Global 2D digital image correlation for motion estimation in a finite element framework: a variational formulation and a regularized, pyramidal, multi-grid implementation

In this study, the inverse problem of reconstructing the in-plane (2D) displacements of a monitored surface through a sequence of two-dimensional digital images, severely ill-posed in Hadamard’s sense, is deeply investigated. A novel variational formulation is presented for the continuum 2D digital image correlation problem, and critical issues such as semi-coercivity and solution multiplicity are discussed by functional analysis tools. In the framework of a Galerkin, finite element discretization of the displacement field, a robust implementation for 2D digital image correlation is outlined, aiming to attenuate the spurious oscillations which corrupt the deformation scenario, especially when very fine meshes are adopted. Recourse is made to a hierarchical family of grids linked by suitable restriction and prolongation operators and defined over an image pyramid. Multi-grid cycles are performed ascending and descending along the pyramid, with only one Newton iteration per level irrespective of the tolerance satisfaction, as if the problem were linear. At each level, the conventional least-square matching functional is herein enriched by a Tychonoff regularization provision, preserving the solution against an unstable response. The algorithm is assessed on the basis of both synthetic and truly experimental image pairs.