INTRODUCTION

Clustering is a process of grouping similar elements used for image segmentation. Medical Image segmentation is a challenging task and a preprocessing step in medical diagnosis, planning of surgical operations and in radiation treatments. Breast Cancer is one of the major causes of cancer deaths in women (Sheshadri and Kandaswamy, 2006). Breast Cancer and Lung cancer can be determined from ultrasound and x-rays. Segmentation of such images is helpful for the early detection of cancer and effective treatment.

Many segmentation methods are proposed for medical-image data are either direct applications or extensions of approaches from computer vision. Image-segmentation algorithms can be classified in many ways (Haralick and Shapiro, 1985; Sarkar and Boyer, 1993; Zucker, 1976; Zhang et al., 2008) have presented a number of image classification methods. Three broad classes of segmented algorithms are manual, semiautomatic and automatic. Reviews of algorithms from each class can be found in the studies of Clarke et al. (1995) and Suetens et al. (1993). In order to overcome the limitations of the traditional clustering algorithms many heuristic optimization algorithms such as Genetic algorithms (Kharrousheh et al., 2011), Particle Swarm Optimization (PSO) (Saadi et al., 2010), Differential Evolution algorithms are proposed. Tlelo-Cuautle et al. (2010) have studied the significance of evolutionary algorithms in parameter optimization. Genetic Algorithms are found to be effective in image segmentation (Maulik, 2009; Mahi and Izabatene, 2011). Almost all GAs are semiautomatic and requires either some parameters in advance or domain knowledge such as number of segments, shape, texture etc. Differential Evolution is a recent simple evolutionary algorithm for optimization (Adeyemo and Otieno, 2009). DE shares many features of the Genetic Algorithms. DE has won the third place in the International contest on Evolutionary Computation (Peng and Wang, 2010). DE is a very fast and robust algorithm and performs well on synthetic, natural and multi-spectral satellite images when compared with the other evolutionary population-based algorithms like Genetic Algorithm (GA) and PSO etc. (Das et al., 2009; Liu and Li, 2011). Automatic Clustering using Differential Evolution (ACDE), is a variant of DE. ACDE is a DE/rand/1/Either-or algorithm and determines the nearly optimal clusters automatically (Das et al., 2008). There is no research found in medical image segmentation using ACDE. This study is made an attempt to extract the shape of the tissues in medical images automatically using ACDE. The performance of the ACDE in extracting shapes of tissues has been studied using 5 breast cancer images and two lung images. The experimental results have shown that ACDE can determine the appropriate shapes by pixel intensity.

DIFFERENTIAL EVOLUTION SCHEME

Differential Evolution (DE) is the stochastic, population-based optimization algorithm (Price et al., 2005). It was developed to optimize real (float) parameters of a real valued function. DE Algorithms can be used in search of the global optimum, when objective function is non-differentiable, non-continuous, non-linear, noisy, flat, multi-dimensional, with many local minima. DE also can be used when a problem is difficult or impossible to solve analytically and the parameters are real (float) variables.

The difference between GA and DE is, GA relies on uniform crossover and DE uses a non uniform crossover (Otieno and Adeyemo, 2010). The parameter vector is called an individual (or chromosome, or genome). The objective function is also called the fitness function. Figure 1 presents that the general scheme is the same for all EAs:

Procedure
Input: Randomly initialized position and velocity of the particles: x_i(0).

New offsprings are created using two main DE operators known as crossover and mutation.

AUTOMATIC CLUSTERING USING DIFFERENTIAL EVOLUTION

The classical DE (Price et al., 2005), is a population-based global optimization algorithm that uses a floating-point real-coded) representation.

Generation of F and Cr: The ith individual vector (chromosome) of the population at time-step generation) t has d components (dimensions), i.e.:

For each individual vector Z_k(t) that belongs to the current population, DE randomly samples three other individuals, i.e., Z_i(t), Z_j(t) and Z_m(t), from the same generation (for distinct k, i, j and m). It then calculates the (component wise) difference of Z_i(t) and Z_j(t), scales it by a scalar F (usually [0, 1]) and creates a trial offspring U_i (t+1) by adding the result to Z_m(t). Thus, for the nth component of each vector:

Cr [0, 1] is a scalar parameter of the algorithm, called the crossover rate.

Fig. 1:	General evolutionary algorithm scheme

If the new offspring yields a better value of the objective function, it replaces its parent in the next generation; otherwise, the parent is retained in the population, i.e.:

where, f(•) is the objective function to be maximized. To improve the convergence properties of DE, we have tuned its parameters in two different ways here. In the original DE, the difference vector (Z_i(t)-Z_j(t)) is scaled by a constant factor F. The usual choice for this control parameter is a number between 0.4 and 1. We propose to vary this scale factor in a random manner in the range (0.5, 1) by using the relation:

where, r and (0, 1) is a uniformly distributed random number within the range [0, 1]. The mean value of the scale factor is 0.75.

Pseudo code of the ACDE: The pseudo code for the complete ACDE algorithm is given here.

Step 1: Initialize each chromosome to contain K number of randomly selected cluster centers and K (randomly chosen) activation thresholds in [0, 1].

Step 2: Find out the active cluster centers in each chromosome.

Step 3: For t = 1 to t_max do

Where:

Step 4: Report as the final solution the cluster centers and the partition obtained by the globally best chromosome (one yielding the highest value of the fitness function) at time t = t_max.

We have used CS measure as the fitness function. Chou et al. (2004) have proposed the CS measure for evaluating the validity of a clustering scheme. The centroid of a cluster is computed by averaging the elements that belong to the same cluster using:

where, is the distance between X_i and X_j i.e., the difference of two pixel intensities and is the distance between two cluster centroids. CS measure is a function of the ratio of the sum of within-cluster distance to between-cluster distance. The cluster configuration that minimizes CS is taken as the optimal number of clusters, k.

EXPERIMENTAL RESULTS

The performance of ACDE has been studied using ultrasound breast cancer tissue images. The extracted shapes of tissues are compared with shape specified by a radiologist and with the chain optimization (Rahnamayan and Mohamad, 2010). The ultrasound images are taken from (Rahnamayan and Mohamad, 2010). The Fig. 2 shows the sample of original ultrasound images, tissue segmentation (green) as specified by the radiologist and the corresponding tissue segmentation (red) results of Chain optimization (Rahnamayan and Mohamad, 2010).

As the ACDE results different clusters in different independent runs, the Table 1 shows different segmented regions for a given medical image. The experimental results demonstrated that the extracted shape of tissues is insensitive to the number of clusters.

Canny edge detection method is applied to the clustering solutions of ACDE to identify the segments and the segmented regions for sample images are included in the Fig. 2.

Table 1:	Original Images and corresponding output images of ACDE
*Segments determined out of total given segments

Fig. 2(a-c):	(a) Four test ultrasound breast cancer images, (b) The green segmentation has been performed by a radiologist and the red one by the chain optimization (5) and (c) Segmented regions from ACDE

The resultant segments of ACDE are compared with the chain optimization and with the segments identified by radiologists. The results of ACDE demonstrated that the ACDE can determine the shape of the tissues as specified by the radiologist.