A fast and memory-efficient algorithm for smooth interpolation of polyrigid transformations: application to human joint tracking

The log Euclidean polyrigid registration framework provides a way to smoothly estimate and interpolate poly-rigid/affine transformations for which the invertibility is guaranteed. This powerful and flexible mathematical framework is currently being used to track the human joint dynamics by first imposing bone rigidity constraints in order to synthetize the spatio-temporal joint deformations later. However, since no closed-form exists, then a computationally expensive integration of ordinary differential equations (ODEs) is required to perform image registration using this framework. To tackle this problem, the exponential map for solving these ODEs is computed using the scaling and squaring method in the literature. In this paper, we propose an algorithm using a matrix diagonalization based method for smooth interpolation of homogeneous polyrigid transformations of human joints during motion. The use of this alternative computational approach to integrate ODEs is well motivated by the fact that bone rigid transformations satisfy the mechanical constraints of human joint motion, which provide conditions that guarantee the diagonalizability of local bone transformations and consequently of the resulting joint transformations. In a comparison with the scaling and squaring method, we discuss the usefulness of the matrix eigendecomposition technique which reduces significantly the computational burden associated with the computation of matrix exponential over a dense regular grid. Finally, we have applied the method to enhance the temporal resolution of dynamic MRI sequences of the ankle joint. To conclude, numerical experiments show that the eigendecomposition method is more capable of balancing the trade-off between accuracy, computation time, and memory requirements.