Polygonal approximation based on coarse-grained parallel genetic algorithm

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Abstract

This paper proposes to apply coarse-grained parallel genetic algorithm (CGPGA) to solve polygonal approximation problem. Chromosomes are used to represent digital curves and genes correspond to points of curves. This method divides the whole population into several subpopulations, each of which performs evolutionary process independently. After every migration interval number of generations, these subpopulations exchange their information with each other. Inspired by the designing theory of ensemble learning in machine learning, this paper further improves the basic CGPGA through adopting different but effective genetic algorithms, respectively, in different subpopulations. Both the diversity among different subpopulations and the accuracy in each individual subpopulation are ensured. Experimental results, based on four benchmark curves and four real image curves extracted from the lake maps, show that the basic CGPGA outperforms the used genetic algorithm, and further the improved CGPGA (ICGPGA) is more effective than the basic CGPGA, in terms of the quality of best solutions, the average solutions, and the variance of best solutions. Especially for those larger approximation problems, the ICGPGA is more remarkably superior to some representative genetic algorithms.

Introduction

The representation of 2-D shape is a crucial issue in the fields of image processing, pattern recognition and machine learning. Particularly, polygonal approximation is an important representative for this problem. High-quality polygonal approximation can capture the essence of shape curves and provide a compact representation for them. Thus, feature extraction for subsequent image analysis is easier to implement. Furthermore, the storage memory of shape contours’ information will be largely reduced via only saving a few of important points of curves in the process of polygonal approximation.

The problems concerning on polygonal approximation can be roughly classified into two categories: (1) Given the number of vertices of approximated polygon, the aim is to obtain the polygon that has the minimum approximation error, compared to the original shape curve. (2) Given the error bound, the aim is to obtain the polygon that has the minimum number of vertices, and to make the error between the original shape curve and the approximating polygon not exceed the tolerance of error.

Over recent decades, numerous methods have been proposed to solve the two types of polygonal approximation problems, such as (1) Sequential method [1], [2], [3], [4], [5], [6], [7]; (2) Spilt method, merge method, and split-and-merge method [8], [9], [10], [11], [12]; (3) Dominant and angle detection method [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25]; (4) Newton’s method [26]; (5) Iteration method [27]; (6) k-means method [28], [29]; (7) Dynamic programming [30], [31], [32], [33]; (8) Bio-inspired intelligent algorithms [34], [35], [36], [37], [38], [39].

Among all these methods, Ray and Ray [2] propose to obtain the polygon of object curve through making the line segment maximize and meanwhile making the approximation error minimize. Rosenfeld and Johnston [18] give a definition of k-cosine, which is seen as the approximation curvature between each break point and its forward and backward points. Also, Ray and Ray [15] use k-cosine to get the region of support of each point in the curve. Borrowing from Newton’s method, Pavlidis [26] uses integral square error as cost function to determine the positions of dominant points. Additionally, Perez and Vidal [32] apply dynamic programming to solve the polygonal approximation problem.

According to the above discussions, we conclude that the methods from (1) to (6) are local search methods, and they only take the local information of curve points into consideration, so that they are easy to fall into local optimal solutions and are not feasible in many complex cases. The method (7) is a global search method and is able to obtain accurate approximate results. However, its computational cost is fairly high because it carries out an exhaustive search in the space of solutions. The method (8) utilizes computational intelligence techniques to solve the polygonal approximation problem, such as ant colony optimization (ACO) [38], particle swarm optimization (PSO) [39] and GA-based methods [40], [34], [35], [36], [37]. These intelligent optimization methods have a good trade-off between the running time and the quality of approximate polygon, so that they attract many researchers’ attention.

This paper focuses on GA-based methods, which have global search ability and which are effective in solving NP-hard problems. Yin [40] first introduces the GA to handle the polygonal approximation problem. After that, some researchers attempted to improve Yin’s method. Sun and Huang [36] add a data reduction preprocess before starting the GA algorithm, give a different definition of fitness function and make changes for crossover and mutation operators. In addition, Ho and Chen [34] introduce a novel orthogonal array to improve crossover operator, and their crossover operator generates offspring by best combining the better genes of their parents instead of random selecting genes from their parents. Moreover, Wang and Shu [41] propose a new method, called SMSR, which combines GA and split-and-merge algorithm. The SMSR has not only the advantage of GA’s global search but also the merit of split-and-merge’s local optimal search. Furthermore, it also reduces the running time by employing a simple operator which uses traditional split-and-merge method to repair all infeasible individuals’ chromosomes.

GAs are efficient to solve NP-hard problems, especially combinatorial optimization ones, however, as the scale of optimization problems becoming larger and larger and the intensity of computation becoming higher and higher, GAs are also confronted with some problems in many practical applications. Naturally, parallel genetic algorithms (PGA) are designed to deal with the more complex optimization tasks. PGA further speeds up the evolutionary process of GAs, keeps the diversity of population, and reduces the probability of premature [42], [43], [44], [60], [61], [62].

The basic idea of CGPGA is to divide the whole population into several subpopulations and to execute a specific GA on each subpopulation in parallel. After every migration interval number of generations, the several top best individuals of every subpopulation are gathered together and migrated into every subpopulation. Inspired by the designing idea of ensemble learning in machine learning, this work further proposes to improve the basic coarse-grained parallel genetic algorithm (CGPGA) for solving the polygonal approximation problem, via executing different types of GAs in different subpopulations respectively, instead of conducting the same type of GAs in each subpopulation.

Ensemble learning which combines outputs from multiple individual learners is one of the most important techniques for improving classification or regression accuracy in machine learning [45], [46], [56], [57], [58], [59]. Generally speaking, a good ensemble of learning machines requires that the base learners should be as more accuracy as possible, and as more diverse as possible. Therefore, in the improved CGPGA, the different subpopulations need to adopt different but effective GAs, such as the GA of Sun and Huang [36], the EEA of Ho and Chen [34], and the SMSR of Wang and Shu [41]. Thus, the accuracy and diversity of subpopulations can be ensured.

The rest of this paper is organized as follows. Section 2 firstly describes the problem on polygonal approximation, and then gives a brief review on traditional GAs and three representative GA-based optimization methods for solving this problem. Section 3 presents the CGPGA method and its improved version. Section 4 shows experimental results and performance comparisons, and the last section concludes this paper.

Section snippets

The formulation of the polygonal approximation problem

For the convenience of the following description, some definitions, similar to the discussion in [41], are first given.

Definition 1

A closed digital curve with n points is defined to be D= {p1, p2,…, pn} and it is a circular sequence, pn+i = pi.

Definition 2

Let pi,pj=pi,pi+1,,pj be the arc from point pi along D to point pj and pipj be the chord that joints point pi and point pj.

Definition 3

The approximation error between pi,pj and pipj can be defined as follows:Lij=e(pi,pj,pipj) =pkpi,pjd2(pk,pipj)where d2pk,pipj is the

Polygonal approximation based on coarse-grained parallel genetic algorithm (CGPGA)

According to the above discussion, several different types of genetic algorithms have been designed to solve the polygonal approximation problem. Each of them has its own specific advantages. Naturally, a promising thought is to combine these different genetic algorithms together and integrate different merits of different genetic algorithms to further enhance the optimization ability of genetic algorithms for polygonal approximation.

An alternative designing method is to carry out several

Experiments and discussions

In the experiments, two sets of curves are used to test the performance of different methods. One set is composed of four synthesized benchmarks used widely in literatures [10], [34], [35], [36], [37], [38], [39], [40], [41], including leaf-shape curve with 120 points, chromosome-shape curve with 60 points, semicircle-shape curve with 102 points, and Fig. 8-shape curve with 45 points (see Fig. 6). The other set contains four famous lake curves used in [41]: Arlington lake curve from US with 133

Conclusions

Coarse-grained parallel genetic algorithm (CGPGA) and its improved version (ICGPGA) are applied to solve the polygonal approximation problem. CGPGA or ICGPGA classifies the whole population into several subpopulations. Different from CGPGA, ICGPGA uses different evolutionary methods in different subpopulations. The improved design makes ICGPGA not only have the advantages of CGPGA but also inherit the merits of different evolutionary methods in its different subpopulations. Such design is also

Contribution

  • (1)

    This paper focuses on GA-based methods, which have global search ability and which are effective in solving NP-hard problems.

  • (2)

    Our method is easy to fall into local optimal solutions and are not feasible in many complex cases

Declaration of Competing Interest

There is no conflict of interest

Acknowledgments

This work is supported by Hebei Science and technology support program (17210104D), Science and technology research project of Hebei higher education institutions (ZD2015099), and High-level Talents Subsidy Project in Hebei Province (A2016002015).

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