E fficient computational tools for the statistical analysis of shape and asymmetryof 3D point sets

In this thesis, we propose a set of tools for the statistical quantification and comparison of asymmetry within a population or between two populations of structures represented by 3D point sets. These tools are designed in a coherent framework relying on a modeling of the point sets as mixtures of probability density functions depending on parameters describing for example a reflection or a 3D deformation. The estimation of the parameters best superposing the mixtures is performed with variants of the Expectation-Maximisation algorithm. Particularly, we propose algorithms allowing i) the estimation of a symmetry plane of a point set, ii) the estimation of a symmetry surface of a point set, iii) the estimation of a non-linear deformation best superposing 2 point sets and iv) the building of statistical shape models from a set of point sets. Then, we show how to combine these methods to build statistical asymmetry maps (mean, standard deviation, significant asymmetries, significant differences, {\it etc}) and global measurements characterising the asymmetry of the populations under study. Finally, we use our tools to compare the cortical asymmetries of 2 Situs Inversus subjects (congenital condition in which the major visceral organs are mirrored from their normal positions) with those of a control population.