In urban scenes, many of the surfaces are planar and bounded by simple shapes. In a laser scan of such a scene, these simple shapes can still be identified. We present a one-parameter algorithm that can identify point sets on a plane for which a rectangle is a fitting boundary. These rectangles have a guaranteed density: no large part of the rectangle is empty of points. We prove that our algor...

Existing theories on 3D surface reconstruction impose strong constraints on feasible object shapes and often require error-free measurements. Moreover these theories can often only be applied to binary segmentations, i.e. the separation of an object from its background. We use the Delaunay complex and α-shapes to prove that topologically correct segmentations can be obtained under much more rea...

The Delaunay triangulation is generated from a points set and a structuring element of type disc. In the Delaunay triangulation definition, replacing the disc by a planar-connected region (we call a molecule), which is a union of a fixed number of discs, allows construction of what we name the molecular graphs. In a finite points set, the molecular graphs record the empty regions which are iden...

For a given point set S in a plane, we develop a distributed algorithm to compute the α−shape of S. α−shapes are well known geometric objects which generalize the idea of a convex hull, and provide a good definition for the shape of S. We assume that the distances between pairs of points which are closer than a certain distance r > 0 are provided, and we show constructively that this informatio...

There is a growing interest for the problem of reconslructing thc shape of an object from multiple range images. Several methods, based on heuristics, have been described in the literature. We propose the use of alpha-shapes, which allow us to give a formal characterization of the reconstruction problem and to prove that, when certain sampling requirements are satisfied, the reconstructed alpha...

The quality of 3D object transformation very often relies on the quality of the matching between the two objects. Geometric considerations regarding scale, rotation and translation, and user defined constraints are used to control the matching. Nevertheless, more complex measures regarding volumetric or topological similarity between the shapes are more difficult to define even manually. Choosi...

Computing a polygon defining a set of planar points is a classical problem of modern computational geometry. In laboratory experiments we demonstrate that a concave hull, a connected α-shape without holes, of a finite planar set is approximated by slime mould Physarum polycephalum. We represent planar points with sources of long-distance attractants and short-distance repellents and inoculate a...

Extracting individual buildings and determining their footprints have been extensively studied towards 3D Building reconstruction. Though most previous works show promising results, it is yet a nontrivial task, especially in dense residential areas. This paper discusses a methodology for resolving this issue. The proposed approach starts with separating ground and nonground LiDAR points. In the...

Existing theories on shape digitization impose strong constraints on feasible shapes and require error-free measurements. We use Delaunay triangulation and α-shapes to prove that topologically correct segmentations can be obtained under much more realistic conditions. Our key assumption is that sampling points represent object boundaries with a certain maximum error. Experiments on real and gen...