Séminaire de Mathématique-Biologie
# Geometric Morphometrics for Motion Analysis of Biological Soft Tissues

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Europe/Paris

Amphithéâtre Léon Motchane (IHES)
### Amphithéâtre Léon Motchane

#### IHES

Le Bois-Marie
35, route de Chartres
91440 Bures-sur-Yvette

Description

Biological soft tissues can suffer for very large size and shape changes during their motion, and the ability to quantify and compare the movements of organisms is a central focus of many studies in biology: our goal is to compare the motion of a same organ from different individuals, as example, the motion of the beating heart from different humans.

Geometric Morphometrics began with the seminal contributions of D.G. Kendall, gathered in the book Shape and Shape Theory (1999) where it has been proposed to sample the configuration of a body through the m coordinates of k landmarks; thus a configuration is represented with a point in an m x k space, called the Configuration Space.

Then, to effectively represent a shape, it must be defined a quotient of the Configuration Space with respect to a Lie Group of transformations, a group that is selected according to specific needs.

Here, we discuss different possible choices of quotient spaces and their implications: dealing with soft tissues the differences between individuals can be very large, and thus the crucial point for quantifying differences in the motion is to filter inter-individual shape differences.

We proposed to solve this problem by performing a data centering in the so-called Size-and-Shape Space, by means of the Riemannian parallel transport. Theoretical considerations, together with analysis on real heart data from human left ventricle suggest that when configurations differ for small Procrustes Distances, the parallel transport can be well approximated by a Euclidean translation, after a hierarchical alignment.

Geometric Morphometrics began with the seminal contributions of D.G. Kendall, gathered in the book Shape and Shape Theory (1999) where it has been proposed to sample the configuration of a body through the m coordinates of k landmarks; thus a configuration is represented with a point in an m x k space, called the Configuration Space.

Then, to effectively represent a shape, it must be defined a quotient of the Configuration Space with respect to a Lie Group of transformations, a group that is selected according to specific needs.

Here, we discuss different possible choices of quotient spaces and their implications: dealing with soft tissues the differences between individuals can be very large, and thus the crucial point for quantifying differences in the motion is to filter inter-individual shape differences.

We proposed to solve this problem by performing a data centering in the so-called Size-and-Shape Space, by means of the Riemannian parallel transport. Theoretical considerations, together with analysis on real heart data from human left ventricle suggest that when configurations differ for small Procrustes Distances, the parallel transport can be well approximated by a Euclidean translation, after a hierarchical alignment.

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