Contactless estimation of continuum displacement and mechanical compressibility from image series using a deep learning based framework

TL;DR

The problem addressed is non-contact estimation of material compressibility from optical image series, which is hindered by the computational cost of traditional non-rigid registration and constitutive modelling. The authors propose an end-to-end framework that couples a CNN-based non-rigid image registration module with a regression network that maps reconstructed displacement to a global Poisson's ratio $ν$, trained on synthetic data generated from the 2D Lamé-Navier PDE across discrete $ν$ values. The key findings show that the DL pipeline delivers accurate, high-throughput ν estimation and outperforms a local PDE-based ν estimation, even when displacement fields are imperfect or noisy, with Grad-CAM analyses indicating reliance on higher-order flow features such as vortices. This approach offers a practical path to rapid, contactless characterisation of material compressibility from image data and can be extended to real materials and more complex spatial distributions with physics-informed enhancements and cross-validation.

Abstract

Contactless and non-invasive estimation of mechanical properties of physical media from optical observations is of interest for manifold engineering and biomedical applications, where direct physical measurements are not possible. Conventional approaches to the assessment of image displacement and non-contact material probing typically rely on time-consuming iterative algorithms for non-rigid image registration and constitutive modelling using discretization and iterative numerical solving techniques, such as Finite Element Method (FEM) and Finite Difference Method (FDM), which are not suitable for high-throughput data processing. Here, we present an efficient deep learning based end-to-end approach for the estimation of continuum displacement and material compressibility directly from the image series. Based on two deep neural networks for image registration and material compressibility estimation, this framework outperforms conventional approaches in terms of efficiency and accuracy. In particular, our experimental results show that the deep learning model trained on a set of reference data can accurately determine the material compressibility even in the presence of substantial local deviations of the mapping predicted by image registration from the reference displacement field. Our findings suggest that the remarkable accuracy of the deep learning end-to-end model originates from its ability to assess higher-order cognitive features, such as the vorticity of the vector field, rather than conventional local features of the image displacement.

Figures (8)

• Figure 1: The schematic representation of the U-Net architecture used in the Image Registration scheme.
• Figure 2: Overview of the data flow and the computational framework for estimation of Poisson's ratio from image series (from left to right): FDM is used to compute the reference displacement $u(x)$ for different boundary value problems and values of Poisson's ratio; source image is warped using the FDM-computed displacements to generate target image $T(x)=S(x+u)$; image registration is used to recover the displacement field $u'(x)$ from a pair of source and target (i.e. warped source) images; Poisson's ratio of unseen image pairs is predicted from the displacement $u'(x)$ using a DNN model which was trained on the reference set of displacements and corresponding values of Poisson's ratio.
• Figure 3: An example of a linear elastic BVP and corresponding FDM solutions for two distinctive values of Poisson's ratio $\nu=0$ and $\nu=0.49$ corresponding to high and low compressible material approximations, respectively: (left) a BVP with prescribed boundary displacements, (middle) FDM solution for the Poisson's ratio $\nu=0$ (high compressible material model), (right) FDM solution for the Poisson's ratio $\nu=0.49$ (low compressible material model). For better visualization, displacement vectors are shown for every fifth pixel of the 128x128 image.
• Figure 4: An example of image warping using displacements computed as a FDM solution of linear elastic BVPs: (top, left) source (static) image ($S(x)$), (top, right) FDM-computed displacement field $u$, (bottom, left) target image computed as deformed version of the source images using the FDM displacement $T(x)=S(x+u)$, (bottom, right) overlay of source and target images.
• Figure 5: Comparison of image displacements computed using FDM vs. image registration: (top, left): displacement field computed as an FDM solution of linear elastic BVP, (top, right) displacement reconstructed using deep learning based image registration for a pair of source and target images, (bottom, left) overlay of displacements obtained from FDM (red arrows) vs. displacement reconstructed using image registration (blue arrows), (bottom, right) visualization of the angle between FDM-computed and DNN-reconstructed displacements as a function of displacement magnitude.