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Figure 9.5: Abnormal ground truth overlays. The top row in each row pair is the left carotid artery overlayed with the ground truth tracing of the inner lumen wall, and the bottom row is the corresponding image of the right carotid artery. Some lumens have a circular shape, while others have an elliptical shape.
9.2.1 Ground Truth Tracing and Data Collection
Figures 9.5–9.9 show abnormal images of the left and right carotid arteries overlayed with the ground truth tracing. In each pair of rows, the top row is the left carotid arteries and the bottom row is the right carotid arteries. The ground truth tracing is the boundary of the inner lumen wall. Figure 9.9 shows the normal images of the same left and right carotid arteries. The normal images show
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Figure 9.6: Abnormal ground truth overlays. The top row in each row pair is the left carotid artery overlayed with the ground truth tracing of the inner lumen wall, and the bottom row is the corresponding image of the right carotid artery. Some lumens have a circular shape, while others have an elliptical shape.
lumens that are more circular than those of the abnormal images, and there is no constriction of the arteries. Ground truth tracing was done using the MATLAB program MRI GUI ver 1.2. For each lumen boundary the image was zoomed in and points were plotted by the user around the inner wall boundary. The points were splineﬁtted to 20 points.
The imaging parameters (TR/TE/TI/NEX/thickness/FOV/ETL) are as following: T1W:1RR/7.1 ms/500 ms/2/3 mm/12–14 cm/21; PDW: 2RR/7.1 ms/600 ms/
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Figure 9.7: Abnormal ground truth overlays. The top row in each row pair is the left carotid artery overlayed with the ground truth tracing of the inner lumen wall, and the bottom row is the corresponding image of the right carotid artery. Some lumens have a circular shape, while others have an elliptical shape.
2/3 mm/12–14 cm/31; T2W: 2RR/68 ms/600 ms/2/3 mm/12–14 cm/31. Matrix for all images were 256 × 192. Voxel size was 0.5 × 0.5 × 3 mm3. For bright blood 3D TOF images: TR/TE/ﬂip angle/thickness/FOV = 20 ms/3.4 ms/25/2.0 mm/18. Zeroﬁlled Fourier transform was used to create voxels of 0.35 × 0.35 × 1 mm3 for 3D TOF imaging. The factors which affect MR plaque quality are (a) random patient motion, including (b) obese patients who may have deeper carotid arteries, (c) incomplete ﬂow suppression, and (d) artery wall pulsation.
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Figure 9.8: Abnormal ground truth overlays. The top row in each row pair is the left carotid artery overlayed with the ground truth tracing of the inner lumen wall, and the bottom row is the corresponding image of the right carotid artery. Some lumens have a circular shape, while others have an elliptical shape.
9.3Three Pixel Classiﬁcation Algorithms: MRF, FCM, and GSM
9.3.1 MarkovianBased Segmentation Method
The algorithm consisted of running the pixel classiﬁcation approach using Markov random ﬁeld with mean ﬁeld (see Zhang [149]). Here, the image segmentation was posed as a classiﬁcation problem where each pixel is assigned to one of K image classes. Suppose the input image was y = {yi, j , (i, j) L}, where yi, j is a pixel, i.e., a 3D vector, and L was a square lattice. Denote the segmentation as z = {zi, j , (i, j) L}. Here, zi, j is a binary indicator vector of dimension K , with only one component being 1 and the others being 0. For
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where and were model parameters. In this work, we assume that the pixels in y are conditionally independent given z, i.e.,
log p(y  z, ) = log p(yi, j  zi, j , ). (9.2)
i, j
Furthermore, we assume that conditioned on zi, j , the pixel yi, j has a multivariate Gaussian density, i.e., for k = 1, 2, . . . , K,





e− 21 (yi, j 
− 
mk)T C−1 
(yi, j 
− 
mk) 


p(y 
z 
e 
, ) 
= 

k 


, 
(9.3) 


(2π )3/2Ck1/2 



i, j  
i, j = 
k 






where ek is a K dimensional binary indicator vector with the kth component being 1. From this, = {mk, Ck}kK=1 contained the mean vectors and covariance matrices for the K image classes. For z, we have adopted an MRF model with a Gibbs’ distribution [149]:
p(z 
 
) 
= 
1 
e−βE(z), 
(9.4) 








Z 


where 










1 




E(z) = 







1 − 2zit, j zk,l 
(9.5) 
2 










i, j (k,l) Ni, j 

is the energy function which decreased (causing p(z  ) to increase) when neighboring pixels were classiﬁed into the same class (the set of the neighbors of (i, j) is denoted as Ni, j ).
Since was generally not sensitive to particular images, it was set manually here. , on the other hand, was directly dependent on the input image and hence had to be estimated during the segmentation process. In this work, as it in [149], this was achieved by using the EM algorithm [177], which amounts to iterating between the following two steps:
1. E step: Compute
ˆ ( p) 
ˆ 
( p) 
. 
Q(  
) = log p(y  z, ) + log p(z  )  y, 

2. M step: Update parameter estimate
( p+1) = arg max Q(  ( p)).
Here · represents the expectation, or mean, and the superscript p denotes the pth iteration. This translated into the following formulas for updating the
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parameter estimates: 















zi(,pj) 
8 
= zi, j 
zi, j f (zi, j ) 







(9.6) 

7 
= 


( p) 













i,7j zi,8j 









mˆ ( p+1) 

i, j 
zi, jk 
yi, j 
, 









k 



z7( p) 
k y8 i, j 

mˆ ( p+1) yi, j 
mˆ ( p+1) T 



ˆ ( p 1) 
= 
7 
i, jk 
82 

− 

i, j zi, j32 
− k 
3 




i, j 



k 





Ck + 













, 
(9.7) 









7 
k 8 




where k = 1, 2, . . . , K, zi, jk is the kth component of zi, j , and f (zi, j ) a “mean ﬁeld” probability distribution (see Zhang [149]).
These formulas, in addition to providing the estimate of , also produced a
segmentation. Speciﬁcally, at each iteration, z( p) was interpreted as the prob
i, jk
ability that yi, j was assigned to class k. Hence, after a sufﬁcient number of iterations, we can obtain the segmentation z for each (i, j) L by
zi, j = ek0 if k0 = arg max1≤k≤K zi, jk . 
(9.8) 
In this way, the EM procedure described above generated the segmentation as a byproduct, and therefore provided an alternative to the MAP solution of Eq. (9.1) For those interested in the application of MRF in medical imaging, see Kapur [150], who recently developed a brain segmentation technique which was an extension to the EM work of Wells et al. [174] by adding the Gibbs’ model to the spatial structure of the tissues in conjunction with a meanﬁeld (MF) solution technique, called Markov random ﬁeld (MRF) technique (for details on MRF, see Li [151] and Geman and Geman [179]). Thus the technique was named expectation maximization–mean ﬁeld (EM–MF) technique. By Gibbs’ modeling of the homogeneity of the tissue, resistance to thermal noise in the images was obtained. The image data and intensity correction were coupled by an external ﬁeld to an Isinglike tissue model, and the MF equations were used to obtain posterior estimates of tissue probabilities. This method was more general to the EMbased method and is computationally simple and an inexpensive relaxationlike update. For other work in the area of MRF for brain segmentation see Held et al. [152].
Pros and Cons of MRF with Scale Space. The major advantages of MRFbased classiﬁcation is (1) addition of Gibbs’ Model: Kapur et al. model the a priori assumptions about the hidden variables as a Gibbs’ random ﬁeld. Thus the prior probability is modeled using the following physicsbased analogous Gibbs’
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1 
1 

E( f ) 



equation: P( f ) = 

exp(− 

)E( f )), where Z = f exp( − T 
) and P( f ) is the 

Z 
κ T 

probability of the conﬁguration 
f , T is the temperature of the system, κ is the 
Boltzmann constant, and Z is the normalizing constant. The major disadvantages of MRFbased classiﬁcation include the following: (1) The computation time would be large if the number of classes is large. In such cases, one needs to use the multiresolution techniques to speed up the computation. (2) The positions of the initial clusters are critical in solving the MRF model for the convergence. Here, the initial cluster was computed using K means algorithm which was a good starting point. However, a more robust method is desirable.
9.3.1.1 Implementation of MRF System
Figure 9.10 shows the main MRF classiﬁcation system implementation. Input is a perturbed gray scale image with left and right lumens. The number of classes, the initial mean, and the error threshold are inputs given to the system. The result is a classiﬁed image with multiple class regions in it, including multiple classes in the lumen region. Figure 9.11 shows the MRF system in more detail. Given the initial center, the number of classes K , the Markov parameters of mean, variance, and covariance, and the perturbed image, the current cluster center is calculated. Using the EM algorithm, new parameters are solved and new cluster

Gray Scale Perturbed Image with Multiple Lumens 

K (classes) 

Classification Process 

Error 
Initial Mean 
Threshold 
Classified Image with Multiple Lumens and Each Lumen Having
Multiple Class Regions
Figure 9.10: Markov random ﬁelds (MRF) classiﬁcation process. Input is a gray scale perturbed image with left and right lumens. The number of classes, the initial mean, and the error threshold are inputs given to the system. The result is a classiﬁed image with multiple class regions in it.
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# of Classes (K) 
Perturbed Image 
Markov Parameters: Mean, 

Variance, Covariance 






Initial Center 
Current Cluster Center 



Current Cluster Center 

EM Algorithm for Solving Φ
New Cluster Centers
Update
Compare Error Between Two Cluster Centers
Error Difference Between Cluster Centers
Is Error less than Threshold?
NO
YES
Segmented Image
Figure 9.11: The Markov random ﬁelds (MRF) segmentation system. Given the initial center, the number of classes K , the Markov parameters of mean, variance, and covariance, and the perturbed image, the current cluster center is calculated. Using the expectationmaximization (EM) algorithm, new parameters are solved and new cluster centers are computed. The error between the previous cluster center and the recently calculated cluster center are compared, and the process is repeated if the error is not less than the error threshold. After the iterative process is ﬁnished, the output is a segmented image.
centers are computed. The error between the previous cluster center and the recently calculated cluster center are compared, and the process is repeated if the error is not less than the error threshold. After the iterative process is ﬁnished, the output is a segmented image.
9.3.2 FuzzyBased Segmentation Method
In this step, we classiﬁed each pixel. Usually, the classiﬁcation algorithm expects one to know how many classes (roughly) the image would have. The number of classes in the image would be the same as the number of tissue types. A pixel could belong to more than one class, and therefore we used the fuzzy membership function to associate with each pixel in the image. There are several