Hisamoto Hiyoshi

TWO GENERALIZATIONS OF AN INTERPOLANT BASED ON VORONOI DIAGRAMS

Recently, the authors found an interpolant using Voronoi diagrams that differs from Sibsons interpolant. This paper generalizes our interpolant in two directions: one is to general-dimensional data, and the other is to data distributed continuously on curves. The Minkowskis theorem is used as the basic principle in generalization.

Improving continuity of Voronoi-based interpolation over Delaunay spheres

There are two types of the discontinuity of the original Voronoi-based interpolants: one appears on the data sites and the other on the Delaunay spheres. Some techniques are known for reducing the first type of the discontinuity, but not for the second type. This is mainly because the second type of the discontinuity comes from the coordinate systems used for the interpolants. This paper proposes a sequence of new coordinate systems, called the kth-order standard coordinates, for all nonnegative integers k, and shows that the interpolant generated by the kth-order standard coordinates have C^k continuity on the Delaunay spheres. The previously known Voronoi-based interpolants coincide with the cases k=0 and k=1. Hence, the standard coordinate systems constructed in this paper can reduce the second type of the discontinuity as much as we want. In addition, this paper derives a formula for the gradient of the standard coordinates.

Closed Curve Reconstruction from Unorganized Sample Points

This paper considers the problem for reconstructing closed curves from unorganized sample points, where the curve to be reconstructed consists of a finite number of pairwise disjoint components of simple closed curves. This paper formulates this problem as a linear integer programming problem, and proposes an algorithm, called ZERO-ONE, based on it. In addition, two other heuristic algorithms called PASTE and SCISSORS are proposed for reducing computational time. Experimental results show that these algorithms yield exact polygonal reconstruction if the density of the sample points is high enough. In addition, it is shown that SCISSORS can be modified for the reconstruction problem of curves with boundaries.

Improving the Global Continuity of the Natural Neighbor Interpolation

The natural neighbor interpolation is a potential interpolation method for multidimensional data. However, only globally C1 interpolants have been known so far. This paper proposes a globally C2 interpolant, and write it in an explicit form. When the data are supplied to the interpolant from a third-degree polynomial, the interpolant can reproduce that polynomial exactly. The idea used to derive the interpolant is applicable to obtain a globally Ck interpolant for an arbitrary non-negative integer k. Hence, this paper gets rid of the continuity limitation of the natural neighbor interpolation, and thus leads it to a new research stage.

Optimization-based approach for curve and surface reconstruction

This paper introduces an optimization-based approach for the curve reconstruction problem, where piecewise linear approximations are computed from sets of points sampled from target curves. In this approach, the problem is formulated as an optimization problem. To be more concrete, at first the Delaunay triangulation for the sample points is computed, and a weight is assigned with each Delaunay edge. Then the problem becomes minimization or maximization of the total weights of the edges that constitute the reconstruction. This paper proposes one exact method and two approximate methods, and shows that the obtained results are improved both theoretically and empirically. In addition, the optimization-based approach is further extended to three dimensions, where surfaces are to be reconstructed, and the quality of the reconstructions is examined.

STABLE COMPUTATION OF NATURAL NEIGHBOR INTERPOLATION

Natural neighbor interpolation is a scattered data interpolation method based on Voronoi diagrams. A class of natural neighbor interpolants are defined in an integral form which is not suitable for computation. This paper demonstrates that these interpolants can be computed solely with algebraic operations and conditional branches. In addition, it is noted that special care is required to achieve stable computation, and a procedure is proposed for stably calculating natural neighbor interpolation with computers. Its usability is ascertained by surface drawings of natural neighbor interpolation.

Greedy Beta-Skeleton in Three Dimensions

In two dimensions, the beta-skeleton is one of the most practical methods for reconstructing smooth curves from a given unorganized point set because of its provable guarantee. In three dimensions, however, the beta-skeleton cannot be used for surface reconstruction because it may contain unwanted holes omnipresently, no matter how high sampling density is. To overcome this difficulty, an extension of the beta-skeleton, called greedy beta-skeleton, is proposed. It is shown by computational experiments that the unwanted holes do not appear in the greedy beta-skeleton even when the dimension is three. In addition, the greedy beta-skeletons are computed for several practical inputs, and their fairness is examined. Computation results for some variants of the greedy beta-skeleton are also reported.

An Interpolant Based on Line Segment Voronoi Diagrams

This paper considers the interpolation for multi-dimensional data using Voronoi diagrams. Sibson’s interpolant is well-known as an interpolation method for discrete data using Voronoi diagrams. Recently, Gross and Farin extended Sibson’s interpolant to continuous data distributed over polygons and circles. On the other hand, the authors recently found another interpolation method for discrete data. This paper outlines the authors’ interpolation method briefly, and extends it to data distributed over line segments. For this purpose, we utilize line segment Voronoi diagrams. The resulting interpolant is expressed in terms of the integrations of the data function over the given line segments.

A sequence of generalized coordinate systems based on Voronoi diagrams and its application to interpolation

This paper presents a general framework for constructing a variety of multi-dimensional interpolants based on Voronoi diagrams. This framework includes previously known methods such as Sibsons interpolant and Laplaces interpolant; moreover it contains infinitely many new interpolants. Computational experiments suggest that the smoothness can be improved by the proposed generalization. In addition, this framework also includes the piecewise linear interpolant over the Delaunay triangulation, which is a finite-element interpolant. This fact suggests that already established techniques in the finite element method might be brought into the research of the Voronoi-based approach. Hence this framework gives a new and promising direction of research on interpolation based on Voronoi diagrams.

Generalization of an interpolant using Voronoi diagrams in two directions

Recently the authors found a local coordinate property based on the planar Voronoi diagram that is simpler than the famous Sibsons local coordinates (R. Sibson, 1980; 1981), and proposed an interpolant using this property. The paper generalizes this property to general dimensions. The proof given in the paper enables us to use more general Voronoi diagrams, e.g., Laguerre Voronoi diagrams and numerically distributed Voronoi diagrams obtained by topology oriented algorithms. The paper also generalizes the authors interpolant to continuously distributed data sites.