From Diffusion MRI to Brain Connectomics - Modeling in Computational Biology and Biomedicine - page 208

Fig. 6.8 Diffusion Spectrum Imaging & Q-Ball Imaging. ( a ) DSI diffusion PDFs from [ 71 ]. Cor-
ticospinal tract (orientations sup.-inf.) and pontine decussation ( left-right ). ( b ) ODFs, estimated
from an analytical q-ball approach, such that they represent the angular marginal distributions of
the true and unknown EAPs (biological rat phantom)

6.4.2.2 Q-Ball Imaging: Emphasizing the Anisotropic Diffusion

Orientation Information

Q-Ball Imaging (QBI) was proposed by Tuch [ 67 , 68 ] spurred by the facts that DSI
had severe acquisition requirements, and that the DSI result of interest wasn't the
estimated EAP itself, but rather its radial projection—the ODF, which emphasized
angular details. His idea was to retrieve the same angular result with reduced
acquisition requirements. His initial attempt was the model based multi-tensor
approach which was stricken with instabilities induced by the assumed model.
Therefore, he proposed QBI, a model free method that sampled q-space only on
a sphere or q-shell with fixed q-radius with high angular resolution.
QBI like DSI is based on the q-space formalism and shows promising results,
although like DSI, in practice it cannot satisfy the NGP condition [ 67 ]. However,
QBI became a forerunner to a plethora of q-space methods that attempted to
reconstruct the EAP or its characteristics from partial sampling of the q-space. QBI,
itself maps spherical acquisitions in q-space to the ODF—a spherical function in
real space.
QBI is based on the Funk Radon transform (FRT), which is a mapping from a
sphere to a sphere

G : S 2 → S 2 . To a point on the sphere, called the pole, the FRT
of a spherical function f , assigns the value of the integral of the spherical function
along the equator on the plane that has for normal the vector connecting the centre
of the sphere to the pole:

u T w

G [ f ( u )]( u )= f ( u ) δ ( ) d w ,

S 2

S 2 . Using the Fourier slice theorem, Tuch was able to show that the
FRT of the signal acquired on a q-sphere was equal to the ODF in Eq. ( 6.24 ) blurred

where u , w ∈