Muthuganapathy Ramanathan

In this paper, we propose a Delaunay triangulation based algorithmic framework for the 2D point set type discernment and its reconstruction. The point set discernment deals with determining whether the given point set S consists of only the points sampled along the boundary of an object (referred to as boundary sample) or S is distributed across the object (referred to as dot-pattern). In general, the existing approaches deal with the reconstruction of an a priori known input type - boundary sample or dot-pattern. Our approach works on both sets of input data type by first identifying the input type that a given point set belongs to. To distinguish the point set type, we introduce the notion of Petal-Ratio(PR) which captures the ratio of the longest edge to the smallest edge emanating from a vertex in Delaunay triangulation. Further, we employ the Petal-Ratio as the essence to design simple reconstruction algorithms pertaining to each input type that unveil the shape of the object hidden in the given point set. The reconstruction algorithms start with computing the Delaunay triangulation of the given point set and for each vertex, the Petal-Ratio is calculated. Following this, the edges are removed based on the value of the PR for every vertex to arrive at the boundaries. Theoretical analysis of determining appropriate values of PRs under ϵ- sampling and minimal reach sampling models have been presented. Unlike many other algorithms, our approach is non-parametric and non-feature specific that can capture features like disconnected components, multiple holes even with the presence of outliers, self-intersection (only for boundary sample) and open curves (only for boundary sample). Moreover, our algorithms take only one pass to reconstruct the hole boundaries as well as outer boundary irrespective of the object features like the number of holes and the number of components. The comparative analysis shows that our algorithms perform equally well or better than their counterparts for the objects with contrasting features. We also demonstrate that the proposed idea can be easily extended to surface reconstruction.

Given a set of points S∈R2, reconstruction is a process of identifying the boundary edges that best approximates the set of points. In general, the set of points can either be derived from only the boundaries of the curves (called as boundary sample) or can be derived from both boundary and interior of the curves (called as dot pattern). Most of the existing algorithms focus towards reconstruction from only boundary samples, termed as curve reconstruction. Unfortunately many of them don't reconstruct when the input is of dot pattern type (called as shape reconstruction). In this paper, we propose an input-independent non-parametric algorithm for reconstruction that works for both dot patterns as well as boundary samples. The algorithm starts with computing the Delaunay triangulation of the given point set. An edge between a pair of triangles is marked for removal when the circumcenters lie on the same side of the edge. Further, we also propose additional criterion for removing edges based on characterizing a triangle by the distance between its circumcenter and incenter. To maintain a manifold output, a degree constraint is employed. The proposed approach requires only a single pass to capture both inner and outer boundaries irrespective of the number of objects/holes. Moreover, the same criterion has been employed for both inner and outer boundary detection. The experiments show that our approach works well for a variety of inputs such as multiple components, multiple holes etc. Extensive comparisons with state-of-the-art methods for various kinds of point sets including varying the sampling density and distribution show that our algorithm is either better or on par with them. Theoretical discussions on the algorithm have also been presented using ϵ-sampling and r-sampling. Limitations of the algorithm are also discussed.