Any dimension polygonal approximation based on equal errors principle

Abstract

In this paper, we present an algorithm based on equal errors principle, which solves the general version of polygonal approximation problem (GPA). The approximation nodes of GPA can be selected anywhere on the original polygonal curve, while the “classical” (usually used) polygonal approximation problem (PA) demands the vertices to be a subset of the original polygon vertices. Although the proposed algorithm does not guarantee global minimum distortion, in many cases almost optimal solutions are achieved. Moreover, it is very flexible on changes of error criteria and on curve dimension yielding an alternative and in many cases approximations with lower error (by relaxing the simplification constraint) than the optimal PA methods with about the same computation cost.

Introduction

The polygonal approximation is an important topic in the area of pattern recognition, computer graphics and computer vision. A huge number of applications like object recognition, computational cartography, signal summarization and compression are based on polygonal approximation. The polygonal approximation process saves memory space, reduces the rendering time on graphics applications and gives a more compact representation.

Given an N-vertex polygonal curve P in the n-dimensional space ${\mathfrak{R}}^n$, the curve approximation P consists in computation of another M-vertex polygonal curve in the n-dimensional space ${\mathfrak{R}}^n$ that approximates the original curve, according to a predefined error criterion. Let P = {p₁, p₂, … , p_N} and $P^{\prime }=\{p_1^{\prime }\text{,}{p}_2^{\prime }\text{,}\dots \text{,}{p}_M^{\prime }\}$ be the set of the vertex points of the given polygonal curve and its approximation, respectively. According to the general polygonal approximation problem (GPA), the vertices of P′ are an ordered subsequence of the curve points along P (Fig. 1a), for which it is not required to be a subset of P vertices as the PA demands (Fig. 1b). Therefore, under this constraint relaxation the solutions of GPA problem give approximations of the polygonal curve P with possibly lower error that than the error of the solutions of PA problem. In addition, it holds that $p_1^{\prime }=p_1$ and $p_M^{\prime }=p_N$.

Different error criteria have been proposed for polygonal approximation problems. One of the most used is the tolerance zone criterion (Imai and Iri, 1988, Chen and Daescu, 1998). Let $\overline{p_k^{\prime }{p}_{k+1}^{\prime }}$, k ∈ {1, 2, … , M − 1} be a segment of P′ and $S=[p_k^{\prime }\text{,}{p}_m\text{,}{p}_{m+1}\text{,}\dots \text{,}{p}_{m+s}\text{,}{p}_{k+1}^{\prime }]$ be the corresponding subcurve of P. Under this criterion, the error between the segment $\overline{p_k^{\prime }{p}_{k+1}^{\prime }}$ and S is defined as the maximum distance in an L_h (h ∈ {1, 2, ∞}) metric between $\overline{p_k^{\prime }{p}_{k+1}^{\prime }}$ and each point on the subcurve S. Another frequently used error criterion is the local integral square error (LISE) (Ray and Ray, 1994, Chung et al., 2002). Under this criterion, the error between the segment $\overline{p_k^{\prime }{p}_{k+1}^{\prime }}$ and S is defined as the sum of squared Euclidean distances from each vertex point of subcurve S. Finally, according to these error criteria the approximation error between P′ and P is defined as the maximum error between the segments of P′ and their corresponding subcurves of P. The polygonal approximation problem can be formulated in two ways:

• The problem of minimum error (min − ε), where the approximation error is minimized given the number of segments M.

• The problem of minimum number of segments (min − #), where the approximation error is bounded (ε) and the goal is to find the minimum number of segments (M) that gives error lower than the given error.

The problem of approximating a polygonal curve P (PA) has been studied extensively during the last two decades (Heckbert and Garland, 1997, Kolesnikov, 2003, Weibel, 1997). The methods, that have been developed, solve the problem by approximating the original polygonal curve P by another polygonal curve P′ under the constraint that the P′ vertex sequence is an ordered subsequence of the vertices along P. There are two well-known approaches for solving this problem: graph–theoretical and dynamic programming. Graph–theoretical methods generate directed acyclic graph on the vertices of P, and then compute the shortest path in the graph (Imai and Iri, 1988, Chan and Chin, 1996, Katsaggelos et al., 1998). Dynamic programming generates the solution for the problem using results of the smaller problem instances (Perez and Vidal, 1994, Salotti, 2001). Concerning the 2-D min − # problem and the min − ϵ problem under the tolerance zone criterion, the lowest computation cost method (Chan and Chin, 1996) has cost O(N²) and O(N² log N), respectively. The memory requirements can be reduced to O(N) (Chen and Daescu, 1998). The 3-D and 4-D polygonal approximation problems require near-quadratic time and sub-cubic time, respectively (Barequet et al., 2002). When the L₁ or L_∞ metrics are used, the time requirements for min − # problem and the min − ϵ problem are reduced to O(N²) and O(N² log N) in any dimensional space (Barequet et al., 2002), respectively. Under the LISE criterion, the 3-D min − # problem and the min − ϵ problem can be solved (Chung et al., 2002) in O(N²) and O(N² log N), respectively. The performance of polygonal approximation algorithms can be measured under variations in scale parameters and data (Rosin, 1997). The equal error principle has been used in PA (Heckbert and Garland, 1997) getting good approximations. In (Sarkar et al., 2003), the proposed Equal Error Tree algorithm, which is a hierarchical polygon approximation method, is generally better than the Arc Tree and compares well with the Strip Tree. The polygonal approximation of curves by a polygon is a similar problem with GPA.

The rest of the paper is organized as follows: Section 2 presents the proposed GPA algorithm. The experimental results and comparisons with the existing PA methods are given in Section 3. Finally, conclusions and discussion are provided in Section 4.