  We propose a new methodology to estimate the 3D displacement field of
deformable objects from video sequences using standard monocular cameras. We
solve in real time the complete (possibly visco-)hyperelasticity problem to
properly describe the strain and stress fields that are consistent with the
displacements captured by the images, constrained by real physics. We do not
impose any ad-hoc prior or energy minimization in the external surface, since
the real and complete mechanics problem is solved. This means that we can also
estimate the internal state of the objects, even in occluded areas, just by
observing the external surface and the knowledge of material properties and
geometry. Solving this problem in real time using a realistic constitutive law,
usually non-linear, is out of reach for current systems. To overcome this
difficulty, we solve off-line a parametrized problem that considers each source
of variability in the problem as a new parameter and, consequently, as a new
dimension in the formulation. Model Order Reduction methods allow us to reduce
the dimensionality of the problem, and therefore, its computational cost, while
preserving the visualization of the solution in the high-dimensionality space.
This allows an accurate estimation of the object deformations, improving also
the robustness in the 3D points estimation.
