Appendix.lyx

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\begin_body

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
appendix
\end_layout

\end_inset


\end_layout

\begin_layout Section
Appendix
\end_layout

\begin_layout Subsection
Data and sequences
\end_layout

\begin_layout Standard
Summary of the acquisition protocols used in this thesis.
\begin_inset Newline newline
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
noindent
\end_layout

\end_inset

 
\series bold
101 and 117 gradients at MRC-CBU
\series default
.
 3T scanner (TIM Trio, Siemens) with a 12 channel coil, using Siemens advanced
 diffusion work-in-progress sequence, and STEAM 
\begin_inset CommandInset citation
LatexCommand cite
key "merboldt1992diffusion,MAB04"

\end_inset

 as the diffusion preparation method.
 The field of view was 
\begin_inset Formula $240\times240\, mm^{2}$
\end_inset

, matrix size 
\begin_inset Formula $96x96$
\end_inset

, and slice thickness 
\begin_inset Formula $2.5mm$
\end_inset

 (no gap).
 55 slices were acquired to achieve full brain coverage, and the voxel resolutio
n was 
\begin_inset Formula $2.5\times2.5\times2.5\, mm{}^{3}$
\end_inset

.
 Two sampling schemes were considered: a 
\begin_inset Formula $102$
\end_inset

-point half grid acquisition(TR=
\begin_inset Formula $8,200\, ms$
\end_inset

, TE=
\begin_inset Formula $69\, ms$
\end_inset

) with a maximum b-value of 
\begin_inset Formula $4,000\, s/mm^{2}$
\end_inset

, and a single shell acquisition using 
\begin_inset Formula $118$
\end_inset

 non-collinear gradient directions (TR=
\begin_inset Formula $7,000\, ms$
\end_inset

, TE=
\begin_inset Formula $47ms$
\end_inset

) and a b-value of 
\begin_inset Formula $1,000\, s/mm^{2}$
\end_inset

.
 The two acquisition schemes were matched for total acquisition time (
\begin_inset Formula $14\, min\,37s$
\end_inset

), voxel resolution, and bandwidth.
\begin_inset Newline newline
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
noindent
\end_layout

\end_inset

 
\series bold
101 gradients at MRC-CBU
\series default
.
 3T scanner (TIM Trio, Siemens) with a 32 channel coil, using Siemens advanced
 diffusion work-in-progress sequence, and STEAM 
\begin_inset CommandInset citation
LatexCommand cite
key "merboldt1992diffusion,MAB04"

\end_inset

 as the diffusion preparation method.
 The field of view was 
\begin_inset Formula $240\times240\, mm^{2}$
\end_inset

, matrix size 
\begin_inset Formula $96\times96$
\end_inset

, and slice thickness 
\begin_inset Formula $2.5\, mm$
\end_inset

 (no gap).
 55 slices were acquired to achieve full brain coverage, and the voxel resolutio
n was 
\begin_inset Formula $2.5\times2.5\times2.5\, mm{}^{3}$
\end_inset

.A 102-point half grid acquisition with a maximum b-value of 
\begin_inset Formula $4,000\, s/mm^{2}$
\end_inset

 was used.
 The total acquisition time was
\begin_inset Formula $14\, min\,21s$
\end_inset

 with TR=
\begin_inset Formula $8,200ms$
\end_inset

 and TE=
\begin_inset Formula $69ms$
\end_inset

.
\begin_inset Newline newline
\end_inset


\series bold
 
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
noindent
\end_layout

\end_inset

 
\series bold
257 gradients at EPFL
\series default
.
 3T scanner (TIM Trio, Siemens) with a 32 channel coil.
 The field of view was 
\begin_inset Formula $210\times210\, mm^{2}$
\end_inset

, matrix size 
\begin_inset Formula $96\times96$
\end_inset

, and slice thickness 
\begin_inset Formula $3\, mm$
\end_inset

.
 44 slices were acquired and the voxel resolution was 
\begin_inset Formula $2.2\times2.2\times3.0\, mm{}^{3}$
\end_inset

.A 258-point half grid acquisition scheme with a maximum b-value of 
\begin_inset Formula $8011\, s/mm^{2}$
\end_inset

 (DSI515) was used.
 The total acquisition time was 
\begin_inset Formula $34\, min$
\end_inset

 with TR=
\begin_inset Formula $8200\, ms$
\end_inset

 and TE=
\begin_inset Formula $165\, ms$
\end_inset

.
\end_layout

\begin_layout Subsection
The Cosine Transform 
\begin_inset CommandInset label
LatexCommand label
name "sub:The-cosine-transform"

\end_inset


\end_layout

\begin_layout Standard
Here we provide a simple derivation of the cosine transform:
\begin_inset Formula $\int_{0}^{\infty}\cos(st)g(t)dt$
\end_inset

 where 
\begin_inset Formula $g(t)$
\end_inset

 defined on 
\begin_inset Formula $[t,\infty)$
\end_inset

.
\end_layout

\begin_layout Standard
Let 
\begin_inset Formula $f(t)$
\end_inset

 be an even function, 
\begin_inset Formula $f(t)=f(-t)$
\end_inset

, defined for 
\begin_inset Formula $-\infty<t<\infty$
\end_inset

.
 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{eqnarray*}
F(s) & = & \int_{-\infty}^{\infty}f(t)e^{its}dt\\
 & = & \int_{0}^{\infty}f(t)e^{its}dt+\int_{-\infty}^{0}f(t)e^{its}dt\\
 & = & \int_{0}^{\infty}f(t)e^{its}dt-\int_{-\infty}^{0}f(-t)e^{-its}dt\\
 & = & \int_{0}^{\infty}f(t)e^{its}dt+\int_{0}^{\infty}f(t)e^{-its}dt\\
 & = & \int_{0}^{\infty}f(t)[e^{its}+e^{-its}]dt\\
 & = & \int_{0}^{\infty}f(t)[\cos(its)+i\sin(its)+\cos(its)-i\sin(its)]dt\\
 & = & 2\int_{0}^{\infty}f(t)cos(st)dt
\end{eqnarray*}

\end_inset


\begin_inset Formula $\noindent$
\end_inset


\begin_inset space ~
\end_inset

In the third row above we replaced 
\begin_inset Formula $t\rightarrow-t$
\end_inset

 in the second integral.
 If we want to express the Fourier transform of a symmetric function as
 an integral over the whole space we have 
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\begin_inset Formula $F(s)=\int_{-\infty}^{\infty}f(t)\cos(st)dt$
\end_inset

.
\end_layout

\begin_layout Subsection
Fourier Transform of 
\begin_inset Formula $P(\mathbf{r})r^{2}$
\end_inset


\begin_inset CommandInset label
LatexCommand label
name "sub:Fourier-transform-ofPr2"

\end_inset


\end_layout

\begin_layout Standard
From Fourier analysis we know that if 
\begin_inset Formula $E(\mathbf{q})$
\end_inset

 is the Fourier transform function of 
\begin_inset Formula $P(\mathbf{r})$
\end_inset

 then
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{eqnarray*}
\mathfrak{F\mathrm{(xP(\mathbf{r}))}} & = & i\frac{\partial E(\mathbf{q})}{\partial\mathbf{q}_{x}}\\
\mathfrak{F\mathrm{(x^{2}P(\mathbf{r}))}} & = & -\frac{\partial^{2}E(\mathbf{q})}{\partial\mathbf{q}_{x}^{2}}
\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\align left
where 
\begin_inset Formula $\mathfrak{F\mathrm{()}}$
\end_inset

 is the Fourier transform.
 By writing the second equation for 
\begin_inset Formula $y$
\end_inset

 and 
\begin_inset Formula $z$
\end_inset

 and summing them all together we obtain 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{eqnarray*}
\mathfrak{F\mathrm{(r^{2}P(\mathbf{r}))}} & = & -\frac{\partial^{2}E(\mathbf{q})}{\partial\mathbf{q}_{x}^{2}}-\frac{\partial^{2}E(\mathbf{q})}{\partial\mathbf{q}_{y}^{2}}-\frac{\partial^{2}E(\mathbf{q})}{\partial\mathbf{q}_{z}^{2}}=-\nabla^{2}E(\mathbf{q})
\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Subsection
Radial projection of a symmetric function 
\begin_inset CommandInset label
LatexCommand label
name "sub:Radial-projection-of-symmetric"

\end_inset


\end_layout

\begin_layout Standard
Let 
\begin_inset Formula $f:\mathbb{R}^{3}\rightarrow\mathbb{R}$
\end_inset

 be a symmetric function with the 3D Fourier transform function 
\begin_inset Formula $\hat{f}(\mathbf{q})$
\end_inset

 and 
\begin_inset Formula $\hat{\mathbf{u}}$
\end_inset

 be an arbitrary unit vector.
 We will show that 
\begin_inset Formula $\intop_{0}^{\infty}f(r\hat{\mathbf{u}})dr=\frac{1}{8\pi^{2}}\int\int_{\hat{\mathbf{u}}^{\bot}}\hat{f}(\mathbf{q})qdqd\phi$
\end_inset

 where 
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\begin_inset Formula $\hat{\mathbf{u}}^{\bot}$
\end_inset

 is the plane perpendicular to 
\family default
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\color inherit

\begin_inset Formula $\hat{\mathbf{u}}$
\end_inset

.
\end_layout

\begin_layout Standard
Without loss of generality, we align 
\begin_inset Formula $\hat{\mathbf{u}}$
\end_inset

with the z-axis having 
\begin_inset Formula $\hat{\mathbf{z}}=\hat{\mathbf{u}}$
\end_inset

.
 Using the Dirac delta function (make use of Lebesgue integral) we can now
 write
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{eqnarray*}
\int_{0}^{\infty}f(r\hat{\mathbf{z})}dr & = & \int_{0}^{\infty}f(0,0,z)dz\\
 & = & \frac{1}{2}\int\int\int_{\mathbb{R}^{3}}f(x,y,z)\delta(x)\delta(y)dxdydz
\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
The factor 
\begin_inset Formula $1/2$
\end_inset

 is required because we need the integral only in the positive half of the
 z-axis, and the function is symmetric.
 Let us define 
\begin_inset Formula $g(x,y,z)\equiv\delta(x)\delta(y)$
\end_inset

.
 For the two functions 
\begin_inset Formula $f,g:\mathbb{R}^{3}\rightarrow\mathbb{R}$
\end_inset

 with Fourier transform functions 
\begin_inset Formula $\hat{f}(\mathbf{q})$
\end_inset

 and 
\begin_inset Formula $\hat{g}(\mathbf{q})$
\end_inset

, Parseval's theorem states that 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{eqnarray*}
\int\int\int_{\mathbb{R}^{3}}f(x,y,z)g^{*}(x,y,z)dxdydz & =\quad\quad\quad\quad\quad\quad\quad\quad
\end{eqnarray*}

\end_inset


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\lang british

\begin_inset Formula 
\begin{eqnarray*}
(2\pi)^{-3}\int\int\int_{\mathbb{R}^{3}}f(q_{x},q_{y},q_{z})\hat{g}^{*}(q_{x},q_{y},q_{z})dq_{x}dq_{y}dq_{z}
\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
Furthermore, 
\begin_inset Formula $\hat{g}(q_{x},q_{y},q_{z})=2\pi\delta(q_{z})$
\end_inset

 and replacing it in the above equations leads to 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{eqnarray*}
\int_{0}^{\infty}f(r\hat{\mathbf{z})} & = & \int\int\int_{\mathbb{R}^{3}}\frac{1}{2}f(x,y,z)g(x,y,z)dxdydz\\
 & = & \frac{1}{2(2\pi)^{3}}\int\int\int_{\mathbb{R}^{3}}f(q_{x},q_{y},q_{z})2\pi\delta(q_{z})dq_{x}dq_{y}dq_{z}\\
 & = & \frac{1}{8\pi^{2}}\int_{-\infty}^{\infty}\hat{f}(q_{x},q_{y},0)dq_{x}dq_{y}
\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Subsection
The Tensor in GQI
\begin_inset CommandInset label
LatexCommand label
name "sub:Tensor-in-GQI"

\end_inset


\end_layout

\begin_layout Standard
We now apply this formulation under the assumption that the diffusion voxel
 can be represented by a single tensor model.
 Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "eq:W"

\end_inset

 can be written in the form
\begin_inset Formula 
\begin{eqnarray}
S(\mathbf{q}) & = & S_{0}exp(-b\mathbf{q}^{T}D\mathbf{q})\label{eq:Tensor}
\end{eqnarray}

\end_inset


\end_layout

\begin_layout Standard
where 
\begin_inset Formula $S_{0}$
\end_inset

 is the image when b-value is equal to 0, 
\begin_inset Formula $b$
\end_inset

 is the b-value for a specific direction and 
\begin_inset Formula $D$
\end_inset

 is a 
\begin_inset Formula $3x3$
\end_inset

 matrix, known as the diffusion tensor.
 Then from Eq.
\begin_inset space ~
\end_inset

 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:Q2S_complex"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:Tensor"

\end_inset

 the Fourier transform of 
\begin_inset Formula $S$
\end_inset

 is equal to 
\end_layout

\begin_layout Standard
\align center
\begin_inset Formula 
\begin{eqnarray}
Q(\mathbf{R}) & = & \int S_{0}\exp(-b\mathbf{q}^{T}D\mathbf{q})\exp(-j2\pi\mathbf{q}\cdot\mathbf{R})d\mathbf{q}\label{eq:FourierW}
\end{eqnarray}

\end_inset


\end_layout

\begin_layout Standard
The same equation in its triple integral form can be written as 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{eqnarray}
Q(R) & =S_{0} & \iiint\exp(-b\sum_{i=1}^{3}q_{i}^{2}\lambda_{i}-j2\pi\sum_{i=1}^{3}q_{i}R_{i})dq_{1}dq_{2}dq_{3}\nonumber \\
 & =S_{0} & \iiint\prod_{i=1}^{3}exp(-bq_{i}^{2}\lambda_{i}-j2\pi q_{i}R_{i})dq_{1}dq_{2}dq_{3}\nonumber \\
 & =S_{0} & \prod_{i=1}^{3}\int\exp(-bq_{i}^{2}\lambda_{i}-j2\pi q_{i}R_{i})dq_{i}\nonumber \\
 & =S_{0} & \prod_{i=1}^{3}\int\exp(-b\lambda_{i}[q_{i}^{2}+\frac{j2\pi R_{i}}{b\lambda_{i}}q_{i}])dq_{i}\nonumber \\
 & =S_{0} & \prod_{i=1}^{3}\int\exp\{-b\lambda_{i}[(q_{i}+\frac{j\pi R_{i}}{b\lambda_{i}})^{2}+\frac{\pi^{2}R_{i}^{2}}{b^{2}\lambda_{i}^{2}}]\}\nonumber \\
 & =S_{0} & \prod_{i=1}^{3}\int\exp\{-b\lambda_{i}(q_{i}+\frac{j\pi R_{i}}{b\lambda_{i}})^{2}\}exp\{-\frac{\pi^{2}R_{i}^{2}}{b\lambda_{i}}\}\label{eq:NextQ}
\end{eqnarray}

\end_inset


\end_layout

\begin_layout Standard
In that stage we can make use of the formula 
\begin_inset Formula $\int\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp(-\frac{(x-\mu)^{2}}{2\sigma^{2}})dx=1$
\end_inset

.
 Now we can see that 
\begin_inset Formula $b\lambda_{i}=1/2\sigma^{2}$
\end_inset

 and 
\begin_inset Formula $\mu$
\end_inset

 corresponds to 
\begin_inset Formula $\mu=-jR_{i}/b\lambda_{i}$
\end_inset

.
 Therefore, Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "eq:NextQ"

\end_inset

 can now be written as 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{eqnarray}
Q(\mathbf{R}) & =S_{0} & \prod_{i=1}^{3}\sqrt{\frac{\pi}{b\lambda_{i}}}\exp(-\frac{\pi^{2}R_{i}^{2}}{b\lambda_{i}})\nonumber \\
 & =S_{0} & \left(\frac{\pi}{b}\right)^{3/2}\frac{1}{\sqrt{\prod_{i=1}^{3}\lambda_{i}}}\exp(-\frac{\,\,\pi^{2}}{b}\mathbf{R}^{T}D^{-1}\mathbf{R})\label{eq:TensorQ}
\end{eqnarray}

\end_inset


\begin_inset Formula $\noindent$
\end_inset


\begin_inset space ~
\end_inset

where 
\begin_inset Formula $D$
\end_inset

 is the diffusion tensor.
 We can replace the displacement vector 
\begin_inset Formula $\mathbf{R}$
\end_inset

 with a scalar value 
\begin_inset Formula $L$
\end_inset

 and a unit vector 
\begin_inset Formula $\hat{u}$
\end_inset

 i.e.
 
\begin_inset Formula $\mathbf{R}=L\hat{\mathbf{u}}$
\end_inset

 and from Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "eq:TensorQ"

\end_inset

 we can replace 
\begin_inset Formula $\frac{2\pi^{2}}{b}\mathbf{\hat{u}}^{T}D^{-1}\mathbf{\hat{u}}$
\end_inset

 with 
\begin_inset Formula $k$
\end_inset

 and 
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\begin_inset Formula $S_{0}\left(\frac{\pi}{b}\right)^{3/2}\frac{1}{\lambda_{1}\lambda_{2}\lambda_{3}}$
\end_inset


\family default
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\color inherit
\lang english
 with 
\begin_inset Formula $\alpha$
\end_inset

.
 Using that last change of variables we can now write 
\begin_inset Formula 
\begin{eqnarray}
\psi_{Q}(\mathbf{r},\mathbf{\hat{u}}) & = & \intop_{0}^{L_{\Delta}}Q(\mathbf{r},L\mathbf{\hat{u}})dL\nonumber \\
 & = & \alpha\intop_{0}^{L_{\Delta}}\exp(-L^{2}\frac{k}{2})dL\label{eq:TensorQ2}
\end{eqnarray}

\end_inset


\end_layout

\begin_layout Standard
Setting 
\begin_inset Formula $m=\sqrt{k}L$
\end_inset

 and using the derivation for the error function Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "eq:TensorQ2"

\end_inset

 illustrates the remarkable result that we can calculate analytically the
 spin ODF for Gaussian diffusion using the cumulative distribution function
 
\begin_inset Formula $CDF$
\end_inset

.
 
\begin_inset Newline newline
\end_inset


\begin_inset Formula 
\begin{eqnarray}
\psi_{Q}(\mathbf{\hat{u}}) & = & \frac{\alpha}{\sqrt{k}}\intop_{0}^{\sqrt{k}L_{\Delta}}e^{-m^{2}/2}dm\\
 & = & \alpha\sqrt{\frac{2\pi}{k}}\left[CDF(\sqrt{k}L_{\Delta})-\frac{1}{2}\right]\label{eq:spinodf_cdf}
\end{eqnarray}

\end_inset


\end_layout

\begin_layout Standard
This can be used as a check to compare the approximated or sampled spin
 ODF that is derived in 
\begin_inset CommandInset citation
LatexCommand cite
key "Yeh2010"

\end_inset

 with Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "eq:spinodf_cdf"

\end_inset

 for the case of gaussian diffusion.
 
\end_layout

\begin_layout Standard
What is also very interesting is to try to derive what the normalization
 factor should be for the spin ODF in Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "eq:TensorQ2"

\end_inset

.
 Because calculating a spherical integral from Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "eq:spinodf_cdf"

\end_inset

 seems at the moment very complicated we first work with the simpler gaussian
 diffusion ODF derived by Tuch 
\begin_inset CommandInset citation
LatexCommand cite
key "Tuch2004"

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\psi_{p_{\Delta}}=\frac{1}{Z}\sqrt{\frac{\pi\tau}{\mathbf{u}^{T}D^{-1}\mathbf{u}}}\label{eq:tuchs_gaussian_odf}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{eqnarray}
\frac{Z}{\sqrt{\pi\tau}} & = & \iintop_{S^{2}}(\mathbf{u}^{T}D^{-1}\mathbf{u})^{-\frac{1}{2}}d\mathbf{u}\label{eq:spherical_integral_Z}
\end{eqnarray}

\end_inset


\end_layout

\begin_layout Standard
Let us now define 
\begin_inset Formula $f_{D}(\mathbf{u})=(\mathbf{u}^{T}D^{-1}\mathbf{u})^{-\frac{1}{2}}$
\end_inset

.
 From 
\begin_inset CommandInset citation
LatexCommand cite
key "olver2010nist"

\end_inset

 (19.31.2) we know that we can calculate the following integral on the entire
 space (eq.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:hole_space_known"

\end_inset

).
 By expanding it in polar form we can find the surface integral required
 in Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "eq:spherical_integral_Z"

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{eqnarray}
MHG & = & \iiint f_{D}(x)e^{-|x|^{2}}dx\label{eq:hole_space_known}\\
 & = & \intop_{0}^{\infty}\left[\iint f_{D}(ru)du\right]e^{-r^{2}}r^{2}dr,\qquad x=ru\:(polar)\label{eq:hole_space_polar}
\end{eqnarray}

\end_inset


\end_layout

\begin_layout Standard
However, we know that 
\begin_inset Formula $f_{D}(r\mathbf{u})=((r\mathbf{u})^{T}D^{-1}(r\mathbf{u}))^{-\frac{1}{2}}=r^{-1}(\mathbf{u}^{T}D^{-1}\mathbf{u})^{-\frac{1}{2}}$
\end_inset

 .
 Therefore, 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{eqnarray*}
MHG & = & \intop_{0}^{\infty}\left[\iint f_{D}(u)du\right]e^{-r^{2}}rdr\\
 & = & \iint f_{D}(u)du\intop_{0}^{\infty}re^{-r^{2}}dr\\
 & = & \frac{1}{2}\iint f_{D}(u)du
\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
Consequently, 
\begin_inset Formula $\frac{Z}{\sqrt{\pi\tau}}=2MHG$
\end_inset

 where MHG is the multivariate hyper-geometric function with 
\begin_inset Formula $\mu=-\frac{1}{2}$
\end_inset

,
\begin_inset Formula $B=I$
\end_inset

,
\begin_inset Formula $n=3$
\end_inset

 and 
\begin_inset Formula $\lambda_{1},\lambda_{2},\lambda_{3}$
\end_inset

 the eigenvalues of 
\begin_inset Formula $\mathbf{A}$
\end_inset

 derived from 
\begin_inset CommandInset citation
LatexCommand cite
key "olver2010nist"

\end_inset

 (19.31.2) & (19.16.9).
 Therefore, 
\begin_inset Formula 
\begin{eqnarray*}
\frac{Z}{\sqrt{\pi\tau}} & = & 2MHG\\
 & = & \frac{2\pi^{\frac{3}{2}}\Gamma(1)}{\sqrt{det(I)}\Gamma(\frac{3}{2})}R_{-\frac{1}{2}}(\frac{1}{2},\frac{1}{2},\frac{1}{2};\lambda_{1},\lambda_{2},\lambda_{3})
\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
and 
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none

\begin_inset Formula $R_{-\frac{1}{2}}=\frac{1}{2}\intop_{0}^{\infty}t^{0}(t+\lambda_{1})^{-\frac{1}{2}}(t+\lambda_{2})^{-\frac{1}{2}}(t+\lambda_{3})^{-\frac{1}{2}}dt$
\end_inset

 with 
\begin_inset Formula $\alpha=\frac{1}{2}$
\end_inset

 and 
\begin_inset Formula $\alpha'=1$
\end_inset

.
\end_layout

\begin_layout Standard
Given 
\begin_inset Formula $\lambda_{1},\lambda_{2},\lambda_{3}$
\end_inset

 we can integrate numerically or even possibly analytically.
 For the isotropic case the integral simplifies to 
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none

\begin_inset Formula $\frac{1}{2}\intop_{0}^{\infty}t^{0}(t+\lambda)^{-\frac{3}{2}}dt=\frac{1}{\sqrt{\lambda}}$
\end_inset

 and for the cylindrical case (
\begin_inset Formula $\lambda_{2}=\lambda_{3}$
\end_inset

) to 
\begin_inset Formula $\frac{1}{2}\intop_{0}^{\infty}(t+\lambda_{1})^{-\frac{1}{2}}(t+\lambda_{2})^{-1}dt$
\end_inset

.
\end_layout

\begin_layout Subsection
Affinity Propagation
\begin_inset CommandInset label
LatexCommand label
name "sub:Affinity-Propagation"

\end_inset


\end_layout

\begin_layout Standard
Affinity propagation (AP) is a recent 
\begin_inset Formula $O(N^{2})$
\end_inset

 clustering method invented by Frey et al.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "frey2007clustering"

\end_inset

 and Dueck et al.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "dueck2009affinity"

\end_inset

 which is inspired by loopy belief propagation 
\begin_inset CommandInset citation
LatexCommand cite
key "pearl1988probabilistic"

\end_inset

 and other recent innovations in graphical models and more specifically
 is an instance of the max-sum algorithm in factor graphs.
 For the completeness of this thesis and because AP is a relatively new
 algorithm we give a short description of the AP in this section.
 AP is an exemplar based clustering method where the center of a cluster
 is a real data point (exemplar) as in k-medoids, and k-centres rather than
 an average virtual point as in k-means.
 AP starts by simultaneously considering all data points as potential exemplars.
 Each data point is a node in a network and AP recursively transmits real-valued
 messages along the edges of the network until a good set of exemplars and
 corresponding clusters emerges.
 
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout

[th!]
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
	filename last_figures/affinity_propagation_ok2.png
	lyxscale 40
	scale 40
	rotateOrigin center

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Simple example of affinity propagation (AP) at work where it can precisely
 identify 
\begin_inset Formula $4$
\end_inset

 different normal distributions with means 
\begin_inset Formula $(1,1),\;(-1,-1),\:(1,-1),\,(-2,2)$
\end_inset

 and standard deviation 
\begin_inset Formula $0.5$
\end_inset

.
 You can see the exemplars - most representative actual points - with thicker
 dots perfectly aligned with the means.
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "Fig:AP_2d"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
AP takes as input a collection of similarities between data points, where
 the similarity 
\begin_inset Formula $S(i,k)$
\end_inset

 indicates how well the data point with index 
\begin_inset Formula $k$
\end_inset

 is suited to be the exemplar for data point 
\begin_inset Formula $i$
\end_inset

.
 In order to understand AP we can think just for the moment that we try
 to cluster 2D data points and each similarity is expressed as the negative
 Euclidean distance 
\begin_inset Formula $S(i,k)=-||x_{i}-x_{k}||^{2}$
\end_inset

 (see Fig.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Fig:AP_2d"

\end_inset

) therefore 
\begin_inset Formula $S$
\end_inset

 for the moment is the negative complete squared distance matrix.
 Rather than requiring the number of clusters to be prespecified, AP adds
 a real number (preference weights) to the diagonal elements of 
\begin_inset Formula $S$
\end_inset

, one for each data point so that larger values of 
\begin_inset Formula $S(k,k)$
\end_inset

 are more likely to become exemplars.
 For simplicity, we can choose the 
\begin_inset Formula $median(S)$
\end_inset

 as the common preference weight for all points; in this way we do not enforce
 any 
\emph on
a priori
\emph default
 information for one point to be an exemplar any more than any other point.
 For some applications this could be an appropriate requirement.
 There are two different messages exchanged between points: (1) responsibilities
 
\begin_inset Formula $R(i,k)=S(i,k)-{\displaystyle \max_{k':k'\neq k}}[S(i,k')+A(i,k')]$
\end_inset

 and (2) availabilities which are 
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
initially 
\begin_inset Formula $A(i,k)=0$
\end_inset

 and then equal to
\family default
\series default
\shape default
\size default
\emph default
\bar default
\noun default
\color inherit

\begin_inset Formula 
\begin{equation}
\forall i,k:\: A(i,k)=\begin{cases}
{\displaystyle \sum_{i':i'\neq i}} & \max[0,\: R(i',k)],\: for\:\: k=i\\
\min & \left[0,\: r(k,k)+{\displaystyle \sum\max_{i':i'\notin\{i,k\}}[0,r(i',k)]}\right],\: for\:\: k\neq i
\end{cases}
\end{equation}

\end_inset

A very interesting fact is the way we get the final exemplars using AP.
 After the messages have converged, there are two ways you can identify
 exemplars: 
\end_layout

\begin_layout Enumerate
For data point 
\begin_inset Formula $i$
\end_inset

, 
\begin_inset Formula $if\:\: R(i,i)+A(i,i)>0$
\end_inset

, then data point 
\begin_inset Formula $i$
\end_inset

 is an exemplar.
 
\end_layout

\begin_layout Enumerate
For data point 
\begin_inset Formula $i$
\end_inset

, 
\begin_inset Formula $if\:\: R(i,i)+A(i,i)>R(i,j)+A(i,j)$
\end_inset

, for all 
\begin_inset Formula $i$
\end_inset

 not equal to 
\begin_inset Formula $j$
\end_inset

, then data point 
\begin_inset Formula $i$
\end_inset

 is an exemplar.
\end_layout

\begin_layout Standard
Therefore, the availabilities and responsibilities are added to identify
 exemplars.
 For point 
\begin_inset Formula $i$
\end_inset

, the value of 
\begin_inset Formula $k$
\end_inset

 that maximizes 
\begin_inset Formula $A(i,k)+R(i,k)$
\end_inset

 either identifies 
\begin_inset Formula $i$
\end_inset

 as an exemplar if 
\begin_inset Formula $k=i$
\end_inset

, or identifies the data point that is the exemplar for point 
\begin_inset Formula $i$
\end_inset

.
 The message passing procedure is terminated either after a fixed number
 of iterations, or after changes in the messages stay low, or local decisions
 stay constant; also the messages are damped - combining previous with current
 message - to avoid numerical oscillations.
 
\end_layout

\begin_layout Standard
Of course, when we need to calculate distances between many points then
 the distance matrix becomes too big for the available memory.
 In that case, if we are lucky and the data sets are sparse then we can
 use AP on sparse matrices.
 When the data sets are not sparse, as it is the case with tractographies,
 we need to reduce the dimensionality of the data sets and this is why QB
 can be very handy.
 The complete algorithm for AP is given in Alg.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "alg:AP"

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Float algorithm
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout

[th!]
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
textbf{Input} Similarity/affinity matrix $S$ where the diagonal elements
 of $S(k,k)$ indicate the a priori preference for $k$ to be chosen as an
 exemplar 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
textbf{Output} Clustering $CAP 
\backslash
leftarrow 
\backslash
{c_{1},...,c_{k},...,c_{|CAP|}
\backslash
}$, where a cluster $c 
\backslash
leftarrow (I,
\backslash
mathbf{e},N) $
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

$
\backslash
forall i,k:
\backslash
: A(i,k)
\backslash
leftarrow R(i,k) 
\backslash
leftarrow 0$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

$S 
\backslash
leftarrow S+n$ 
\backslash
# remove degeneracies
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

$d 
\backslash
leftarrow 0.5$ 
\backslash
# set damping factor
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

last
\backslash
_iter $
\backslash
leftarrow$ $100$ 
\backslash
# last iteration 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

$
\backslash
textbf{For}$ iter $=1$ to last
\backslash
_iter $
\backslash
textbf{Do}$ 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{2em} $R_{old} 
\backslash
leftarrow R$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{2em} $
\backslash
forall i,k:
\backslash
: R(i,k) 
\backslash
leftarrow S(i,k)-max_{k':k'
\backslash
neq k}[S(i,k')+A(i,k')]$ 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{2em} $R 
\backslash
leftarrow (1-d)R+d*R_{old}$ 
\backslash
# dampen responsibilities 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{2em} $A_{old} 
\backslash
leftarrow A$ 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{2em} 
\backslash
# update availabilities 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{2em} $
\backslash
forall i,k:
\backslash
: A(i,k) 
\backslash
leftarrow 
\backslash
begin{cases} 
\backslash
sum_{i':i'
\backslash
neq i} & max[0,
\backslash
: R(i',k)],
\backslash
: k=i
\backslash

\backslash
 
\backslash
min & 
\backslash
left[0,
\backslash
: 	R(k,k)+{
\backslash
displaystyle 
\backslash
sum
\backslash
max_{i':i'
\backslash
notin
\backslash
{i,k
\backslash
}}[0,R(i',k)]}
\backslash
right],
\backslash
: k
\backslash
neq i
\backslash
end{cases}$ 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{2em} $A 
\backslash
leftarrow (1-d)A+dA_{old}$ 
\backslash
# dampen availabilities 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

$
\backslash
forall i, I_{e} 
\backslash
leftarrow argmax
\backslash
: S(i,I_{d})$ 
\backslash
# find indices of exemplars 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

$I_{e}(I_{d}) 
\backslash
leftarrow 1:size
\backslash
: (I_{d})$ 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

$L 
\backslash
leftarrow I_{d}(I_{e})$ 
\backslash
# assign labels 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

$C_{AP} 
\backslash
leftarrow 
\backslash
{c_{0},...,c_{k},...,c_{|C_{AP}|}
\backslash
}$ 
\backslash
# clustering output
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
# where a cluster $c 
\backslash
leftarrow (I,
\backslash
mathbf{e},N)$ holds the AP exemplars $
\backslash
mathbf{e}$,
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
# the indices $I$ of the cluster elements and $N$ the number of elements
 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
caption{Affinity Propagation}
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "alg:AP"

\end_inset


\end_layout

\end_inset


\end_layout

\end_body
\end_document