Image Processing Reference
In-Depth Information
(a)
(b)
(c)
Fig. 8.1 Volume rendering ensembles of orientation distribution functions highlights regions that
are included in most ensemble members. Subfigures a - c compare the uncertainty resulting from
different fiber configurations and measurement setups. Images provided by the authors of [ 28 ]. a
b =
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000, SNR
=
5, b b =
7
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000, SNR
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10, c b =
7
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000, SNR
=
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to the data, and using the result to parameterize a probability distribution in a heuristic
manner. This often assumes that the fiber distribution is related to a sharpened version
of the diffusivity profile [ 38 ], sometimes regularized by a deliberate bias towards the
direction of the previous tracking step [ 4 , 19 ]. Programmable graphics hardware
accelerates the sampling of such models, and enables immediate visualization of the
results [ 40 ]. Parker et al. [ 45 ] present two different fiber distribution models that
are parameterized by measures of diffusion anisotropy. Subsequent work allows for
multimodal distributions that capture fiber crossings, and uses the observed variation
of principal eigenvectors in synthetic data to calibrate model parameters [ 44 ].
In contrast to these techniques, which use the model parameters from a single
fit, a second generation of probabilistic tractography methods estimates the poste-
rior distribution of fiber model parameters, based on the full information from the
measurements, which includes fitting residuals. Behrens et al. [ 3 ] do so in an objec-
tive Bayesian framework, which aims at making as few assumptions as possible, by
choosing noninformative priors. They have later extended the “ball-and-stick” model
that underlies their framework to allow for multiple fiber compartments [ 2 ].
Bootstrapping estimates the distribution of anisotropymeasures [ 43 ] or fiber direc-
tions [ 31 , 55 ] by repeated model fitting, after resampling data from a limited number
of repeated scans. This has been used as the foundation of another line of probabilistic
tractography approaches [ 35 , 39 ]. As an alternative to estimating the amount of noise
from repeated measurements, wild bootstrapping takes its noise estimates from the
residuals that remain when fitting a model to a single set of measurements [ 67 ]. This
has been proposed as an alternative to repetition-based bootstrapping for cases where
only a single acquisition is available [ 32 ]. Residual bootstrapping [ 12 ] builds on the
same basic idea, but allows for resampling residuals between gradient directions,
by modeling the heteroscedasticity in them. It has not only been combined with the
diffusion tensor model, but also with constrained deconvolution, which allows for
multiple fiber tractography [ 27 ].
To visualize the distributions estimated by bootstrapping, Jiao et al. [ 28 ] volume
render ensembles of orientation distribution functions (ODFs). As shown in Fig. 8.1 ,
this highlights regions included in most ensemble members, representing the most
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