Left ventricular motion reconstruction based on elastic vector splines

In medical imaging it is common to reconstruct dense motion estimates, from sparse measurements of that motion, using some form of elastic spline (thin-plate spline, snakes and other deformable models, etc.). Usually the elastic spline uses only bending energy (second-order smoothness constraint) or stretching energy (first-order smoothness constraint), or a combination of the two. These elastic splines belong to a family of elastic vector splines called the Laplacian splines. This spline family is derived from an energy minimization functional, which is composed of multiple-order smoothness constraints. These splines can be explicitly tuned to vary the smoothness of the solution according to the deformation in the modeled material/tissue. In this context, it is natural to question which members of the family will reconstruct the motion more accurately. We compare different members of this spline family to assess how well these splines reconstruct human cardiac motion. We find that the commonly used splines (containing first-order and/or second-order smoothness terms only) are not the most accurate for modeling human cardiac motion.