Biomedical Engineering Reference
In-Depth Information
results shown in Figs. 7.6d, 7.6e, and 7.6f, respectively. Root-mean-squared-error
(RMSE) for Figs. 7.6c-7.6f are 1.9725, 1.9711, 2.1439, and 2.2497 voxels, respec-
tively. The RMSE obtained in a RaG approximation of a digital region is usually
much smaller than the tolerance used to approximate a digital shape by a trian-
gular mesh. Figures 7.6c-7.6f appear very similar, but from the RMSE obtained,
we see that their local geometries are somewhat different. The shape in Fig. 7.6f
is much smoother and rounder than the shape in Fig. 7.6c.
At the standard deviation that matches the level of detail in a shape, the
smallest surface-fitting error is obtained. This minimum error can be determined
by a steepest-descent algorithm. However, since the given region is known to
contain errors, finding the surface that is very close to the region may not be
of particular interest. Currently, after the control points of an approximating
surface are determined, the user interactively varies the smoothness (standard
deviation) of the surface and views the obtained surface as well as the associating
RMSE. In this manner, the standard deviation of Gaussians can be interactively
selected to reproduce a desired level of details in a constructed shape.
7.2.4
Shape Editing
Once the result of an automatic segmentation is represented by a free-form
parametric surface, the surface can be revised to a desired geometry by appro-
priately moving its control points. In the system we have developed, an obtained
surface is overlaid with the original volumetric image. Then, by going through
different image slices along one of the three orthogonal directions, the user vi-
sually observes the intersection of the surface with the image slices and verifies
the correctness of the segmentation. When an error is observed, one or more of
the control points are appropriately moved to correct the error. As the control
points are moved, the user will observe changes in the surface immediately.
An example of shape editing by the proposed method is shown in Fig. 7.7.
Figure 7.7a shows the surface approximating a brain tumor within the original
volumetric image. The user selects a number of control points using a small
sphere that is attached to the cursor and whose center lies in the image slice
being reviewed. By placing the cursor near the area where an error has occurred
in one of the slices and pressing the mouse button, the sphere is activated and the
control points falling in the sphere are selected. By changing the radius of sphere,
the number of control points selected for movement are changed. Control points
