Chapter 2 Methodologies
2.4 Bayesian Information Criterion for Model Selection
Bayesian information criterion (BIC) is a criterion for model selection developed by Schwarz (1978). It is similar to Akaike information criterion (AIC) (Akaike, 1974). The difference between BIC and AIC is that the penalty of the BIC for additional parameters is stronger than the penalty of the AIC. They are defined as follows:
where p is the number of free parameters, L is maximized value of likelihood function for the estimated model, and N is the number of data. The values of AIC and BIC are the same when the sample size N 7.389. When the sample size is greater than 7.389, the penalty term in BIC is greater than the AIC. Since the size of the image data are larger than 7.389, we will use BIC to select model. Given several estimated models, the model with the lower value of BIC is preferred. Hence, the number of clusters for GMM can be determined by the model with the minimum value of BIC.
intensity on images. CT, however, has a better outlining effect. The profiles of organs can be observed very clearly on CT images. We can combine the advantages of PET and CT. And we take weighed sum of PET and CT images to form one image as follow.
FusionPET 1 CT, 0, 1 .
In this study, let 0.5. That is, we take the average of PET and CT images to form fusion images. In addition, before we fused PET and CT images, we normalized the images between 0 and 1.
2.6 F-measure
In a statistical classification task, F-measure is a measure of testing accuracy (van Rijsbergen, 1979). It combines the precision (or we called Predicted Positive Value , PPV) and recall (or we called True Positive Rate, TPR) to form a single measure. Precision could be seen a measure of exactness and recall could be seen a measure of completeness. They are defined as following.
Table 2.6.1: The 2 2 contingency table of the real and estimated result.
Real result
True False
True True Positive (TP) False Positive (FP) Estimated result
False False Negative (FN) True Negative(TN)
Precision TP ,
The F-measure could be interpreted as the harmonic mean of precision and recall. It has the best performance when the value is 1 and worst performance when the value is 0.
2.7 Procedure
In this study, we have five procedures to segment the images shown as follows. And we will compare the accuracy of these methods by phantom study on Chapter 3.
Procedure 1: PET + GMM
Use one-dimensional GMM to segment PET images only.
Procedure 2: Fusion + GMM
Use one-dimensional GMM to segment fusion images of PET and CT.
Procedure 3: GMM (PET & CT)
Use two-dimensional GMM to segment PET/CT images.
Procedure 4: 1st order spatial dependence in GMM (PET & CT)
Use the 1st order spatial dependence in GMM to segment PET/CT images.
Procedure 5: 2nd order spatial dependence in GMM (PET & CT)
Use the 2nd order spatial dependence in GMM to segment PET/CT images.
Figure 2.7.1: The procedure of segmenting images. Plot the kernel density curve
using these data Input the image data
Reduce the data to one-dimensional by taking the
Determine the initial parameters and initial number
of clusters of one-dimensional GMM
Classify the data by one-dimensional GMM parameters and initial number
of clusters of multi-dimensional GMM
Classify the data by multi-dimensional GMM
and select the model that minimizes BIC
Multi-dimensional data
Chapter 3 Phantom Study
The torso phantom data were provided by Dr. Tai-Been Chen from I-Shou University. It is designed for comparing the accuracy of different segmenting methods in this study. The information of phantom container is shown below. The region of the simulated heart container is injected a large amount of medicine. This makes the image appear in high intensity on PET images.
Figure 3.1: The information of phantom container.
PET and CT both have 114 slices where pixel size is 256 256 in the phantom experiment. The PET images which contain high intensity identify the area with high activity in the simulator. Different segmenting methods are used to segment the region of the highest activity in the simulator. The aim of this phantom study is to compare the accuracy of the highest region by these different segmenting methods. The following is the 43th slice of PET and CT images.
Figure 3.2: (A) The 43th slice of 114 PET images. (B) The 43th slice of 114 CT images.
The region of high activity can be observed in the PET image. From CT image, the profile and organ can be observed very clearly. Therefore, we can find the boundary of the simulated heart container from CT image. Because the region of the boundary of simulated heart container is injected a large amount of medicine, the true highest activity region can be selected by CT images’ good outlining effect. It is shown as follow:
Figure 3.3: (A) The boundary of simulated heart container can be found from CT images. (B) The region of the true highest activity of 43th slice of CT image.
A B
A B
After obtaining the true highest activity region, we can compare the accuracy by F-measure (van Rijsbergen, 1979). First, the GMM is used to segment PET images. The following is kernel density curve of PET image data.
0 500 1000 1500 2000 2500 3000 3500
0.0000.0020.0040.0060.008
Kernel density for the PET data
intensity
Density
-20 -15 -10 -5 0 5 10
0.000.050.100.150.20
Kernel density for the PET data
log(intensity)
Density
Figure 3.4: (A) The kernel density curve of PET image (B) Take log of data and plot density curve. These peaks are more obvious.
By the kernel density curve from Figure 3.4, the initial values of GMM can be determined. And the model with minimum value of BIC can be selected as k 6 (the number of clusters) from Figure 3.5 shown as follow.
738000740000742000744000746000748000
PET
BIC
(A) (B)
The results of segmenting PET image by GMM shown as following. It obviously overestimates the real highest region.
Figure 3.6: k 6, The result of segmenting PET image by GMM with KDE.
Next, the fusion image of PET and CT is segmented by GMM. The following is kernel density curve of fusion image data.
0.0 0.2 0.4 0.6
05101520
Kernel density for the Fusion data
intensity
Density
-5 -4 -3 -2 -1 0
0.00.51.01.5
Kernel density for the Fusion data
log(intensity)
Density
Figure 3.7: (A) The kernel density curve of fusion image of PET and CT. (B) Take log of data and plot density curve. These peaks are more obvious.
(A) (B)
The initial values of GMM can be determined by the kernel density curve from Figure 3.7. And the model with minimum value of BIC can be selected as k 8 from Figure 3.8 shown as follow.
6 7 8 9 10
-413500-412500-411500-410500
Fusion
clusters (k)
BIC
Figure 3.8: The BIC values of different clusters for GMM (Fusion). k is the number of clusters.
The result of segmenting fusion image by GMM is shown as following. It is better than the result of only implementing segmenting PET image at the highest region.
Finally, the two-dimensional GMM is fitted to the PET and CT data. And the 1st order and the 2nd order spatial dependence are used to in GMM as shown Figure 2.3.2 and Figure 2.3.3 on Chapter 2. The followings are the BIC values of different clusters of these three
1st spatial dependence in GMM(PET & CT)
clusters (k)
BIC
10 11 12 13 14 15
-11300000-11260000-11220000
2nd spatial dependence in GMM(PET & CT)
clusters (k)
BIC
Figure 3.10: The BIC values of different clusters. (A) GMM(PET & CT). (B) 1st spatial dependence in GMM(PET & CT). (C) 2nd spatial dependence in GMM(PET & CT).
(A)
(B) (C)
From Figure 3.10, the cluster sizes can be selected k 13 for GMM(PET & CT), and k 14 for the 1st and 2nd spatial dependence GMM. These clustering results and accuracy are shown as following.
Figure 3.11: (A) k 13, the result of GMM (PET & CT). (B) k 14, the result of 1st spatial dependence GMM(PET & CT). (C) k 14, the result of 2nd spatial dependence GMM (PET & CT).
A
B C
Table 3.1: Comparison for the accuracy of the highest activity region of different methods by F-measure.
Recall (TRR) Precision (PPV) F-measure
PET + GMM 1.0000 0.5256 0.6891
Fusion + GMM 0.9682 0.9346 0.9511
GMM (PET&CT) 0.8962 0.9976 0.9442
1st order spatial GMM (PET&CT) 1.0000 0.7307 0.8444 2nt order spatial GMM (PET&CT) 0.9322 0.9692 0.9503
The comparison from Table 3.1 shows that the results of Fusion + GMM, GMM (PET&CT), 1st order spatial GMM (PET&CT) and 2nd order spatial GMM (PET&CT) all have higher accuracy than PET + GMM. Although GMM (PET&CT), the 1st and 2nd order spatial GMM (PET&CT) have slightly lower F-measure values than Fusion + GMM, they provided more information for the data. We can find the correlation of PET and CT on the regions of interesting. The correlation of the result of PET and CT could be used to detect the association pattern between the pixels of these two images. The correlation is positive as the area of a PET image has high (or low) isotope radioactivity and the area of a CT image has high (or low) X-ray absorption overlap very much. The correlation is negative as the area of a PET image with high (or low) isotope radioactivity and the area of a CT image with low (or high) X-ray absorption overlap very much. In addition, the correlations of neighbor points on PET and CT images also can be obtained by using spatial dependence in GMM. The following is some correlations of PET and CT on the result of using 2nd order spatial dependence in GMM.
Figure 3.12: The red regions show different correlations of PET and CT on the result of using 2nd order spatial dependence in GMM. (A) The correlation is -0.7575. (B) The correlation is 0.3704. (C) The correlation is -0.7540.
From Figure 3.12, the correlations of (A) and (C) are negative. It represents that the area of the PET image with high (or low) isotope radioactivity and the area of the CT image with low (or high) X-ray absorption overlap very much.
(C) A
B C
Chapter 4 Conclusions
The information of CT images is used when segmenting PET images in this study.
Besides segmenting fusion images of PET/CT and taking two-dimensional GMM fit the PET and CT data, we also used spatial dependence in GMM. These clustering results on the highest activity region all have better performance than the result of segmenting PET images only in Chapter 3. Besides the correlation of PET and CT, we also can find the correlations between neighbor points of PET and CT on the regions of interesting by using spatial dependence in GMM. Therefore, the model using spatial dependence is preferred in this phantom experiment since it provides more information for the data.
For further investigation, these methods can be tested for the stability of the performances with more phantom studies. Furthermore, it also could be applied in empirical study and judged by medical experts in the future.
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