The FCM (fuzzy

Clustering unlabeled data into the most homogeneous groups is a problem that has received extensive attention in many application domains [

Even though the standard FCM algorithm has demonstrated its accuracy in segmenting different kinds of images, it is still inefficient in the presence of noise, where its performance gradually decreases as the image noise increases. This problem is due to the lack of spatial information. To enhance the robustness and the efficiency of the standard FCM algorithm and make it strong enough in the presence of noise, lots of researchers have modified it in different ways; some have modified the objective function, while the others have used different distance metrics. In fact, Pham [

To deal with the intensity inhomogeneity in MRI images, Ahmed et al. [

By combining the main ideas of FCM_S1, FCM_S2, and EnFCM and incorporating the local spatial and the gray information together, Cai et al. [

In the same context of improving the standard FCM by including the spatial information, Chuang et al. [

To improve the robustness to noise of the FCM_S and

In order to deal with noise in MRI images, Ji et al. [

So far, all the aforementioned extensions of the standard FCM have succeeded to different extents in dealing with noise. However, they all share the major drawback of adjusting empirical parameters (

In addition to the inherited advantages from FCM_S, FCM_S1, and FCM_S2, the proposed algorithms come up with valuable ones. At first, they are all fully free of the empirical parameters. Second, they control the tradeoff between noise elimination and detail preservation automatically. Third, the RFCMLGI algorithm is noise type-independent. Finally, all the algorithms are easy to be implemented, because the new factor

The fuzzy clustering is always defined as the process of grouping, with uncertainty, unlabeled data into the most homogeneous groups or clusters as much as possible [

The fuzzy

The minimization of the objective function presented in (

In order to improve the standard FCM and deal with the intensity inhomogeneities in MRI images, Ahmed et al. [

It is noteworthy that the neighborhood information appears in both updating functions (

Like the standard FCM and FCM_S algorithms, the FCM_S1 and FCM_S2 algorithms minimize iteratively the objective function (

Even though the FCM_S, FCM_S1, and FCM_S2 have shown their strength in handling noise, adjusting the parameter

The new factor

2D square window. (a) The central pixel is noisy. (b) The central pixel is not noisy.

The RFCMLGI algorithm clusters data by minimizing iteratively the following objective function and under the previous condition

This optimization problem will be solved using Lagrange multiplier:

By taking the first derivative of

Solving (

As

Thus,

Substituting

This time, we take the first derivative of

Solving for

It is noticeable that the factor

As in FCM_S1 and FCM_S2,

The major advantages of the proposed algorithms are summarized as follows:

They are fully free of the empirical parameters.

Controlling the tradeoff between noise elimination and detail preservation is automatically made.

They are easy to be implemented.

The first version of RFCMLGI is noise type-independent.

In this section, we present some experimental results to show the efficiency of the proposed algorithms RFCMLGI, RFCMLGI_1, and RFCMLGI_2 compared to four other fuzzy clustering algorithms: FCM, FCM_S, FCM_S1, and FCM_S2. Thus, several experiments were performed on synthetic and real images and under different types and levels of noise. The clustering parameters were fixed as follows:

To evaluate quantitatively the segmentation results, we use the segmentation accuracy (SA) defined as follows:

First, we apply all the algorithms to a synthetic image corrupted by different levels of Gaussian and “Salt and Pepper” noise, respectively. This image is composed of 250 × 250 pixels spanning into three classes with three gray level values taken as 0, 100, and 200; thus,

Segmentation accuracies (SA, in %) of seven algorithms on synthetic image.

Algorithm | Gaussian | Salt and Pepper | ||||
---|---|---|---|---|---|---|

9% | 12% | 15% | 9% | 12% | 15% | |

FCM | 91.24 | 88.73 | 86.55 | 97.89 | 97.76 | 97.50 |

FCM_S | 98.23 | 97.94 | 97.58 | 98.21 | 98.18 | 98.10 |

RFCMLGI | | | | | | |

FCM_S1 | 98.79 | 98.61 | 98.29 | 98.73 | 98.58 | 98.55 |

RFCMLGI_1 | | | | | | |

FCM_S2 | 98.93 | 98.72 | 98.33 | 98.89 | 98.79 | 98.77 |

RFCMLGI_2 | | | | | | |

Segmentation results on synthetic image. (a) Original image corrupted by 15% of Gaussian noise. (b) FCM result. (c) FCM_S result. (d) FCM_S1 result. (e) FCM_S2 result. (f) RFCMLGI result. (g) RFCMLGI_1 result. (h) RFCMLGI_2 result.

Segmentation results on synthetic image. (a) Original image corrupted by 15% of Salt and Pepper noise. (b) FCM result. (c) FCM_S result. (d) FCM_S1 result. (e) FCM_S2 result. (f) RFCMLGI result. (g) RFCMLGI_1 result. (h) RFCMLGI_2 result.

From the visual results presented in Figures

From the numerical results depicted in Table

The segmentation accuracy decreases as the level of noise increases for all the algorithms except for the RFCMLGI under both types of noise and RFCMLGI_2 under Salt and Pepper noise.

For each type and level of noise, the proposed algorithms RFCMLGI, RFCMLGI_1, and RFCMLGI_2 outperformed the FCM_S, FCM_S1, and FCM_S2, respectively. And, more specifically, the segmentation accuracies produced by the RFCMLGI are more or less similar, which means that this algorithm is less dependent on the noise type.

Under Salt and Pepper noise, the segmentation accuracies performed by the RFCMLGI_2 are equal and tend towards the maximum, which proves the convenience of this algorithm to segment images corrupted by Salt and Pepper noise.

Under Gaussian noise, RFCMLGI has the best performance.

Based on the previous remarks, we conclude that the proposed algorithms surpassed the FCM_S and its two variants. In addition, if the type of noise is unknown the RFCMLGI is the best choice.

To validate our methods, we test them on two real images and compare their results with the best results of the FCM_S, FCM_S1, and FCM_S2 that are obtained by seeking the value of

We use the trial-and-error method to select the best values of

This image was corrupted by 30% of Gaussian and Salt and Pepper noise, respectively. Even though this image contains two objects, we fixed

Segmentation result on

Segmentation result on

In Figures

Under Salt and Pepper noise (Figure

In terms of detail preserving, we notice clearly (from Figures

As has been concluded in the previous section, the RFCMLGI algorithm is the most convenient one when noise is a priori unknown.

To show the effect of our algorithms on images with mixed noise, we use the “

Segmentation result on

From Figure

Globally, in the experimental results presented in this section we found that the proposed algorithms RFCMLGI, RFCMLGI_1, and RFCMLGI_2 performed better than the FCM_S, FCM_S1, and FCM_S2, respectively, and the RFCMLGI had the best performance. Even though in some cases RFCMLGI_1 and RFCMLGI_2 performed closely to the FCM_S1 and FCM_S2, they remain better because they are free of any parameter selection and they control the effect of the neighboring term automatically.

The standard FCM and its extensions FCM_S, FCM_S1, and FCM_S2 have the same time complexity which is O(HWC) [

In order to furnish a fuzzy clustering algorithm that is fully free of empirical parameters and noise type-independent, this work extended the FCM_S and its two variants to three algorithms based on a new factor that uses the local spatial and the gray level information to calculate the weight of the neighboring term. Generally, all the proposed algorithms RFCMLGI, RFCMLGI_1, and RFCMLGI_2 proved their efficiency on synthetic and real images. More specifically, the RFCMLGI algorithm surpassed considerably the others where it showed its noise type-independence and its ability to retain fine details.

In spite of their fruitful results, the proposed algorithms need to be improved in the running times point of view, where computing the factor

The authors declared that there is no conflict of interests regarding the publication of this paper.