Geometric Modeling

Two options are usually available for viewing the scalar volume datasets, direct volume rendering[14,15] and volume segmentation[16] combined with conventional surface rendering. The first option, direct volume rendering, is only capable of supplying images of the data. While this method may provide useful views of the data, it is well-known that it is difficult to construct the exact transfer function that highlights the desired structures in the volume dataset[17]. Our approach instead focuses on extracting geometric models of the structures embedded in the volume datasets. The extracted models may be used for interactive viewing, but the segmentation of geometric models from the volume datasets provides a wealth of additional benefits and possibilities. The models may be used for quantitative analysis of the segmented structures, for example the calculation of surface area and volume; quantities that are important when studying how these structures change over time. The models may be used to provide the shape information necessary for anatomical studies and computational simulation, for example EEG/MEG modeling within the brain[18]. Creating separate geometric models for each structure allows for the straightforward study of the relationship between the structures, even though they come from different datasets. The models may also be used within a surgical planning/simulation/VR environment[19], providing the shape information needed for collision detection and force calculations. The geometric models may even be used for manufacturing real physical models of the structures[20]. It is clear that there are numerous reasons to develop techniques for extracting geometric models from diffusion tensor volume datasets.

The most widely used technique for extracting polygonal models from volume datasets is the Marching Cubes algorithm[21]. This technique creates a polygonal model that approximates the iso-surface embedded in a scalar volume dataset for a particular iso-value. The surface represents all the points within the volume that have the same scalar value. The polygonal surface is created by examining every ``cube`` of eight volume grid points and defining a set of triangles that approximates the piece of the iso-surface within the space bounded by the eight points. While the Marching Cubes algorithm is easy to understand and straightforward to implement, applying it directly to raw volume data from scanners can produce undesirable results, as seen in top row images in Figures6, 6. The algorithm is susceptible to noise and can produce many unwanted triangles that mask the central structures in the data. In order to alleviate this problem, we utilize a deformable model approach to smooth the data and remove the noise-related artifacts. Many types of deformable models have been proposed for extracting structures from volumes[16,22]. We utilize level set models as they have been shown to be flexible and effective for segmentation[23,24,26,27,28].Level set methods produce active deformable surfaces that may be directed to conform to features in a volume dataset while simultaneously applying a smoothing operation based on local surface curvature[28]. Most importantly, they easily change topology during deformation and have no fixed parameterization, allowing them to represent complex shapes.