  Shape-from-Template (SfT) methods estimate 3D surface deformations from a
single monocular RGB camera while assuming a 3D state known in advance (a
template). This is an important yet challenging problem due to the
under-constrained nature of the monocular setting. Existing SfT techniques
predominantly use geometric and simplified deformation models, which often
limits their reconstruction abilities. In contrast to previous works, this
paper proposes a new SfT approach explaining 2D observations through physical
simulations accounting for forces and material properties. Our differentiable
physics simulator regularises the surface evolution and optimises the material
elastic properties such as bending coefficients, stretching stiffness and
density. We use a differentiable renderer to minimise the dense reprojection
error between the estimated 3D states and the input images and recover the
deformation parameters using an adaptive gradient-based optimisation. For the
evaluation, we record with an RGB-D camera challenging real surfaces exposed to
physical forces with various material properties and textures. Our approach
significantly reduces the 3D reconstruction error compared to multiple
competing methods. For the source code and data, see
https://4dqv.mpi-inf.mpg.de/phi-SfT/.
