Image Processing Reference
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(a)
(b)
(c)
(d)
(e)
(f )
FIGURE 13.6
Vector and tensor summation. Two vectors, (a) and (b), and their sum
(c). Two diffusion tensors, (d) and (e), of rank close to 1 visualized as ellipsoids with
eigenvectors forming principal axes. The summation of the 2 tensors gives a rank-2
tensor (f ). (From Westin, C.-F., Maier, S., Mamata, H., Nabavi, A., Jolesz, F., and
Kikinis, R. (2002). Processing and visualization of diffusion tensor MRI.
Med. Image
Anal.
6(2): 93-108.)
output has more degrees of freedom than the input tensors and describes the plane
in which diffusion is present. In this sense, averaging of tensors is different from
averaging a vector field. The average of a set of vectors gives the “mean event,”
whereas the average of a set of tensors gives the “mean event” and the “range of
the present events.”
Figure 13.7
shows a 2-D example illustrating the effect of Gaussian filtering
of a diffusion tensor field. The filtered areas that contain inconsistent data give
a result of almost-round ellipses (upper right half of the image). Moreover,
Gaussian filtering results in more stable estimates of the field directionality in
the areas where there is a clear bias in one direction (lower left).
The macrostructural measure achieved (by averaging the tensor field using an
isotropic mask) is essentially a feature extraction method rather than a restoration
method; the latter aims at reducing the noise level in the data. Although our method
does remove noise, the incorporation of more advanced regularization methods
[14,18] should be explored, if noise reduction is the main target. Anisotropic filter
masks are preferable because they reduce the risk of blurring edges. However, using
an anisotropic mask for the macrostructural measure would limit its purpose, i.e.,
the description of the organization within an area. If the signal changes due to edges
inside the local area of interest, this should be reflected in the measure.
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