In this paper, we present a novel framework for analyzing and segmenting point-sampled 3D objects. Our algorithm computes a decomposition of a given point set surface into meaningful components, which are delimited by line features and deep concavities. Central to our method is the extension of the scale-space theory to the three-dimensional space to allow feature analysis and classification at different scales. Then, a new surface classifier is computed and used in an anisotropic diffusion process via partial differential equations (PDEs). The algorithm avoids the misclassifications due to fuzzy and incomplete line features. Our algorithm operates directly on points requiring no vertex connectivity information. We demonstrate and discuss its performance on a collection of point sampled 3D objects including CAD and natural models. Applications include 3D shape matching and retrieval, surface reconstruction and feature preserving simplification.