also, when multiple objects are measured within one sample chamber, the processing of the phase images is simplified—there is no need to subtract the background wavefront or perform additional deconvolution operations.

2.3.

Measurement Procedure

In order to perform calibration and measurement using the active tomographic microscope with optically controlled projections, a few steps must be performed. This is shown briefly in Fig. 5.

Fig. 5

Measurement procedure for look-up table (LUT)-calibrated measurement.

In the approach presented in this paper, the preprocessing part of the measurement is extended. The key part of the preprocessing is calculating Zernike polynomials followed by preparation of calibrated LUT containing a blazed grating for each projection. The effect of the superposition of the correction wavefront and the blazed grating is presented in Fig. 6. The maximum intensity of the created pattern is adjusted to match the required $2\pi$ phase shift.

Fig. 6

An example of a LUT frame for a 36-pixel blazed grating ($\theta =28\mathrm{deg}$): (a) precalibration blazed grating and (b) calibrated LUT frame, modified with reference wavefront.

The result of implementation of the regular and corrected LUT cells is presented in Fig. 7. The wavefront has been successfully corrected over the sample area and most of the SLM. It can be noticed that errors caused by the borders of the SLM are not corrected well. In addition, apparently the correction wavefront might not fully match the real aberrated wavefront. This might occur if the SLM border is not perfectly parallel to the camera plane. For this reason, a calibration method needs to be proposed in the future. Also, placing an amplitude mask in the plane conjugate with the SLM surface would reduce the diffraction caused by the border of the modulator.

Fig. 7

Wrapped phase of a $23.5\mu \mathrm{m}$ PMMA microsphere: (a) measurement with a 15-pixel blazed grating and (b) measurement with a calibrated LUT frame—15-pixel blazed grating modified with a reference wavefront.

2.4.

Tomographic Reconstruction

The illumination architectures described above do not provide data from full tomographic angular range and therefore require a special approach. According to the diffraction slice theorem, these architectures result in empty areas in the Fourier domain,²¹ which in turn strongly influence reconstruction resolution, which becomes anisotropic. Anisotropic resolution is not the only artifact present in the reconstruction. Another is a distorted geometry of the reconstructed object—the shape is elongated in the direction in which no projections were recorded.^37,38 In general, the problem of tomographic reconstruction becomes even more problematic, which in this case means that from the same set of initial data, an almost infinitely large set of solutions with a relatively small residuum may be calculated.⁴⁹ A number of algorithms that compensate for the partial lack of input data have already been developed.⁵⁰ The state-of-the-art approaches utilize TVM,^51–54 a technique derived from compressed sensing. TV minimization is a very powerful regularizer that makes use of the fact that some objects’ gradient is sparse. By applying the TVM iteratively in the reconstruction process, one can limit the number of possible solutions of the tomographic reconstruction, thus it is possible to retrieve full information about the object. By definition, the TVM is applicable to piecewise constant samples only. Herein, we propose a novel technique, namely the TVIC. It allows applying the TVM technique to reconstruct nonpiecewise constant samples such as biological cells. In this approach, in the first step, a piecewise constant reconstruction of the sample is calculated by means of Chambolle–Pock algorithm.⁵⁵ Despite the fact that inner structures of the reconstructed sample are erroneous (by applying Chambolle–Pock algorithm we assumed piecewise constancy to a nonpiecewise constant specimen), the external geometry is retrieved correctly with very good precision. This reconstruction is binarized and treated as an external geometry mask. The knowledge about external geometry is now a powerful prior, which strongly improves the final results. In our approach, this prior is applied iteratively to the algebraic reconstruction algorithm.⁵⁶ Other priors are nonnegativity constraint and smoothness of the refractive index distribution. The latter prior is implemented in the form of a 3-D median filter of the $3\times 3\times 3$ size.

When algebraic reconstruction techniques are applied to 3-D tomography, the size of the system matrix $A$ from the equation $Ax=p$, in which $x$ is the reconstructed image and $p$ is the input data (sinogram), is always a problem. In our case, the system matrix $A$ has a size of approximately $3,600,000\times 1,000,000$, which would normally mean more than 10 TB of data. This is why we utilize the ASTRA Toolbox,⁵⁷ which makes use of the fact that a major part of this matrix consists of zeros and so the matrix may be created with a reasonable amount of memory. It also allows the implementation of custom TV regularizers.

The algorithm results in a reconstruction with partially improved quality of inner structures of the specimen and with precisely retrieved external geometry. During the main reconstruction stage, the piecewise constancy was not assumed, making it applicable to a wide range of samples.

2.5.

Sample Preparation

One of the two samples prepared for the experiment in this paper was a microsphere made of PMMA. The microspheres were produced by microParticles GmbH. The diameter of the sphere was $23.5\mu \mathrm{m}$ and the refractive index for $\lambda =632.8\mathrm{nm}$ is $n_{\text{sphere}}=1.489$, and it was immersed in index-matching liquid (Cargille) with refractive index $n_{\text{immersion}}=1.517$. The object was placed between two 0.17 mm coverslips separated with #0 coverslip (0.08-mm thick).

As the second sample, the C2C12 mouse myoblast cell line⁵⁸ was prepared. The C2C12 cells were maintained in DMEM high glucose medium with l-glutamine and sodium pyruvate (Gibco) supplemented with 10% fetal bovine serum (Sigma–Aldrich) and 1% penicillin/streptomycin mixture (Life Technologies) at 37°C in a humidified 95% air and 5% ${\mathrm{CO}}_2$ atmosphere. Cells were passaged at 70% to 80% of confluence. For the experiment, cells were seeded on autoclaved 0.17 mm coverslips.

Cells on coverslips were rinsed briefly with cold $1\times$ phosphate buffer saline (PBS) (Pracownia Chemii Ogólnej IIITD PAN) and fixed in 4% paraformaldehyde (Sigma–Aldrich) for 10 min at room temperature. Next, the coverslips were rinsed with cold PBS buffer ($2\times 5\min$ at room temperature). Finally, the coverslips were stored in $1\times$ PBS buffer supplemented with 0.02% sodium azide (Sigma–Aldrich). For the measurement, coverslips with cells were supported on two #0 coverslip spacers and covered with a #1 coverslip. For the measurement, the cells were immersed in water ($n_{\mathrm{w}}=1.3319$ at $\lambda =632.8\mathrm{nm}$ and $T=20°\mathrm{C}$).

3.1.

Tomographic Reconstruction of a Calibrated Model

In order to test the performance of the algorithm as well as compare the proposed scanning scenarios, a well–defined object, i.e., polymer microsphere, was measured. Two illumination architectures were used in the measurement, as described in Sec. 2.1: single-axis (linear) illumination and conical illumination. The acquired data were reconstructed with TVIC algorithm. It should be noted that the calibrated object is piecewise constant, but during the second stage of the reconstruction process piecewise constancy was not assumed. However, the piecewise constant microsphere still allows verification of the quality of the reconstruction in terms of shape and refractive index changes. Figures 8 and 9 present the reconstruction of the refractive index distribution in the microsphere based on the data captured with the single-axis illumination and the conical illumination scenarios, respectively. The color scale represents the absolute refractive index values calculated using the a priori information of the refractive index of the background (certified index-matching liquid).

Fig. 8

Central cross section of a three-dimensional (3-D) reconstruction of the refractive index of a microsphere reconstructed from data gathered using the linear illumination scanning measurement scenario.

Fig. 9

Central cross section of a 3-D reconstruction of refractive index of the microsphere reconstructed from data gathered using the CSM scenario.

Figures 8 and 9 show that TVIC algorithm provides correct external geometry of the measured object. There is also visible improvement in the reconstruction of the inner structure when these results are compared with results calculated with SART+ATV algorithm.^26,37 However, it is clearly visible in Fig. 8(b) that the shape of the microsphere obtained with LSM is distorted along the direction of the optical axis, which is even more apparent in Fig. 10(a).

Fig. 10

The comparison of the microsphere 3-D refractive index distribution reconstruction profiles in two directions ($X$ and $Z$) for (a) LSM scenario and (b) CSM scenario.

The asymmetry of the microsphere measured with the circular scanning scenario (along the $X$- and $Z$-axes) is in this case less than 1.7% with a slight elongation of the reconstruction in the direction of optical axis. We consider this to be a very good result and, based on this, have decided to use only a CSM scenario. Also, the distribution of the refractive index in the $Z$ direction is much more uniform and similar to the $X$ profile for the circular scanning measurement. In the case of the linear measurement, the inner structure of the microsphere was affected by the character of scanning and could be reduced if the linear scanning was performed in two directions. The reconstructed value of the refractive index in the center of the PMMA microsphere lies within the uncertainty usually estimated for limited-angle tomographic systems.³²

3.2.

Tomographic Reconstruction of a Biological Sample

In this section, we present the second step of the experiment, in which fixed adherent cells from the C2C12 line were measured. Even though we did not measure living biological samples, our technique is equally suitable for such objects. The result of the measurement is shown in Fig. 11. In the reconstruction slices presented in Fig. 11(b), the region of highest refractive index values can be identified as the nucleus of the C2C12 cell, and the nuclear envelope is visible. The regions of higher refractive index in the nucleus are also visible, which could correspond to the nucleoli. We find our result to be in agreement with DHM-based studies on the refractive index of the intracellular structures of adherent cells.⁵⁹

Fig. 11

Tomographic reconstruction of a single C2C12 cell, measured using the CSM scenario and scaled to absolute refractive index values: (a) slice through the sample in the specimen plane ($XY$), (b) slice through the sample containing the optical axis ($XZ$), (c) $YZ$ slice in 3-D (Video 1, MPEG, 1.4 MB) [URL: http://dx.doi.org/10.1117/1.JBO.20.11.111216.1], and (d) $XZ$ slice in 3-D.

The reconstructed cell example is artifact free and does not exhibit any significant error at the borders, which suggests that the algorithm performs better for nonpiecewise constant objects than the piecewise constant calibration object. However, to fully characterize the reconstruction quality and accuracy, a well-defined 3-D phase phantom would be very useful.