Edge weights in the graph cuts method

The proposed method construct a graph at each level and apply the graph cuts algorithm to extract the brain region. In the constructed graph, neighboring voxels have a connected edge, which is denoted by n-link. Besides, each voxel has edges, denoted by t-link, con-nected to two specific vertices: foreground terminal S and background terminal T as shown in Figure 2.7. There are two kinds of n-links that need to be assigned with weights, one is the inter-n-link and the other is the intra-n-link. Inter-n-link connects voxels in consecutive slices, shown as black lines in Figure 2.7, and the intra-n-link connects voxels in the same slice, shown as orange lines in Figure 2.7. For segmentation of foreground and background, we need to assign edge weight for each edge in the constructed graph and apply graph cut

Figure 2.5: The results of segmented white matter tissue in coronal slice. The subject is in the second IBSR data set. Top row are the results of the initial brain extraction. Bottom row are the results of the segmented white matter tissue.

Figure 2.6: The result of deleting non-WM voxels with gray values higher than estimated WM. Top row: the results of initial brain extraction. Bottom row: the result of deleting the voxels with gray values higher than estimated WM.

of the graph cut, and A(x_i, x_j)/B(x_i) is the function to assign the weight of n-link/t-link, respectively.

The finest level has different equation of edge weights from other levels. To assign the weight of n-link at coarser levels, the edge weight is assigned as:

A(x_i, x_j) = w_n× exp(−(I_xi − I_xj)²

2α² ) , (2.2)

where wnis a parameter representing the importance of n-link, Ixi and Ixj are the intensity of two adjacent voxels, x_i and x_j, connected by this link. If the difference between I_xi and I_xj is large, then the weight of this link becomes small. Thus, the cut path is more likely to go through this link.

The weight of n-link at the finest level is assigned as:

A(x_i, x_j) = w_n× exp(−(I_xi− I_xj)²

2α² ) + w_d× (D(x_i) + D(x_j)

2 ) . (2.3)

In the finest level, we add a distance term D(x) to Eq. (2.2). The D(x) is defined as the least-squares distance from x to the contour of the extracted brain which is upsampled from the coarser level to the finest level. In this work we use a library which is based on kd-trees and box-decomposition trees for approximate nearest neighbor searching (ANN) [23] to estimate the distance term.

The parameter value of w_n and w_d for inter slice is half of the value in intra slice, in order to keep the relationship of different slices but not too strong.

For the weight of t-link connected to foreground terminal is defined as:

B(xi) = wt× P_F(x_i)

P_F(x_i) + P_B(x_i) , (2.4) where w_t stands for the importance of this term in contrast to n-link. For each voxel x, PF(xi) and PB(xi) are its likelihood functions to foreground seeds and background seeds, respectively:

The σ_f is the standard deviation of foreground seeds, whereas and σ_b is set as twice the standard deviation of background seeds for increasing the separability of foreground and background by increasing the background range. The µ_f and µ_b are the average of fore-ground seeds and backfore-ground seeds. If voxel I_xi is more likely a background seed, then the t-link connected to the foreground terminal is much easier to be cut, then x has more chance to be classified as a background. The weights of foreground seeds connecting to foreground terminal are set to be infinity and they will be classified as foreground. Simi-larly, t-link connected to background terminal is defined as:

B(x_i) = w_t× P_B(x_i)

PF(xi) + PB(xi) . (2.7)

The weights of background seeds connecting to background terminal are set to be infin-ity and they will be classified as background. The edge weight calculation functions from Eq. (2.1) to Eq. (2.7) were modified from those in [11]. We added a distance term in Eq.

(2.3) at the original resolution level. We use the graph cuts algorithm proposed in [22] and the implementation can be found in http://www.cs.ucl.ac.uk/staff/V.Kolm

ogorov/software.html.

Figure 2.7: Demonstration of n-link and t-link. Blue lines are the t-link connecting to background terminal. Green lines are the t-link connecting to foreground terminal. Orange lines are the intra slice n-link (link in the same slice). Black lines are the inter slice n-link (link in different slices).

2.3.2 Multiresolutional graph cuts for brain extraction

For the purpose of multiresolutional graph cut, we downsample the V^L(the highest res-olution) at level L into coaser levels, V^L−1, V^L−2, . . . , V⁰. The equation of downsampling The V_x,y,z^L−l is the intensity at coordinate (x, y, z) and at level L − l. The number of voxels at L − (l − 1) level are 8 times of voxels at L − l level.

Graph cuts at the coarsest level

To get the shape of brain, we start running graph cut with V⁰, which is at the lowest resolution. At this level, the shape of brain is much easier to obtain, because of the brighter voxels that are not belonging to brain were blurred. On the other hand, the brighter voxels

(WM) belonging to the brain area are set as foreground seeds, which always be classified as foreground. We use the region growing method, describe in Section 2.1, to segment the WM tissue in V⁰ as foreground seeds, as voxels overlaid with warm colors in the middle row of Figure 2.8. On the other hand, we use the voxels with non-zero gray values and below the lowest threshold as background seeds, as shown as voxels overlaid with cool colors in the middle row of Figure 2.8. After the graph cut of the entire brain was completed, we get a foreground object, O⁰, as show in bottom row of Figure 2.8. Then erode O⁰ one voxel to get a smaller volume, E⁰, as a constraint to the middle resolution level.

Graph cuts at the middle levels

The graph cuts at middle levels is to refine the extracted brain from the lowest resolution level. At middle dimension level l, 0 < l < L, to get a pre-determined foreground area we first upsample the eroded brain region, E^l−1, at coarser level l − 1 to level l the pre-determined foreground area E^l. The upsampling equation of the eroded volume is defined as:

E_x,y,z^0l = E^l−1x

2,^y₂,^z₂ , (2.9)

where E_x,y,z^0l is the intensity at coordinate (x, y, z) and at level l. The weights of t-link connecting to foreground terminal in E^0l, as shown in third row of Figure 2.9, are set to be infinity. This ensures E^0l must be classified as foreground. The pre-determined foreground area can reduce the short-cut problem and prevent cut at WM/GM boundary.

Considering local gray value distribution, we partition V^l into 2^l× 2^l× 2^lcubes (each dimension is divided into smaller overlapping region, as shown in Figure 2.10. Then we perform graph cuts algorithm for each cube individually. Each cube has 1/4 in length overlapped with its adjacent cubes, this reduce the inconsistent border between two cubes.

Each cube has its own foreground and background seeds. We estimate the three thresholds

is defined as E^l. For the overlapped region between neighboring cubes, we integrate the results by logical OR operation.

Graph cuts at the finest level

Graph cuts applied at the finest level is to get the final extracted brain. At the finest level, the pre-determined foreground area is the upsampled E^L−1, as shown in the third row of Figure 2.11. We partition V^Linto 2^L× 2^L× 2^Lcubes individually. Then we perform graph cut with each cube. The way of finding foreground seeds and background seeds is the same as in the middle levels, as shown in the second row of Figure 2.11. We also upsampling O^L−1to this level, called O^0L, as shown in top row cool colors of Figure2.11. The equation of upsampling O^L−1is defined as:

O^0L_x,y,z =

P1

k=−1O^L−1x 2,^y₂,^z₂+k

3 , (2.10)

where O_x,y,z^0L is the intensity at coordinate (x, y, z) and at level L. Different from middle and the coarsest levels, we add a distance term: wd× (^D(xi)+D(x₂ ^j)), in Eq. (2.3). The distance term D(x) is defined as the least-squares distance from x to contour of O^0L. The reason is that, the brighter voxels belonging to non-brain region are similar to foreground seeds, so they might be classified as foreground. But in the middle and the coarsest levels, those brighter voxels are blurred, hence those voxels can be judged as background seeds. There-fore, we use the contour of O^0L as a factor of the edge weight in this level because when the n-link is closer to the contour, it should has more chance to be cut. For the overlapped

region between neighboring cubes, we integrate the results by logical OR operation.

2.4 Postprocessing to fill holes that dark brain region be