A GPU-based elastic shape registration approach in implicit spaces

Abstract

In this paper, we present a GPU-based implementation of an elastic shape registration approach in implicit spaces. Shapes are represented using signed distance functions, while deformations are modeled by cubic B-splines. In a variational framework, an incremental free form deformation strategy is adopted to handle smooth deformations through an adaptive size control lattice grid. The grid control points are estimated by a closed-form solution which avoids the gradient descent iterations. However, even this solution is very far from real time. We show in detail that such an algorithm is computationally expensive with a time complexity of $${\mathbf O} (NCP_xNCP^2X^2Y^2)$$ where $$NCP_x$$ and NCP are the grid lattice resolution parameters in the shape domain of size $$X\times Y$$. Moreover, the problem becomes more time-consuming with the increase in the number of control points because this requires the execution of the incremental algorithm several times. The closed-form solution was implemented using eight different GPU techniques. Our experimental results demonstrate speedups of more than $$150{\times}$$ compared to the $$\texttt {C}$$ implementation on a CPU.