Elastic Shape-from-Template with Spatially Sparse Deforming Forces

Current Elastic SfT (Shape from Template) methods are based on ℓ
₂
-norm minimization. None can accurately recover the spatial location of the acting forces since ℓ
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 ℓ
₀
-norm to minimize the cardinal of non-zero components of the deforming forces. The second relaxed formulation uses an ℓ
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 non-zero 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.