We describe our experience with a new algorithm for the reconstruction of surfaces from unorganized sample points in R³. The algorithm is the first for this problem with provable guarantees. Given a “good sample” from a smooth surface, the output is guaranteed to be topologically correct and convergent to the original surface as the sampling density increases. The definition of a good sample is itself interesting: the required sampling density varies locally, rigorously capturing the intuitive notion that featureless areas can be reconstructed from fewer samples. The output mesh interpolates, rather than approximates, the input points. Our algorithm is based on the three-dimensional Voronoi diagram. Given a good program for this fundamental subroutine, the algorithm is quite easy to implement.

We give a simple combinatorial algorithm that computes a piecewise-linear approximation of a smooth surface from a finite set of sample points. The algorithm uses Voronoi vertices to remove triangles from the Delaunay triangulation. We prove the algorithm correct by showing that for densely sampled surfaces, where density depends on "local feature size", the output is topologically valid and convergent (both pointwise and in surface normals) to the original surface. We describe an implementation of the algorithm and show example outputs. 1 Introduction The problem of reconstructing a surface from scattered sample points arises in many applications such as computer graphics, medical imaging, and cartography. In this paper we consider the specific reconstruction problem in which the input is a set of sample points S drawn from a smooth two-dimensional manifold F embedded in three dimensions, and the desired output is a triangular mesh with vertex set equal to S that faithfully represen...

We present an algorithm that provably reconstructs a curve in the framework introduced by Amenta, Bern and Eppstein. The highlights of the algorithm are: (i) it is simple, (ii) it requires a sampling density better than previously known, (iii) it can be adapted for curve reconstruction in higher dimensions straightforwardly. 1 Introduction We consider the problem of curve reconstruction that takes a set of sample points on a smooth closed curve C, and requires to produce a geometric graph G having exactly those edges that connect sample points adjacent in C. Obviously, given only the samples, it is not always possible to compute G unless some additional conditions are satisfied by the input. Amenta, Bern and Eppstein [1] proposed a framework based on local feature size under which they show two graphs, crust and fi-skeleton, coincide with G if the points are sufficiently sampled. Some of the other effective approaches include ff-shapes by [6] which is analyzed later by [3], r-reg...

We give a simple algorithm for surface reconstruction from a set of point samples in R , using only one three-dimensional Voronoi diagram computation. We also give a fairly simple proof that the reconstruction is topologically correct when the input is a sufficiently dense sample from a smooth surface.

Recent results on curve reconstruction are described. Reconstruction of a curve from sample points ("connect-the-dots") is an important problem studied now for twenty years. Early efforts, primarily by researchers in computer vision, pattern recognition, and computational morphology, relied on ad hoc heuristics (e.g., my own [OBW87]). The heuristics were placed on a firmer footing with ff-shapes [EKS83] and fi-skeletons [KR85] and other structures, whose underlying proximity graphs were later shown to support accurate reconstruction from uniformly sampled curves [FMG95, Att98, BB97]. User selection of the ff or fi parameter is still necessary. A breakthrough was achieved by Amenta, Bern, and Eppstein [ABE98], who designed two algorithms that guarantee correct reconstruction of smooth closed curves even with (sufficiently dense) nonuniform samples, and which lift the burden of selecting a parameter. One of their algorithms computes what they call the crust , a subgraph of the complete...

Many problems in document image analysis can be couched in geometric terms. We outline recent advances in computational geometry that contribute to many aspects of the document analysis process and we provide pointers to a selection of the computational geometry literature where the most relevant results can be found. 1 Introduction Document image analysis (DIA) is concerned with the automatic transfer by machine of visual two-dimensional documents, most commonly consisting of printed pages from books, magazines or newspapers. Maps and engineering drawings constitute another class of common documents. The first class of problems have much in common with optical character recognition (OCR) and both with computer vision. On the other hand DIA is a special case of computer vision and therefore its special properties give rise to special sub-problems such as text-block isolation and textline-orientation inference. Furthermore these special properties allow the tailoring of more gener...

Natural neighbour coordinates and natural neighbour interpolation have been introduced by Sibson for interpolating multivariate scattered data. In this paper, we consider the case where the data points belong to a smooth surface S, i.e. a (d-1)-manifold of R^d. We show that the natural neighbour coordinates of a point X belonging to S tends to behave as a local system of coordinates on the surface when the density of points increases. Our result does not assume any knowledge about the ordering, connectivity or topology of the data points or of the surface. An important ingredient in our proof is the fact that a subset of the vertices of the Voronoi diagram of the data points converges towards the medial axis of S when the sampling density increases.

The problem of efficiently reconstructing the shape from a set of scattered points on the plane has been recently mentioned as one of the emerging challenges in the computational geometry and computational topology fields [2]. Namely, efficient algorithms that can extract a shape from a discrete sample of points have immediate applications in areas like visual perception, computer vision and pattern recognition, geography and cartography, data mining, and biology, to cite a few (see, e.g., [9]). A typical approach to extracting a shape from a given set P of points is to compute a proximity graph of P , i.e. a geometric graph whose vertices are elements of P and where two points are connected by an edge if they are deemed close according to some proximity measure. It is the definition of closeness which determines different types of proximity graphs on the same point set. We give here only a few examples. The Gabriel graph [3] is a proximi...