The goal of lung extraction is to separate the voxels corresponding to lung tissue from those belonging to the surrounding anatomical structures. We assume that each slice consists of two types of pixels: lung and other tissues (e.g., chest, ribs, and liver). The problem in lung segmentation is that there are some tissues in the lung such as arteries, veins, bronchi, and bronchioles having gray level close to the gray level of the chest. Therefore, in this application if we depend only on the gray level we lose some of the lung tissues during the segmentation process. Our proposed model which depends on estimating parameters for two processes (high-level process and low-level process) is suitable for this application because the proposed model not only depend on the gray level but also takes into consideration the characterization of spatial clustering of pixels into regions.

We will apply the approach that was described in Section 9.2.4 on lung CT. Figure 9.4 shows a typical CT slice for the chest. We assume that each slice consists of two types of tissues: lung and other tissues (e.g., chest, ribs, and liver). As discussed above, we need to estimate parameters for both low-level process and high-level process. Table 9.1 presents the results of applying the

Parameter |
ßin |
ßir2 |
°iT2 |
nIT1 |
nIT2 | |

Value |
59.29 |
139.97 |
177.15 |
344.29 |
0.25 |
0.758 |

dominant Gaussian components extracting algorithm described in 9.2.5.1. Figure 9.5 shows the empirical density for the CT slice shown in Fig. 9.4 and the initial estimated density (which represented the two dominant Gaussian components in the given CT). The Levy distance between the two distribution functions which represented the densities shown in Fig. 9.5 is 0.09. This value is large and this means there is a mismatch between empirical pem(y) and pj (y). Figure 9.6 shows the error and absolute error between pem(y) and pj (y).

After we apply sequential EM algorithm to \Z (y)|, we get that the number of normal components that represent \Z (y)\ is 10 as shown in Fig. 9.7. Figure 9.8

0.45

0.35

0.45

0.35

0.25

0.15

Figure 9.7: Conditional expectation Q(n) and the error function (e(n)) versus the number of Gaussians approximating the scaled absolute deviation in Fig. 9.6.

0.25

0.15

10 11 n

Figure 9.7: Conditional expectation Q(n) and the error function (e(n)) versus the number of Gaussians approximating the scaled absolute deviation in Fig. 9.6.

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Figure 9.9: 12 Gaussian components which are used in density estimation.

Figure 9.9: 12 Gaussian components which are used in density estimation.

shows the estimated density for \Z(y)\. Figure 9.9 shows all Gaussian components which are estimated after using dominant Gaussian components extracting algorithm and sequential EM algorithms. Figure 9.10 shows the estimated density for the CT slices shown in Figure 9.4. The Levy distance between the distributions Pes(y) and Pem(y) is 0.0021 which is smaller compared to the Levy distance between the distributions Pem(y) and PI(y).

Now we apply components classification algorithm on the ten Gaussian components that are estimated using sequential EM algorithm in order to determine which components belong to lung tissues and which components belong to chest tissues. The results of components classification algorithm show that the minimum risk equal to 0.004 48 occurs at threshold Th = 108 when Gaussian components 1, 2, 3, and 4 belong to lung tissues and component 5, 6, 7, 8, 9, and 10 belong to chest tissues. Figure 9.11 shows the estimated density for lung tissues and estimated density for chest and other tissues that may appear in CT.

The next step of our algorithm is to estimate the parameters for high-level process. A popular model for the high-level process is the Gibbs Markov mode, and we use the Bayes classifier to get initial labeling image. After we run Metropolis algorithm and GA to determine the coefficients of potential function E(x), we get

Figure 9.12: (a) Segmented lung using the proposed algorithm, error = 1.09%. (b) Output of segmentation algorithm by selecting parameters for high-level process randomly, error = 1.86%. (c) Segmented lung by radiologist.

Figure 9.12: (a) Segmented lung using the proposed algorithm, error = 1.09%. (b) Output of segmentation algorithm by selecting parameters for high-level process randomly, error = 1.86%. (c) Segmented lung by radiologist.

the following results: a = 1, 01 = 0.89, 02 = 0.8, 03 = 0.78, 04 = 0.69, 05 = 0.54, 06 = 0.61, 07 = 0.89 , 08 = 0.56, and 09 = 0.99.

The result of segmentation for the image shown in Fig. 9.4 using these parameters is shown in Fig. 9.12. Figure 9.12(a) shows the results of proposed algorithm. Figure 9.12(b) shows output of the Metropolis algorithm by selecting parameters randomly. Figure 9.12(c) shows the segmentation done by a radiologist.

As shown in Fig. 9.12(a) the accuracy of our algorithm seems good if it is compared with the segmentation of the radiologist. Figure 9.13 shows comparison between our results and the results obtained by iterative threshold method which was proposed by Hu and Hoffman [23]. It is clear from Fig. 9.13 that the

99 99

Figure 9.13: (a) Original CT, (b) segmented lung using the proposed model, (c) segmented lung using the iterative threshold method, and (d) segmented lung by radiologist. The errors with respect to this ground truth are highlighted by red color.

Figure 9.14: (a) Generated Phantom, (b) ground truth image (black pixel represent lung area, and gray pixels represent the chest area), and (c) segmented lung using the proposed approach (error 0.091). The errors with respect to this ground truth are highlighted by red color.

Figure 9.14: (a) Generated Phantom, (b) ground truth image (black pixel represent lung area, and gray pixels represent the chest area), and (c) segmented lung using the proposed approach (error 0.091). The errors with respect to this ground truth are highlighted by red color.

proposed algorithm segments the lung without causing any loss of abnormality tissues if it is compared with the iterative threshold method. Also, in order to validate our results we create a phantom which has the same distribution as lung and chest tissues. This phantom is shown in Fig. 9.14. One of the advantages of this phantom is that we know its ground truth. It is clear from Fig. 9.14 that the error between segmented lung and ground truth is small and this shows that the proposed model is accurate and suitable for this application.

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