Skip to content

HTTPS clone URL

Subversion checkout URL

You can clone with HTTPS or Subversion.

Download ZIP

Comparing changes

Choose two branches to see what's changed or to start a new pull request. If you need to, you can also compare across forks.

Open a pull request

Create a new pull request by comparing changes across two branches. If you need to, you can also compare across forks.
base fork: Garyfallidis/didaktoriko
base: 328a3c168e
...
head fork: Garyfallidis/didaktoriko
compare: f35238fe16
Checking mergeability… Don't worry, you can still create the pull request.
  • 2 commits
  • 3 files changed
  • 0 commit comments
  • 1 contributor
Showing with 2,141 additions and 1,463 deletions.
  1. +813 −135 Chapter_2.lyx
  2. +15 −15 Chapter_4.lyx
  3. +1,313 −1,313 diffusion.bib
View
948 Chapter_2.lyx
@@ -44,6 +44,10 @@
\paperpagestyle default
\tracking_changes false
\output_changes false
+<<<<<<< HEAD
+\author ""
+=======
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\author ""
\author ""
\author "Ian Nimmo-Smith,,,"
@@ -60,6 +64,10 @@ Overview
\end_layout
\begin_layout Standard
+<<<<<<< HEAD
+Between one to two thirds of imaging voxels in the human brain's white matter
+ are thought to contain multiple fibre bundle crossings
+=======
Between one
\change_deleted 2 1331648613
-third
@@ -74,12 +82,11 @@ thirds of imaging voxels in the human brain's white matter are thought to
contain multiple fibre bundle crossings
\change_inserted 2 1331820052
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset space ~
\end_inset
-\change_unchanged
-
\begin_inset CommandInset citation
LatexCommand cite
key "Behrens2007NeuroImage"
@@ -87,14 +94,15 @@ key "Behrens2007NeuroImage"
\end_inset
, in which case the Diffusion Tensor model proposed by Basser et al.
+<<<<<<< HEAD
+=======
\change_inserted 2 1331811507
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset space ~
\end_inset
-\change_unchanged
-
\begin_inset CommandInset citation
LatexCommand cite
key "Basser1994"
@@ -102,11 +110,15 @@ key "Basser1994"
\end_inset
breaks down.
+<<<<<<< HEAD
+ High Angular Resolution Diffusion Imaging (HARDI) methods
+=======
High Angular Resolution Diffusion Imaging (HARDI)
\change_inserted 2 1331811491
methods
\change_unchanged
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset CommandInset citation
LatexCommand cite
key "Tuch2002"
@@ -143,6 +155,12 @@ key "barmpoutis2009regularized"
and many more reconstruction methods have been proposed to overcome the
limitations of the Diffusion Tensor.
+<<<<<<< HEAD
+ These methods can be divided into those which need specific acquisition
+ parameterizations, and those which can be used independently of q-space
+ structure.
+ For instance, for Q-ball Imaging
+=======
These methods can be divided into those wh
\change_inserted 2 1331649178
ich
@@ -163,6 +181,7 @@ For instance,
\change_inserted 2 1331811517
for Q-ball Imaging
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset space ~
\end_inset
@@ -173,6 +192,10 @@ key "Tuch2004"
\end_inset
+<<<<<<< HEAD
+ sampling needs to be on one or more spherical grids, and in Generalized
+ Q-sampling Imaging (GQI)
+=======
\change_deleted 2 1331649444
they have to be
@@ -234,12 +257,17 @@ in
\change_deleted 2 1331649553
like in Generalized Q-sampling Imaging (GQI)
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset CommandInset citation
LatexCommand cite
key "Yeh2010"
\end_inset
+<<<<<<< HEAD
+, requires sampling on a Cartesian grid; by contrast DTI can be used independent
+ly of q-space structure.
+=======
and those which can be used independently of q-space structure like
\change_inserted 2 1331649560
by contrast
@@ -249,10 +277,26 @@ by contrast
can be used independently of q-space structure
\change_unchanged
.
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
A further division considers the level of model assumptions for the diffusion
process.
Although all methods have some underlying assumptions we generally separate
them in model-based and model-free.
+<<<<<<< HEAD
+ Model-based methods like the Single Tensor or Multi Tensor require a number
+ of parameters to be fitted.
+ By contrast for model-free methods fitting is not necessary and the directional
+ity of the underlying tissue can be approximated by some re-parametrization
+ or re-transformation of the signal.
+ The latter is usually more efficient than fitting models with many parameters
+ which typically call for iterative methods.
+\end_layout
+
+\begin_layout Standard
+This chapter presents and evaluates and compares different model-free methods
+ for the reconstruction of orientation distribution functions using diffusion
+ MRI data sampled on a Cartesian lattice in
+=======
Model-based methods like the Single Tensor or Multi Tensor
\change_deleted 2 1331650589
necessitate the fitting from a few to many
@@ -299,6 +343,7 @@ document
\change_unchanged
different model-free methods for the reconstruction of orientation distribution
functions using diffusion MRI data sampled on a Cartesian lattice in
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\series bold
q
\series default
@@ -309,6 +354,10 @@ q
and a family of methods is defined called the Equatorial Inversion Transform
(EIT).
The EIT is a new way to represent and reconstruct the diffusion signal.
+<<<<<<< HEAD
+ Our results show that it can perform better or as well as the current state-of-
+the art methods i.e.
+=======
Our results show that it can perform better or as
\change_deleted 2 1331543232
good
@@ -316,6 +365,7 @@ good
well
\change_unchanged
as the current state-of-the art methods i.e.
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
DSI and GQI.
\end_layout
@@ -370,6 +420,9 @@ Here
\begin_inset Formula $RF$
\end_inset
+<<<<<<< HEAD
+ represents the complex RF signal measured at spatial wave number
+=======
\change_inserted 2 1331651296
represents
@@ -377,6 +430,7 @@ represents
is
\change_unchanged
the complex RF signal measured at spatial wave number
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset Formula $\mathbf{k}$
\end_inset
@@ -393,6 +447,9 @@ is
\begin_inset Formula $\Delta$
\end_inset
+<<<<<<< HEAD
+ is the time between diffusion gradients,
+=======
is the
\change_deleted 2 1331651348
diffusion time scale of the sequence
@@ -400,6 +457,7 @@ diffusion time scale of the sequence
time between diffusion gradients
\change_unchanged
,
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset Formula $P_{\Delta}$
\end_inset
@@ -412,11 +470,15 @@ time between diffusion gradients
\series default
+<<<<<<< HEAD
+ is the voxel coordinate, and
+=======
is the voxel coordinate
\change_inserted 2 1331651363
,
\change_unchanged
and
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset Formula $\mathbf{r}$
\end_inset
@@ -447,6 +509,13 @@ S(\mathbf{v},\mathbf{q}) & = & \int\rho(\mathbf{v})P_{\Delta}(\mathbf{v},\mathbf
\end_layout
\begin_layout Standard
+<<<<<<< HEAD
+For the rest of the chapter we consider each voxel independently and assume
+ intra-voxel spatial homogeneity so we can drop explicit reference to
+\begin_inset Formula $\mathbf{v}$
+\end_inset
+
+=======
For the rest of the chapter
\change_deleted 2 1331651498
we can drop
@@ -469,10 +538,14 @@ we assume that the formulation is the same for every voxel
\change_deleted 2 1331651543
\change_unchanged
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
and
\begin_inset Formula $\Delta$
\end_inset
+<<<<<<< HEAD
+..
+=======
\change_inserted 2 1331651558
.
@@ -483,6 +556,7 @@ and
.
\change_inserted 2 1331820697
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
(We note in passsing that the shape of
\begin_inset Formula $P_{\Delta}$
\end_inset
@@ -492,9 +566,8 @@ and
\end_inset
.
- We will not pursue this matter further here.)
-\change_unchanged
- We can also replace the spin density
+ We will not pursue this matter further here.) We can also replace the spin
+ density
\family roman
\series medium
\shape up
@@ -564,27 +637,29 @@ P(\mathbf{r}) & = & S_{0}^{-1}\int S(\mathbf{q})\exp(-i2\pi\mathbf{q}\cdot\mathb
\end_layout
\begin_layout Standard
+<<<<<<< HEAD
+=======
\change_inserted 2 1331555449
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset ERT
status open
\begin_layout Plain Layout
+<<<<<<< HEAD
+=======
\change_inserted 2 1331555391
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\backslash
noindent
-\change_unchanged
-
\end_layout
\end_inset
-
-\change_unchanged
-or diffusion propagator.
+ or diffusion propagator.
It was shown by Wedeen et al.
\begin_inset CommandInset citation
@@ -595,22 +670,31 @@ key "Wedeen"
that the dMRI signal is positive for any type of spin motion without net
flux (i.e.
+<<<<<<< HEAD
+=======
\change_inserted 2 1331555778
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset ERT
status open
\begin_layout Plain Layout
+<<<<<<< HEAD
+=======
\change_inserted 2 1331555778
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
~
-\change_unchanged
-
\end_layout
\end_inset
+<<<<<<< HEAD
+spin displacements due to thermal molecular agitation) or other random fluxes
+ such as intravoxel incoherent motion.
+ Under this assumption we can replace the complex signal
+=======
\change_deleted 2 1331555726
@@ -630,6 +714,7 @@ reference "eq:P"
\change_unchanged
the complex signal
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset Formula $S$
\end_inset
@@ -637,8 +722,11 @@ reference "eq:P"
\begin_inset Formula $|S|$
\end_inset
+<<<<<<< HEAD
+=======
\change_inserted 2 1331651746
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
in eq.
(
\begin_inset CommandInset ref
@@ -647,9 +735,7 @@ reference "eq:P"
\end_inset
-)
-\change_unchanged
-
+)
\begin_inset Formula \begin{eqnarray}
P(\mathbf{r}) & = & S_{0}^{-1}\int|S(\mathbf{q})|\exp(-i2\pi\mathbf{q}\cdot\mathbf{r})d\mathbf{r}\label{eq:P_modulus}\end{eqnarray}
@@ -660,6 +746,9 @@ P(\mathbf{r}) & = & S_{0}^{-1}\int|S(\mathbf{q})|\exp(-i2\pi\mathbf{q}\cdot\math
\begin_layout Standard
The modulus of the signal coincides with the output of the standard MRI
+<<<<<<< HEAD
+ scanners as dMRI and that simplifies the acquisition procedure.
+=======
scanners as
\change_deleted 2 1331651891
DWI
@@ -667,12 +756,21 @@ DWI
dMRI
\change_unchanged
and that simplifies the acquisition procedure.
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
It represents the density of the average relative spin displacement in
a voxel.
In other words,
\begin_inset Formula $P(\mathbf{r})\, d\mathbf{r}$
\end_inset
+<<<<<<< HEAD
+ is a measure of the probability that a spin in a chosen voxel , during
+ the experimental mixing time
+\begin_inset Formula $\Delta$
+\end_inset
+
+, would make a vector displacement
+=======
is a measure of the probability
\change_deleted 2 1331651979
for
@@ -698,14 +796,19 @@ to make
would make
\change_unchanged
a vector displacement
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset Formula $\mathbf{r}$
\end_inset
.
We can visualize the propagator for every voxel as a 3D density volume
+<<<<<<< HEAD
+ (see Fig.
+=======
(
\change_inserted 2 1331653300
see Fig.
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset CommandInset ref
LatexCommand ref
@@ -713,6 +816,18 @@ reference "Flo:beautiful-triple-crossing"
\end_inset
+<<<<<<< HEAD
+)).
+\end_layout
+
+\begin_layout Standard
+In the classical Q-ball acquisition, at each location, diffusion-weighted
+ images are acquired for
+\begin_inset Formula $N=515$
+\end_inset
+
+ or fewer values of
+=======
)
\change_deleted 2 1331653241
SEE Figure Introduction
@@ -742,6 +857,7 @@ less
fewer
\change_unchanged
values of
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\series bold
q
\series default
@@ -778,11 +894,15 @@ with
\end_inset
values.
+<<<<<<< HEAD
+ Often to obtain data for the complete grid of
+=======
\change_inserted 2 1331652873
\change_unchanged
Often to obtain data for the complete grid of
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset Formula $515$
\end_inset
@@ -794,6 +914,14 @@ Often to obtain data for the complete grid of
\begin_inset Formula $515$
\end_inset
+<<<<<<< HEAD
+ diffusion weighted volumes) the overall acquisition time would be too long
+ and a smaller number of unique
+\series bold
+q
+\series default
+-vectors are employed for just a single hemisphere usually between
+=======
diffusion weighted volumes)
\change_deleted 2 1331652557
is very time consuming
@@ -815,6 +943,7 @@ only at
for just
\change_unchanged
a single hemisphere usually between
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset Formula $102$
\end_inset
@@ -830,6 +959,11 @@ key "Kuo"
\end_inset
.
+<<<<<<< HEAD
+ This is valid because the underlying self-diffusion process is symmetric
+ and so the signal is symmetric, therefore the vectors can be mapped on
+ the other hemisphere to create the full q-space.
+=======
This is valid because
\change_inserted 2 1331652985
the underlying self-diffusion process is symmetric and so
@@ -840,6 +974,7 @@ the signal is symmetric
\change_unchanged
therefore the vectors can be mapped on the other hemisphere to create the
full q-space.
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\end_layout
@@ -883,12 +1018,17 @@ This defines the orientation density function (ODF) for DSI which measures
\begin_layout Standard
\lang british
+<<<<<<< HEAD
+Note at this point that in order to find the ODF we have to create first
+ the diffusion propagator by
+=======
Not
\change_deleted 2 1331653050
ic
\change_unchanged
e at this point that in order to find the ODF we have to create first the
diffusion propagator by
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\lang english
applying
\lang british
@@ -896,39 +1036,46 @@ applying
\lang english
Yeh et al.
+<<<<<<< HEAD
+=======
\change_inserted 2 1331811943
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset space ~
\end_inset
-\change_unchanged
-
\begin_inset CommandInset citation
LatexCommand cite
key "Yeh2010"
\end_inset
+<<<<<<< HEAD
+ proposed a direct way to calculate a slightly different ODF using the Cosine
+ transform.
+=======
proposed a direct
\change_deleted 2 1331653062
analytical
\change_unchanged
way to calculate a slightly different ODF using the Cosine transform.
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\end_layout
\begin_layout Standard
\align block
In order to represent the average propagator in the scale of spin quantity
Yeh et al.
+<<<<<<< HEAD
+=======
\change_inserted 2 1331811951
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset space ~
\end_inset
-\change_unchanged
-
\begin_inset CommandInset citation
LatexCommand cite
key "Yeh2010"
@@ -991,6 +1138,9 @@ S(\mathbf{q}) & = & \int Q(\mathbf{r})\exp(i2\pi\mathbf{q}\cdot\mathbf{r})d\math
\noun default
\color inherit
\lang english
+<<<<<<< HEAD
+We can apply the Fourier transform again to eq.
+=======
We can apply the Fourier transform again
\change_inserted 2 1331653530
to
@@ -998,6 +1148,7 @@ to
in
\change_unchanged
eq.
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:W2Q"
@@ -1105,11 +1256,16 @@ where
the more detailed the SDF will be but also more noisy.
This ODF was used as the basis of the analysis of the GQI method.
+<<<<<<< HEAD
+ It provides a simple direct analytical solution of the ODF which can be
+ written in a simple matrix form
+=======
It provides a simple direct analytical solution of
\change_inserted 2 1331654548
t
\change_unchanged
he ODF which can be written in a simple matrix form
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset Formula \begin{eqnarray*}
\bm{\psi}_{GQI}= & \mathbf{s}\cdot\mathtt{sinc}((6D\cdot G\circ\mathbf{b}\circ\mathbb{1})\cdot G)\lambda/\pi\end{eqnarray*}
@@ -1157,6 +1313,9 @@ denotes the Hada-mard product,
\begin_inset Formula $M$
\end_inset
+<<<<<<< HEAD
+-dimensional vector with components corresponding to the selected directions
+=======
-dimensional vector
\change_inserted 2 1331654525
with components corresponding to the
@@ -1169,15 +1328,20 @@ direction
\change_inserted 2 1331654534
s
\change_unchanged
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset Formula $\hat{\mathbf{u}}$
\end_inset
+<<<<<<< HEAD
+ on the ODF sphere,
+=======
\change_inserted 2 1331654561
on the ODF sphere
\change_unchanged
,
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset Formula $\mathbf{s}$
\end_inset
@@ -1295,6 +1459,9 @@ This equation can be similarly implemented with a simple matrix transform
\begin_layout Standard
\align left
+<<<<<<< HEAD
+and has not to date been published with real or simulated data sets.
+=======
and
\change_deleted 2 1331654625
until today it hasn't
@@ -1302,13 +1469,14 @@ until today it hasn't
has not to date
\change_unchanged
been published with real or simulated data sets.
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\end_layout
\begin_layout Standard
The addition of the spin density plays a very important role on normalizing
the ODF and providing scalar or vector metrics for the analysis of dMRI
data sets.
- GQI similarly with DSI expects the q-vectors to sit on a cubic lattice
+ GQI, similarly to DSI, expects the q-vectors to sit on a cubic lattice
within a sphere.
However, because of the direct analytical formulation of the GQI ODFs;
the creation of the volumetric grid with the signal values is not necessary.
@@ -1421,11 +1589,15 @@ The Fourier transform of
\begin_inset Formula $\nabla^{2}$
\end_inset
+<<<<<<< HEAD
+ is the Laplacian operator (for proof see section
+=======
is the Laplacian operator (for proof see
\change_inserted 2 1331812204
section
\change_unchanged
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset CommandInset ref
LatexCommand ref
reference "sub:Fourier-transform-ofPr2"
@@ -1468,6 +1640,8 @@ reference "sub:Radial-projection-of-symmetric"
\begin_layout Standard
From eq.
+<<<<<<< HEAD
+=======
\change_inserted 2 1331812611
@@ -1475,6 +1649,7 @@ From eq.
\change_unchanged
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:ODF_DSI"
@@ -1561,6 +1736,15 @@ key "aganj2010reconstruction"
and we propose here that it can be directly calculated in a cubic grid
using the standard 3D discrete Laplacian filter which is given by the 3D
kernel defined by the following
+<<<<<<< HEAD
+\begin_inset Formula $3\times3\times3$
+\end_inset
+
+ array:
+\end_layout
+
+\begin_layout Standard
+=======
\change_inserted 2 1331713841
\begin_inset Formula $3\times3\times3$
@@ -1602,6 +1786,7 @@ three planes
\begin_layout Standard
\change_inserted 2 1331713951
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset Formula \[
\left[\left(\begin{array}{ccc}
0 & 0 & 0\\
@@ -1622,6 +1807,12 @@ three planes
\begin_layout Standard
\noindent
\align block
+<<<<<<< HEAD
+This is a filter commonly used for image processing.
+ From now on when we use the Laplacian operator in order to measure the
+ directionality of the diffusion signal we will call this reconstruction
+ method Diffusion Nabla Imaging as nabla-squared
+=======
This is a f
\change_inserted 2 1331714112
ilter
@@ -1636,6 +1827,7 @@ unction
-squared
\change_unchanged
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\family roman
\series medium
\shape up
@@ -1755,6 +1947,10 @@ name "Flo:single_fibre_spherical_grid"
\begin_layout Standard
These are two very important geometric properties of the signal that we
+<<<<<<< HEAD
+ can try to exploit to its limit by calculating equatorial integrals in
+ order to identify the directionality of the signal.
+=======
can try to exploit
\change_deleted 2 1331654855
at the
@@ -1769,6 +1965,7 @@ limit
\change_unchanged
by calculating equatorial integrals in order to identify the directionality
of the signal.
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\end_layout
@@ -1792,11 +1989,16 @@ reference "eq:ODF_DNI"
\end_inset
+<<<<<<< HEAD
+ where an equatorial integral creates a connection between the real ODF
+ and the signal.
+=======
where an equatorial integral
\change_deleted 2 1331654929
which
\change_unchanged
creates a connection between the real ODF and the signal.
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
The Funk-Radon Transform (FRT) used by
\family default
\series default
@@ -1819,6 +2021,13 @@ ction sphere.
\end_layout
\begin_layout Standard
+<<<<<<< HEAD
+With EIT the most important goal is to try to identify the orientational
+ variation in the signal in the most accurate way by generating a spherical
+ density.
+ However it is possible to calculate as well the classical ODF as defined
+ by Wedeen et al.
+=======
With EIT the most important goal is to try to identify the
\change_deleted 2 1331654986
correct directionality
@@ -1835,6 +2044,7 @@ real
classical
\change_unchanged
ODF as defined by Wedeen et al.
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset CommandInset citation
LatexCommand cite
@@ -2305,6 +2515,9 @@ reference "Tab:EIT_table"
\end_inset
+<<<<<<< HEAD
+ we see that by choosing different functions for
+=======
we see that
\change_inserted 2 1331655182
by cho
@@ -2320,6 +2533,7 @@ for
of
\change_unchanged
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset Formula $F$
\end_inset
@@ -2327,6 +2541,9 @@ of
\begin_inset Formula $O$
\end_inset
+<<<<<<< HEAD
+ we can generate both old and new distribution functions on the sphere.
+=======
\change_inserted 2 1331655187
we
@@ -2338,6 +2555,7 @@ both old and
older well known or
\change_unchanged
new distribution functions on the sphere.
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
With
\begin_inset Formula $F(E(q))=-\nabla^{2}(E(q))$
\end_inset
@@ -2363,6 +2581,10 @@ older well known or
\begin_inset Formula $O(q)=1$
\end_inset
+<<<<<<< HEAD
+ then this is similar to the Funk Radon Transform (used in Qball imaging)
+ but applied to multiple spherical shells.
+=======
then this is similar
\change_inserted 2 1331655217
to
@@ -2371,6 +2593,7 @@ with
\change_unchanged
the Funk Radon Transform (used in Qball imaging) but applied to multiple
spherical shells.
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
However, we can also try to use different functions like
\begin_inset Formula $F(E(q))=-\nabla^{4}(E(q))$
\end_inset
@@ -2380,6 +2603,11 @@ with
\end_inset
which can potentially increase the amount of directional information beyond
+<<<<<<< HEAD
+ that of the standard ODFs.
+ Before starting investigating the realms of EIT we should first give a
+ short overview of other methods commonly found in the literature.
+=======
tha
\change_inserted 2 1331655274
t of
@@ -2394,6 +2622,7 @@ n
These are grid-based, mostly non-grid based and usually parametric.
\change_unchanged
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\end_layout
\begin_layout Subsection
@@ -2430,6 +2659,11 @@ key "descoteaux2010multiple"
\end_inset
.
+<<<<<<< HEAD
+ HYDI acquires the signal values on five concentric spherical q-space shells,
+ then interpolates onto a cubic grid and applies the standard Fourier transform
+ in the same way as DSI.
+=======
HYDI acquires the signal values on
\change_inserted 2 1331655431
five
@@ -2442,6 +2676,7 @@ on
\change_unchanged
to a cubic grid and applies the standard Fourier transform in the same way
as DSI.
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
In mq-DPI the EAP is calculated by solving Laplace's equation for the diffusion
signal using a real and symmetric modified spherical harmonic basis.
The EAP can be found analytically by the inversion of a linear system using
@@ -2476,6 +2711,14 @@ key "ozarslan2006resolution"
\begin_inset Formula $P(r_{0}\mathbf{\hat{u}})$
\end_inset
+<<<<<<< HEAD
+, the probability of finding a particle initially at the origin at the point
+
+\begin_inset Formula $r_{0}\mathbf{\hat{u}}$
+\end_inset
+
+, using spherical harmonics.
+=======
, the probability of finding
\change_inserted 2 1331655486
a
@@ -2495,6 +2738,7 @@ the
,
\change_unchanged
using spherical harmonics.
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
Not surprisingly there is a relationship connecting CSA with DOT which
is
\lang british
@@ -2552,6 +2796,10 @@ key "JansonsPAS2003"
\begin_layout Standard
The first reference of using spherical harmonic expansions with diffusivity
+<<<<<<< HEAD
+ profiles, which are now quite common in the literature, was by Alexander
+ et al.
+=======
profiles, which are now
\change_deleted 2 1331655602
very
@@ -2565,6 +2813,7 @@ favorable
common
\change_unchanged
in the literature, was by Alexander et al.
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset CommandInset citation
LatexCommand cite
key "alexander2002detection"
@@ -2572,11 +2821,15 @@ key "alexander2002detection"
\end_inset
.
+<<<<<<< HEAD
+ Q-ball imaging was introduced by Tuch
+=======
Q
\change_inserted 2 1331653788
-
\change_unchanged
ball imaging was introduced by Tuch
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset CommandInset citation
LatexCommand cite
key "Tuch2004"
@@ -2592,12 +2845,17 @@ key "Tuch2004"
\end_inset
is a normalization constant.
+<<<<<<< HEAD
+ It was later provided for Q-Ball imaging a fast and analytical solution
+ using spherical harmonics (SH) and Laplace-Beltrami regularization
+=======
It was later provided for Q
\change_inserted 2 1331655610
-
\change_unchanged
Ball imaging a fast and analytical solution using spherical harmonics (SH)
and Laplace-Beltrami regularization
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset CommandInset citation
LatexCommand cite
key "Descoteaux2007"
@@ -2650,17 +2908,24 @@ key "yeh2011estimation"
\end_layout
\begin_layout Standard
+<<<<<<< HEAD
+On Tensor related methods we have the classical Single Tensor
+=======
On Tensor related methods we have the classical Single
\change_inserted 2 1331655845
\change_unchanged
Tensor
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset CommandInset citation
LatexCommand cite
key "Basser1994"
\end_inset
+<<<<<<< HEAD
+, Sticks and Ball
+=======
, Sticks
\change_inserted 2 1331655853
and
@@ -2668,6 +2933,7 @@ key "Basser1994"
&
\change_unchanged
Ball
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset CommandInset citation
LatexCommand cite
key "Behrens2007NeuroImage"
@@ -2704,11 +2970,15 @@ key "barmpoutis2009regularized"
.
In addition there are also model based methods which try to calculate non-Gauss
+<<<<<<< HEAD
+ian properties, for example the Kurtosis Tensor
+=======
ian properties
\change_inserted 2 1331655874
,
\change_unchanged
for example the Kurtosis Tensor
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset CommandInset citation
LatexCommand cite
key "jensen2005diffusional"
@@ -2893,11 +3163,16 @@ reference "Flo:sEITvsEIT"
In this document whenever we use the prefix s in front of a method that
will mean that this was calculated with the standard EIT algorithm.
For example if standard EIT is used for DNI we will write sDNI or sEITL.
+<<<<<<< HEAD
+ (The 'L' stands for 'Lapacian' or 'Nabla'.) Of course sDNI and sEITL are
+ equivalent.
+=======
\change_inserted 2 1331658366
(The 'L' stands for 'Lapacian' or 'Nabla'.)
\change_unchanged
Of course sDNI and sEITL are equivalent.
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\end_layout
\begin_layout Subsubsection
@@ -3514,6 +3789,10 @@ name "Alg:PeakFinding"
\end_layout
\begin_layout Standard
+<<<<<<< HEAD
+We have used a triangulation of the unit sphere (which we refer to simply
+ as 'sphere') obtained by triangular subdivision of a regular icosahedron.
+=======
We have
\change_deleted 2 1331656962
a constructed
@@ -3539,6 +3818,7 @@ obtained by triangular subdivision of a regular icosahedron.
\change_deleted 2 1331657315
which
\change_unchanged
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
It is symmetric over the z-axis, i.e.
\begin_inset space ~
\end_inset
@@ -3560,15 +3840,15 @@ key "Yeh2010"
\end_inset
for GQI reconstructions.
- Every face contains the
+ Every face (triangle) corresponds to a list of the
\begin_inset Formula $3$
\end_inset
- indices which indicate at the
+ indices of the
\begin_inset Formula $3$
\end_inset
- different points that create the triangle (face) of the sphere.
+ vertices on the sphere.
The idea here is that we can travel from face to face and nullify all points
on a face which are lower that the higher value of the face.
At the end only local maxima will survive the procedure.
@@ -3754,6 +4034,9 @@ smoothing
\end_inset
to reduce the effect of noise in the real data sets.
+<<<<<<< HEAD
+ DSI uses a Hanning filter and then avoids sampling from low values in
+=======
DSI
\change_deleted 2 1331657425
is using
@@ -3773,6 +4056,7 @@ ing
s
\change_unchanged
sampling from low values in
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\series bold
r
\series default
@@ -3799,6 +4083,11 @@ key "aganj2010reconstruction"
\begin_layout Standard
All these approaches smooth and calculate the ODFs simultaneously.
+<<<<<<< HEAD
+ We propose something different namely that the ODF can be calculated initially
+ and then smoothed.
+ For example using the operator shown below in matrix form
+=======
We propose something different
\change_inserted 2 1331812844
namely t
@@ -3826,6 +4115,7 @@ or example
following
\change_unchanged
operator shown below in matrix form
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset Formula \[
W=\exp(\frac{U\cdot U^{T}}{\sigma})\]
@@ -3988,6 +4278,10 @@ status collapsed
An example of spherical angular Gaussian smoothing applied with different
smoothing factors on the distribution function of a triple-fibre crossing
on the left.
+<<<<<<< HEAD
+ The simulation was used using a Sticks and Ball model with diffusivity
+ value
+=======
The simulation was used using
\change_inserted 2 1331812952
a
@@ -4005,6 +4299,7 @@ Ball
)
\change_unchanged
model with diffusivity value
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset Formula $0.0015$
\end_inset
@@ -4042,17 +4337,18 @@ name "Flo:Smoothing_Example"
\begin_layout Standard
We can easily see in Fig.
+<<<<<<< HEAD
+=======
\change_deleted 2 1331813237
\change_inserted 2 1331813241
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset space ~
\end_inset
-\change_unchanged
-
\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:Smoothing_Example"
@@ -4063,6 +4359,9 @@ reference "Flo:Smoothing_Example"
\begin_inset Formula $\sigma$
\end_inset
+<<<<<<< HEAD
+, small noisy peaks, as seen in the center of the unsmoothed spherical function,
+=======
\change_inserted 2 1331658462
,
@@ -4077,10 +4376,14 @@ small noisy peaks
\change_inserted 2 1331658497
,
\change_unchanged
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
can be easily removed.
However, with too much smoothing even the longer peaks can lose their definitio
n.
This spherical operator can help to set the trade-off between noise and
+<<<<<<< HEAD
+ signal and it can also simplify the peak finding process, i.e.
+=======
signal and it can also simplify the peak finding process
\change_inserted 2 1331658513
,
@@ -4091,17 +4394,23 @@ n.
\change_inserted 2 1331658613
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset space ~
\end_inset
-
-\change_unchanged
finding the underlying primary fibre directions as this problem is much
easier on smooth surfaces.
\end_layout
\begin_layout Standard
Finally, we believe that by decoupling the smoothing from the reconstruction
+<<<<<<< HEAD
+ step we have an important advantage: we can reduce the effect of the noise
+ to our data more strongly and independently.
+ Many spherical operators can be added as plugins independently of the reconstru
+ction phase, and these can work with any function on the sphere.
+ For example Eq.
+=======
\change_deleted 2 1331679005
phase
@@ -4151,14 +4460,18 @@ can work with any function on the sphere.
For example Eq.
\change_inserted 2 1331658656
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset space ~
\end_inset
+<<<<<<< HEAD
+=======
\change_deleted 2 1331658651
\change_unchanged
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:spherical_gaussian_angular_smoothing"
@@ -4175,6 +4488,10 @@ Comparisons and Results
\begin_layout Standard
Validation of reconstruction and tractography algorithms is not straightforward
due to the lack of relevant gold standards.
+<<<<<<< HEAD
+ Simulated voxels and digital phantoms are a useful way to overcome this
+ difficulty and test new methods.
+=======
Simulated voxels and digital phantoms
\change_inserted 2 1331678900
are
@@ -4182,6 +4499,7 @@ are
is
\change_unchanged
a useful way to overcome this difficulty and test new methods.
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
After the simulation results we also show results with real human data
sets.
\end_layout
@@ -4294,6 +4612,8 @@ S_{i}=S_{0}((1-\sum_{j=1}^{N}f_{j})\exp(-b_{i}d)+\sum_{j=1}^{N}f_{j}\exp(-b_{i}d
\end_inset
+<<<<<<< HEAD
+=======
\change_deleted 2 1331682834
\end_layout
@@ -4303,25 +4623,25 @@ S_{i}=S_{0}((1-\sum_{j=1}^{N}f_{j})\exp(-b_{i}d)+\sum_{j=1}^{N}f_{j}\exp(-b_{i}d
\align block
\change_inserted 2 1331682919
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset ERT
status open
\begin_layout Plain Layout
+<<<<<<< HEAD
+=======
\change_inserted 2 1331682300
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\backslash
noindent
-\change_unchanged
-
\end_layout
\end_inset
-
-\change_unchanged
-where
+ where
\begin_inset Formula $\theta_{ij}$
\end_inset
@@ -4388,6 +4708,11 @@ where
\begin_layout Standard
A comparison method/metric is needed in order to evaluate the new/old reconstruc
+<<<<<<< HEAD
+tion methods discussed in this chapter.
+ The standard procedure is to calculate the similarity between the measured
+ and simulated ground truth data sets.
+=======
tion methods discussed in this
\change_deleted 2 1331679144
document
@@ -4415,6 +4740,7 @@ ground
\change_unchanged
data sets.
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
We want to calculate the angular precision of the ODFs from simulations
derived from eq.
@@ -4434,6 +4760,9 @@ reference "eq:sticks_ball_eq"
\end_inset
when there is no match i.e.
+<<<<<<< HEAD
+ angular distance is ),
+=======
angular distance is
\change_deleted 2 1331679537
maximum (
@@ -4451,12 +4780,11 @@ maximum (
one
\change_deleted 2 1331679563
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset Formula $1$
\end_inset
-
-\change_unchanged
- fibre is matched (
+ when one fibre is matched (
\begin_inset Formula $0^{\circ}$
\end_inset
@@ -4464,6 +4792,13 @@ one
\begin_inset Formula $2$
\end_inset
+<<<<<<< HEAD
+ when two fibres are matched and
+\begin_inset Formula $3$
+\end_inset
+
+ when three fibres are matched.
+=======
when
\change_deleted 2 1331679601
@@ -4489,6 +4824,7 @@ two
three
\change_unchanged
fibres are matched.
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
In table
\begin_inset CommandInset ref
LatexCommand ref
@@ -4751,6 +5087,10 @@ name "Tab:AS_behaviour"
\begin_layout Standard
\align block
+<<<<<<< HEAD
+If our ground truth set consists of
+\begin_inset Formula $g=[(1,0,0),(0,1,0)]=[g_{0},g_{1}]$
+=======
If our
\change_deleted 2 1331680734
@@ -4767,6 +5107,7 @@ ground truth
\change_unchanged
set consists of
\begin_inset Formula $g=[(1,0,0),(0,1,0)]$
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\end_inset
and the measured set consists of
@@ -4779,7 +5120,7 @@ ground truth
.
If the measured set was
-\begin_inset Formula $m=[(0,\nicefrac{\sqrt{2}}{2},\nicefrac{\sqrt{2}}{2})]$
+\begin_inset Formula $m=[(0,\nicefrac{\sqrt{2}}{2},\nicefrac{\sqrt{2}}{2})]=[m_{0}]$
\end_inset
then AS
@@ -4798,6 +5139,8 @@ ground truth
.
This is because according to the AS definition we have
+<<<<<<< HEAD
+=======
\change_deleted 2 1331681582
AS(
\begin_inset Formula $g$
@@ -4818,13 +5161,17 @@ ground truth
.
\change_inserted 2 1331681624
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\end_layout
\begin_layout Standard
\align block
+<<<<<<< HEAD
+=======
\change_inserted 2 1331681632
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\family roman
\series medium
\shape up
@@ -4834,7 +5181,7 @@ ground truth
\noun off
\color none
\lang british
-\begin_inset Formula $AS(g,m)=\max(|g[0]\cdot m[0]|,|g[1],m[0]|)=\nicefrac{\sqrt{2}}{2}.$
+\begin_inset Formula $AS(g,m)=\max(|g_{0}\cdot m_{0}|,|g_{1},m_{0}|)=\nicefrac{\sqrt{2}}{2}.$
\end_inset
@@ -4843,8 +5190,11 @@ ground truth
\begin_layout Standard
\align block
+<<<<<<< HEAD
+=======
\change_inserted 2 1331681737
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\family roman
\series medium
\shape up
@@ -4859,29 +5209,31 @@ status open
\begin_layout Plain Layout
+<<<<<<< HEAD
+=======
\change_inserted 2 1331681737
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\backslash
noindent
-\change_unchanged
-
\end_layout
\end_inset
+<<<<<<< HEAD
+=======
\change_deleted 2 1331681632
\change_inserted 2 1331681743
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset space ~
\end_inset
-
-\change_unchanged
If
-\begin_inset Formula $g=[(1,0,0),(0,1,0)]$
+\begin_inset Formula $g=[(1,0,0),(0,1,0)]=[g_{0},g_{1}]$
\end_inset
and
@@ -4889,6 +5241,8 @@ If
\end_inset
then
+<<<<<<< HEAD
+=======
\change_deleted 2 1331681582
AS(
\begin_inset Formula $g,m$
@@ -4901,13 +5255,17 @@ AS(
.
\change_inserted 2 1331681577
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\end_layout
\begin_layout Standard
\align block
+<<<<<<< HEAD
+=======
\change_inserted 2 1331680918
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\family roman
\series medium
\shape up
@@ -4917,12 +5275,16 @@ AS(
\noun off
\color none
\lang british
-\begin_inset Formula $AS(g,m)=\max(|g[0]\cdot m[0]|+|g[1]\cdot m[1]|,|g[0]\cdot m[1]|+|g[1]\cdot m[0]|)=2.$
+\begin_inset Formula $AS(g,m)=\max(|g_{0}\cdot m_{0}|+|g_{1}\cdot m_{1}|,|g_{0}\cdot m_{1}|+|g_{1}\cdot m_{0})=2.$
\end_inset
+<<<<<<< HEAD
+
+=======
\change_deleted 2 1331681785
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\end_layout
\begin_layout Standard
@@ -4970,11 +5332,15 @@ from
\begin_inset Formula $90^{\circ}$
\end_inset
+<<<<<<< HEAD
+ and then rotate themuniformly around
+=======
and then rotate them
\change_deleted 2 1331813253
\change_unchanged
uniformly around
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset Formula $200$
\end_inset
@@ -4984,17 +5350,18 @@ uniformly around
\end_inset
simulated ODFs and the results are shown in Fig.
+<<<<<<< HEAD
+=======
\change_deleted 2 1331813270
\change_inserted 2 1331813270
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset space ~
\end_inset
-\change_unchanged
-
\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:2_100"
@@ -5067,16 +5434,22 @@ name "Flo:2_100"
\begin_layout Standard
\align block
We can easily observe in Fig.
+<<<<<<< HEAD
+=======
\change_inserted 2 1331813294
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset space ~
\end_inset
+<<<<<<< HEAD
+=======
\change_deleted 2 1331813293
\change_unchanged
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:2_100"
@@ -5084,6 +5457,9 @@ reference "Flo:2_100"
\end_inset
that EITL2 can resolve more accurately fibre crossings at low angles and
+<<<<<<< HEAD
+ continues to perform well even at higher angles
+=======
continue
\change_inserted 2 1331731203
s
@@ -5097,10 +5473,19 @@ perform
ing decently
\change_unchanged
well even at higher angles
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset Formula $>50^{\circ}$
\end_inset
.
+<<<<<<< HEAD
+ EITL performs better than DSI, GQI, GQI2 and EITS at low angles and very
+ well at high angles as well.
+ GQI2 performs better than DSI, GQI, and ETS.
+ It is also impressive that EITS can have such a good performance although
+ it is such a simple operation.
+ In summary we see from the graphs that EITL2
+=======
\change_deleted 2 1331731257
Then
@@ -5131,74 +5516,84 @@ that EITL2
>
\change_inserted 2 1331714795
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset Formula $>$
\end_inset
-
-\change_unchanged
EITL
+<<<<<<< HEAD
+=======
\change_deleted 2 1331714798
>
\change_inserted 2 1331714798
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset Formula $>$
\end_inset
-
-\change_unchanged
GQI2
+<<<<<<< HEAD
+=======
\change_deleted 2 1331714804
>
\change_inserted 2 1331714804
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset Formula $>$
\end_inset
-
-\change_unchanged
DSI
+<<<<<<< HEAD
+=======
\change_deleted 2 1331714807
>
\change_inserted 2 1331714807
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset Formula $>$
\end_inset
-
-\change_unchanged
GQI
+<<<<<<< HEAD
+=======
\change_deleted 2 1331714810
>
\change_inserted 2 1331714810
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset Formula $>$
\end_inset
-
-\change_unchanged
EITS where
+<<<<<<< HEAD
+=======
\change_deleted 2 1331714816
>
\change_inserted 2 1331714816
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset Formula $>$
\end_inset
-
-\change_unchanged
means better average angular similarity.
The same pattern takes place even when we increase the noise level see
for example Fig.
+<<<<<<< HEAD
+=======
\change_inserted 2 1331813305
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset space ~
\end_inset
+<<<<<<< HEAD
+=======
\change_deleted 2 1331813304
\change_unchanged
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:2_20"
@@ -5377,16 +5772,22 @@ name "Flo:3_100"
\begin_layout Standard
\align block
The results of the 3-fibre crossings shown in Fig.
+<<<<<<< HEAD
+=======
\change_inserted 2 1331813312
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset space ~
\end_inset
+<<<<<<< HEAD
+=======
\change_deleted 2 1331813312
\change_unchanged
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:3_100"
@@ -5400,6 +5801,10 @@ reference "Flo:3_20"
\end_inset
+<<<<<<< HEAD
+ were very similar to those of the 2-fibre crossings; EITL2 performed better
+ at low angles with a bit reduced performance at high angles and EITL performed
+=======
were very similar with those of the 2-fibre crossings; EITL2
\change_deleted 2 1331731411
scores higher
@@ -5412,6 +5817,7 @@ doing
\change_inserted 2 1331731433
performed
\change_unchanged
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
better with low angles than the rest of the methods and also having high
accuracy on larger angles.
\end_layout
@@ -5471,6 +5877,15 @@ name "Flo:3_20"
\begin_layout Standard
\align block
+<<<<<<< HEAD
+These summary plots give strong evidence that both DNI (EITL) and in general
+ EIT can be used to accurately generate spherical distribution functions
+ for the determination of the directional information of the diffusion signal
+ and that these can do better or similar to the current state-of-the-art
+ grid-based reconstruction methods i.e DSI and GQI.
+ Also the addition of noise did not affect the determination of the fibre
+ directions considerably.
+=======
These
\change_deleted 2 1331731491
concentrative
@@ -5497,6 +5912,7 @@ t
manage to
\change_unchanged
affect the determination of the fibre directions considerably.
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\end_layout
\begin_layout Standard
@@ -5510,10 +5926,13 @@ Furthermore, we can also see that GQI2 can do better than DSI, GQI and that
\begin_inset Formula $36$
\end_inset
+<<<<<<< HEAD
+=======
\change_deleted 2 1331731558
\change_unchanged
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
, GQI:
\begin_inset Formula $\text{\lambda=1.2}$
\end_inset
@@ -5552,6 +5971,9 @@ Furthermore, we can also see that GQI2 can do better than DSI, GQI and that
\end_layout
\begin_layout Standard
+<<<<<<< HEAD
+In these tests, EIT and fast EIT produced very similar results.
+=======
\change_inserted 2 1331733688
In these tests, EIT and fast EIT produced very similar results
@@ -5560,27 +5982,37 @@ Finally, we should stress that we have never seen any considerable differences
between spherical functions created using the standard or fast EIT
\change_unchanged
.
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
For example a simple test for the
\begin_inset Formula $3$
\end_inset
-fibre case as seen in Fig.
+<<<<<<< HEAD
+=======
\change_inserted 2 1331813327
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset space ~
\end_inset
+<<<<<<< HEAD
+=======
\change_deleted 2 1331813324
\change_unchanged
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:FastvsStandardEITL"
\end_inset
+<<<<<<< HEAD
+ shows that there is close agreement between the two methods i.e.
+=======
\change_deleted 2 1331731903
can
@@ -5595,12 +6027,15 @@ s
\change_inserted 2 1331731917
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset space ~
\end_inset
-
-\change_unchanged
their results are nearly equivalent.
+<<<<<<< HEAD
+ Therefore we can conclude that the fast EIT is an acceptable approximation
+ of the standard EIT.
+=======
Therefore we can conclude that the fast EIT is
\change_inserted 2 1331731874
an acceptable
@@ -5608,6 +6043,7 @@ an acceptable
great
\change_unchanged
approximation of the standard EIT.
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\end_layout
\begin_layout Standard
@@ -5684,6 +6120,9 @@ key "Correia2009"
\end_inset
+<<<<<<< HEAD
+ which supported only paths with analytically calculated derivatives.
+=======
wh
\change_inserted 2 1331734126
ich
@@ -5697,6 +6136,7 @@ semi-circular functions
paths
\change_unchanged
with analytically calculated derivatives.
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\end_layout
\begin_layout Standard
@@ -5812,6 +6252,17 @@ S_{vox}=\sum_{i=1}^{K}\Delta S_{i}\label{eq:digital_phantom_signal}\end{equation
\end_layout
\begin_layout Standard
+<<<<<<< HEAD
+In addition, we can generate simulations of more than one fibre by generating
+ a single volume for every orbit and then add them all together to create
+ complex configurations in the final volume.
+ This is acceptable, under the assumption that the diffusion is Gaussian
+ in all compartments, because the diffusion signal is additive i.e.
+ the signal of a crossing of two fibres is equal to the sum of the the signals
+ of the individual fibres.
+ In this way we can simulate phantoms with Multi Tensor based diffusion
+ signals as that described in eq.
+=======
In addition, we can generate simulations of more than one fibre
\change_deleted 2 1331733268
s
@@ -5833,6 +6284,7 @@ is
\change_unchanged
way we can simulate phantoms with Multi Tensor based diffusion signals
as that described in eq.
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:multitensor"
@@ -5850,6 +6302,12 @@ reference "eq:multitensor"
\begin_layout Standard
The method we use to create these digital phantoms offers the opportunity
to simulate partial volume effects.
+<<<<<<< HEAD
+ If partial volume effects are not desired then we need to normalize by
+ dividing by the number of fibre elements for each voxel (This function
+ is implemented in dipy.sims.phantom.orbital_phantom).
+ In Fig.
+=======
If partial volume effects are not desired then we need to normalize
\change_deleted 2 1331733418
with
@@ -5864,12 +6322,11 @@ by dividing by
\change_inserted 2 1331733432
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset space ~
\end_inset
-\change_unchanged
-
\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:cool_phantoms"
@@ -5959,6 +6416,10 @@ Results with digital phantoms
\begin_layout Standard
With the purpose of comparing and visualizing the differences between the
+<<<<<<< HEAD
+ reconstruction methods described in this chapter a digital phantom of two
+ crossing bundles was created.
+=======
reconstruction methods described in this
\change_deleted 2 1331733480
document
@@ -5974,6 +6435,7 @@ a digital phantom of two crossing bundles
was created
\change_unchanged
.
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
The bundles are crossing at an angle of
\begin_inset Formula $90^{\circ}$
\end_inset
@@ -5982,6 +6444,9 @@ a digital phantom of two crossing bundles
The digital phantom was generated using the method described in the previous
section.
Here we describe the basic steps: (a) We first represented the first bundle
+<<<<<<< HEAD
+ as a discrete straight path starting from point
+=======
as a discrete straight
\change_deleted 2 1331734351
orbit
@@ -5989,6 +6454,7 @@ orbit
path
\change_unchanged
starting from point
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset Formula $(-1,-1,0)$
\end_inset
@@ -6001,6 +6467,10 @@ path
\end_inset
time steps.
+<<<<<<< HEAD
+ (b) We scaled, centred and radially expanded this path so that it fits
+ a volume of size
+=======
(b) We scaled, centred and radially expanded this
\change_deleted 2 1331734363
orbit
@@ -6008,16 +6478,21 @@ orbit
path
\change_unchanged
so that it fits a volume of size
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset Formula $64\times64\times64$
\end_inset
.
This volume corresponds to the diffusion volume without any weighting.
+<<<<<<< HEAD
+ (c) We then applied the weightings for all the following volumes corresponding
+=======
(c)
\change_inserted 2 1331734396
\change_unchanged
We then applied the weightings for all the following volumes corresponding
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
to non-zero b-values.
(d) We replicated the same procedure for the other bundle which initially
started as an orbit from position
@@ -6030,14 +6505,15 @@ We then applied the weightings for all the following volumes corresponding
.
(e) We added the two volumes together to create an 'x' shape (see Fig.
+<<<<<<< HEAD
+=======
\change_inserted 2 1331733544
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset space ~
\end_inset
-\change_unchanged
-
\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:x-shape-thin-tensor"
@@ -6045,14 +6521,15 @@ reference "Flo:x-shape-thin-tensor"
\end_inset
,
+<<<<<<< HEAD
+=======
\change_inserted 2 1331733548
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset space ~
\end_inset
-\change_unchanged
-
\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:x-shape-fat-tensor"
@@ -6088,6 +6565,10 @@ key "Tuch2002ThesisMIT"
\end_inset
.
+<<<<<<< HEAD
+ Two sets of simulation experiments were performed each using a tensors
+ of different shapes.
+=======
Two sets of simulation experiments were performed each using a
\change_inserted 2 1331762769
tensors of
@@ -6099,6 +6580,7 @@ type of Tensor
shapes
\change_unchanged
.
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\end_layout
\begin_layout Standard
@@ -6149,17 +6631,18 @@ Results with an 'x' shape digital phantom.
In this figure we can easily perceive that GQI is very similar to EITS,
GQI2 is very similar to EITL and DSI is very similar to EITL.
In Fig.
+<<<<<<< HEAD
+=======
\change_deleted 2 1331732858
\change_inserted 2 1331732858
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset space ~
\end_inset
-\change_unchanged
-
\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:x-shape-thin-tensor-zoomed"
@@ -6192,17 +6675,18 @@ name "Flo:x-shape-thin-tensor"
\begin_layout Standard
In the first experiment shown in Fig.
+<<<<<<< HEAD
+=======
\change_deleted 2 1331732865
\change_inserted 2 1331732865
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset space ~
\end_inset
-\change_unchanged
-
\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:x-shape-thin-tensor"
@@ -6210,14 +6694,15 @@ reference "Flo:x-shape-thin-tensor"
\end_inset
,
+<<<<<<< HEAD
+=======
\change_inserted 2 1331732877
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset space ~
\end_inset
-\change_unchanged
-
\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:x-shape-thin-tensor-zoomed"
@@ -6243,7 +6728,10 @@ reference "Flo:x-shape-thin-tensor-zoomed"
.
In the second experiment shown in Fig.
-
+\begin_inset space ~
+\end_inset
+
+
\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:x-shape-fat-tensor"
@@ -6316,6 +6804,12 @@ reference "Flo:x-shape-fat-tensor"
\end_layout
\begin_layout Standard
+<<<<<<< HEAD
+Furthermore, we can easily see that GQI is mostly similar to EITS, GQI2
+ is very similar to EITL and DSI is mostly similar to EITL.
+ That DSI ODFs are very similar to EITL ODFs is to be expected as the two
+ methods create theoretically the same real ODFs.
+=======
Furthermore, we can easily see that GQI is mostly similar
\change_inserted 2 1331801109
to
@@ -6346,6 +6840,7 @@ to
to be
\change_unchanged
expected as the two methods create theoretically the same real ODFs.
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
Remarkably, EITL can create these ODFs without using the Fourier Transform
neither using any filter or thresholds in r-space which are necessary in
DSI.
@@ -6374,16 +6869,22 @@ status collapsed
\begin_layout Plain Layout
Same as previous Fig.
+<<<<<<< HEAD
+=======
\change_inserted 2 1331813361
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset space ~
\end_inset
+<<<<<<< HEAD
+=======
\change_deleted 2 1331813361
\change_unchanged
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:x-shape-thin-tensor"
@@ -6442,16 +6943,6 @@ Showing the spherical distribution functions (DSI, GQI, GQI2, EITL, EITL2,
single tensors along the direction of the phantom.
On the crossing area we see a dual tensor effect in every voxel.
Every single tensor compartment had the following eigenvalues
-\begin_inset Note Note
-status collapsed
-
-\begin_layout Plain Layout
- 0.0014,0.0001, 0.0001
-\end_layout
-
-\end_inset
-
-
\begin_inset Formula $\lambda_{\parallel}=1.7\times10^{-3}$
\end_inset
@@ -6502,16 +6993,22 @@ name "Flo:x-shape-fat-tensor"
\begin_layout Standard
Fig.
+<<<<<<< HEAD
+=======
\change_inserted 2 1331813390
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset space ~
\end_inset
+<<<<<<< HEAD
+=======
\change_deleted 2 1331813388
\change_unchanged
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:x-shape-thin-tensor"
@@ -6558,16 +7055,22 @@ status collapsed
\begin_layout Plain Layout
A zoomed version of previous Fig.
+<<<<<<< HEAD
+=======
\change_inserted 2 1331813406
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset space ~
\end_inset
+<<<<<<< HEAD
+=======
\change_deleted 2 1331813405
\change_unchanged
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:x-shape-fat-tensor"
@@ -6619,6 +7122,11 @@ reference "Flo:x-shape-fat-tensor-zoomed"
we do not use any amount of smoothing as used in DSI (through hanning filter),
GQI, GQI2 (through sampling length) and it is extraordinary that we obtain
so well defined distributions.
+<<<<<<< HEAD
+ If we want to apply some weighting / smoothing / denoising in EIT-based
+ methods that is simply possible through the spherical angular smoothing
+ approach described in section
+=======
If we want to apply some weighting
\change_inserted 2 1331714996
@@ -6637,6 +7145,7 @@ smoothing
\change_unchanged
denoising in EIT-based methods that is simply possible through the spherical
angular smoothing approach described in section
+>>>>>>> 328a3c168e7f04efe7152640a044d11debeed4f2
\begin_inset CommandInset ref
LatexCommand ref
reference "sub:Spherical-Angular-Smoothing"
@@ -6686,6 +7195,9 @@ Results with real data sets
\end_layout
\begin_layout Standard
+<<<<<<< HEAD
+We want to compare reconstruction methods on Cartesian grid-based acquisitions
+=======
\change_deleted 2 1331801324
Apparently w