Abstract
Current Elastic SfT (Shape from Template) methods are
based on ?2-norm minimization. None can accurately recover the spatial location of the acting forces since ?2-norm
based minimization tends to find the best tradeoff among
noisy data to fit an elastic model. In this work, we study
shapes that are deformed with spatially sparse set of forces.
We propose two formulations for a new class of SfT problems dubbed here SLE-SfT (Sparse Linear Elastic-SfT). The
First ideal formulation uses an ?0-norm to minimize the cardinal of non-zero components of the deforming forces. The
second relaxed formulation uses an ?1-norm to minimize
the sum of absolute values of force components. These new
formulations do not use Solid Boundary Constraints (SBC)
which are usually needed to rigidly position the shape in
the frame of the deformed image. We introduce the Projective Elastic Space Property (PESP) that jointly encodes
the reprojection constraint and the elastic model. We prove
that filling this property is necessary and sufficient for the
relaxed formulation to: (i) retrieve the ground-truth 3D deformed shape, (ii) recover the right spatial domain of nonzero deforming forces. (iii) It also proves that we can rigidly
place the deformed shape in the image frame without using
SBC. Finally, we prove that when filling PESP, resolving
the relaxed formulation provides the same ground-truth solution as the ideal formulation. Results with simulated and
real data show substantial improvements in recovering the
deformed shapes as well as the spatial location of the deforming forces