Signal Processing - May 2016 - 47
Disconnect
Join
Remove
Create
(a)
(b)
(c)
(d)
FIGURe 14. Two tractography methods. (a) The local method grows a fiber track step by step, according to local fiber information. (b) The global method
improves a set of fiber tracks by disconnecting, joining, removing, and creating fibers with the aim of minimizing an energy function that matches fiber
tracks and the FOD image through a generative model. (c) Fiber bundles simulated with a local tractography method seeded in a ball region. (d) An
example of fiber tracks simulated by Daducci et al.'s [47] global tractography method. In (c) and (d), fiber tracks are color coded with the directional
vector from their start point to their end point, with the red, green, and blue colors for the left-right, back-front, and up-down directions, respectively.
[(d) reprinted from [47] with permission.)
combination of spherical functions disperses probability density on the sphere and hence reduces its angular focus. Cheng et
al. [41] proposed a hyperspherical manifold of FODs. Because
8u ! S 2 F (u) du = 1, its square-rooted function f = F $ 0 satisfies 8u ! S 2 f 2 (u) du = 1, or equivalently, the norm of f is one.
This implies that f resides on a unit sphere in a hyperspace, a
well-studied manifold. On this manifold, the distance between
two FODs is the length of the great arc connecting them, and
the interaction force is in the tangent plane of the hypersphere.
Rotation-induced spherical manifold
Although the hyperspherical manifold reduces the angular
dispersion of FODs, it does not directly handle the main
cause of dispersion: the FODs' difference in orientation. As
neuronal fibers extend in the brain, they do not take straight
paths, but turn gradually to make curves. Li et al. [40] proposed a rotation-induced manifold to directly handle this ori-
entational difference. The rotation-induced manifold treats
tangent vectors on the hyperspherical surface differently,
that is, whichever can be realized by rotation is associated
with a parameter m rot, and whichever purely reflects shape
differences is associated with a constant one. This unequal
treatment does not change the topology, but defines a new
Riemannian metric. By setting m rot smaller than one, the
FODs' orientational difference is underweighted, and consequently FODs with similar shapes but different orientations
are squeezed closer. If m rot equals zero, then the manifold
reduces to a quotient space completely discarding differences
caused by rotation. Figure 13 compares the rotation-induced
and hyperspherical manifolds, showing that manifold structures can lead to very different processing effects. Reisert and
Kiselev [42] have provided a framework for embedding fiber
continuity into FOD reconstruction. It is interesting to utilize
these FOD manifolds in their framework.
IEEE Signal Processing Magazine
|
May 2016
|
47
Table of Contents for the Digital Edition of Signal Processing - May 2016
Signal Processing - May 2016 - Cover1
Signal Processing - May 2016 - Cover2
Signal Processing - May 2016 - 1
Signal Processing - May 2016 - 2
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Signal Processing - May 2016 - 4
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Signal Processing - May 2016 - 120
Signal Processing - May 2016 - Cover3
Signal Processing - May 2016 - Cover4
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