Probabilistic Multi-Shape Representation Using an Isometric Log-Ratio Mapping

Several sources of uncertainties in shape boundaries in medical images have motivated the use of probabilistic labeling approaches. Although it is well-known that the sample space for the probabilistic representation of a pixel is the unit simplex, standard techniques of statistical shape analysis (e.g., principal component analysis) have been applied to probabilistic data as if they lie in the unconstrained real Euclidean space. Since these techniques are not constrained to the geometry of the simplex, the statistically feasible data produced end up representing invalid (out of the simplex) shapes. By making use of methods for dealing with what is known as compositional or closed data, we propose a new framework intrinsic to the unit simplex for statistical analysis of probabilistic multi-shape anatomy. In this framework, the isometric log-ratio (ILR) transformation is used to isometrically and bijectively map the simplex to the Euclidean real space, where data are analyzed in the same way as unconstrained data and then back-transformed to the simplex. We demonstrate favorable properties of ILR over existing mappings (e.g., LogOdds). Our results on synthetic and brain data exhibit a more accurate statistical analysis of probabilistic shapes.