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This paper presents a novel automatic image segmentation method based on the theory of active contour models and estimation of distribution algorithms. The proposed method uses the univariate marginal distribution model to infer statistical dependencies between the control points on different active contours. These contours have been generated through an alignment process of reference shape priors, in order to increase the exploration and exploitation capabilities regarding different interactive segmentation techniques. This proposed method is applied in the segmentation of the hollow core in microscopic images of photonic crystal fibers and it is also used to segment the human heart and ventricular areas from datasets of computed tomography and magnetic resonance images, respectively. Moreover, to evaluate the performance of the medical image segmentations compared to regions outlined by experts, a set of similarity measures has been adopted. The experimental results suggest that the proposed image segmentation method outperforms the traditional active contour model and the interactive Tseng method in terms of segmentation accuracy and stability.

Automatic image segmentation is an important and challenging problem in computer vision and medical image analysis. The objective of image segmentation is to separate objects of interest from a given image based on different attributes such as shape, color, intensity, or texture. In recent years, several techniques have been reported for this purpose including graph cut [

The Active Contour Model is an energy-minimizing spline curve composed of discrete control points called snaxels. The curve is attracted towards features as edges of a target object through the evaluation of internal and external forces. The classical implementation of ACM is prone to be trapped into local minima problem and it is also highly sensitive to initialization of the control points because they require being close to the target object; otherwise failure of convergence will occur.

Since ACM was introduced by [

The population-based methods are an effective way to solve discrete optimization problems. Recently, a new approach known as Estimation of Distribution Algorithms (EDAs) from the family of Evolutionary Algorithms has begun to attract more attention for solving global optimization problems with a fast convergence. EDAs are stochastic methods that incorporate statistical knowledge to solve optimization problems [

In this paper, we introduce a novel automatic image segmentation method based on the theory of Active Contour Model and Estimation of Distribution Algorithms. The proposed method uses the Univariate Marginal Distribution Algorithm (UMDA) to infer statistical dependencies between snaxels belonging to a population, in order to increase the exploration and exploitation capabilities regarding the classical Active Contour Model. To establish the initial positioning of the proposed method, a shape prior and the alignment process proposed in [

The remainder of this paper is organized as follows. In Section

In this section, the fundamentals of the Active Contour Model and the Univariate Marginal Distribution Algorithm are described in detail.

The traditional Active Contour Model, also known as snake, is a parametric curve that can move within a spatial image domain where it was defined [

This energy function is composed of two different energies: the external energy

Since the classical ACM has the drawbacks of initialization and local minima problem, Chan and Vese [

These three energies are iteratively evaluated, until the difference between the previous and current segmented area becomes stable. Although the Chan and Vese method is suitable to solve the initialization disadvantage of the classical ACM, this method is prone to be trapped into local minima. An alternative to overcome this drawback is to use population-based methods such as Estimation of Distribution Algorithms, which are described in the following Section

Estimation of Distribution Algorithms [

In the second step a joint probability

According to the above description, the UMDA algorithm can be implemented through the following.

Establish

Select a subpopulation

Compute the univariate marginal probabilities

Generate

Stop if the convergence criterion is satisfied (e.g., stability or number of generations); otherwise, repeat steps (2)–(5).

The proposed image segmentation method based on Active Contour Model and the Univariate Marginal Distribution Algorithm is described in Section

Due to the two main shortcomings of the traditional ACM discussed above, the Univariate Marginal Distribution Algorithm has been adopted to solve the local minima drawback by building probabilistic models, and the initialization disadvantage is addressed by using scaled templates obtained from an alignment process. Since the methodology of the proposed method makes it possible to apply the UMDA strategy directly in the segmentation task, the advantages of low computational time, efficiency, and robustness are inherently acquired. The procedure of the proposed method is illustrated in Figure

Process of the proposed image segmentation method.

The first step of the proposed method consists of the construction of a shape template through the alignment of a training set of selected reference images, which leads to differences in position, direction, and scale. The alignment process is performed using the technique developed in [

In the second step of the method, a preprocessing stage is required, where we first remove the noise from the test image by using a 2D median filter (3

The procedure of the proposed image segmentation method is described as follows.

Align reference shapes according to [

Perform maximum mutual information to positioning the template.

Initialize number of active contours

Initialize the UMDA parameters: number of generations, number of binary bits, and maximum distance

Generate

For each population

apply restriction of the search space to ignore improper individuals;

evaluate each individual in fitness function derived from the Euclidean distance map;

select a subpopulation of individuals according to selection method;

compute the probabilistic model (univariate marginal distribution);

generate

stop if the convergence criterion is satisfied (e.g., stability or number of generations); otherwise go to step (a).

To evaluate the performance of the proposed method on medical images, Jaccard and Dice indices have been adopted to analyze the segmentation results between the regions obtained by computational methods and the regions outlined by experts.

The Jaccard index

In both indices, when the regions

In Section

In this section, the proposed method is applied firstly, on the segmentation of the hollow core in microscopic images of photonic crystal fibers, and secondly, to segment the human heart and ventricular areas from computed tomography and magnetic resonance images. The computational implementations presented in this section are performed using the GNU Compiler Collection (C++) version 4.4.5 running on Debian GNU/Linux 6.0, Intel Core i3 with 2.13 GHz and 4 GB of memory.

Figure

Hollow core photonic crystal fiber: (a) microscopic test image, (b) Euclidean distance map of test image, (c) segmentation result of classical ACM, (d) segmentation result of interactive Tseng method, and (e) segmentation result of proposed method.

Figure

Hollow core photonic crystal fiber: (a) microscopic test image, (b) Euclidean distance map of test image, (c) segmentation result of classical ACM, (d) segmentation result of interactive Tseng method, and (e) segmentation result of proposed method.

Figure

CT images (human heart segmentation): (a) regions outlined by experts, (b) results of classical ACM, (c) results of interactive Tseng method, and (d) segmentation results of proposed method.

From the dataset of computed tomography images of the human heart described above, in Table

Average similarity measure with the Jaccard and Dice indices among the regions segmented by the traditional ACM, interactive Tseng method, our proposed method, and the regions outlined by experts of the CT dataset.

Comparative studies | Similarity measure | |
---|---|---|

Jaccard index ( |
Dice index ( | |

ACM versus experts | 0.6981 | 0.8222 |

Tseng versus experts | 0.7647 | 0.8666 |

Our method versus experts | 0.8367 | 0.9111 |

In Figure

MR images (ventricular area segmentation): (a) regions outlined by experts, (b) results of classical ACM, (c) results of interactive Tseng method, and (d) segmentation results of proposed method.

According to the human ventricular area dataset of MR images previously described, in Table

Average similarity measure with the Jaccard and Dice indices among the ventricular areas segmented by the traditional ACM, interactive Tseng method, our proposed method, and the regions outlined by experts of the MR dataset.

Comparative studies | Similarity measure | |
---|---|---|

Jaccard index ( |
Dice index ( | |

ACM versus experts | 0.6666 | 0.8000 |

Tseng versus experts | 0.7857 | 0.8800 |

Our method versus experts | 0.9230 | 0.9600 |

The use of the Univariate Marginal Distribution Algorithm in the proposed method provides robustness, accuracy and stability in the segmentation problem. Although the computational time of the optimization process is appropriate regarding the comparative computational methods, the proposed image segmentation method improve the segmentation results avoiding the local minima and sensitivity of initialization disadvantages of the classical active contour model.

In this paper, a novel image segmentation method based on the theory of active contour models and estimation of distribution algorithms has been proposed. This segmentation method has introduced two important advantages with respect to different interactive segmentation techniques: firstly, the automatic initialization by using scaled shape templates obtained from an alignment process, in order to overcome the sensitivity to initial contour position and secondly, the incorporation of statistical information of the control points for addressing the local minima problem. Moreover, this proposed method was applied to segment the hollow core in microscopic images of photonic crystal fibers and it was also used to segment the human heart and ventricular areas from CT and MR images. The experimental results demonstrated that the proposed method can lead to more accuracy and efficiency than the traditional active contour model and the interactive Tseng method. In addition, the experimental results have also shown that the exploitation and exploration capabilities of the proposed method, are highly efficient for different applications according to the evidence showed by the set of similarity measures within a competitive computational time.

This research has been supported by the National Council of Science and Technology of México (CONACYT) under Grant no. 241224-218157. The authors would like to thank for the partial funding provided to this work through the projects: DAIP-UG 01/12 and CONCyTEG GTO-2012-C03-195247. The authors wish to thank the cardiology department of the Mexican Social Security Institute UMAE T1 Leon, for the clinical advice and for kindly providing the sources of cardiac CT images. The authors wish to thank the Auckland MRI Research Group, University of Auckland, and Cardiac Atlas Website for the valuable collaboration supplying them the sources of Magnetic Resonance Imaging.