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region with highest density. The MVE searching methods normally use randomly selected elements in V as the ellipsoids’ initial centers. After inflating each of these ellipsoids until h elements are covered, the one with minimum volume is selected as a searched mode. Then we can remove the elements associated with this mode from dataset V and a new search is repeated until all the cluster centers are found. Although many approaches based on this multivariate locations estimator have shown its success in various applications [88], experimental results indicate that the performance of MVE decreases when the number of modes is greater than 10 [86]. The reason for this phenomenon is that the density definition in MVE presumes the multivariate normal distribution. Therefore, in the case of multiple modes, where no mixture of Gaussian distribution appears, MVE model will not be able to give an accurate description.

Another type of cluster searching techniques is called nonparametric estimation. The advantage is that they require no prior knowledge of the form of the density function in the search process. They can be applied to arbitrary distribution dataset. There are two main categories of methods for nonparametric density estimation, Parzen window and k-nearest neighbors. For the Parzen window approach, the kernel type needs to be given before it is applied. In the k-nearest neighbor method, the number of neighbors must be assigned in advance. Therefore, both require additional prior information. In addition, they are hard to optimally initialize.

To overcome these problems and provide more robust cluster estimation, a framework was proposed by Dorin [83]. It is based on the mean-shift algorithm, a nonparametric procedure to estimate density gradients. Although this method claims to avoid the drawbacks of most existing approaches, it still has the following weaknesses: (1) The initialization scheme cannot guarantee that all the cluster centers are under consideration in the search process because of its random tessellation selection and (2) because of the static size of the search sphere, the approaching speed may be slowed and the accuracy of cluster center estimation may be affected.

In this study, the data to be processed is MCW MR image, in which the distribution of data can vary arbitrarily between subjects. Since it is impossible to obtain the forms of the underlying density function, a nonparametric technique has to be employed for multivariate location estimation. To overcome the problems in Dorin’s method, we first apply a preestimation of cluster distribution to guarantee that all the typical cluster centers are considered in the initial center

set. In addition, a dynamic sphere size is introduced to improve the searching performance.

8.4.3.1.3 Mean Shift Density Estimator. The mean shift algorithm was proposed by Fukunaga and Hosterler in 1975 as a “very intuitive” [89] estimator for data density. In 1995, Cheng generalized this algorithm and conducted a more rigorous study [90]. In this section, we will review its estimation of the density gradient in a uni-modal situation.

Assume f (v) is the probability density function of a d-dimensional variable v. Suppose that centered with vector r, a sphere, Sv , with radius r is established. For any given vector within this sphere, y, the expected distance to sphere center v is

where VSv = c · r D represents the sphere volume. Based on Eqs. (8.53) and (8.54),

Expand the LHS of Eq. (8.55), the expected center of the sphere E[v | v Sv ] and v have the following relation:

For a given sphere, Eq. (8.56) shows that the difference vector between the local estimation of cluster center and sphere center is proportional to f (v), the gradient of the density function at v. It is also reciprocal to f (v). When it approaches the mode, f (v) is generally large in value and f (v) is small due to the slow increase, so a small mean-shift vector is applied. Compared to traditional density gradient searching techniques [86], in which only f (v) is considered, the dynamically adjusted step size used in this method is more

accurate. Additionally, the mean-shift vector is always guaranteed to point to the direction of the maximum density. However, in the case when v is far away from the mode, the density change is often very small (close to uniform) within the small sphere under consideration. This may lead to the mean-shift algorithm failing to predict the correct direction and step size. Therefore, some measures need to be taken if the initial x is not at a location close to the mode.

8.4.3.2 Dynamic Mean-Shift Density Estimation

In the density estimation process, the mean-shift vector is very important to the search speed and accuracy of result. In previous methods [83], the size of the local estimation sphere is always fixed. This may not work well in the following two situations: (1) In a location which is far from the mode where the density distribution is relatively uniform, and if the sphere size is not large enough, the mean-shift vector may be misled and (2) when the search progress is close to the mode, the mean-shift vector needs to be sensitive enough to catch the local change. Therefore, if the sphere volume is too large, the local information cannot play the determinant role in the vector calculation and hence may affect the accuracy of center searching.

To solve these problems, we propose a dynamic search range with q levels: r1, . . . , rq , where ri < ri+1. The values of r1 and rq come from prior analysis of MCW MR image data. In the search process, the initial radius starts from rq (the largest radius). If the mean-shift vector is over the stopping threshold

Tstop, it moves to the next location with the same sphere radius as the previous position. If the mean-shift vector is lower than Tstop, it uses the next smaller radius to calculate the mean-shift vector again. Once the mode is found, a small perturbation is applied [83] and the procedure is repeated to avoid a local maximum.

Given the initial center of a sphere at location v and starting with sphere radius index i = q, the proposed dynamic mean-shift algorithm is implemented in the following procedures:

1.Based on x and r = ri, compute the mean-shift vector vms.

2.If vms ≥ Tstop, v = v + vms and repeat step 1; else if i = 1, i = i − 1 and repeat step 1 until convergence.

3.If perturbation has not been applied, add a small vector vpert to the converged result and repeat steps 1 and 2, where |vpert| = |r1| and its direction is randomly selected.

8.4.3.3 DMC-Based Segmentation

8.4.3.3.1 Cluster Center Searching. Based on the enhanced means shift

density estimation, the steps for cluster searching are as follows:

1.Apply the Quad-Tree procedure to partition each contrast weighting image and assign a vector at the center of each region as a member of the initial center set X = {xi, i = 1, . . . , n}. In addition to the two constraints in previous method [83], the minimum neighboring points’ distance and sphere density—a more strict constraint is applied on the pixel location in each contrast weighting image—the distance of the corresponding pixels’ locations to any two elements xi and x j , cannot be too close.

2.For each point in the center set, apply the dynamic mean-shift search described in section 8.4.3.2 to find the candidate cluster centers in the d- dimensional data space.

3.Partition the center set elements into subsets in which the distances between points are within Tsub. Merging each subset by calculating the mean of elements in it, we arrive at a new center set Y = {yi, i = 1, . . . , p}.

4.To validate the cluster centers, the constraint on the valley between every two elements in the center set, yp and yq , is applied [83]. Each point, for example yr after a fixed interval along the line linking yp and yq is checked and the corresponding density f (xr ) is estimated with Epanechnikov ker-

a valley is detected. If no valley is detected between yp and yq , the one with lower density will be removed.

Input MCW MR Images

Create Data Set

Create Initial Center Set

Dynamic Mean Shift

Data Set Dispatching

Spatial Constraints

Output Segmentation

Figure 8.19: Processing flow chart of the proposed MCW MR image segmentation algorithm.

in which Li and µLi are assumed to be the label and mean of neighboring region.

UN and UE are the clique energy functions [92]. A MAP criterion is applied to find the most reliable neighboring region to merge into.

8.4.3.4 MCW MR Images Segmentation

Two categories of experiments are designed to apply the multispectral segmentation technique on MCW MR image. One compares the effectiveness of multiple contrast weightings in the MR image segmentation, and the other verifies the partition accuracy of typical plaque tissues.

To show the advantage of using multiple contrast weighting techniques over single contrast weighting, 20 cases of ex vivo MR images were obtained using T1W, PDW, and TOF. In each case, the described MCW segmentation approach was applied to the three contrast-weighted images with the result IMCW. The single contrast weighting image in each case was also segmented using the same approach of assigning the dimension of vector space to one; the results are ISCW1, ISCW2, and ISCW3. All other parameters remained the same. Preliminary work had indicated that composite segmentation might disclose information not available with single segmentation, this again proved to be the case. Figure 8.19 compares MCW to single contrast weighting segmentation results, whereby (a), (c), and (e) are the original images obtained with T1W, PD2, and TOF,

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Figure 8.20: An example of MCW MR image segmentation. (a), (c), (e) The original contrast weighting images at the same location. (b), (d), (f) The corresponding single contrast weighing segmentations. (g) (color) Result with the proposed multiple contrast weighting segmentation algorithm.

respectively, and (b), (d), and (f) are the corresponding segmentation results using single contrast weighting only. Three distinctive differences are labeled in the images. Arrow 1 points to a region poorly segmented by the single T1W and TOF images. Arrow 2 shows a region that loses all detail in the TOF segmentation. Arrow 3 points to an area that by PDW segmentation shows no detail. However, the MCW-segmented image (Fig. 8.20(g)) retrieves and reveals these details by considering all contrast weightings in the segmentation process. Of 20 cases analyzed, 17 showed distinct differences occurring at more than two

locations between the MCW and SCW methods. Lower image quality (blurring, poor contrast, and imaging artifacts) was responsible for the poor segmentation result in the three remaining cases.

The second category of experiments was used to verify the partitioning of typical tissues of interest of the atherosclerotic plaque. Again we applied the above algorithm to ex vivo MR images of endarterectomy specimens. Using histology as the gold standard, we compared each segmented MR image to the best corresponding histology section. The carotid bifurcation was used as a landmark for location registration. On sections distant from the bifurcation, lumen size and shape and distinctive regions of calcification were used for matching. A coordinate system of eight segments was generated and applied to the matched histology and MR images. Figure 8.20 contains a pair of sample images.

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Figure 8.21: An example of MCW MR images verification (CA: calcium, LM: loose matrix, NEC: necrotic debris). (a) (color) Segmentation of MCW MR images. (b) (color) Outlined corresponding histology section.

Figure 8.21(a) is a MCW segmented MR image overlaid with the eight sector coordinate system. Figure 8.21(b) is the corresponding histology section stained with Mallory’s Trichrome. Tissues of interest were outlined and labeled prior to matching.

A preliminary study of 22 matched MRI histology sections from eight patients were analyzed with results shown in Table 8.5. Typical tissues of the atherosclerotic plaque such as calcification, fibrous matrix, and mixed necrotic cores appear to have good agreement with histology. For those improperly matched cases, besides the inaccuracy of segmentation algorithm, the following may also be part of the reasons that affect the comparison results: (i) low-image quality (noise involved in the imaging process) and (ii) the deformation of plaques in the making process of histology section, including shrinkage. These are beyond the study of our research. However, further refinement of our technique may allow for better detection of the less discrete tissues such as loose matrix, speckled calcification, and intraplaque hemorrhage.

8.4.4 Conclusion

In this section, we investigated the segmentation algorithm based on multiple dimensional MRF model and clustering-based solutions, and introduced an effective approach for MCW MR image segmentation. This technique is based on mean-shift density estimation algorithm and was carefully designed to overcome the drawbacks in other existing methods. Experimental comparisons with histology section have demonstrated its successful performance.

For the processing speed of the proposed DMC-based approach, the same 50 multiple contrast weighing MR images with different image size were also

Table 8.6: Average segmentation time of DMC and mMRF

used for testing. The comparison of the average segmentation time for DMC and mMRF approaches are shown in Table 8.6 which indicates that DMC uses much less time than mMRF.

In the validation experiments with histology sections, we can also note that poor image quality can reduce the accuracy of the proposed method by reading the cases that showed little agreement with histology. One hypothesis of this problem is that the poor separation of data in the vector space V makes the segmentation method unable to distinguish the different clusters.

8.5Semiautomatic Detection of Fibrous Cap Status

Detection of fibrous cap status is crucial for understanding the disease status and prognosis of atherosclerosis. At the same time, fibrous cap segmentation is difficult because of resolution issues, registration issues, and the presence of artifacts. Hence a different approach is required to implement semiautomatic detection of fibrous cap status.

8.5.1 Importance of Fibrous Cap Detection

The development of a lipid core marks the development of an intimal xanthomata or fibrous streak into an atherosclerotic plaque. A thin layer of smooth muscle cells form a covering called the fibrous cap (FC) over the lipid core and separate it from the lumen [93]. Rupture of the FC in advanced lesions leads to thrombosis or intraplaque hemorrhage. Inflammatory destabilization of the