In this section, we describe the application of the MFCM segmentation on MRI images having intensity inhomogeneity. Spatial intensity inhomogeneity induced by the radio frequency (RF) coil in magnetic resonance imaging (MRI) is a major problem in the computer analysis of MRI data [24-27]. Such inhomogeneities have rendered conventional intensity-based classification of MR images very difficult, even with advanced techniques such as nonparametric, multichannel methods [28-30]. This is due to the fact that the intensity inhomogeneities appearing in MR images produce spatial changes in tissue statistics, i.e. mean and variance. In addition, the degradation on the images obstructs the physician's diagnoses because the physician has to ignore the inhomogeneity artifact in the corrupted images [31].

The removal of the spatial intensity inhomogeneity from MR images is difficult because the inhomogeneities could change with different MRI acquisition parameters from patient to patient and from slice to slice. Therefore, the correction of intensity inhomogeneities is usually required for each new image. In the last decade, a number of algorithms have been proposed for the intensity inhomogeneity correction. Meyer et al. [32] presented an edge-based segmentation scheme to find uniform regions in the image followed by a polynomial surface fit to those regions. The result of their correction is, however, very dependent on the quality of the segmentation step.

Several authors have reported methods based on the use of phantoms for intensity calibration. Wicks et al. [26] proposed methods based on the signal produced by a uniform phantom to correct for MRI images of any orientation. Similarly, Tincher et al. [33] modeled the inhomogeneity function by a second-order polynomial and fitted it to a uniform phantom-scanned MR image. These phantom approaches, however, have the drawback that the geometry relationship of the coils and the image data is typically not available with the image data. They also require the same acquisition parameters for the phantom scan and the patient. In addition, these approaches assume the intensity corruption effects are the same for different patients, which is not valid in general [31].

The homomorphic filtering approach to remove the multiplicative effect of the inhomogeneity has been commonly used due to its easy and efficient implementation [29,34]. This method, however, is effective only on images with relatively low contrast. Some researchers [33,35] reported undesirable artifacts with this approach.

Dawant et al. [35] used operator-selected reference points in the image to guide the construction of athin-plate spline correction surface. The performance of this method depends substantially on the labeling of the reference points. Considerable user interactions are usually required to obtain good correction results. More recently, Gilles et al. [36] proposed an automatic and iterative B-spline fitting algorithm for the intensity inhomogeneity correction of breast MR images. The application of this algorithm is restricted to MR images with a single dominant tissue class, such as breast MR images. Another polynomial surface fitting method [37] was proposed based on the assumption that the number of tissue classes, the true means, and standard deviations of all the tissue classes in the image are given. Unfortunately, the required statistical information is usually not available.

A different approach used to segment images with intensity inhomogeneities is to simultaneously compensate for the shading effect while segmenting the image. This approach has the advantage of being able to use intermediate information from the segmentation while performing the correction. Recently, Wells et al. [28] developed a new statistical approach based on the EM algorithm to solve the bias field correction problem and the tissue classification problem. Guillemaud et al. [38] further refined this technique by introducing the extra class "other." There are two main disadvantages of this EM approach. First, the EM algorithm is extremely computationally intensive, especially for large problems. Second, the EM algorithm requires a good initial guess for either the bias field or the classification estimate. Otherwise, the EM algorithm could be easily trapped in a local minimum, resulting in an unsatisfactory solution [31].

Another approach based on the FCM [40, 41] clustering technique has been introduced lately [42-44]. FCM has been used with some success in image segmentation in segmenting MR images [42,47,50]. Xu et al. [42] proposed a new adaptive FCM technique to produce fuzzy segmentation while compensating for intensity inhomogeneities. Their method, however, is also computationally intensive. They reduced the computational complexity by iterating on a coarse grid rather than the fine grid containing the image. This introduced some errors in the classification results and was found to be sensitive to a considerable amount of salt and pepper noise [43].

To solve the problem of noise sensitivity and computational complexity of the Pham and Prince method, we will generalize the MFCM algorithm to segment MRI data in the presence of intensity inhomogeneities.

The observed MRI signal is modeled as a product of the true signal generated by the underlying anatomy and a spatially varying factor called the gain field:

where Xk and Yk are the true and observed intensities at the kth voxel, respectively, Gk is the gain field at the kth voxel, and N is the total number of voxels in the MRI volume.

The application of a logarithmic transformation to the intensities allows the artifact to be modeled as an additive bias field [28]

where xk and yk are the true and observed log-transformed intensities at the kth voxel, respectively, and pk is the bias field at the kth voxel. If the gain field is known, it is relatively easy to estimate the tissue class by applying a conventional intensity-based segmenter to the corrected data. Similarly, if the tissue classes are known, we can estimate the gain field, but it may be problematic to estimate either without the knowledge of the other. We will show that by using an iterative algorithm based on fuzzy logic, we can estimate both.

9.4.4.2 Bias Corrected Fuzzy C-means (BCFCM) Objective Function

Substituting Eq. 9.36 into Eq. 9.25, we have c N a c N I \

Jm = EE upk \\Vk - Pk - «ill2 + E II yr - Pr - Vi||2 •

Formally, the optimization problem comes in the form min Jm subject to U e U• (9.38)

9.4.4.3 BCFCM Parameter Estimation

The objective function Jm can be minimized in a fashion similar to the MFCM algorithm. Taking the first derivatives of Jm with respect to uik, vi, and pk and setting them to zero results in three necessary but not sufficient conditions for Jm to be at a local extrema. In the following subsections, we will derive these three conditions.

9.4.4.4 Membership Evaluation

Similar to the MFCM algorithm, the constrained optimization in Eq. 9.38 will be solved using one Lagrange multiplier

where Dik = \\y - Pk - Vi\\2 and Yi = (E y N II M- - Pr - «i||2). The zero-gradient condition for the membership estimator can be written as

Djk+NR Yj J

9.4.4.5 Cluster Prototype Updating

Taking the derivative of Fm w.r.t. vi and setting the result to zero, we have

Nr Vr eNk

Solving for we have vi =

9.4.4.6 Bias Field Estimation

In a similar fashion, taking the derivative of Fm w.r.t. and setting the result to zero we have d

Since only the kth term in the second summation depends on jik, we have c ß

Differentiating the distance expression, we obtain vYlUik - ßkJ2Up UikVi i=1

Thus, the zero-gradient condition for the bias field estimator is expressed as fl* — V £c=1 UikVi

9.4.4.7 BCFCM Algorithm

The BCFCM algorithm for correcting the bias field and segmenting the image into different clusters can be summarized in the following steps:

Step 1. Select initial class prototypes (vj^x. Set 1 to equal and very small values (e.g. 0.01).

Step 2. Update the partition matrix using Eq. 9.40.

Step 3. The prototypes of the clusters are obtained in the form of weighted averages of the patterns using Eq. 9.42.

Step 4. Estimate the bias term using Eq. 9.46.

Repeat steps 2-4 till termination. The termination criterion is as follows

where 11 11 is the Euclidean norm, V is a vector of cluster centers, and e is a small number that can be set by the user.

In this section, we describe the application of the BCFCM segmentation to synthetic images corrupted with multiplicative gain, as well as digital MR phantoms [51] and real brain MR images. The MR phantoms simulated the appearance and image characteristics of the T1 weighted images. There are many advantages of using digital phantoms rather than real image data for validating segmentation methods. These advantages include prior knowledge of the true tissue types and control over image parameters such as mean intensity values, noise, and intensity inhomogeneities. We used a high-resolution T1 weighted phantom with in-plane resolution of 0.94 mm2, Gaussian noise with a = 6.0, and 3D linear shading of 7% in each direction. All of the real MR images shown in this section were obtained using a General Electric Signa 1.5 T clinical MR imager with the same in-plane resolution as the phantom. In all the examples, we set the parameter a (the neighbors effect) to be 0.7, p = 2, NR = 9 (a 3 x 3 window centered around each pixel), and e = 0.01. For low SNR images, we set a = 0.85. The choice of these parameters seems to give the best results.

Figure 9.15(a) shows a synthetic test image. This image contains a two-class pattern corrupted by a sinusoidal gain field of higher spatial frequency. The test image is intended to represent two tissue classes, while the sinusoid represents an intensity inhomogeneity. This image was constructed so that it would be difficult to correct using homomorphic filtering or traditional FCM approaches. As shown in Fig. 9.15(b), FCM algorithm was unable to separate the two classes, while the BCFCM and EM algorithms have succeeded in correcting and classifying the data as shown in Fig. 9.15(c). The estimate of the multiplicative gain

Figure 9.15: Comparison of segmentation results on a synthetic image corrupted by a sinusoidal bias field. (a) The original image, (b) FCM results, (c) BCFCM and EM results, and (d) bias field estimations using BCFCM and EM algorithms: this was obtained by scaling the bias field values from 1 to 255.

Figure 9.15: Comparison of segmentation results on a synthetic image corrupted by a sinusoidal bias field. (a) The original image, (b) FCM results, (c) BCFCM and EM results, and (d) bias field estimations using BCFCM and EM algorithms: this was obtained by scaling the bias field values from 1 to 255.

using either BCFCM or EM is presented in Fig. 9.15(d). This image was obtained by scaling the values of the bias field from 1 to 255. Although the BCFCM and EM algorithms produced similar results, BCFCM was faster to converge to the correct classification, as shown in Fig. 9.16.

Figures 9.17 and 9.18 present a comparison of segmentation results between FCM, EM, and BCFCM, when applied on T1 weighted MR phantom corrupted with intensity inhomogeneity and noise. From these images, we can see that

40 50

Figure 9.16: Comparison of the performance of the proposed BCFCM algorithm with EM and FCM segmentation when applied to the synthetic two-class image shown in Fig. 9.15(a).

40 50

Figure 9.16: Comparison of the performance of the proposed BCFCM algorithm with EM and FCM segmentation when applied to the synthetic two-class image shown in Fig. 9.15(a).

traditional FCM was unable to correctly classify the images. Both BCFCM and EM segmented the image into three classes corresponding to background, gray matter (GM), and white matter (WM). BCFCM produced slightly better results than EM due to its ability to cope with noise. Moreover, BCFCM requires far less number of iterations to converge compared to the EM algorithm. Table 9.2 depicts the segmentation accuracy (SA) of the three mentioned method when applied to the MR phantom. SA was measured as follows:

Number of correctly classified pixels

Total number of pixels

SA was calculated for different SNR. From the results, we can see that the three methods produced almost similar results for high SNR. BCFCM method, however, was found to be more accurate for lower SNR.

Figure 9.17: Comparison of segmentation results on a MR phantom corrupted with 5% Gaussian noise and 20% intensity inhomogeneity: (a) original T1 weighted image, (b) using FCM, (c) using EM, and (d) using the proposed BCFCM.

Figure 9.17: Comparison of segmentation results on a MR phantom corrupted with 5% Gaussian noise and 20% intensity inhomogeneity: (a) original T1 weighted image, (b) using FCM, (c) using EM, and (d) using the proposed BCFCM.

Figure 9.18: Comparison of segmentation results on an MR phantom corrupted with 5% Gaussian noise and 20% intensity inhomogeneity: (a) original T1 weighted image, (b) using FCM, (c) using EM, and (d) using the proposed BCFCM.

Figure 9.18: Comparison of segmentation results on an MR phantom corrupted with 5% Gaussian noise and 20% intensity inhomogeneity: (a) original T1 weighted image, (b) using FCM, (c) using EM, and (d) using the proposed BCFCM.

Segmentation Method |
SNR | ||

13 db |
10 db |
8 db | |

FCM |
98.92 |
86.24 |
78.9 |

EM |
99.12 |
93.53 |
85.11 |

BCFCM |
99.25 |
97.3 |
93.7 |

Figure 9.19 shows the results of applying the BCFCM algorithm to segment a real axial-sectioned T1 MR brain. Strong inhomogeneities are apparent in the image. The BCFCM algorithm segmented the image into three classes corresponding to background, GM, and WM. The bottom right image shows the estimate of the multiplicative gain, scaled from 1 to 255.

Figure 9.20 shows the results of applying the BCFCM for the segmentation of noisy brain images. The results using traditional FCM without considering the neighborhood field effect and the BCFCM are presented. Notice that the BCFCM segmentation, which uses the the neighborhood field effect, is much less fragmented than the traditional FCM approach. As mentioned before, the relative importance of the regularizing term is inversely proportional to the SNR of MRI signal. It is important to note, however, that the incorporation of spatial constraints into the classification has the disadvantage of blurring some fine details. There are current efforts to solve this problem by including contrast information into the classification. High contrast pixels, which usually represent boundaries between objects, should not be included in the neighbors.

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