Figure 1.
T2-weighted MR image showing a degenerative spondylolisthesis (red arrow) and a lumbar stenosis (green arrow).
Figure 2.
Vertebral anatomy: (a) illustrates the anatomy of a vertebra from a coronal view (adopted from [5]), (b) shows a sagittal T2-weighted MRI slice.
The green arrow in the enlargement points to an area inside the vertebral body, whereas the red arrow points to the cortical bone, the outer boundary.
Figure 3.
Object/background transition regions (red arrows).
(a) shows a homogenous object/background transition. In (b), the spinal canal (CSF) makes up parts of the vertebral body's outer boundaries.
Figure 4.
Illustration of voxel labeling for the foreground (Ls) and the background (Lt).
Figure 5.
Profile of two cube faces intersected by three rays (a) and a cubic voxel subset (b).
Figure 6.
Illustration of the different kinds of edges.
(a) i-links: z-edges (black), xy-edges (blue). (b) o-links: s-links(green), t-links(red). (c) whole graph.
Figure 7.
Illustration of the penalty effect.
(a) shows a network without i-links. (b) shows a network with an i-link. The red line depicts a minimal cut.
Figure 8.
Illustration of the z-edges principle.
(a) shows a ray without z-edges: The minimal s-t-cut (red) cuts the ray twice with a capacity of 0. (b) shows the same ray with z-edges. The ray is only cut once. The capacity of the minimal s-t-cut is ∞ +5. (c) shows z-edges, embedded into an MR image.
Figure 9.
Adverse effects on segmentation results (2-dimensional view).
(a) shows an overrun in the upper part due to a violation of condition (3). (b) shows a segmentation result affected by an outlier which causes a violation of condition (4). The cut happens too close to the seed point (not shown) in the middle of the vertebra because there is a light area similar to the spinal canal.
Figure 10.
In (b), w is applied on the s-weights in (a): The cut, with a capacity of ∞ +2.5, now happens closer to the seed point. Note that the same cut in (b) would have cost ∞ +10 whereas the cut depicted in (b) has a capacity of only ∞ +5.
Figure 11.
Courses of w(i,11) (green) and w(i,15) (red).
The upper part illustrates that w(i,k) reflects the position of the voxel pi on a ray consisting of k uniformly distributed voxels. Note that that w is only partially defined for the natural numbers but that Cube-Cut never calls w with an argument in the undefined scope.
Figure 12.
A feasible surface and intersecting rays (transformed in x-direction for a better visibility).
The green node depicts an outlier as it would violate the smoothness constraint Δx = 1 if classed with the surface voxels.
Figure 13.
Illustration of the xy-edges principle.
(a) shows a minimal cut (thick lines) and the two possible continuations (dashed lines) within the boundaries of a smoothness constraint Δ = 1. All other cuts would have a capacity greater than 7·∞. (b) shows the only possible continuation within the boundaries of a Δ-value of 0, where the cut has a capacity of 3·∞.
Figure 14.
Topology of xy-edges for Δ = 0 (a) and Δ = 1 (b).
Figure 15.
Segmentation result for Δ = 0 (left) and Δ = 2 (right).
Figure 16.
3D segmentation result (left and middle image) and 2D segmentation result with the user-defined seed point in blue (rightmost image).
Figure 17.
Superimposition of a manual segmentation result and a Cube-Cut segmentation result.
Figure 18.
Typical user initialization of GrowCut for this study.
The Editor module is used to mark parts of the vertebra (green) and the background (yellow) in an axial, sagittal and coronal plane.
Table 1.
Direct comparison of manual slice-by-slice and Cube-Cut segmentation results for ten vertebrae via the Dice Similarity Coefficient (DSC).
Table 2.
Direct comparison of manual slice-by-slice and GrowCut segmentation results for ten vertebrae via the Dice Similarity Coefficient (note: the cases 1–10 correspond to Table 1).
Figure 19.
Vertebra segmentation results (red) for a graph that has been constructed with a spherical template from a user-defined seed point (blue).
The left image shows the segmentation result when the density of the rays/sampled nodes and the delta value are set to very large values. When these values are smaller the graph cut prefers a more spherical/elliptical segmentation result (middle). The rightmost image shows the extreme case where the delta value was set to zero. There the graph cut has to come back with a perfect sphere and the only variation is the size of the sphere which depends on the gray values.
Figure 20.
Corresponding 3D results of Figure 19, where a graph has been constructed with a spherical template for vertebra segmentation.
The left image shows the 3D segmentation result (yellow) when the density of the rays/sampled nodes and the delta value are set to very large values. When these values are smaller the graph cut prefers a more spherical/elliptical segmentation result (middle). The rightmost image shows the extreme case where the delta value was set to zero, which resulted into a perfect sphere.
Figure 21.
Sagittal 2D-view on Cube-Cut segmentation result (left, red), GrowCut segmentation result (center, white) and reference image (right).
The GrowCut algorithm detects false boundaries in the pedicles-region.
Figure 22.
"Vertices" of the vertebral body's outer boundaries were not detected accurately (green circles).
The upper image shows a 3D segmentation result, the lower images show 2D overlaps of manual (red) and automatic (white) segmentation results.
Figure 23.
The size of the cube (red) in the left image is too small to segment the vertebra, because a graph that is constructed inside this cube does not cover the border of the vertebra and the s-t-cut will lie inside the vertebra.
In contrast, the size of the cube in the right image is sufficient, because a graph that is constructed inside this cube will also cover the vertebra’s border and thus is able to segment it.