Regularization of dielectric tensor tomography using total variation HERVE HUGONNET 12 SEUNGWOO SHIN 3 AND YONGKEUNPARK124

2025-04-26 0 0 2.05MB 7 页 10玖币

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Regularization of dielectric tensor tomography using total

variation

HERVE HUGONNET,1,2 † SEUNGWOO SHIN ,3 † AND YONGKEUNPARK1,2,4,*

1Department of Physics, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 34141, South Korea

2KAIST Institute for Health Science and Technology, KAIST, Daejeon 34141, South Korea

3Department of Physics, University of California at Santa Barbara, Santa Barbara, CA 93106, USA

4Tomocube Inc., Daejeon 34109, South Korea

†These authors equally contributed to the work.

*yk.park@kaist.ac.kr

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 present the generalization of total

variation regularization to three-dimensional tensor distributions. In particular, demonstrate the reduction of artifacts when

applied to dielectric tensor tomography.

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. 1a), 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].

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–Saxton 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 use of such methods. Other minimization-based techniques, such

as total variation (TV) regularization [24, 25] and other optimization methods using the image gradient [26-30] 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; however, their applicability to tensor data remains to be demonstrated. The last class of regularization uses deep

learning to fill in missing data [31].

Here we propose and demonstrate the generalization of TV regularization to 3D tensor distributions. In particular, we focus

on applying TV to DTT measurements, presenting that component-wise regularization of tensor data from DTT measurements.

We show that using TV can greatly reduce missing cone artifacts: improving both principal RI and crystalline structure direction

information.

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 [32]. 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 symmetric 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 [32].

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, leading 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, 33].

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.

DTT reconstruction is based on polarization-dependent light-field measurements obtained using off-axis holographic

microscopy (Fig. 1b). For a given illumination wave vector

ill



and polarization

 

, the field

 

transmitted through the sample

is recorded. The illumination was then slightly tilted using

ill

k∆

to obtain a third field. Using three fields, the dielectric tensor



can then be expressed as

11 1

22 2

2 [ ]( )

( ) 2 [ ]( )

(1 )

2 [ ]( )

z ill

ill

z ill

ik p k k p

r ik p k k p

ir k p

ik p k k

εψ

−−

−





−⋅ +







= −⋅ +





− ⋅∆





−⋅ +





     

      

  

   



Using

( )

xyz

k kkk=



, the Fourier space coordinate system. This process is repeated for several illumination angles to

populate the Fourier space (Fig. 1d) [1, 34].

2.1 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 a missing cone

problem [35]. The name of the missing cone problem comes from the fact that non-accessible spatial frequencies form a cone

in the Fourier space (Figs. 1c−d). The missing cone problem is a common problem in tomographic measurements with a limited

illumination angle [20, 36] and also occurs in X-ray computed tomography [37], three-dimensional transmitted electron

microscopy [38], optical diffraction tomography [39-42], and widefield fluorescence imaging [43, 44].

Various computational methods, called regularization, have been developed to fill in the missing information. A known

constraint of a sample was 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 [45].

Image plane

PBS

DMD

LCR Condenser

Sample

Objective

HWP BS Pol

Camera

Pol

Camera

Laser

(b)

Dielectric tensor tomography

Polarization microscope

Polarizer Intensity

Field Dielectric tensor

(a)

1.53 1.557

3µm

(c)

Polarization

Computational step

Light propagation

Fourier plane

Missing cone

(d)

-6

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RegularizationofdielectrictensortomographyusingtotalvariationHERVEHUGONNET,1,2†SEUNGWOOSHIN,3†ANDYONGKEUNPARK1,2,4,*1DepartmentofPhysics,KoreaAdvancedInstituteofScienceandTechnology(KAIST),Daejeon34141,SouthKorea2KAISTInstituteforHealthScienceandTechnology,KAIST,Daejeon34141,SouthKorea3DepartmentofP...

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Regularization of dielectric tensor tomography using total variation HERVE HUGONNET 12 SEUNGWOO SHIN 3 AND YONGKEUNPARK124

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