A synergy of the sine-cosine algorithm and particle swarm optimizer for improved global optimization and object tracking

Abstract

Due to its simplicity and efficiency, a recently proposed optimization algorithm, Sine Cosine Algorithm (SCA), has gained the interest of researchers from various fields for solving optimization problems. However, it is prone to premature convergence at local minima as it lacks internal memory. To overcome this drawback, a novel Hybrid SCA-PSO algorithm for solving optimization problems and object tracking is proposed. The $P_{best}$ and $G_{best}$ components of PSO (Particle Swarm Optimization) is added to traditional SCA to guide the search process for potential candidate solutions and PSO is then initialized with $P_{best}$ of SCA to exploit the search space further. The proposed algorithm combines the exploitation capability of PSO and exploration capability of SCA to achieve optimal global solutions. The effectiveness of this algorithm is evaluated using 23 classical, CEC 2005 and CEC 2014 benchmark functions. Statistical parameters are employed to observe the efficiency of the Hybrid SCA-PSO qualitatively and results prove that the proposed algorithm is very competitive compared to the state-of-the-art metaheuristic algorithms. The Hybrid SCA-PSO algorithm is applied for object tracking as a real thought-provoking case study. Experimental results show that the Hybrid SCA-PSO-based tracker can robustly track an arbitrary target in various challenging conditions. To reveal the capability of the proposed algorithm, comparative studies of tracking accuracy and speed of the Hybrid SCA-PSO based tracking framework and other trackers, viz., Particle filter, Mean-shift, Particle swarm optimization, Bat algorithm, Sine Cosine Algorithm (SCA) and Hybrid Gravitational Search Algorithm (HGSA) is presented.

Introduction

Global optimization problems are continually unavoidable in current engineering and science fields. Mathematically, an optimization problem can be expressed as

$$\stackrel{Minimize}{x\in {\Re }^n}{f}_i\left(x\right),\left(i=\mathrm{1,2},\dots ,M\right)$$

$$\text{s}.\text{t}{h}_j\left(x\right)=0,\left(j=\mathrm{1,2},\dots ,J\right)$$

$$g_k\left(x\right)\le 0,\left(k=\mathrm{1,2},\dots K\right)$$

Here $f_i\left(x\right),g_k\left(x\right)$ and $h_j\left(x\right)$ are design vector functions.

$$x={\left(x_1,x_2,\dots ,x_n\right)}^T$$

Here the factor $x_i$ of $x$ are termed decision or design variables, and they are real discrete, continuous or combination of the two.

The $f_i\left(x\right)$ functions where $i=\mathrm{1,2},\dots ,M$ are termed cost or objective functions, and $M=1$ case, there is a single objective. The area covered by the design variables is called the search space or design space ${\Re }^n$, while the area formed by the cost function results is termed as the response or solution space. The inequalities for $g_k$ and equalities for $h_j$ are termed as constraints.

Recently, nature-inspired stochastic optimization algorithms have received much attention. Such optimization mostly mimics individual/social behaviour of a group of animal or natural phenomena. Such algorithms start the optimization process by creating a set of random solutions and improve them as candidate solutions for a particular problem. Due to the superior performance of such techniques compared to mathematical optimization approaches, the application of stochastic optimization methods can be found in different fields. Despite the symbolic amount of newly recommended algorithms in optimization field, there is an essential query here is why we need other optimization algorithms. This query can be answered referring to NFL (No Free Lunch) [1] theorem. It logically substantiates that no one can recommend a method for solving all optimization problems. In other words, it cannot be assured that the achievement of an algorithm in solving a definitive set of problems could solve every optimization problem of different nature and type. This theorem, therefore, allows researchers to propose new optimization techniques or improve the current algorithms for solving a wider range of problems.

Some of the most popular and well-known algorithms are particle swarm optimization (PSO) [2], genetic algorithm (GA) [3], differential evolution algorithm (DE) [4], ant colony optimization (ACO) [5], Artificial Bee Colony (ABC) [6], Harmony Search (HS) [7], Teaching Learning Based Optimization (TLBO) [8,9], Colliding Bodies Optimization (CBO) [10], Soccer League Competition algorithm (SLC) [11], Exchange Market Algorithm (EMA) [12], Mine Blast Algorithm (MBA) [13], Social based Algorithm (SBA) [14], Sine Cosine Algorithm (SCA) [15], Crow search algorithm (CSA) [16], Moth-flame optimization algorithm (MFO) [17], Whale Optimization Algorithm (WOA) [18], gravitational search algorithm (GSA) [19], Grey wolf optimizer (GWO) [20], Biogeography-based optimization algorithm (BBO) [21], Ant Lion Optimizer (ALO) [22], Multi-Verse Optimizer (MVO) [23], Ions motion optimization (IMO) algorithm [24], Dragonfly algorithm (DA) [25], and Grasshopper Optimization Algorithm (GOA) [26].

The literature shows that hybridizing stochastic optimization algorithms is one of the ways for designing superior algorithms and using the advantages of multiple algorithms when solving optimization problems [[27], [28], [29], [30], [31], [32]].

Object tracking is one of the most important components in computer vision and it has gained significant attention from the research community over the past decade. Even though object tracking has been studied for more than a few decades and significant evolution has been made in recent years [[33], [34], [35], [36], [37], [38]], it remains a challenging problem. The reason is that object tracking has found its way into many real-world applications, including visual surveillance, video communication and compression, navigation, display technology, high-level video analysis, traffic control, metrology, video editing, augmented reality and human-computer interfaces to medical imaging [39,40], and so on. However, object tracking is a challenging task because it often suffers from difficulties in handling complex factors in real world scenarios, e.g. illumination variation, shape deformation, partial occlusion, camera motion, etc. Object tracking in the image can be regarded as a process of searching for the most similar candidate region of the target by an efficient target representation [41]. Therefore, a robust appearance model and an efficient search strategy are of crucial importance for a tracker.

A novel optimization algorithm called Hybrid Sine-Cosine Algorithm with Particle swarm optimization algorithm (SCA-PSO) is proposed in this paper for solving optimization problems and object tracking. The fundamental motivation for designing the Hybrid SCA-PSO was to utilize the memory-enabled behaviour of PSO in the memory-less approach of SCA; i.e., SCA does not keep track of the path of any individual particle in its memory. The proposed algorithm is applied for object tracking as a real thought-provoking case study.

The organization of this paper is as follows. The basics of SCA and PSO are briefly introduced in Section 2. The proposed algorithm is presented in Section 3. Section 4 deals with the evaluation of proposed algorithm using Twenty-three classical, 10 standard, ten CEC 2005 and CEC 2014 benchmark functions. Application of Hybrid SCA-PSO for object tracking is introduced in Section 5 and is compared with Particle Filter (PF), Mean-shift (MS), Particle swarm optimization (PSO), Bat algorithm (BA), Sine Cosine Algorithm (SCA) and Hybrid Gravitational Search Algorithm (HGSA) is presented. Finally, the conclusion is given in Section 6.