Multi-scale analysis of discrete point sets

Presents the shape of a sparse point set S in R² . A crucial step in finding the shape of a sparse point set is the definition of its boundary. This boundary is a graph indicating a relation among the elements of S. No well defined definition of such a boundary is found in literature. For continuous point sets this problem does not exist as the boundary has a unique definition. The authors pose general criteria a boundary definition should satisfy and show that the α-graph satisfies those criteria. The boundary is a function of the scale parameter α. The authors further show that the α-graph has a strong relation with mathematical morphology. As an application the use of the α-graph in the multi-scale recognition of industrial objects is shown