Image histogram thresholding based on multiobjective optimization

Abstract

The thresholding process based on the optimization of one criterion only does not work well for a lot of images. In many cases, even when equipped with the optimal value of the threshold of its single criterion, the thresholding program does not produce a satisfactory result. In this paper, we propose to use the multiobjective optimization approach to find the optimal thresholds of three criteria: the within-class criterion, the entropy and the overall probability of error criterion. In addition we develop a new variant of simulated annealing adapted to continuous problems to solve the Gaussian curve-fitting problem. Some examples of test images are presented to compare our segmentation method, based on the multiobjective optimization approach, with that of four competing methods: Otsu method, Gaussian curve fitting-based method, valley-emphasis-based method and two-dimensional Tsallis entropy-based method. From the viewpoints of visualization, object size and image contrast, our experimental results show that the thresholding method based on multiobjective optimization performs better than the competing methods.

Introduction

The image segmentation process is defined as the extraction of the important objects from an input image. Since the segmentation is considered by many authors to be an essential component of any image analysis system, that problem has received a great deal of attention; thus any attempt to survey completely the literature in this paper would take up too much space. However, surveys of most segmentation methods may be found in [1], [2], [3], [4].

Image thresholding is definitely one of the most popular segmentation approaches to extract objects from images, and more particularly infrared images (e.g. [1], [4]), for the reason that it is straightforward to implement. It is based on the assumption that the objects can be distinguished by their gray levels. The optimal thresholds are those permitting the distinction of different objects from each other or different objects from the background (e.g. [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]). The automatic fitting of this threshold is one of the main challenges of image segmentation.

A typical technique of image segmentation based on histogram thresholding consists in the separation of different gray level classes in an optimal way according to some a posteriori criterion [6], [7], [8], [9], [10], [11], [12], [13], [14]. For instance, in the case of bi-level thresholding, the histogram of the image is usually assumed to have one valley between two peaks, representing the background and the objects respectively. There are two main approaches to the problem of locating the intensity threshold: parametric and non-parametric techniques. The difference between these approaches lies in the estimation of the parameters of the histogram. The non-parametric methods are more robust than the parametric ones. Conversely, the non-parametric methods are computationally costly when they are extended to multilevel thresholding.

Multiobjective optimization (MO) (also known as multicriterion, or Pareto optimization) extends the optimization theory by permitting several design objectives to be optimized simultaneously. A MO problem is solved in a way similar to the conventional single-objective (SO) problem. The goal is to find a set of values for the design variables that simultaneously optimizes several objective (or cost) functions. In general, the solution obtained through a separate optimization of each objective (i.e. SO optimization) does not represent a feasible solution of the multiobjective problem.

Several methods have been devised for solving MO problems, including hierarchical optimization, weighting objectives, distance functions, goal programming, constraint methods, min–max optimization, and many others (e.g. [15], [16], [17], [18], [19]). These methods fall into two broad categories: (i) methods which attempt to optimize each criterion in turn, subject to constraints derived from the optimization of previously optimized criteria, and (ii) methods which attempt to minimize a SO function that combines the design objectives in some prescribed functional form (often referred to as a global criterion) [18].

Unfortunately, part of MO theory deals with continuous design variables and is not directly applicable to discrete, combinatorial problems such as the application at hand. In this paper, we exploit the power of simulated annealing (SA) for searching within vast combinatorial state spaces to find the optimum of the image segmentation multiobjective functions and the optimal thresholds. The method allows traditional objectives of image segmentation such as the following: the regions must be uniform, homogenous; the interiors of the regions must be simple; adjacent regions must present significantly different values for uniform characteristics; the intra-class variance, entropy and the overall probability of error must be optimized.

The development of optimization techniques has been flourishing during the last decade. Many optimization methods such as genetic algorithms [14] and recently particle swarm optimization [9] have been applied to image segmentation problems. Authors in [9] proposed a hybrid Nelder–Mead particle swarm optimization method to handle the objective function of Gaussian curve fitting (GCF) for multilevel thresholding. However, curve fitting is unusually time-consuming for multilevel thresholding and when the optimal threshold is not located at the intersection of the Gaussian curves it may never be found. In [14] the authors proposed an adaptive segmentation system using different MO approaches to evaluate the segmentation quality based on genetic and hybrid genetic algorithms applied to an outdoor TV imagery segmentation problem. The authors showed that the combined segmentation quality measure achieved a good segmentation result. However, the segmentation result depends on the segmentation algorithm that is employed and the executing times are far from real-time processing rates. In [13] the authors used a variant of Mumford–Shah's general model to threshold images. However, the fitting of the regularization parameter is not discussed and the method takes about 120 s for a test image of 256×256 pixels.

The goal of our work is to show that the segmentation techniques based on the combination of some criteria gives a good segmentation result and increases the ability to apply the same technique to a wide variety of images. We also show that combining criteria helps to overcome the weaknesses of these criteria when used separately.

In this paper, a new multilevel image thresholding technique that combines the flexibility of multiobjective fitness functions with the power of SA for searching vast combinatorial state spaces is proposed.

SA was used for both histogram GCF and optimizing the image segmentation multiobjective criterion in order to find the optimal thresholds. A variant of SA handling many continuous variables, called enhanced simulated annealing (ESA), was also used [20], to solve the continuous optimization problem of fitting the image histogram to a sum of Gaussians.

The outline of the paper is as follows: we begin in the next section by introducing the problem of multilevel image thresholding as a multiobjective problem and giving the principle of the plain aggregating approach to solve a multiobjective problem. The mathematical formulation of the different criteria is given in the first part of this section; in the second part, we present a version of SA that we used to minimize the objective function; this section ends with the sketch description of the multilevel image thresholding algorithm. In Section 3, we present a complete description of our thresholding algorithm, where each step of the algorithm is developed in detail. In Section 4, we illustrate the obtained results through the proposed image thresholding algorithm. The paper ends with a brief concluding section.