A non-parametric approach to shape reconstruction from planar point sets through Delaunay filtering☆
Introduction
Given a finite set of points, , 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 (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 -shape [8], -shape [9] and -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, -shape family of a brain point set consists of a set of different but structurally close shapes for different values of 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 be a set of points in the Euclidean plane and be its convex hull. , , and denote, respectively, the Voronoi diagram of , the Delaunay triangulation of , the interior of and the boundary of . Let , denote the Euclidean distance between two points , .
In geometry, a -simplex is defined as the non-degenerate convex hull of geometrically distinct points, where [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.