The isometric log-ratio transform for probabilistic multi-label anatomical shape representation

Sources of uncertainty in the boundaries of structures in medical images have motivated the use of probabilistic labels in segmentation applications. An important component in many medical image segmentation tasks is the use of a shape model, often generated by applying statistical techniques to training data. Standard statistical techniques (e.g., principal component analysis) often assume data lies in an unconstrained vector space, but probabilistic labels are constrained to the unit simplex. If these statistical techniques are used directly on probabilistic labels, relative uncertainty information can be sacrificed. A standard method for facilitating analysis of probabilistic labels is to map them to a vector space using the LogOdds transform. However, the LogOdds transform is asymmetric in one of the labels, which skews results in some applications. The isometric log-ratio (ILR) transform is a symmetrized version of the LogOdds transform, and is so named as it is an isometry between the Aitchison geometry, the inherent geometry of the simplex, and standard Euclidean geometry. We explore how to interpret the Aitchison geometry when applied to probabilistic labels in medical image segmentation applications. We demonstrate the differences when applying the LogOdds transform or the ILR transform to probabilistic labels prior to statistical analysis. Specifically, we show that statistical analysis of ILR transformed data better captures the variability of anatomical shapes in cases where multiple different foreground regions share boundaries (as opposed to foreground-background boundaries).