Elsevier

Pattern Recognition Letters

Volume 26, Issue 13, 1 October 2005, Pages 2052-2062
Pattern Recognition Letters

An intelligent modified fuzzy c-means based algorithm for bias estimation and segmentation of brain MRI

https://doi.org/10.1016/j.patrec.2005.03.019Get rights and content

Abstract

The segmentation of magnetic resonance images (MRI) is a challenging problem that has received an enormous amount of attention lately. Many researchers have applied various techniques however fuzzy c-means (FCM) based algorithms have produced better results compared to other methods. In this paper, we present a modified FCM algorithm for bias (also called intensity in-homogeneities) estimation and segmentation of MRI. Normally, the intensity in-homogeneities are attributed to imperfections in the radio-frequency coils or to the problems associated with the image acquisition. Our algorithm is formulated by modifying the objective function of the standard FCM and it has the advantage that it can be applied at an early stage in an automated data analysis before a tissue model is available. The proposed method can deal with the intensity in-homogeneities and Gaussian noise effectively. We have conducted extensive experimental and have compared our results with other reported methods. The results using simulated images and real MRI data show that our method provides better results compared to standard FCM-based algorithms and other modified FCM-based techniques.

Introduction

There are many kinds of image processing techniques used as diagnostic imaging modalities and amongst them the most popular is MRI (Bushong, 1996). The advantages of MRI are its high spatial resolution and soft-tissue contrast and that is why MRI is widely used in many medical applications (Chaozhe and Jiang, 2003). A typical MRI analysis of a patient involves vast amounts of data, thus it is time consuming as manual segmentation is normally done for many slices of images. Therefore, there is a need for computer analysis of MRI such as precise delineation of tumors and reliable, reproducible segmentation of images. In segmenting MRI data, we have mainly three considerable difficulties: noise, partial volume effects (where more than one tissue is inside a pixel volume) and intensity in-homogeneity. The bias field (intensity in-homogeneity) is induced by the radio-frequency coil in MRI and is a major problem in computer-based analysis of MRI data. Although MRI images may appear visually uniform, such in-homogeneities can cause serious misclassifications when intensity-based segmentation techniques are used (Ahmed et al., 2002). Ideally, for any given set of MRI, the intensity values of the voxels of any given tissue class should be constant, or, in view of the partial volume effect, it should correspond to a Gaussian distribution. With small standard deviation and differentiation between white and gray matter in the brain, it should be easy to segment it since these tissue exhibit distinct signal intensities. However, in practice, spatial intensity in-homogeneities can be as much as 30% of image amplitudes and thus causes the distribution of signal intensities associated with these tissue classes to overlap significantly (Ahmed et al., 2002, Chaozhe and Jiang, 2003). Therefore, the correction of spatial intensity in-homogeneity has been regarded as a necessary requirement for robust automated segmentation of MRI.

In the last decade, a number of algorithms have been proposed for the intensity in-homogeneity correction (Bushong, 1996, Axel et al., 1987, Listerud et al., 1989, Wicks et al., 1993, Tincher et al., 1993, Lai and Fang, 1998, Moyher et al., 1995, Meyer et al., 1995). Early methods for bias field estimation and correction used prior acquisition of a phantom image to empirically measure the bias field in-homogeneity (Axel et al., 1987, Listerud et al., 1989). Wicks et al. (1993) proposed methods based on the signal produced by a uniform phantom to correct MRI of any orientation problems. Furthermore, Tincher et al. (1993) modeled in-homogeneity function by a second-order polynomial and fitted it to a uniform phantom-scanned MRI. However, all such approaches based on a prior phantom acquisition have the drawback that the geometry relationship of the coils and the image data is normally not available (Ahmed et al., 2002, Chaozhe and Jiang, 2003). They also require the same acquisition parameters for the phantom scan and the patient. In addition, these approaches assume that the intensity corruption effects are the same for different patients, which is not valid in general (Lai and Fang, 1998, Moyher et al., 1995).

Meyer et al. (1995) presented an edge-based segmentation scheme to find uniform regions in the image followed by a polynomial surface fit for those regions. The result of their correction is, however, very dependant on the quality of the segmentation step. Dawant et al. (1993) developed a two-step approach for estimation of bias field. In this approach first “reference points” are selected for at least one tissue class (they used white matter) throughout the image, then a thin-plate spline is “least-squared” and fitted to the reference point data. They suggest the coefficient of variations as a measure for the degree of restoration. The selection of reference points is either done manually, or by a tissue classification algorithm after a partial classification. They found that the expert selection of reference points can give better results than automatic selection; and also it is prone to errors when points are mislabeled (Ahmed et al., 2002). The homomorphic filtering approach to remove the multiplicative effect of in-homogeneity has also been commonly used due to its easy and efficient implementation (Johnston et al., 1996, Brinkmann et al., 1998). This method assumes that the frequency spectrum of the bias field and the image structures are well separated, but this assumption is generally not valid for MRI (Tincher et al., 1993, Dawant et al., 1993).

Wells et al. (1996) developed a statistical approach based on the expectation–maximization (EM) algorithm to estimate the bias field and the tissue classification. Guillemaud and Brady (1997) improved this approach by introducing an extra class called “others” with a non-Gaussian probability distribution. This new extra class is intended to gather all the pixels which are far from the Gaussian distributions of the identified tissue classes. Held et al. (1997) refined the adaptive segmentation algorithm. They used Marko random field (MRF) (Zhang et al., 2001) as a convenient means for introducing context or neighborhood effect in the classification (Li, 1995, Geman and Geman, 1984). Leemput et al., 1999a, Leemput et al., 1999b developed a generalized EM algorithm (GEM) for bias field correction. They used the variable parameters of Gaussian which changes in the algorithm. Marroquin et al. (2002) developed a new Bayesian method for automatic segmentation of brain MRI. They use a variant of the EM algorithm making the whole procedure computationally efficient. However, all of the above statistical approaches need good initialization of tissue class. The large pathology-induced abnormalities in brain shape will thus make the initialization unacceptable (Chaozhe and Jiang, 2003). Sled et al. (1998) developed non-parametric non-uniform intensity normalization (N3) approach; however this approach only corrects the intensity in-homogeneity in the MRI and does not include the segmentation procedure.

Another approach based on the fuzzy c-means (FCM) (Bezdek, 1980, Bezdek and James, 1999) clustering technique has been used for image segmentation in general (Qing et al., 1992, Kettaf et al., 1996) and also in segmenting MR images (Brandt et al., 1994, Hall et al., 1992, Yoon et al., 1999, Clark et al., 1994). Li et al. (1993) proposed an “over-segmentation” method and then merge or split these separated parts based on the prior tissue knowledge. However, “over-segmentation” method is very time consuming and the strategy of merge and split methods is also not reliable (Chaozhe and Jiang, 2003). Pham and Prince (1999) proposed a new adaptive FCM technique to produce fuzzy segmentation while compensating the intensity in-homogeneities. Their method, however, is also computationally intensive. They reduced the computational complexity by iterating on a coarse grid rather on the fine grid containing the image. This introduced some errors in the classification results and was found to be sensitive to noise (Ahmed et al., 2002). To solve the problem of noise sensitivity and computational complexity of Pham and Prince, 1999, Ahmed et al., 2002 present a modified FCM algorithm for bias field estimation and segmentation of MRI. They propose a modification to the standard FCM objective function by introducing a term that allows the labeling of a voxel to be influenced by the labels in its immediate neighborhood. Their method also has many problems. They use zero-gradient condition for the bias-field estimator but this produces misclassification. Also the modification to accommodate neighborhood effect blurs the edge of tissues and hence produces the segmenting errors (Chaozhe and Jiang, 2003).

In this paper, we present a modified fuzzy c-means (FCM) algorithm for intensity in-homogeneities estimation and segmentation of MR images. Our algorithm is formulated by modifying the objective function of the standard FCM and uses a special spread method to obtain a smooth and slowly varying bias field, which is also used in classification of tissues This method has the advantage that it can be applied at an early stage in an automated data analysis before a tissue model is available. The bias field can deal with the intensity in-homogeneities and Gaussian noise effectively. We have conducted extensive experiments and the results on simulated and real MRI data show that this method provides better results compared to standard FCM algorithms and other modified FCM algorithms.

The rest of the report is organized as follows: in Section 2, the traditional fuzzy c-means method is reviewed and the MRI signal model is presented. We also show how we change the FCM to fit the model and compare it with previous modified FCM algorithms. In Section 3, the segmentation results are discussed and compared with other reported techniques. In Section 4, the paper is concluded.

Section snippets

FCM algorithm and our modified method

The c-means (or k-means) families are the best known and well developed families of batch clustering models because they are “least square” models. The aim of a partitioned clustering algorithm given a set of data X = {x1,  , xn}, xi  {Rd} is to divide it into c self-similar groups (c  2). These groups (clusters) form a c-partition of X. A real (c × n) matrix U can be used to represents the results of a cluster analysis of X by interpreting uik as the degree to which xk belongs.

Results and discussion

To validate the performance of our method, in this section we describe the results of our modified FCM algorithm on simulated and real MRI data. The real MRI data is from internet brain segmentation repository (IBSR) (ISBR, 2004). The simulated 3-D MRI data is obtained from the BrainWeb Database at the McConnell Brain Imaging Center of the Montreal Neurological Institute (MNI), McGill University (2004).

Conclusions

We have developed a modified FCM algorithm for automatic segmentation and intensity correction of MR brain images. The algorithm was formulated by modifying the objective function of the standard FCM algorithm to compensate for intensity in-homogeneity. We use a mean spread filter to smooth the bias field for obtaining the correct classification. We also use a fuzzy filter for removing the noise. Our algorithm avoids some of the restrictive model assumptions and initialization that are

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