Polygonal approximation of digital planar curves through vertex betweenness
Introduction
Shape is one of the most important visual attributes of an object, and it has an important role in shape analysis and representation. However, most of the shape information is redundant and could be discarded without damaging the original shape representation. According to [1], points of high curvature are the most important source of information regarding a shape. The human visual system is capable of recognizing a shape by its higher curvature points. In fact, these points are important features for pattern recognition tasks, as they provide efficient data reduction while retaining crucial shape information [24].
The work of Attneave has motivated the development of methods to reduce the information of a shape using polygonal approximation. The main objective of polygonal approximation methods is to build a representation of the original shape by using a series of straight lines. This is a very popular representation of shapes due to their simplicity, locality, generality, and compactness [14]. Among the advantages of its use in the computer vision area are the easy detection of man-made objects (such as, the recognition of numerals on car number plates [11]), as well as the low complexity of feature extraction for algorithms of image analysis and matching [6], [15].
Basically, the methods to obtain a polygonal approximation can be split into two approaches: (i) to obtain an approximation of the original shape with an error smaller than a maximum error allowed, using as few line segments as possible (here included our approach) or (ii) to obtain an approximation with a fixed number of line segments using an error as smaller as possible. For both approaches, various methods have been developed over the years. In [27], [22], [25], [19], the curvature is used to detect the dominant points of a shape curve. A hierarchical approach to detect these points can be found in [4], [12]. The shape curve can be iteratively reduced by splitting and merging segments according to a pre-specified tolerance [20], [21]. There are also methods which deal with the polygonal approximation as an optimization problem [8], [29], [28], [7], [5], as well as methods that find a subset of the curve points as dominant points [16], [13], [18].
In this paper, we present a graph-based approach to compute the polygonal approximation. Given a closed curve, our proposed approach creates a graph where each vertice corresponds to a point in this curve. Then, we select the vertices which present higher vertex betweenness to compose a feasible polygonal approximation. The vertex betweenness is a measurement of the importance of each vertice in the graph according to the number of shortest paths passing through it. This measurement enable us to select the vertices which correspond to a high transitivity region of the graph. These high transitivity regions are similar to the dominant points of a shape curve detected by methods such as the curvature [27], [22], [25], [19]. To accomplish this task, we propose a modified version of the Bellman–Ford algorithm [3], which considers the polygonal approximation error during the calculus of the shortest path. We also propose a path optimization step similar to split–merge methods [20], [21] to remove non-representative points of the polygonal approximation without or with a small increase in approximation error.
The remainder of this paper is organized as follows. Section 2 describes the problem of the polygonal approximation while Section 3 describes the proposed method in detail. In Section 4 we present the experimental results and the discussions. Finally, the conclusions are presented in Section 5.
Section snippets
Problem definition
The polygonal approximation is a classic problem in shape representation. Given a clockwise ordered curve S = {s1, s2, … , sn}, where si = {xi, yi} is the Cartesian coordinate of a point in the curve, n is the number of points and the sequence of points is circular, i.e., s1 is considered as the succeeding point of sn, the aim of the polygonal approximation is to find , with S∗ ⊂ S and 3 ⩽ m ⩽ n, where S∗ is a circular sequence of m points , arranged in ascending order, which
Proposed method
The polygonal approximation of contour S is usually defined as a set of selected points, which describes a polygon and best represents S. However, if we consider S as a graph, and each point of S as a vertex in this graph, the problem of the polygonal approximation can be understood as finding a path in this graph which length is smaller than ϵ, where ϵ is the maximum error allowed for the polygonal approximation. Therefore, in our proposed approach, we consider contour S as a graph G(V, E, A, W),
Experimental results
This section presents the computational experiments performed in the proposed method, as well as the numerical results achieved. To accomplish this task, three synthesized curves (Fig. 6) were used. These curves are broadly used in the literature as benchmark curves for polygonal approximation problems [4], [21], [25], [29], [13], [18], [16].
An important aspect of the proposed method is graph G used to compute the vertex betweenness. Two vertices are connected by a directed edge only if the
Conclusions
In this paper, we presented a novel graph based approach to compute a polygonal approximation of a shape contour. In a graph, these points correspond to a high transitivity region of the graph. By using the vertex betweenness, we measured the importance of each vertex in the graph according to the number of shortest paths where each vertex occurs. By selecting the vertices with higher vertex betweenness, a polygon which preserves the main characteristics of the contour is achieved. These
Acknowledgement
Odemir M. Bruno gratefully acknowledges the financial support of CNPq (National Council for Scientific and Technological Development, Brazil) (Grant #306628/2007-4 and #484474/2007-3).
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