Segmentation

In this section we demonstrate the application of our methods to the segmentation of DT-MRI data of the human head. We use a high resolution data set from a normal volunteer which contains $60$ slices each of $128$x$128$ pixels resolution. The raw data is sampled on a regular uniform grid.

We begin by generating two scalar volume datasets based on the invariants described in Section 2. The first scalar volume dataset ( $\mathcal{V}_1$) is formed by calculating the trace ($C_1$) of the tensor matrix for each voxel of the diffusion tensor volume. It provides a single number that characterizes the total diffusivity at each voxel within the sample. Higher values signify greater total diffusion irrespective of directionality in the region represented by a particular voxel. A slice from this volume can be seen in Figure 6 (left). The second scalar volume dataset ( $\mathcal{V}_2$) is formed by calculating $(C_1,C_2,C_3)$ invariants for each voxel and combining them into $C_a$. 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 Figure 6 (right). The measure $C_a$ 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 level set methods to extract smoothed models from the two derived scalar volumes. Our level set segmentation approach consists of defining a set of suitable pre-processing techniques for initialization and selecting/tuning different feature-extracting terms in the level set equation to produce a surface deformation. Within our segmentation framework a variety of operations are available in each stage. A user must ``mix-and-match'' these operations in order to produce the desired result. We only describe those operations needed to produce the models in this paper. A more detailed description of our segmentation methods may be found in [28].

Because level set models move using gradient descent, they seek local solutions, and therefore the results are strongly dependent on the initialization, i.e., the starting position of the surface. Thus, one controls the nature of the solution by specifying an initial model from which the surface deformation process proceeds. We are able to computationally construct reasonable initial estimates directly from the input data by combining a variety of techniques.

The first step involves filtering the input data with a low-pass Gaussian filter ( $\sigma \approx 0.5$) to blur the data and thereby reduce noise. This tends to distort shapes, but the initialization need only be approximate. Next, the volume voxels are classified for inclusion/exclusion in the initialization based on the filtered values of the input data ( $k\approx 7.0$ for $\mathcal{V}_1$ and $k\approx 1.3$ for $\mathcal{V}_2$). For grey scale images, such as those used in this paper, the classification is equivalent to high and low thresholding operations. These operations are usually accurate to only voxel resolution, but the deformation process will achieve sub-voxel results. The final step before the actual level set deformation consist of performing a set of topological or logical operations on the voxels to ``clean up'' the initialization surface. This allows for the removal of undesired internal and external structures, which is extremely helpful to obtain simple models. It includes unions or intersections of voxel sets to create the better initializations. Specifically the topological operations consist of connected-component analyses (e.g. flood fill) to remove small pieces or holes from objects.

The initialization described above positions the model near the desired solution while retaining certain properties such as consistent geometry, connectivity, etc. Given this rough initial estimate, the level set surface deformation process, as described in Section 3.1, moves the surface model toward specific features in the data.

Figures 6 and 6 present two models that we extracted from DT-MRI volume datasets using our techniques. Figure 6 contains segmentations from volume $\mathcal{V}_1$, the measure of total diffusivity. The image in the first row shows a marching cubes iso-surface using an iso-value of 7.5. In the bottom we have extracted just the ventricles from $\mathcal{V}_1$. 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 the level set method described in Section 3, using a $\beta$ value of 0.2.

Figure 6 again provides the comparison between direct iso-surfacing and and level set model, but on the volume $\mathcal{V}_2$. The image in the top-left corner is a marching cubes iso-surface using an iso-value 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 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 6 contains three models extracted from $\mathcal{V}_2$ at different values of smoothing parameter $\beta$ 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 iso-surface of value 1.3. This was followed by a level set deformation using a $\beta$ value of 0.2. The result of this segmentation is presented on the bottom-left side of Figure 6. The top-right side of this figure presents a model extracted from $\mathcal{V}_2$ using an initial iso-surface of value 1.4 and a $\beta$ value of 0.5. The result chosen as the ``best'' by our scientific/medical collaborators is presented on the bottom-right side of Figure 6. This model is produced with an initial iso-surface of 1.3 and a $\beta$ 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.

We can also bring together the two models extracted from datasets $\mathcal{V}_1$ and $\mathcal{V}_2$ into a single image. Figure 6 demonstrates that we are able to isolate different structures in the brain and show their proper spatial inter-relationship. 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 data set of a different volunteer. This data set has 20 slices of the 256x256 resolution. We generated the anisotropy measure volume $\mathcal{V}_2^2$ and performed the level-set model extraction using the same iso-values and smoothing parameters as for $\mathcal{V}_2$. The results are shown in Figure 6.

 

Figure 4

Figure 4. Segmentation using isotropic measure $\mathcal{V}_1$ for the first DT-MRI dataset. The first row is the marching cubes iso-surface with 7.5.iso-value. The second row is the result of flood-fill algorithm applied to the same volume and used for level set initialization. The third row is the final level set model.

 

Figure 5

 

 

Figure 5. Model segmentation from volume $\mathcal{V}_2$. Top left image is an iso-surface of value 1.3, used for initialization of the level set. Clockwise, are the results of level set development with corresponding $\beta$ values of 0.2, 0.4 and 0.5.

 

Figure 6

 

 Figure 6. 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 $\mathcal{V}_1$ fits into model extracted from anisotropic measure dataset $\mathcal{V}_2$.

 

Figure 7

Figure 7. Segmentation using anisotropic measure $\mathcal{V}_2$ from the second DT-MRI dataset. The first row is the marching cubes iso-surface with iso-value 1.3. The second row is the result of flood-fill algorithm applied to the volume and used for level set initialization. The third is the final level set model.