Local and global energies for shape analysis in medical imaging

Abstract

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 three‐fold 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. A new approach is here presented in order to calculate the bending energy on the body alone (body bending energy) and 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 be preserved 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 discussed in order to better represent the relationship between the Γ‐energy, the strain energy, and the so‐called bending energy densities.

Current Options for Visualization of Local Deformation in Modern Shape Analysis Applied to Paleobiological Case Studies

“…gives the decay index Varano et al (2018), i.e., the ratio between bending energy computed on whole R m and on the body only. Recently, Varano et al (2019) proposed the construction of the body bending energy matrix B in order to restrict it exclusively within the physical boundaries of objects involved in the deformation analysis. The bending energy can be considered as a pseudo-distance as it vanishes for globally affine transformations.…”