NeST: Neural Stress Tensor Tomography by leveraging 3D Photoelasticity

TL;DR

This work introduces NeST, an analysis-by-synthesis approach for reconstructing 3D stress tensor fields as neural implicit representations from polarization measurements, and paves the way for scalable non-destructive 3D photoelastic analysis.

Abstract

Photoelasticity enables full-field stress analysis in transparent objects through stress-induced birefringence. Existing techniques are limited to 2D slices and require destructively slicing the object. Recovering the internal 3D stress distribution of the entire object is challenging as it involves solving a tensor tomography problem and handling phase wrapping ambiguities. We introduce NeST, an analysis-by-synthesis approach for reconstructing 3D stress tensor fields as neural implicit representations from polarization measurements. Our key insight is to jointly handle phase unwrapping and tensor tomography using a differentiable forward model based on Jones calculus. Our non-linear model faithfully matches real captures, unlike prior linear approximations. We develop an experimental multi-axis polariscope setup to capture 3D photoelasticity and experimentally demonstrate that NeST reconstructs the internal stress distribution for objects with varying shape and force conditions. Additionally, we showcase novel applications in stress analysis, such as visualizing photoelastic fringes by virtually slicing the object and viewing photoelastic fringes from unseen viewpoints. NeST paves the way for scalable non-destructive 3D photoelastic analysis.

Figures (18)

• Figure 1: Overview of our formulation. We develop a formulation to render stress-induced birefringence from the stress tensor field distribution (a) in a 3D transparent object. The stress field at each point changes the polarization state of a ray passing through that point (Sec. \ref{['sec:birefringence']}, Fig. \ref{['fig:projected_stress_tensor']}). We integrate these polarization changes for all points along the ray to obtain an equivalent Jones matrix (Sec. \ref{['sec:integrated_photoelasticity']}, Fig. \ref{['fig:equivalence_theorem']}). These Jones matrices and consequently the underlying stress tensors are then measured through our multi-axis polariscope capture setup (b) (Sec. \ref{['sec:polariscope']}, Fig. \ref{['fig:render_multiple_views']}). In (c), we visualize the multi-axis polariscope measurements rendered from our formulation for the stress field in (a).
• Figure 2: Modeling stress induced birefringence. Each point in the specimen under stress corresponds to a $3 \times 3$ Cartesian stress tensor (a). A ray passing through this point encodes information about the $2 \times 2$ projection of the stress tensor on a plane perpendicular to the ray (b). The projected stress tensor can be represented by principle stress directions corresponding to solely normal stress (c). Difference in principle stress values results in a phase difference that corresponds to a change in the polarization of the light ray (d).
• Figure 3: Modeling integrated photoelasticity. By ray marching, we sample points along the queried ray (a). We model each segment between the sampled points, $t_{i}$ and $t_{i+1}$ as a retarder with retardance $\delta_i$ and slow-axis orientation $\theta_i$ (b). From the Poincaré equivalence theorem, stack of any number of retarders is equivalent to a single retarder with parameters $(\delta_{\textit{eq}},\theta_{\textit{eq}})$ followed by a rotator with rotation $\gamma_{\mathrm{eq}}$ (c).
• Figure 4: General vs linear 3D photoelasticity model. The proposed general 3D photoelasticity formulation involves converting projected stress tensors at each point on the ray to Jones matrices (first phase wrap) that are then integrated using matrix multiplications (then aggregate). We also derive a first-order approximation of this model that involves first summing the projected stress tensors (first aggregate) and then converting the aggregated stress tensor to Jones matrix (then phase wrap)
• Figure 5: Multi-axis polariscope capture setup By varying yaw-pitch rotations of the specimen and varying rotations of quarter waveplates and polarizers (a), we capture the intensity measurements (b). These measurements exhibit fringes which encode the projections of the underlying stress tensor field.