Lagrangian Strain Tensor Computation with Higher Order Variational Models

The reliable estimation of the Lagrangian stress tensor from an image sequence is a challenging problem in mechanical engineering. Since this tensor involves first order motion derivatives, it appears tempting to estimate the optical flow field with a highly accurate variational model and compute its derivatives afterwards. In this paper we explain why this idea is inappropriate due to lower order smoothness assumptions and the ill-posedness of differentiation. As a remedy, we propose a variational framework that performs higher order regularisation of the optical flow field and directly computes the Lagrangian stress tensor from the image measurements. Due to its recursive structure, this framework is very generic. It can incorporate smoothness assumptions of arbitrary high order and allows to compute derivatives of any desired order in a stable way. With a biaxial tensile experiment with an elastomer we demonstrate that our novel approach gives substantially better results for the Lagrangian stress tensor than computing derivatives of the optical flow field. Moreover, it also outperforms a frequently used commercial software that marks the state-of-the-art for Lagrangian stress tensor computation.