**2. Background**

Several polygonal approximation techniques have been proposed in the recent decades. Some of them use various optimization approaches [3, 13, 16–19]. On the other hand, there are other techniques that use local/global geometric features of a curve to influence the process of determining the polygon with minimal number of line segments [9–12, 23–26], and these techniques prove its competence against many real-time datasets. Among these, this section briefly analyzes some of the bench mark techniques.

Prasad [23] proposed a non-parametric framework to detect points of high curvature. The framework uses the maximum deviation incurred between pixels from a digitized boundary as an upper bound to make approximation. The authors proved that the analytical bound can be incorporated by dominant point detection framework to get rid of specification in terms of the tolerable error (for min # approaches) or the number of points (for min ϵ approaches). The authors established the robustness of their framework against scaling invariance as well as noise tolerance. However, there are applications in which the curve needs to be approximated using a specified number of dominant points, which is not possible through this framework. Though the approximation bounded below to digitization value, points detected on the curve seem to be redundant for human visual perception. Prasad [24] used metrics such as precision and reliability as measures to fit the polygon edges. Depending upon the threshold values for these measures, the technique produces coarser or finer approximation. Thus, this technique can flexibly control the degree of smoothness required for an application. And also the paper suggests some performance metrics to quantify the techniques. Parvez [19] obtained the digital boundary using contour extraction techniques. The objective of the method was to produce approximate polygon with minimal error possible. To attain this goal, the method relaxes the criteria that dominant points need not be on the contour. The technique computes neighborhood points for every point *pi* on the contour *Cd* and introduces a new point on the contour provided its presence should reduce the approximation error. The neighborhood points are not the ones computed using 4 connected graph or 8 connected graph; instead, the technique

adaptively defines the width for every point on the curve, and thereby, it obtains the neighborhood points. Fernandez [25] produced symmetric approximation for symmetric contours. The technique obtains first initial point *p1* as the farthest in terms of distance from the centroid of the curve. The next point *p2* is the farthest to *p1*. The method proceeds to find point *p4*, which is farthest from *p2*, and point *p3*, which is farthest from *p1*. Likewise, the technique obtains the all possible line segments such as {*p1*, *p2*}, {*p3*, *p4*}, until the maximum deviation from the curve does not exceed a threshold value that constitutes the boundary point set. The authors demonstrate that their method of choosing initial points ensures symmetricity. The technique then identifies all possible candidate points (*q1*, *q2*, ..., *qm*) from the boundary point set between every two initial points and computes a significant value by ensuring symmetry property. Additionally, the technique presents various thresholding methods to normalize the significant values of the boundary points. Though the technique produces symmetric approximation for symmetric curve, it did not establish geometric invariance. And in real-time data sets, in most of the cases, the points are always distributed asymmetrically on the planar curve. The main objectives of this chapter are to i) present a framework that considers the projection position of a point and thereby invokes the proper criterion measure to compute the contribution; ii) produce output polygon without missing significant points; iii) produce polygon with minimal possible number of points; and iv) present a technique that is reasonably strong enough against rotation invariance. These objectives are achieved and demonstrated through experimentations of the proposed technique using benchmarking data sets.
