Detection of continuous symmetries in 3D objects from sparse measurements through probabilistic neural networks

The traditional approach to the geometrical dimensioning and tolerancing of mechanical components and assemblies essentially relies on definitions by examples. Over the last few years that approach is increasingly being challenged by a unifying and theoretically sound perspective. The technical commission ISO/TC213 devised a very elegant and powerful classification of 3D objects based on their symmetries. The authors embed that classification in a fully fledged probabilistic framework and propose a practical methodology for the statistical recognition of 3D shapes from sparse, noisy measurements. To this purpose we first extend the ISO/TC213 partitioning to probability density functions so as to include the measurement process in the formalism. Then we make use of unsupervised probabilistic neural networks to build a semi-parametric probabilistic model for each class of symmetry. Finally, we rank all competing models against clouds of measured points according to their leave-one-out likelihood.