Local and global energies for shape analysis in medical imaging. (8th January 2020) - Varano, Valerio; Piras, Paolo; Gabriele, Stefano; Teresi, Luciano; Nardinocchi, Paola; Dryden, Ian L.; Torromeo, Concetta; Schiariti, Michele; Puddu, Paolo E.
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. Abstract : In this paper, the new concept of Body Bending Energy (BEB) is introduced to measure the distances between shapes, by means of a suitable matrix named the BEB Matrix. A new Riemannian metric and a new Riemannian connection, on the basis of the BEB matrix, are then introduced in the size‐and‐shape space, and used to decompose the deformations in three meaningful subspaces. This new Riemannian structure is applied to perform shape analysis on data coming from 3D speckle‐tracking‐echocardiography of human left ventricles.