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Abstract
Dielectric tensor tomography reconstructs the three-dimensional dielectric tensors of microscopic objects and provides information about the crystalline structure orientations and principal refractive indices. Because dielectric tensor tomography is based on transmission measurement, it suffers from the missing cone problem, which causes poor axial resolution, underestimation of the refractive index, and halo artifacts. In this study, we study the application of total variation and positive semi-definiteness regularization to three-dimensional tensor distributions. In particular, we demonstrate the reduction of artifacts when applied to dielectric tensor tomography.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Dielectric tensor tomography (DTT) is a recent development in microscopy that enables the reconstruction of the three-dimensional (3D) dielectric tensor distribution of a sample [1]. The dielectric tensor is a measurement of the 3D optical anisotropy, which fundamentally describes the light-matter interaction considering polarization of light, optical anisotropy, and molecular orientations. The intrinsic information of optically anisotropic materials, including the principal refractive indices (RIs) and crystalline structure orientation, can be obtained by diagonalizing the dielectric tensor. The unique ability of DTT to directly reconstruct the dielectric tensor is expected to be utilized in many disciplines, from metrology and soft-matter physics to biology [2–8]. Previous methods such as polarized light microscopy [9] (Fig. 1(a)), polarization-dependent digital holographic microscopy [10–14], polarization-dependent optical diffraction tomography [15–17] or fluorescence-based imaging techniques [18] can only provide limited information about the sample anisotropy or require some assumptions to retrieve the dielectric tensor. On the other hand, DTT can directly access the full dielectric tensor by solving a vectorial wave equation. However, DTT is based on light transmission three-dimensional quantitative phase imaging (QPI) measurements [1,19], which makes it inherently affected by the missing-cone problem. Because of the limited numerical aperture of the used imaging system, some spatial frequencies of a sample cannot be reconstructed, resulting in poor axial resolution, low precision of reconstructed 3D orientations, underestimation of principal refractive indices, and halo artifacts [20].
Fig. 1. (a) Schematic comparison between dielectric tensor tomography and polarization microscopy. (b) Optical setup, HWP: half-wave plate, LCR: liquid crystal rotator, DMD: digital micromirror device, PBS: polarized beam splitter and BS: beam splitter. (c) Effects of TV regularization in the image plane (d) Effects of the regularization in the Fourier plane.
Prior knowledge of a sample can be used to alleviate some of these artifacts. In many imaging systems, an image is known to have a non-negative value, which can be applied using the Gerchberg-Papoulis algorithm [21,22] or in combination with a denoising algorithm [23]. However, in DTT, the off-diagonal components of the dielectric tensor are often negative, depending on the orientations of the crystalline structures, hindering the direct use of this method. An analog approach for a tensor image is the use of positive semi-definiteness constraints, constraining eigenvalues of a matrix to be non-negative [24]. Other minimization-based techniques, such as total variation (TV) regularization [25,26] and other optimization methods using the image gradient [27–32] instead suppress missing cone artifacts by assuming the sparse presence of edges in the data. These methods are promising for the regularization of DTT data. While already demonstrated in other tensorial imaging methods, the regularization was mostly used for denoising purposes [16,24,33–36], while its effect on the missing cone problem was not specifically studied and the demonstration in DTT was not performed. A last class of regularization uses deep learning to fill in missing data [37].
Here we demonstrate the effects of TV and positive semi-definiteness regularization on missing cone artifacts of 3D tensor distributions. In particular, we focus on DTT measurements. We show that using TV or positive semi-definiteness constraints can both greatly reduce missing cone artifacts: improving both principal RI and crystalline structure direction information. We also discuss the strengths and weaknesses of both methods.
2. Methods
2.1 Dielectric tensor tomography
Birefringence is a property of optical materials that causes light to diffract differently depending on the polarization. Birefringence resulting from supramolecular assemblies or crystalline structures can be physically described by a dielectric tensor [38]. The dielectric tensor is a generalized physical quantity from the dielectric constant that expresses the speed of light in a material but is dependent on the polarization. The dielectric tensor is a symmetric matrix and can be decomposed using singular value decomposition into three dielectric constants and three optical axes, with each dielectric constant corresponding to the speed of light for polarization parallel to a given optical axis [38].
As such, six unknown values must be obtained to completely determine the dielectric tensor. However, most optical systems can only change the illumination and detection polarizations, and polarization having two degrees of freedom leads to four independent measurements that are insufficient for fully reconstructing the dielectric tensor. Using redundant spatial frequency information from slightly tilted illumination angles, DTT can reconstruct a full dielectric tensor [1,39].
DTT reconstruction is based on polarization-dependent light-field measurements obtained using off-axis holographic microscopy (Fig. 1(b)). For a given illumination wave vector $\overrightarrow {{k_{ill}}} $ and polarization $\overrightarrow {{p_i}} $, the field $\overrightarrow {{\psi _i}}$ transmitted through the sample is recorded. The illumination was then slightly tilted to a new wave vector $\overrightarrow {{k_{ill}}} + \Delta \overrightarrow {{k_{ill}}} $ to obtain a third field. Using three fields and the symmetry property of the tensor, the dielectric tensor $\overleftrightarrow \varepsilon $ can then be expressed as
Using $\overrightarrow k = {\left( {\begin{array}{ccc} {{k_x}}&{{k_y}}&{{k_z}} \end{array}} \right)^T}$, the Fourier space coordinate system. This process is repeated for several illumination angles to populate the Fourier space (Fig. 1(d)) [1,40].
2.2 Total variation regularization for dielectric tensor tomography
In DTT, although the full dielectric tensor can be retrieved for accessible spatial frequencies, not all frequencies are accessible. Because DTT is based on transmitted light in an optical setup with a limited numerical aperture, it suffers from the missing cone problem [41]. The name of the missing cone problem comes from the fact that non-accessible spatial frequencies form a cone in the Fourier space (Figs. 1(c),(d)). The missing cone problem is a common problem in tomographic measurements with a limited illumination angle [20,42] and also occurs in X-ray computed tomography [43], three-dimensional transmitted electron microscopy [44], optical diffraction tomography [45–48], and widefield fluorescence imaging [49,50].
Various computational methods, called regularization, have been developed to fill in the missing information. A known constraint of a sample is used to predict inaccessible spatial frequency information. One constraint that can be used is edge sparsity, in which the image is supposed to have a limited number of edges. This constraint can be applied using a total variation (TV) algorithm [51].
The TV algorithm minimizes the sum of gradient norm while preserving the fidelity of measured data. This can be mathematically expressed as follows:
We further improve the existing TV algorithm by using a multiscale approach to speed up the computation time. This is particularly important for DTT because of the large size of the tensorial data compared to scalar imaging. We experimentally noticed that TV algorithms are often very fast at removing high-frequency noise, but the suppression of low-frequency missing cone artifacts often requires several iterations. For this reason, we suppressed missing cone artifacts using an under-sampled low-resolution version of the data by using several iterations and then suppressed high-frequency artifacts at full resolution by using only a few iterations. Owing to the lower execution time when using low-resolution data, this strategy reduces the overall computation time while preserving imaging quality.
For the simulation, we chose to use a small regularization constant to only fill up data in the missing cone region, but we used a larger regularization constant for the experiment because it enables faster convergence rates and has a denoising effect on experimental data [25]. We used $\lambda = 5 \cdot {10^{ - 4}}$ for simulations and $\lambda = {10^{ - 2}}$ for experiments; at low resolution, a total of 200 inner and outer iterations were performed in the nested iteration of the FISTA algorithm, whereas at high resolution, 100 inner iterations and 20 outer iterations were used. The execution time of the regularization algorithm for 218 × 218 × 80 × 3 × 3 data on a workstation computer with a graphics processing unit (GTX1070ti, Nvidia) was 69 s.
2.3 Positive semi-definiteness constraint
In DTT, due to the complex orientation of the crystalline structures, the off-diagonal components of the dielectric tensor are often negative. As such, non-negativity regularization cannot be used directly. While non-negativity cannot be used on each tensor element, the principal refractive indexes (RIs) are often known to be bigger than the background RI. This constraint can be expressed as the relative dielectric tensor $\overleftrightarrow {\Delta \varepsilon } = \overleftrightarrow \varepsilon - n_m^2I$ being positive semi-definite.
Here, to apply this constraint, we devised two strategies. The first strategy is analogous to the Gerchberg-Papoulis algorithm iterates between the Fourier space spectrum constraint and an image space positive semi-definite projection, while the second one incorporates the positive semi-definite projection in the TV algorithm. Both strategies need a projection on the positive semi-definite matrix space, which can be defined as [55]:
where A is an arbitrary matrix with eigenvalues λi and eigenvector ui. The function (x)+ leaves positive numbers unchanged and sets negative numbers to 0.2.4 Optical setup
The setup used to demonstrate the proposed method experimentally is shown in Fig. 1(b). Based on a Mach-Zehnder interferometer, the laser beam was first split into a reference beam and a sample beam. A digital micromirror device was used to systematically control the illumination angle of a plane wave [56,57], followed by a liquid crystal retarder to control the polarization of the beam. The sample beam was then projected onto a sample using a water immersion condenser (UPLSAPO 60XW, NA = 1.2, Olympus). Then, the scattered light was collected using an oil immersion objective (PLAPON 60X, NA = 1.42). Finally, the sample beam interfered with the reference beam on two cameras (Lt425R, Lumenera) with an orthogonal analyzer to obtain polarization-dependent fields.
3. Results
3.1 Quantification of reconstruction quality improvement
To evaluate the efficiency of TV regularization in reducing missing cone artifacts, we first simulated light scattering in synthetic liquid crystal distributions. In particular, we chose hybrid aligned nematic (HAN) and twisted nematic (TN) distributions because of their importance in liquid-crystal display technology [58,59].
Simulations were carried out on a 5-µm-height and 5-µm-diameter cylinder containing the aforementioned distributions. The transmitted field was computed using the finite-difference time-domain method (Lumerical FDTD 2020), and tomograms were obtained using the DTT algorithm. The fidelity was measured against the original phantom using three criteria: the orientation angle, axial resolution, and RI value. Reconstructed tomograms are shown in Fig. 2(a) for visual inspection.
Fig. 2. (a) 3D visualization of the different liquid crystal distribution (b) plot of the azimuth angle (i) and RI (ii) for the twisted nematic distribution as a function of depth (c) plot of the altitude angle (i) and RI (ii) for the hybrid aligned nematic distribution as a function of depth. None: no regularization, GT: ground truth, TV: total variation, PSD: Positive semi-definiteness.
First, we examined the crystalline orientation. In the HAN distribution, the altitude angle varies linearly from 0° to 90°, whereas in the TN distribution, the liquid crystal is rotated along the azimuthal angle. Figures 2(b)(i),(c)(i) show the average angle of the crystalline orientation as a function of the vertical position. The expected, non-regularized, positive semi-definiteness constrained and TV-regularized distributions were plotted. Both the positive semi-definiteness constrain and TV regularization improves the orientation fidelity for both distributions. This improvement can be evaluated using the standard deviation of the expected distribution. The HAN standard deviation improved from 13.9° to 13.6° for TV and 9.77° for the semi-definiteness constraint, whereas the TN standard deviation improved from 15.9° to 6.5° for TV and 3.3° for the positive semi-definiteness constraint.
Figures 2(b)(ii),(c)(ii) show the RI distribution at the center of the cylinder as a function of the vertical position. Again, the use of both regularizations corrected RI underestimation and improved the resolution, as seen from the sharp increase in the RI near the cylinder boundary.
Overall, the positive semi-definiteness constraint yielded quantitatively better imaging results. Visual imaging artifacts where similar between the two techniques, probably caused by multiple scattering.
The TN distribution showed great fidelity after regularization. However, the HAN distribution, while showing improvement when using regularization, still suffers from a significant RI underestimation. We believe this to be due to the liquid crystal alignment being parallel to the optical axis, which reduces the SNR and makes the system more sensitive to other issues, such as multiple light scattering.
3.2 Liquid crystal droplet sample
While simulations are important to quantify the fidelity improvement after regularization, applicability to experimental data also needs to be demonstrated. In experiments, imaging noise and system aberration hinder missing cone artifacts removal so one will often need to use new regularization strategies, including bigger regularization constants.
In this study, we set out to image liquid crystal droplets due to their predictable crystalline arrangement. Both radial and bipolar liquid crystal droplets were prepared following the protocol in [1] and were subsequently imaged (Fig. 3). We attempted different regularization strategies using TV with different strengths, positive semi-definiteness constraint and a combination of positive semi-definiteness constrain with a weak TV regularization (Fig. 3).
Fig. 3. TV and positive semi-definiteness regularization effect on experimental data. (a) Bipolar liquid crystal droplet with zoom-in of the point defect of the bottom droplet. (b) Radial liquid crystal droplet with zoom-in on the radial hedgehog point defect.
The reconstructed results showed the expected crystalline structure. In Fig. 3(a), two bipolar droplets are stacked vertically, the poles of the bottom droplets are along the z-axis, while the poles of the top droplet are along the x-axis. In Fig. 3(b), the radial droplet clearly shows the expected radial hedgehog structure at its center. Liquid crystals droplet are ideal structures to study the effect of regularization on missing cone artifacts. Indeed, their structures are well known as large spherical structures and rapidly varying point defects. Here, for each sample, we displayed the reconstructed RI and directors while including a zoom-in view of the point defects (Fig. 3).
When using TV with a strength of λ = 10−2, the high regularization strength enables the full removal of sample elongation and halo effects. However, the strong regularization constant leads to staircase artifacts where the dielectric tensor elements change abruptly between the patches of piece-wise constant values. This leads to an abrupt change in the director orientation which does not correctly display the expected radial orientation structure near the droplets point defects (Fig. 3, zoom-in). If using weaker Total variation with a strength of λ = 10−3 discrete changes in the director orientations are avoided. Due to the weak regularization, halo artifacts do not fully disappear and the RI between the two beads shows abnormally high RI due to image elongation. Finally, the radial hedgehog in (Fig. 3(b)) also shows a slight elongation. When using the positive semi-definiteness constraint, the radial arrangement of point defects is well resolved, showing a smooth radial variation of the director orientation. The sample RI is also increased, hinting at the correction of missing-cone-induced RI underestimation. Unfortunately, similarly to the original Gerchberg-Papoulis algorithm image noise is worsened, which is particularly visible in the background region.
Finally, we combine a weak TV regularization with the positive semi-definiteness constraint. This regularization scheme yielded the best results showing no halo effect, no elongation, and resolving the point defect structures well. When compared to using only the positive semi-definiteness constraint, noise was also significantly reduced.
In conclusion, a combination of the TV and positive semi-definiteness constraint seems to give the best results. Nevertheless, for samples with a RI lower than the background medium, a strong TV regularization might be helpful at the cost of some staircase artifacts. The use of total generalized variation [60] could also help reduce those artifacts.
4. Discussion
We presented that TV and positive semi-definiteness regularization were able to reduce missing cone artifacts in both the simulation and experimental DTT data. Using both the numerical and experimental studies, we demonstrated that the present approach could significantly enhance the reconstruction accuracy and fidelity of DTT by filling up the information in the missing cone. The crystalline direction fidelity, RI underestimation, and axial resolution were improved. Numerical simulations of the HAN and TN liquid crystals showed the highly precise retrieval of dielectric tensor values. Experimental results also showed significantly enhanced axial resolution, signal-to-noise ratio, and clear tomographic visualization of crystalline directions. On experimental data, the synergistic use of both total variation and a positive semi-definiteness constraint demonstrated the most promises by maximizing image quality while avoiding background noise and staircase artifacts. While the applicability of TV and positive semi-definiteness regularizations have been demonstrated on DTT data, the algorithms and concepts presented here can be extended to other tensor imaging modalities, such as diffusion tensor imaging [61,62] or anisotropic Brillouin microscopy [63]. More advanced regularizations [24,34–36] might further improve the reconstruction quality.
Although the present method requires additional processing time compared to unregularized DTT, several approaches may be used to address this speed issue. For example, newer primal-dual algorithms can be used for the optimization [52–54] or high-performance graphic processing units can be employed with parallel computations. Also, deep learning approaches can be used to apply the TV algorithm to avoid time-consuming iteration steps [64,65]. Then, the present method can be further extended to the time-lapse quantitative analyses of various phenomena related to optical anisotropy.
Another limitation of the present method is the presence of speckle noise, resulting from a highly temporally coherent light source and an off-axis Mach-Zehnder interferometric microscopy. Recently, it was shown that a low-coherence light source could be used for reconstructing three-dimensional reconstruction of RIs of a sample by carefully matching the optical path lengths in interferometry [66], deconvolution microscopy with partially coherent light [67], Fourier ptychographic diffraction tomography [68], or space-domain Kramers–Kronig relations [69,70]. If such a tomographic approach was used in terms of a polarization-sensitive states, it could be used to improve DTT image quality [33].
In view of the unique label-free and quantitative imaging contrast capability exhibited by dielectric tensors down to diffraction-limited resolutions, it could potentially be used for the quantitative imaging and analysis of topological defects in liquid-crystal materials [71]. Moreover, the dielectric tensor information has considerable advantages in histopathological applications over standard polarization microscopy or staining methods, particularly in assay time and cost, and it can also provide quantitative tomographic evaluations of collagen fiber structures which is related to tumor metastasis [71]. Going forward, we envision that this synergistic approach of regularization and DTT could have broad applications, possibly in conjunction with newly emerging active soft materials [72].
Funding
Information & Communications Technology Planning & Evaluation (IITP), Korean government (MSIT) (2021-0-00745); KAIST Institute of Technology Value Creation, Industry Liaison Center (G-CORE Project), Ministry of Science and ICT, South Korea (N11230131); KAIST Institute of Technology Value Creation; National Research Foundation of Korea (2015R1A3A2066550, 2022M3H4A1A02074314); Tomocube Inc.; BK21+; KAIST UP.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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