Interpolation

The 3D MR images were interpolated initially to obtain a 3D image with approxi- mately isotropic voxel dimensions using B-spline interpolation [41]. For the 3T MR images, nine additional slices were interpolated in the axial direction between each adjacent pair of original slices to obtain a voxel dimension of 0.2_×0.2_×0.2 mm3.

Table 4.2: Magnetic Resonance Imaging Parameters for T1-weighted Double Inversion Recovery scans at 1.5T and 3.0T.

Acquisition Parameter 1.5T 3.0T

Echo Time, TE (ms) 12 11.4

Recovery Time, TR 1RR 1RR

Receiver Bandwidth, RBW (kHz) 41.67 41.67 Field of View, FOV (cm) 11 11

Thickness (mm) 2 2

Matrix 224_×224 224_×224

Number of Excitations, NEX 3 3 Scan Time (minutes) 8:48 8:48

Fat Saturation Yes Yes

Spacing Overlap 0 0

Number of Slices 16 16

Pulse Sequence FSE FSE

Similarly, for the 1.5T MR images, three additional slices were interpolated in the ax- ial direction between each adjacent pair of original slices to obtain a voxel dimension of 0.5_×0.5_×0.5 mm3.

This step ensures the direct applicability of the classical 3D total-variation func- tion to equally regularize along each spatial direction. However, it is also possible to weight regularization in each spatial direction independently to account for anisotropic images.

4.3.2.2 User interaction

The sole user interaction in our approach was choosing some sampled voxels of the carotid wall, lumen, and background regionson a single transverse slice of the inter- polated input 3D MR image. The voxels were chosen by the user using a paint brush user interface tool, as shown in Fig. 4.4(a)-(d). Such region-based user interaction techniques have been used previously for medical image segmentation [42, 43]. The sampled voxels marked by green, red, and blue correspond to lumen, wall, and back- ground regions, respectively. In our experiments, the transverse slice was chosen to be the furthest from the carotid bifurcation. However, this choice is arbitrary and the user can also choose any other image slice for initialization. The purpose of the sam- pled voxels is threefold: I) the sampled voxels are used to approximate the intensity

(a) (b) (c) (d)

(e) (f) (g) (h)

Figure 4.4: Example user initializations and the 2D AB-LIB segmentations using the proposed continuous max-flow algorithm. The only user interaction in the pipeline is choosing the sampled voxels on a single transverse slice. The voxels marked by green, red, and blue correspond to the lumen, wall and background regions respectively. (a) and (b): Initializations for two 3T MR images, (c) and (d): Initializations for two 1.5T MR images. (e) and (f): 2D Segmentations for two 3T MR images, (g) and (h): 2D segmentations for two 1.5T MR images.

PDF models for distribution matching in the 2D AB-LIB segmentation step; II) they are used as the hard constraints of segmentation regions such that the sampled voxels are fixed to be in their corresponding regions, which can be readily implemented in the introduced continuous max-flow framework (as described below); and III) they are also used as the initial regions for the 2D AB-LIB segmentation step (as described below).

4.3.2.3 2D AB-LIB segmentation and generation of intensity PDF models

A 2D segmentation of the carotid AB and LIB was performed on the same transverse slice that the user initialized, in order to further refine the intensity PDF models

(a) PDF models (b) Estimated PDFs at Iter #1

(c) Final result PDFs (d) Ground truth PDFs

Figure 4.5: Normalized intensity probability density functions (PDF) used in 3D AB-LIB segmentation for Bhattacharyya distance matching for a single 3D image. Gaussian kernel width of seven is used to generated the PDFs.

using the 2D segmentation result. The 2D AB-LIB segmentation was obtained using the proposed coupled surface evolution approach with the inter-surface order constraint. The optimization problem of the coupled surface evolution is solved using the proposed continuous max-flow approach by minimizing the objective function (4.18). Four criteria were used for 2D AB-LIB segmentation: The Bhattacharyya distribution matching (4.7), gradient-based smoothness term (4.9), hard constraints for user marked voxels (4.23), and minimum AB-LIB separation-based prior (4.24). The last two terms are described below.

Hard constraints for user sampled voxels: The sampled voxels (see Fig. 4.4(a)- (d)) by the user were used both as the initial regions and as the hard region constraints for the 2D AB-LIB segmentation step. For example, the blue voxels marked on the background region are fixed to the background region during the whole computation procedure (see Fig.4.4(a)-(d)). Such region-based hard constraints can be easily implemented into the proposed continuous flow-maximization scheme as the following flow capacity constraints, which are standard in a max-flow procedure (see [21] etc.):

D_w(x) = +_∞, D_b(x) = +_∞, x_∈S_l

D_l(x) = +_∞, D_b(x) = +_∞, x_∈S_w (4.23) D_l(x) = +_∞, D_w(x) = +_∞, x_∈S_b

where S_i, i = l, w, b, denote the sampled voxel regions within the lumen, wall, and background regions, respectively. Note that, when x _∈ S_l, the voxel x retains its status in the lumen region, because it would otherwise incur an infinite cost to change its status to be within either the wall or background regions.

Minimum AB-LIB separation-based prior: The lumen region usually appears uniformly in intensity and has well defined image edges at the LIB. However, the wall region is relatively heterogeneous in intensity and has overlapping image intensities with the background region, which is challenging for the segmentation task. We incorporated an anatomically motivated separation of the carotid AB and LIB by the intima-media layer, in order to prevent the AB from collapsing to the LIB, when the carotid AB is weakly defined in the image. Because, the AB and LIB are

separated by the carotid media layer, they are encouraged to have a greater or equal separation distance (d_min) of 0.5 mm from each other [44]. We used hard constraints to implement the minimum AB-LIB separation-based prior, such that the minimum distance between the AB and LIB should be larger than some constant d_min in order to maintain the separation of the AB and LIB. This can be implemented as follows

D_b(y) = +_∞, s.t. d(y,_Rl)< d_min; (4.24) whered(y,_Rl) denotes the Euclidean distance from voxely _{∈ Rw}to the lumen region

Rl.

Using the computed 2D AB-LIB segmentation result, more representative inten- sity PDF model for each region were constructed to aid in the 3D AB-LIB segmen- tations, as shown in Fig. 4.5.