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Figure 8.12: Isotropic C1 (left) and anisotropic Ca (right) tensor invariants for the tensor slice shown in Fig. 8.11.

Figure 8.13: Segmentation from isotropic measure volume V1 for the first DT-MRI dataset. The first row is the Marching Cubes isosurface with iso-value 7.5. The second row is the result of flood-fill algorithm applied to the same volume and used for initialization. The third row is the final level set model.

Figure 8.13: Segmentation from isotropic measure volume V1 for the first DT-MRI dataset. The first row is the Marching Cubes isosurface with iso-value 7.5. The second row is the result of flood-fill algorithm applied to the same volume and used for initialization. The third row is the final level set model.

dataset (V2) is formed by calculating (C1; C2, C3) invariants for each voxel and combining them into Ca. It provides a measure of the magnitude of the anisotropy within the volume. Higher values identify regions of greater spatial anisotropy in the diffusion properties. A slice from the second scalar volume is presented in Fig. 8.12 (right). The measure Ca does not by definition distinguish between linear and planar anisotropy. This is sufficient for our current study since the brain does not contain measurable regions with planar diffusion anisotropy. We therefore only need two scalar volumes in order to segment the DT dataset.

We then utilize our level set framework to extract smoothed models from the two derived scalar volumes. First the input data is filtered with a low-pass Gaussian filter (a « 0.5) to blur the data and thereby reduce noise. Next, the volume voxels are classified for inclusion/exclusion in the initialization based on the filtered values of the input data (k « 7.0 for V1 and k « 1.3 for V2). For grayscale images, such as those used in this chapter, the classification is equivalent to high and low thresholding operations. The last initialization step consists of performing a set of topological (e.g. flood fill) operations in order to remove small pieces or holes from objects. This is followed by a level set deformation that pulls the surface toward local maxima of the gradient magnitude and smooths it with a curvature-based motion. This moves the surface toward specific features in the data, while minimizing the influence of noise in the data.

Figures 8.13 and 8.14 present two models that we extracted from DT-MRI volume datasets using our techniques. Figure 8.13 contains segmentations from volume Vi, the measure of total diffusivity. The top image shows a Marching Cubes isosurface using an isovalue of 7.5. In the bottom we have extracted just the ventricles from V1. This is accomplished by creating an initial model with a flood-fill operation inside the ventricle structure shown in the middle image. This identified the connected voxels with value of 7.0 or greater. The initial model was then refined and smoothed with a level set deformation, using a i value of 0.2.

Figure 8.14 again provides the comparison between direct isosurfacing and and level set modeling, but on the volume V2. The image in the top-left corner is a Marching Cubes isosurface using an isovalue of 1.3. There is significant high-frequency noise and features in this dataset. The challenge here was to isolate coherent regions of high anisotropic diffusion. We applied our segmentation approach to the dataset and worked with neuroscientists from LA Childrens

Figure 8.14: Model segmentation from volume V2. Top left image is an isosurface of value 1.3, used for initialization of the level set. Clockwise are the results of level set development with corresponding i values of 0.2, 0.4, and 0.5.

Hospital, City of Hope Hospital and Caltech to identify meaningful anatomical structures. We applied our approach using a variety of parameter values, and presented our results to them, asking them to pick the model that they felt best represented the structures of the brain. Figure 8.14 contains three models extracted from V2 at different values of smoothing parameter i used during segmentation. Since we were not looking for a single connected structure in this volume, we did not use a seeded flood-fill for initialization. Instead, we initialized the deformation process with an isosurface of value 1.3. This was followed by a level set deformation using a i value of 0.2. The result of this segmentation is presented on the bottom-left side of Fig. 8.14. The top-right side of this figure presents a model extracted from V2 using an initial isosurface of value 1.4 and a i value of 0.5. The result chosen as the "best" by our scientific/medical collaborators is presented on the bottom-right side of Fig. 8.14. This model is produced with an initial isosurface of 1.3 and a i value of 0.4. Our collaborators were able to identify structures of high diffusivity in this model, for example the corpus callosum, the internal capsul, the optical nerve tracks, and other white matter regions.

Figure 8.15: Combined model of ventricles and (semi-transparent) anisotropic regions: rear, exploded view (left), bottom view (right), side view (bottom). Note how model of ventricles extracted from isotropic measure dataset V1 fits into model extracted from anisotropic measure dataset V2.

We can also bring together the two models extracted from datasets V1 and V2 into a single image. They will have perfect alignment since they are derived from the same DT-MRI dataset. Figure 8.15 demonstrates that we are able to isolate different structures in the brain from a single DT-MRI scan and show their proper spatial interrelationship. For example, it can be seen that the corpus callosum lies directly on top of the ventricles, and that the white matter fans out from both sides of the ventricles.

Finally, to verify the validity of our approach we applied it to the second dataset from a different volunteer. This dataset has 20 slices of the 256 x 256 resolution. We generated the anisotropy measure volume V2 and performed the level set model extraction using the same isovalues and smoothing parameters as for V2. The results are shown in Fig. 8.16, and demonstrate the generality of our approach.

Figure 8.16: Segmentation using anisotropic measure V2 from the second DT-MRI dataset. (left) Marching Cubes isosurface with iso-value 1.3. (middle) Result of flood-fill algorithm applied to the volume and used for initialization. (right) Final level set model.
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