Accurate Rendering of Liquid-Crystals and Inhomogeneous Optically Anisotropic Media

Shlomi Steinberg

ACM Transactions on Graphics

Publication date: April 30th, 2020
doi: 10.1145/3381748

Liquid-crystal shells rendered using our method. The shells are formed when a spherical liquid-crystal droplet suspended in liquid is injected with an optically isotropic liquid (see figure 9 in the paper for an illustration of the geometry). The imaging is performed by passing linearly polarized light, emitted from a source at −𝑦, through the shell and then through a polarizer oriented perpendicular to the source polarization (“cross polarized” light) and finally captured by a camera located at +𝑦. The orientation of the polarizers is depicted in the bottom left corner of the figures. The shells were illuminated using blue (480 nm), green (580 nm), red (630 nm) and white (illuminant E) light. The artefacts around the left and right edges of the rendered shells are numerical errors. See the paper for a comparison with captured micrograph photos of the same liquid-crystal shells.

Abstract

We present a novel method for devising a closed-form analytic expression to the light transport through the bulk of inhomogeneous optically anisotropic media. Those optically anisotropic materials, e.g., liquid crystals and elastic fluids, arise in a plethora of established applications and exciting new research, however current state-of-the-art methods of visually deducing their optical properties or rendering their appearance are either lacking or non-existent. We formulate our light transport problem under the context of electromagnetism and derive, from first principles, a differential equation of the transmitted complex wave fields that fully accounts for the complicated interference phenomena that arise. At the core of our proposed rendering framework is a powerful mathematical representation, carefully crafted to enable us to produce highly accurate analytic approximative solutions for the light transport. This approach is previously unused in computer rendering, and our framework is capable of rendering accurately optically anisotropic materials with varying optically properties at orders of magnitude faster than existing methods. We demonstrate a few practical applications of our method and validate it against polarized photos of liquid-crystals as well as numerically against numerical solvers and qualitatively against brute-force renderings.

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Cite

@article{Steinberg_Rendering_Liquid_Crystals_2020,
 	doi = {10.1145/3381748},
 	url = {https://doi.org/10.1145%2F3381748},
 	year = 2020,
 	month = {jun},
 	publisher = {Association for Computing Machinery ({ACM})},
 	volume = {39},
 	number = {3},
 	pages = {1--23},
 	author = {Shlomi Steinberg},
 	title = {Accurate Rendering of Liquid-Crystals and Inhomogeneous Optically Anisotropic Media},
 	journal = {{ACM} Transactions on Graphics}
 }

Overview of the Mathematics

The primary contribution in this paper is a mathematical representation and decomposition of an ordinary differential equation (ODE) that, in the case of the problem studied in this paper, allows the fast computation of the refractive light transport through a liquid-crystal. This approach is rather general and can be used for other problems.

The starting point is the operator-valued ODE

$\frac{\operatorname{d}}{\operatorname{d}y}\psi\left(y\right) = \mathbf{A}\left(y\right)\psi\left(y\right) ~,$

with some initial condition $\psi\left(0\right) = \psi_0$ , and where $\mathbf{A}(y)$ is a complex-valued $2\times 2$ matrix, that is a function of $y$ . We seek a solution $\psi(y)$ . Assume that $\mathbf{A}$ does not commute with itself, i.e. it does not hold that for every $y_1,y_2$ , $\mathbf{A}(y_1)\mathbf{A}(y_2) = \mathbf{A}(y_2)\mathbf{A}(y_1)$ . Otherwise, the solution reduces to a simple matrix exponential, see matrix differential equation.

To proceed we apply powerful tools that arise from the Magnus expansion: Expand the matrix $\mathbf{A}$ as a linear combination of some constant (independent of $y$ ), but otherwise arbitrary matrices, as follows

$\mathbf{A}(y) = \sum_{j=1}^m a_j(y) \mathbf{X}_j ~,$

where $a_j(y)$ are the scalar-valued function coefficients. The solution to the ODE then becomes the matrix exponential

$\psi(y) = e^{\sum_{j=1}^m f_j(y)\mathbf{X}_j} ~.$

The relation between the (known) $a_j$ and (unknown) $f_j$ are given by a rather complicated differential system (see Wei and Norman [1964] or the paper). To devise a simple enough system, we choose the following matrices:

$\mathbf{X}_1 = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} ~, \quad \mathbf{X}_2 = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} ~, \quad \mathbf{X}_3 = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} ~, \quad \mathbf{X}_4 = \mathbf{I} ~.$

Working through the math yields the following (exact) final solution:

$\psi(y) = e^{f_4}\begin{bmatrix} \left(1+f_1f_2\right)e^{f_3} & f_1e^{-f_3} \\ f_2e^{f_3} & e^{-f_3} \end{bmatrix} \tag{1} ~,$

with the scalar ODEs for $f_{1,2,3,4}$ given by

$\begin{aligned} \frac{\operatorname{d}}{\operatorname{d}{y}} f_1(y) &= -a_2(y)f_1(y)^2 + 2a_3(y)f_1(y) + a_1(y) ~, \\ \frac{\operatorname{d}}{\operatorname{d}{y}} f_2(y) &= 2\left[a_2(y)f_1(y) - a_3(y)\right]f_2(y) + a_2(y) ~, \\ \frac{\operatorname{d}}{\operatorname{d}{y}} f_3(y) &= -a_2(y)f_1(y) + a_3(y) ~, \\ \frac{\operatorname{d}}{\operatorname{d}{y}} f_4(y) &= a_4(y) ~. \end{aligned}$

Once expressions for $f_{1,2,3,4}$ are found, $\psi(y)$ is computed directly via Eq. 1. We have effectively reduced the original operator-valued ODE into the set of 4 scalar-valued ODEs above. Solving these scalar ODEs is, in general, a far easier problem compared to the operator-valued ODE we started with: The ODEs for $f_3,f_4$ are separable and reduce to integration, while the ODE for $f_2$ is linear. The only difficulty arises in the quadratic $f_1$ ODE, which is known as the Riccati equation.

In addition, note, that because $\mathbf{A}$ and the scalar functions $f_{1,2,3,4}$ might be complex, the original operator-valued ODE might be highly oscillatory, frustrating numerical approaches and approximative solutions. The representation of the solution as a product of matrix exponentials (Eq. 1) “extracts” some of this oscillatory behaviour into the exponents in Eq. 1, which means that it is reasonable to assume that the scalar-valued ODEs are, in general, “better behaved” than the original problem. This is exactly what happens in the context discussed in the paper, and the scalar ODEs (which are an exact representation of the original problem) are orders-of-magnitude faster to solve numerically.

The analysis above applies to an arbitrary operator-valued ODE with a complex-valued $2\times 2$ matrix $\mathbf{A}$ , and can be extended in a like-manner to higher dimensions.