On the shape of a set of points and lines in the plane

Detailed geometric models of the real world are in increasing demand. LiDAR data is appropriate to reconstruct such models. For urban reconstruction, it works well to reconstruct the individual surfaces of the scene and connect them to form the scene geometry. There are various methods for reconstructing the free-form shape of a point sample on a single surface. However, these methods do not take the context of the surface into account. We present the generalized and guided α-shape: two extensions of the well known α-shape. The generalized α-shape handles lines (generalized points) as if they are infinite collections of points. The guided α-shape handles lines (guides) as indicators for preferred locations for the boundary of the shape. The guided α-shape uses (parts of) these lines as boundary where the points suggest that this is appropriate. We prove that, for an input of n points and m guides, both the generalized and guided α-shape can be constructed in O((n + m) log(n + m)) time. We apply guided α-shapes to urban reconstruction from LiDAR, where neighboring surfaces can be connected conveniently along their intersection lines into adjacent surfaces of a 3D model. We analyze guided α-shapes of both synthetic and real data and show they are consistently better than α-shapes for this application.