Elsevier

Computer-Aided Design

Volume 62, May 2015, Pages 164-175
Computer-Aided Design

A non-parametric approach to shape reconstruction from planar point sets through Delaunay filtering

https://doi.org/10.1016/j.cad.2014.12.002Get rights and content

Highlights

  • A fully automatic algorithm using the structural properties of Delaunay triangles.

  • External boundary as well as an internal hole detection have been addressed.

  • Demonstrate the efficacy with varying point set densities and distributions.

  • Theoretical guarantee of the algorithm has been presented.

Abstract

In this paper, we present a fully automatic Delaunay based sculpting algorithm for approximating the shape of a finite set of points S in R2. The algorithm generates a relaxed Gabriel graph (RGG) that consists of most of the Gabriel edges and a few non-Gabriel edges induced by the Delaunay triangulation. Holes are characterized through a structural pattern called as body-arm formed by the Delaunay triangles in the void regions. RGG is constructed through an iterative removal of Delaunay triangles subjected to circumcenter (of triangle) and topological regularity constraints in O(nlogn) time using O(n) space.

We introduce the notion of directed boundary samples which characterizes the two dimensional objects based on the alignment of their boundaries in the cavities. Theoretically, we justify our algorithm by showing that under given sampling conditions, the boundary of RGG captures the topological properties of objects having directed boundary samples. Unlike many other approaches, our algorithm does not require tuning of any external parameter to approximate the geometric shape of point set and hence human intervention is completely eliminated. Experimental evaluations of the proposed technique are done using L2 error norm measure, which is the symmetric difference between the boundaries of reconstructed shape and the original shape. We demonstrate the efficacy of our automatic shape reconstruction technique by showing several examples and experiments with varying point set densities and distributions.

Introduction

Given a finite set of points, SR2, distributed across a known or unknown object, shape reconstruction deals with the construction of a polygon (could be with holes) that best captures the geometric shape of S (Fig. 1). The problem finds various applications in computer graphics, computer vision, pattern recognition  [1], Geographical Information System  [2], [3], study of physical objects like stars in astronomy  [4], molecular shape description in biology  [5], and target area monitoring in wireless sensor networks  [6].

There exist few challenges in approximating the shape of a point set. First, it is unclear what constitutes the best approximation for the geometric shape of a point set mainly due to a lack of precise mathematical definition for its optimal shape. Second, the point set shapes are highly subjective in nature and often depend on a specific application context or other human cognitive factors. As a consequence, the shapes perceived by humans for a majority of point sets vary and reaching a consensus on the optimum shape is an extremely difficult task. The rich variety of shapes available in the nature and the heterogeneity of point sets further weaken a well-defined formulation of the shape approximation problem  [7].

Delaunay based shape reconstruction approaches such as α-shape  [8], A-shape  [9] and χ-shape  [10] depend on external parameters to construct the geometric shape of a point set. As a result, a family of shapes is generated for a given point set. For example, α-shape family of a brain point set consists of a set of different but structurally close shapes for different values of α as demonstrated in Fig. 2. Though it gives the flexibility of choosing the best shape that suits an application, it has few limitations. For instance, choosing the best shape from a set of similar shapes demands a great deal of human involvement which might induce humanly errors thereby affecting the accuracy of the whole system or application. Moreover, since everyone is cognitively different from each other, reaching a consensus on the desired shape by different individuals becomes a challenging task. Considering these facts, it is often desirable to have an automatic algorithm that constructs the geometric shape of a point set.

In general, two approaches have been adopted to evaluate shape reconstruction results. In one approach, the results are experimentally evaluated against a known original shape with respect to certain error parameters. Error is estimated as the area of symmetric difference between the original shape and the reconstructed result  [10]. The evaluation of the results are made extensive and authentic by carrying out experiments like varying point density and homogeneity in point distribution. In another approach, algorithms are analyzed theoretically for geometric and topological guarantees on the output shape under some input sampling assumptions  [12]. We use both the approaches for evaluating the proposed shape reconstruction algorithm.

Section snippets

Related work

A plethora of work has been proposed for geometric shape reconstruction from a planar point set. Though a full review of all the shape reconstruction algorithms is beyond the scope of this paper, we briefly mention some closely related methods based on Delaunay triangulation filtering and proximity graphs.

In a survey on shape reconstruction  [7], Edelsbrunner elaborates on how different shape reconstruction methods restrict the Delaunay triangulation to arrive at their respective shapes. The

Geometrical definitions and notations

Let S be a set of n points in the Euclidean plane and conv(S) be its convex hull. Vor(S), Del(S), int(conv(S)) and conv(S) denote, respectively, the Voronoi diagram of S, the Delaunay triangulation of S, the interior of conv(S) and the boundary of conv(S). Let d(p,q)=pq, denote the Euclidean distance between two points p, qS.

In geometry, a k-simplex is defined as the non-degenerate convex hull of k+1 geometrically distinct points, v0,v1,,vkRd where kd   [23]. Points, edges, triangles

The algorithm

In computational geometry, sculpting is the process of creating shapes through repeated elimination of triangles from an initial triangular mesh. The proposed algorithm is analogous to Boissonat’s sculpture  [26] method but uses a different sculpting strategy. As opposed to a selection criteria based on the maximum distance in sculpture, we use the combination of circumcenter and circumradius of Delaunay triangle to capture the structure of triangles and consequently the geometric proximity of

Experimental results and discussion

The proposed algorithm is implemented in C++ using computational geometry algorithms library (CGAL)  [11]. A user-friendly GUI that inherits the Delaunay triangulation GUI of the Triangulation_2 package  [11] has also been developed.

We analyzed the effectiveness of our shape approximation algorithm by testing on a variety of 2D point sets of different sampling densities and distributions. Since there does not exist a mathematical definition for what constitutes the optimal shape of a set of

Conclusion

In this paper, we have proposed an automatic shape reconstruction algorithm for planar point sets. Such a non-parametric algorithm is highly desirable as it relieves the end user of the tedious task of tuning an external parameter to construct the geometric shapes of point sets. The approach is more relevant when considering the fact that most of the established shape reconstruction techniques depend on external parameters to define the shapes but provide limited information on how to pick the

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    This paper has been recommended for acceptance by Dr. Vadim Shapiro.

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