GPU-accelerated locally injective shape deformation

We present a highly efficient planar meshless shape deformation algorithm. Our method is based on an unconstrained minimization of isometric energies, and is guaranteed to produce C⁸ locally injective maps by operating within a reduced dimensional subspace of harmonic maps. We extend the harmonic subspace of [Chen and Weber 2015] to support multiply-connected domains, and further provide a generalization of the bounded distortion theorem that appeared in that paper. Our harmonic map, as well as the gradient and the Hessian of our isometric energies possess closed-form expressions. A key result is a simple-and-fast analytic modification of the Hessian of the energy such that it is positive definite, which is crucial for the successful operation of a Newton solver. The method is straightforward to implement and is specifically designed to harness the processing power of modern graphics hardware. Our modified Newton iterations are shown to be extremely effective, leading to fast convergence after a handful of iterations, while each iteration is fast due to a combination of a number of factors, such as the smoothness and the low dimensionality of the subspace, the closed-form expressions for the differentials, and the avoidance of expensive strategies to ensure positive definiteness. The entire pipeline is carried out on the GPU, leading to deformations that are significantly faster to compute than the state-of-the-art.