  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.
