Estimation of mixtures of probabilistic PCA with stochastic EM for the 3D biplanar reconstruction of scoliotic rib cage

In this paper, we present a robust method for estimating the model parameters in a mixture of probabilistic principal component analyzers. This method is based on the stochastic version of the expectation maximization (SEM) algorithm. Parameters of this mixture model are herein used to constrain the 3D reconstruction problem of scoliotic rib cage from a pair of planar and conventional calibrated radiographic images (postero-anterior with normal incidence (I_PA) and lateral (I_LAT)). More precisely, the proposed PPCA mixture model is herein robustly exploited for dimensionality reduction and to get a set of probabilistic prior models associated to each detected class of pathological deformations observed on a representative training scoliotic rib cage population. By using an appropriate likelihood and for each considered class-conditional prior model, the proposed 3D reconstruction is stated as an energy function minimization problem, which is solved with a stochastic optimization algorithm. The optimal 3D reconstruction then corresponds to the class of deformation and parameters leading to the minimal energy. This 3D method of reconstruction has been successfully tested on several biplanar radiographic images, yielding very promising results.