Biomedical Engineering Reference
In-Depth Information
visualization and segmentation. Although eigenvalue/vector computation of the
3
×
3 matrix is not expensive, it must be repeatedly performed for every voxel
in the volume. This calculation easily becomes a bottleneck for large datasets.
For example, computing eigenvalues and eigenvectors for a 512
3
volume re-
quires over 20 CPU min on a powerful workstation. Another problem associated
with eigenvalue computation is stability—a small amount of noise will change
not only the values but also the ordering of the eigenvalues [60]. Since many
anisotropy measures depend on the ordering of the eigenvalues, the calculated
direction of diffusion and classification of tissue will be significantly altered by
the noise normally found in diffusion tensor datasets. Thus it is desirable to
have an anisotropy measure which is rotationally invariant, does not require
eigenvalue computations, and is stable with respect to noise. Tensor invari-
ants with these characteristics were first proposed by Ulug
et al.
[61]. In Sec-
tion 8.5.1 we formulate a new anisotropy measure for tensor field based on these
invariants.
Visualization and model extraction from the invariant 3D scalar fields is
the second issue addressed in this chapter. One of the popular approaches
to tensor visualization represents a tensor field by drawing ellipsoids asso-
ciated with the eigenvectors/values [62]. This method was developed for 2D
slices and creates visual cluttering when used in 3D. Other standard CFD
visualization techniques such as tensor-lines do not provide meaningful re-
sults for the MRI data due to rapidly changing directions and magnitudes
of eigenvector/values and the amount of noise present in the data. Recently
Kindlmann [63] developed a volume rendering approach to tensor field vi-
sualization using eigenvalue-based anisotropy measures to construct transfer
functions and color maps that highlight some brain structures and diffusion
patterns.
In our work we perform isosurfacing on the 3D scalar fields derived from
our tensor invariants to visualize and segment the data [64]. An advantage of
isosurfacing over other approaches is that it can provide the shape information
needed for constructing geometric models, and computing internal volumes and
external surface areas of the extracted regions. There has also been a number
of recent publications [65, 66] devoted to brain fiber tracking. This is a different
and more complex task than the one addressed in this chapter and requires data
with a much higher resolution and better signal-to-noise ratio than the data used
in our study.
