**1. Introduction**

Shape representation and shape classification are efficiently facilitated by polygonal approximation. This approach is popular due to its compact representation and insensitive to noise. These salient features are found useful in many applications [1–8]. The main objective of polygonal approximation is to approximate the shape of a curve using a polygon whose vertices are specified by a subset of points on the curve. These points are referred to as dominant points and are often the points with high curvature. An example is illustrated in **Figure 1**. A digital curve representing the shape of snowflake is displayed in **Figure 1(a)**, and its identified dominant points are shown in **Figure 1(b)**. The anticipated output of polygonal approximation using dominant point can be seen in **Figure 1(c)**. Broadly polygonal/ closed curve approximation of a digital planar curve may be cast as min ε problem or min 6¼ problem. In min ε problem, the techniques derive polygonal approximation with specified number of line segments or dominant points. These techniques

#### **Figure 1.**

*(a) A digital curve representing the shape of a snowflake, (b) initial set of dominant points, and (c) suitable polygonal approximation are shown here.*

ensure that the deviation between the curve and the approximate polygon is minimal, condition to the specified number of dominant points. Min # techniques derive polygonal approximation with a specified error. These techniques generate the approximate polygon with minimal number of dominant points while ensuring the measure of closeness is not larger than the specified error. In recent years, there are many dominant point-based polygonal approximation techniques that were presented in the literature [9–19].

And few older ones can be found in [20–22]. The techniques presented in [9, 10, 12, 20, 21] use reverse polygonization, where instead of detecting the real points the techniques make a search to detect redundant points and delete points iteratively. The methods in [11, 15] use breakpoint suppression, where the techniques apply criterion measure on the finest approximated set of points to suppress the redundant points and make the approximation. The methods in [3, 13, 16, 18] present a solution using dynamic programming, where the techniques make exhaustive search to detect points on curve, thereby making final approximation. The method in [14] makes polygonal approximation by detecting ADSS (Approximate Digital Straight Segment). The method in [17] uses MIP (mixed integer programming) model. The method in [19] uses vertex relocation procedure around neighbors. In this method, while approximating the output curve by detecting the dominant point, the technique allows neighborhood points to become a dominant point provided that new dominant point facilitates in reduction of approximation error. The method in [22] uses split and merge, where the method makes a search to find the points with maximum deviation in the splitting stage using the proposed criterion function and merge all the points identified in the splitting stage using the threshold

#### *Polygonal Approximation of Digital Planar Curve Using Novel Significant Measure DOI: http://dx.doi.org/10.5772/intechopen.92145*

value. Most of the dominant points [9–12] detecting methods use the magnitude of orthogonal projection of a point on the line segments, which connect adjacent high curvature points to influence the process of detecting dominant points. The methods in the literature [9–12, 14, 15, 20, 23] do not address the issue where the projection of point lies beyond its candidate line segment, where the situation may be often anticipated during approximation. The techniques that neglect to check this criterion may miss good curvature points, which are critical for shape representation. The technique proposed in this chapter measures the positions of projections of a point on the curve, thereby invoking different metrics for computing the significant measure of the dominant points. This practice makes the proposed technique to preserve the original shape of the curve even at very minimal number of dominant points. Such characteristic is very essential for compact representation. And it is very essential for object detection and shape classification applications. Especially, the proposed technique can facilitate the parallel manipulators in cutting and milling operations by preserving the actual shape of the target boundary points. The rest of the chapter is organized is as follows: Section 2 presents a brief review of some of the state-of-the-art methods along with an insight into their demerits wherever possible. Section 3 presents the proposed work. Section 4 summarizes the experimental results. Section 5 concludes the chapter.
