Local and Global Energies for Shape Analysis in Medical Imaging.

In a previous contribution a new Riemannian shape space, named TPS space, was introduced to perform statistics on shape data. This space was endowed with a Riemannian metric and a flat connection, with torsion, compatible with the given metric. This connection allows the definition of a Parallel Transport of the deformation compatible with the threefold decomposition in spherical, deviatoric and non affine components. Such a Parallel Transport also conserves the Γ-energy, strictly related to the total elastic strain energy stored by the body in the original deformation. New machinery is here presented in order to calculate the bending energy on the body only (body bending energy) in order to restrict it exclusively within physical boundaries of objects involved in the deformation analysis. The novelty of this new procedure resides in the fact that we propose a new metric to conserve during the TPS direct transport. This allows transporting the shape change more coherently with the mechanical meaning of the deformation. The geometry of the TPS Space is then further developed in order to better represent the relationship between the Γ-energy, the strain energy and the so called bending-energy densities.