On the associated family of Delaunay surfaces

We use the generalised Weierstraßrepresentation of Dorfmeister, Pedit and Wu to obtain the associated family of Delaunay surfaces and derive a formula for the neck size of the surface in terms of the entries of the holomorphic potential.

     

Reconstruction Using Witness Complexes

We present a novel reconstruction algorithm that, given an input point set sampled from an object S, builds a one-parameter family of complexes that approximate S at different scales. At a high level, our method is very similar in spirit to Chew’s surface meshing algorithm, with one notable difference though: the restricted Delaunay triangulation is replaced by the witness complex, which makes our algorithm applicable in any metric space. To prove its correctness on curves and surfaces, we highlight the relationship between the witness complex and the restricted Delaunay triangulation in 2d and in 3d. Specifically, we prove that both complexes are equal in 2d and closely related in 3d, under some mild sampling assumptions.     

A topological sampling theorem for Robust boundary reconstruction and image segmentation

Existing theories on shape digitization impose strong constraints on admissible shapes, and require error-free data. Consequently, these theories are not applicable to most real-world situations. In this paper, we propose a new approach that overcomes many of these limitations. It assumes that segmentation algorithms represent the detected boundary by a set of points whose deviation from the true contours is bounded. Given these error bounds, we reconstruct boundary connectivity by means of Delaunay triangulation and α-shapes. We prove that this procedure is guaranteed to result in topologically correct image segmentations under certain realistic conditions. Experiments on real and synthetic images demonstrate the good performance of the new method and confirm the predictions of our theory.     

A weak characterisation of the Delaunay triangulation

We consider a new construction, the weak Delaunay triangulation of a finite point set in a metric space, which contains as a subcomplex the traditional (strong) Delaunay triangulation. The two simplicial complexes turn out to be equal for point sets in Euclidean space, as well as in the (hemi)sphere, hyperbolic space, and certain other geometries. There are weighted and approximate versions of the weak and strong complexes in all these geometries, and we prove equality theorems in those cases also. On the other hand, for discrete metric spaces the weak and strong complexes are decidedly different. We give a short empirical demonstration that weak Delaunay complexes can lead to dramatically clean results in the problem of estimating the homology groups of a manifold represented by a finite point sample.      

Structure-transport correlation for the diffusive tortuosity of bulk, monodisperse, random sphere packings

The mass transport properties of bulk random sphere packings depend primarily on the bed (external) porosity ε, but also on the packing microstructure. We investigate the influence of the packing microstructure on the diffusive tortuosity τ=D(m)/D(eff), which relates the bulk diffusion coefficient (D(m)) to the effective (asymptotic) diffusion coefficient in a porous medium (D(eff)), by numerical simulations of diffusion in a set of computer-generated, monodisperse, hard-sphere packings. Variation of packing generation algorithm and protocol yielded four Jodrey-Tory and two Monte Carlo packing types with systematically varied degrees of microstructural heterogeneity in the range between the random-close and the random-loose packing limit (ε=0.366-0.46). The distinctive tortuosity-porosity scaling of the packing types is influenced by the extent to which the structural environment of individual pores varies in a packing, and to quantify this influence we propose a measure based on Delaunay tessellation. We demonstrate that the ratio of the minimum to the maximum void face area of a Delaunay tetrahedron around a pore between four adjacent spheres, (A(min)/A(max))(D), is a measure for the structural heterogeneity in the direct environment of this pore, and that the standard deviation σ of the (A(min)/A(max))(D)-distribution considering all pores in a packing mimics the tortuosity-porosity scaling of the generated packing types. Thus, σ(A(min)/A(max))(D) provides a structure-transport correlation for diffusion in bulk, monodisperse, random sphere packings.     

Delaunay triangulation and the convex hull ofn points in expected linear time

An algorithm is presented which produces a Delaunay triangulation ofn points in the Euclidean plane in expected linear time. The expected execution time is achieved when the data are (not too far from) uniformly distributed. A modification of the algorithm discussed in the appendix treats most of the non-uniform distributions. The basis of this algorithm is a geographical partitioning of the plane into boxes by the well-known Radix-sort algorithm. This partitioning is also used as a basis for a linear time algorithm for finding the convex hull ofn points in the Euclidean plane.     

Delaunay Transformations of a Delaunay Polytope

Let P be a Delaunay polytope in ⁿ . Let T(P) denote the set of affine bijections f of ⁿ for which f (P) is again a Delaunay polytope. The relation: fg if f, g differ by an orthogonal transformation and/or a translation is an equivalence relation on T(P). We show that the dimension (in the topological sense) of the quotient set T(P)/ coincides with another parameter of P, namely, with its rank.Let V denote the set of vertices of P and let dp denote the distance on V defined by dp(u, v)=u–v ² for u, vV. Assouad [1] shows that dp belongs to the cone |V|:={d | _u,vV b _u b _v d(u,v) 0 for b ^V with _uV b _u = 1}. Then, the rank of P is defined as the dimension of the smallest face of the cone |V| that contains dp. AMS Subject Classification (1991): 11H06, 52C07.     

Angular invariants and local order in the structures of substances

A method is suggested for analyzing the model structures of crystals by calculating the spherical coordinates of the normals to the simplex faces of the simplicial Delaunay partitioning of a set of points (atoms). The normals to the simplex faces of the Delaunay partitioning of the crystal structure characterize the structure at the local level. An algorithm for constructing the invariant of the crystal structure (crystal module) was considered. Crystals modules were constructed for hexagonal and cubic ices.     

Existence of Delaunay Pairwise Gibbs Point Process with Superstable Component

The present stuffy deals with the existence of Delaunay pairwise Gibbs point process with superstable component by using the well-known Preston theorem. In particular, we prove the stability, the lower regularity, and the quasilocality properties of the Delaunay model.