ISBI_2012.lyx

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\author "" 
\author "Ian Nimmo-Smith,,," 
\author "" 
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\begin_body

\begin_layout Section*
Cartesian Grid Q-space Reconstruction
\end_layout

\begin_layout Section*
Overview
\end_layout

\begin_layout Standard
Between one to two thirds of imaging voxels in the human brain's white matter
 are thought to contain multiple fiber crossings
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "Behrens2007NeuroImage"

\end_inset

, in which case the Diffusion Tensor model proposed by Basser et al.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "Basser1994"

\end_inset

 breaks down.
 High Angular Resolution Diffusion Imaging (HARDI) methods 
\begin_inset CommandInset citation
LatexCommand cite
key "Tuch2002"

\end_inset

 such as Diffusion Spectrum Imaging (DSI) 
\begin_inset CommandInset citation
LatexCommand cite
key "callaghan1988nmr"

\end_inset

, 
\begin_inset CommandInset citation
LatexCommand cite
key "wedeen2005mapping"

\end_inset

 or Higher Order Tensors 
\begin_inset CommandInset citation
LatexCommand cite
key "ozarslan2003generalized"

\end_inset

, 
\begin_inset CommandInset citation
LatexCommand cite
key "barmpoutis2009regularized"

\end_inset

 and many more reconstruction methods have been proposed to overcome the
 limitations of the Diffusion Tensor.
 Although all methods have some underlying assumptions; we generally separate
 them in two categories a) model-based and b) model-free.
 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
For the purpose of the HARDI reconstruction workshop we used GQI2
\begin_inset CommandInset citation
LatexCommand cite
key "Yeh2010"

\end_inset

,
\begin_inset CommandInset citation
LatexCommand cite
key "Garyfallidis_thesis"

\end_inset

 a non-parametric method to find the ODFs 
\begin_inset CommandInset citation
LatexCommand cite
key "Tuch2002"

\end_inset

,
\begin_inset CommandInset citation
LatexCommand cite
key "aganj2010reconstruction"

\end_inset

.
 From these ODFs we can find the number of peaks (number of fiber compartments)
 and if there is only one peak then we fit the Single Tensor in the data
 otherwise we keep the initial GQI2 ODF.
\end_layout

\begin_layout Section
Theory
\end_layout

\begin_layout Standard
According to 
\begin_inset CommandInset citation
LatexCommand cite
key "Callaghan1991OUP"

\end_inset

, 
\begin_inset CommandInset citation
LatexCommand cite
key "callaghan1988nmr"

\end_inset

 using the narrow pulse gradient spin echo (PGSE) sequence of Tanner and
 Stejskal.
 The 
\series bold
k
\series default
-space reconstruction gives us diffusion weighted image data 
\begin_inset Formula $S$
\end_inset

 which reveal the average propagator 
\begin_inset Formula $P_{\Delta}$
\end_inset

 of each voxel according to the following equation 
\begin_inset Formula \begin{eqnarray}
S(\mathbf{v},\mathbf{q}) & = & \int\rho(\mathbf{v})P_{\Delta}(\mathbf{v},\mathbf{r})\exp(i2\pi\mathbf{q}\cdot\mathbf{r})d\mathbf{r}.\label{eq:W}\end{eqnarray}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

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\backslash
noindent
\end_layout

\end_inset

 where
\begin_inset Formula $\Delta$
\end_inset

 is the time between diffusion gradients, 
\begin_inset Formula $P_{\Delta}$
\end_inset

 is the average diffusion propagator (transition probability distribution),
 
\series bold

\begin_inset Formula $\mathbf{v}$
\end_inset


\series default
 is the voxel coordinate, and 
\begin_inset Formula $\mathbf{r}$
\end_inset

 is the diffusion displacement.
 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

and 
\begin_inset Formula $\Delta$
\end_inset

.
 We can also replace the spin density 
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\lang british

\begin_inset Formula $\rho(\mathbf{v})$
\end_inset


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\lang english
 with 
\begin_inset Formula $S_{0}$
\end_inset

 i.e.
 the measured signal without diffusion weighting 
\begin_inset Formula $\mathbf{q}=\mathbf{0}$
\end_inset

.
 Therefore we can write
\begin_inset Formula \begin{eqnarray}
S(\mathbf{q}) & = & S_{0}\int P(\mathbf{r})\exp(i2\pi\mathbf{q}\cdot\mathbf{r})d\mathbf{r}\label{eq:S}\end{eqnarray}

\end_inset


\end_layout

\begin_layout Standard
By applying the 3D Fourier transform in eq.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:S"

\end_inset

 we can reconstruct the average propagator also known as the diffusion spectrum
 
\begin_inset CommandInset citation
LatexCommand cite
key "WWS+08"

\end_inset


\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}\end{eqnarray}

\end_inset


\end_layout

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noindent
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\end_inset

 or diffusion propagator.
 It was shown by Wedeen et al.
 
\begin_inset CommandInset citation
LatexCommand cite
key "Wedeen"

\end_inset

 that the dMRI signal is positive for any type of spin motion without net
 flux (i.e.
\begin_inset ERT
status open

\begin_layout Plain Layout

~
\end_layout

\end_inset

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 
\begin_inset Formula $S$
\end_inset

 with its modulus 
\begin_inset Formula $|S|$
\end_inset

 in eq.
 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:P"

\end_inset

)
\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}

\end_inset


\end_layout

\begin_layout Standard
The modulus of the signal coincides with the output of the standard MRI
 scanners as dMRI and that simplifies the acquisition procedure.
 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

 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 
\begin_inset Formula $\mathbf{r}$
\end_inset

.
 We can visualize the propagator for every voxel as a 3D density volume.
\end_layout

\begin_layout Standard
Since we are mainly interested in the angular structure of the underlying
 tissue, we further simplify the data by taking the weighted radial summation
 of 
\begin_inset Formula $P(\mathbf{r})$
\end_inset


\lang british

\begin_inset Formula \begin{equation}
\psi_{DSI}(\hat{\mathbf{u}})=\int_{0}^{\infty}P(r\hat{\mathbf{u}})r^{2}dr\label{eq:ODF_DSI}\end{equation}

\end_inset

This defines the orientation density function (ODF) for DSI which measures
 the quantity of diffusion in the direction of the unit vector 
\begin_inset Formula $\mathbf{\hat{u}}$
\end_inset

 where 
\begin_inset Formula $\mathbf{r=}r\hat{\mathbf{u}}$
\end_inset

.
\end_layout

\begin_layout Standard

\lang british
Note at this point that in order to find the ODF we have to create first
 the diffusion propagator by 
\lang english
applying
\lang british
 the Fourier transform on the lattice.
 
\lang english
Yeh et al.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "Yeh2010"

\end_inset

 proposed a direct way to calculate a slightly different ODF using the Cosine
 transform.
 In order to represent the average propagator in the scale of spin quantity
 Yeh et al.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "Yeh2010"

\end_inset

 introduced the 
\emph on
spin density function 
\emph default

\begin_inset Formula $Q$
\end_inset

 which is estimated by scaling the average propagator 
\begin_inset Formula $P_{\Delta}$
\end_inset

 with the spin density 
\begin_inset Formula $\rho$
\end_inset

, i.e.
 
\begin_inset Formula $Q(\mathbf{r})=\rho P(\mathbf{r})=S_{0}P(\mathbf{r})$
\end_inset

.
 From eq.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:S"

\end_inset

 we obtain
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\lang british

\begin_inset Formula \begin{eqnarray}
S(\mathbf{q}) & = & \int Q(\mathbf{r})\exp(i2\pi\mathbf{q}\cdot\mathbf{r})d\mathbf{r}\label{eq:W2Q}\end{eqnarray}

\end_inset


\end_layout

\begin_layout Standard
\align block
We can apply the Fourier transform again to eq.
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:W2Q"

\end_inset

 and obtain
\begin_inset Formula \begin{eqnarray}
Q(\mathbf{r}) & = & \int S(\mathbf{q})exp(-i2\pi\mathbf{q}\cdot\mathbf{r})d\mathbf{q}\label{eq:Q2S_complex}\end{eqnarray}

\end_inset

because
\family roman
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\begin_inset Formula $Q(\mathbf{r})$
\end_inset


\family default
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\color inherit
\lang english
 is real and 
\begin_inset Formula $S(\mathbf{q})$
\end_inset

 is symmetric (even), i.e.
 
\begin_inset Formula $S(\mathbf{q})=S(-\mathbf{q})$
\end_inset

, we can use directly the Fourier Cosine transform to calculate
\begin_inset Formula \begin{eqnarray}
Q(\mathbf{r}) & = & \int S(\mathbf{q})cos(2\pi\mathbf{q}\cdot\mathbf{r})d\mathbf{q}\label{eq:cosine_transform}\end{eqnarray}

\end_inset

and obtain the orientation distribution function 
\begin_inset Formula $\psi_{GQI}$
\end_inset

 (ODF) from an unweighted truncated radial projection
\begin_inset Formula \begin{eqnarray}
\psi_{GQI}(\mathbf{\hat{u}}) & = & \intop_{0}^{\lambda}Q(r\mathbf{\hat{u}})dr\label{eq:SDF}\\
 & = & \lambda\int S(\mathbf{q})\mathtt{sinc}(2\pi r\mathbf{q}\cdot\mathbf{\hat{u}})d\mathbf{q}\end{eqnarray}

\end_inset

where 
\begin_inset Formula $\lambda$
\end_inset

 is a constant called the diffusion sampling length.
 This parameter acts like a smoothing factor.
 The higher 
\begin_inset Formula $\lambda$
\end_inset

 the more detailed the ODF will be but also easier affected by noise.
 This ODF was used as the basis of the analysis of the GQI method.
 However, we consentrated to a similar ODF like the one produced using DSI
 where we need to take into consideration the weighted truncated radial
 projection.
 This will give us a different ODF which we symbolize with 
\begin_inset Formula $\psi_{GQI_{2}}$
\end_inset


\begin_inset Formula \begin{eqnarray}
\psi_{GQI2}(\mathbf{\hat{u}}) & = & \intop_{0}^{\lambda}Q(r\mathbf{\hat{u}})r^{2}dr\label{eq:SDF2}\\
 & = & \lambda^{3}\int S(\mathbf{q})H(2\pi r\mathbf{q}\cdot\mathbf{\hat{u}})d\mathbf{q}\end{eqnarray}

\end_inset


\end_layout

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\backslash
noindent
\end_layout

\end_inset

 where 
\begin_inset Formula $H(x)=\begin{cases}
\frac{2\cos(x)}{x^{2}} & +\frac{(x^{2}-2)\sin(x)}{x^{3}},x\neq0\\
 & 1/3\qquad\qquad,x=0\end{cases}$
\end_inset

.
\end_layout

\begin_layout Standard
\align left
This equation can be implemented analytically in a simple matrix form
\begin_inset Formula \begin{eqnarray*}
\bm{\psi}_{GQI2} & = & \mathbf{s}\cdot\mathtt{H}((6D\cdot G\circ\mathbf{b}\circ\mathbb{1})\cdot G)\lambda^{3}/\pi\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\align left
where 
\begin_inset Formula $\cdot$
\end_inset

 denotes standard matrix or vector dot product, 
\family roman
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\lang british

\begin_inset Formula $\circ$
\end_inset

 
\family default
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\lang english
denotes the Hadamard product, 
\begin_inset Formula $\mathbf{y}$
\end_inset

 is 
\begin_inset Formula $\psi_{GQI}$
\end_inset

 as a 
\begin_inset Formula $M$
\end_inset

-dimensional vector with components corresponding to the selected directions
 
\begin_inset Formula $\hat{\mathbf{u}}$
\end_inset

 on the ODF sphere, 
\begin_inset Formula $\mathbf{s}$
\end_inset

 is a vector with all the signal values, 
\begin_inset Formula $D=0.0025$
\end_inset


\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
\lang british
 where 
\begin_inset Formula $D$
\end_inset

 is a constant known as the the free water diffusion coefficient
\family default
\series default
\shape default
\size default
\emph default
\bar default
\noun default
\color inherit
\lang english
, 
\begin_inset Formula $G$
\end_inset

 is the 
\begin_inset Formula $N\times3$
\end_inset

 matrix with the gradient vectors, 
\begin_inset Formula $\mathbf{b}$
\end_inset

 is the 
\begin_inset Formula $N\times1$
\end_inset

 matrix with the b-values and 
\begin_inset Formula $\mathbb{1}$
\end_inset

 is the 
\begin_inset Formula $N\times3$
\end_inset

 incidence matrix where all values are equal to 
\begin_inset Formula $1$
\end_inset

.
 This method has not to date been published with real or simulated data
 sets.
\end_layout

\begin_layout Standard
We use 
\begin_inset Formula $\lambda=3.,3.3,3.5$
\end_inset

 for the provided phantoms with SNR 
\begin_inset Formula $10,20,30$
\end_inset

 respectively.
 In the case where a single peak is found the Single Tensor model is fitted
 using Weighted Least Squares and the standard ODF for that is generated.
 
\end_layout

\begin_layout Section*
Conclusion
\end_layout

\begin_layout Standard
Non-parametric methods have the advantage of representing the signal with
 minimum number of assumptions and without needing any fitting.
 For many years there has been a trend in science to prefer model-based
 rather than model free (non-parametric) methods.
 This is perhaps because model-based can be easier to describe, and more
 readily allow the use of popular Bayesian approaches.
 However, there are some crucial issues with fitting: (a) Usually the interestin
g models have many parameters and that makes fitting very slow.
 (b) Commonly non-linear fitting is needed and accurate fitting is not trivial.
 (c) Often the model does not represent precisely the complexity of the
 real problem.
 (d) The more complex the model, the more difficult to fit 
\begin_inset CommandInset citation
LatexCommand cite
key "rice2006mathematical"

\end_inset

, 
\begin_inset CommandInset citation
LatexCommand cite
key "lee1997bayesian"

\end_inset

, 
\begin_inset CommandInset citation
LatexCommand cite
key "montgomery2001introduction"

\end_inset

.
 The source code for the methods described here can be found in dipy (
\begin_inset ERT
status open

\begin_layout Plain Layout

dipy.org
\end_layout

\end_inset

).
 An exampe of our method can be seen in Fig.
 1.
 GQI2 is a method theoretically identical with the framework of Equatorial
 Inversion Transform (EIT) 
\begin_inset CommandInset citation
LatexCommand cite
key "Garyfallidis_thesis"

\end_inset

 with Laplacian weighting.
 
\end_layout

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\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename hardi_isbi2012/isbi2012.png
	lyxscale 30

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Result showing our reconstruction ODFs with the training data set (SNR 30)
 provided from the organizers of the HARDI Workshop 2012
\end_layout

\end_inset


\end_layout

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\end_layout

\end_inset


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\change_inserted 1 1331819849

\lang british
\begin_inset CommandInset bibtex
LatexCommand bibtex
bibfiles "diffusion"
options "ieeetr"

\end_inset


\change_unchanged

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\end_body
\end_document