Model Properties

Once a user has produced a satisfactory model of the desired segmented structures s/he may perform a number of quantitative geometric calculations on the resulting polygonal model, e.g. total area, volume, and average curvature. Though most of these measures are interesting from the modeling point of view, the volume of the ventricles, for example, can have clinical applications for disorder diagonosis and population comparison.

The models generated in the previous section are represented by triangle meshes consisting of vertices ${\bf v}_i$, connectivities and associated normal vectors. The total surface area of the model can be easily computed by adding the areas $A_i$ of each triangle

$$A = \sum_{i=1}^{N_{poly}} A_i =\sum_{i=1}^{N_{poly}}\frac{1}{2}\vert({\bf v}_i^1-{\bf v}_i^2)\times({\bf v}_i^1-{\bf v}_i^3)\vert,$$ (16)

where ${\bf v}_i^k$ is the $k$'th vertex of triangle $i$. Assuming that all of the extracted models are composed of closed polygonal surfaces, we can compute enclosed volume as a a signed sum of the pyramids with a base composed of the $i$'th triangle and a top vertex places at the origin of the dataset[32]. Then

$$V\approx\frac{1}{6}\sum_{i=1}^{N_{poly}}A_i \cdot \frac{1}{3}({\bf v}_i^1+{\bf v}_i^2+{\bf v}_i^3)\cdot {\bf N}_i;$$ (17)

Table 1 lists values of polygon count, surface area and total volume, for the models extracted from scalar volume datasets ( $\mathcal{V}_1$ and $\mathcal{V}_2$), before and after the level set algorithm is applied to the volumes. We note that the polygon count drops, because of the simplified form of the final extracted triangular mesh. The total surface area decrease is also due to smoothing imposed by the level set model. Volume decrease is partially caused by the removal (i.e. collapse) of small high frequency fragments cluttering the model and partially due to deformations of the model.

Table 1: Total polygon count in the models $N_{poly}$, surface areas $A$ and volumes $V$ and before/after application of the level set smoothing to datasets $\mathcal{V}_1$ and $\mathcal{V}_2$.

Data set	$N_{poly}$	$A(cm^2)$	$V(cm^3)$
$\mathcal{V}_1$	36,620/15,096	188/85	26/22
$\mathcal{V}_2$	142,212/81,488	760/743	98/87