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The decomposition of deformation: new metrics to enhance shape analysis in medical imaging

Varano, Valerio; Piras, Paolo; Gabriele, Stefano; Teresi, Luciano; Nardinocchi, Paola; Dryden, Ian L.; Torromeo, Concetta; Puddu, Paolo E.


Valerio Varano

Paolo Piras

Stefano Gabriele

Luciano Teresi

Paola Nardinocchi

Professor of Statistics

Concetta Torromeo

Paolo E. Puddu


In landmarks-based Shape Analysis size is measured, in most cases, with Centroid Size. Changes in shape are decomposed in affine and non affine components. Furthermore the non affine component can be in turn decomposed in a series of local deformations (partial warps). If the extent of deformation between two shapes is small, the difference between centroid size and m-Volume increment is barely appreciable. In medical imaging applied to soft tissues bodies can undergo very large deformations, involving large changes in size. The cardiac example, analyzed in the present paper, shows changes in m-Volume that can reach the 60%. We show here that standard Geometric Morphometrics tools (landmarks, Thin Plate Spline, and related decomposition of the deformation) can be generalized to better describe the very large deformations of biological tissues, without losing a synthetic description. In particular, the classical decomposition of the space tangent to the shape space in affine and non affine components is enriched to include also the change in size, in order to give a complete description of the tangent space to the size-and-shape space. The proposed generalization is formulated by means of a new Riemannian metric describing the change in size as change in m-Volume rather than change in Centroid Size. This leads to a redefinition of some aspects of the Kendall’s size-and-shape space without losing Kendall’s original formulation. This new formulation is discussed by means of simulated examples using 2D and 3D platonic shapes as well as a real example from clinical 3D echocardiographic data. We demonstrate that our decomposition based approaches discriminate very effectively healthy subjects from patients affected by Hypertrophic Cardiomyopathy.


Varano, V., Piras, P., Gabriele, S., Teresi, L., Nardinocchi, P., Dryden, I. L., …Puddu, P. E. (in press). The decomposition of deformation: new metrics to enhance shape analysis in medical imaging. Medical Image Analysis, 46,

Journal Article Type Article
Acceptance Date Feb 19, 2018
Online Publication Date Feb 21, 2018
Deposit Date Apr 27, 2018
Publicly Available Date Feb 22, 2019
Journal Medical Image Analysis
Print ISSN 1361-8415
Electronic ISSN 1361-8423
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 46
Keywords Geometric Morphometrics, Decomposition of Deformation, Riemannian Metrics, Size and Shape, Left Ventricle Deformation
Public URL
Publisher URL


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