Given a planar point set sampled from an object boundary, the process of approximating the original shape is called curve reconstruction. In this paper, a novel non-parametric curve reconstruction algorithm based on Delaunay triangulation has been proposed and it has been theoretically proved that the proposed method reconstructs the original curve under ε-sampling. Starting from an initial Delaunay seed edge, the algorithm proceeds by finding an appropriate neighbouring point and adding an edge between them. Experimental results show that the proposed algorithm is capable of reconstructing curves with different features like sharp corners, outliers, multiple objects, objects with holes, etc. The proposed method also works for open curves. Based on a study by a few users, the paper also discusses an application of the proposed algorithm for reconstructing hand drawn skip stroke sketches, which will be useful in various sketch based interfaces. © 2016 The Author(s) Computer Graphics Forum © 2016 The Eurographics Association and John Wiley & Sons Ltd. Published by John Wiley & Sons Ltd.

Ramanathan Muthuganapathy

Department of Engineering Design

Computer graphics

Drawing (graphics)

CURVE RECONSTRUCTION

CURVE-RECONSTRUCTION ALGORITHMS

DELAU-NAY TRIANGULATIONS

Descriptors

LINE AND CURVE GENERATION

Object boundaries

Planar point sets

SKETCH BASED INTERFACE

Algorithms

IIT Madras is a public technical and research university located in Chennai, Tamil Nadu. Founded in 1959, it is recognised as an Institute of National Importance.

IIT Madras has been ranked as the top engineering institute in India for four years in a row by the National Institutional Ranking Framework of the MHRD

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IIT Madras

Computer Graphics Forum

We present an incremental Voronoi vertex labelling algorithm for approximating contours, medial axes and dominant points (high curvature points) from 2D point sets. Though there exist many number of algorithms for reconstructing curves, medial axes or dominant points, a unified framework capable of approximating all the three in one place from points is missing in the literature. Our algorithm estimates the normals at each sample point through poles (farthest Voronoi vertices of a sample point) and uses the estimated normals and the corresponding tangents to determine the spatial locations (inner or outer) of the Voronoi vertices with respect to the original curve. The vertex classification helps to construct a piece-wise linear approximation to the object boundary. We provide a theoretical analysis of the algorithm for points non-uniformly (ε-sampling) sampled from simple, closed, concave and smooth curves. The proposed framework has been thoroughly evaluated for its usefulness using various test data. Results indicate that even sparsely and non-uniformly sampled curves with outliers or collection of curves are faithfully reconstructed by the proposed algorithm. © 2018 The Authors Computer Graphics Forum © 2018 The Eurographics Association and John Wiley & Sons Ltd.

Incremental Labelling of Voronoi Vertices for Shape Reconstruction

Reconstruction problem in R2 computes a polygon which best approximates the geometric shape induced by a given point set, S. In R2, the input point set can either be a boundary sample or a dot pattern. We present a Delaunay-based, unified method for reconstruction irrespective of the type of the input point set. From the Delaunay Triangulation (DT) of S, exterior edges are successively removed subject to circle and regularity constraints to compute a resultant boundary which is termed as ec-shape and has been shown to be homeomorphic to a simple closed curve. Theoretical guarantee of the reconstruction has been provided using r-sampling. In practice, our algorithm has been shown to perform well independent of sampling models and this has been illustrated through an extensive comparative study with existing methods for inputs having varying point densities and distributions. The time and space complexities of the algorithm have been shown to be O(nlogn) and O(n) respectively, where n is the number of points in S. © 2015 Elsevier Ltd. All rights reserved.

Computers and Graphics (Pergamon)

A unified approach towards reconstruction of a planar point set

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. © 2014 Elsevier Ltd.

CAD Computer Aided Design

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

Journal	Computer Graphics Forum
Publisher	Blackwell Publishing Ltd
ISSN	01677055
Open Access	No