A liver atlas using the special Euclidean group

Abstract

An atlas is a shape model derived using statistics of a population. Standard models treat local deformations as pure translations and apply linear statistics. They are often inadequate for highly variable anatomical shapes. Non-linear methods has been developed but are generally difficult to implement. This paper proposes encoding shapes using the special Euclidean group SE(3) for model construction. SE(3) is a Lie group, so basic linear algebra can be used to analyze data in non-linear higher-dimensional spaces. This group represents non-linear shape variations by decomposing piecewise-local deformations into rotational and translational components. The method was applied to 49 human liver models that were derived from CT scans. The atlas covered 99% of the population using only three components. Also, the method outperformed the standard method in reconstruction. Encoding shapes as ensembles of elements in the SE(3) group is a simple way of constructing compact shape models.