QAandFA.lyx

#LyX 1.6.5 created this file. For more info see http://www.lyx.org/
\lyxformat 345
\begin_document
\begin_header
\textclass article
\use_default_options true
\language english
\inputencoding auto
\font_roman default
\font_sans default
\font_typewriter default
\font_default_family default
\font_sc false
\font_osf false
\font_sf_scale 100
\font_tt_scale 100

\graphics default
\paperfontsize default
\spacing single
\use_hyperref false
\papersize a4paper
\use_geometry true
\use_amsmath 1
\use_esint 1
\cite_engine basic
\use_bibtopic false
\paperorientation portrait
\leftmargin 2cm
\topmargin 2cm
\rightmargin 2cm
\bottommargin 2cm
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\defskip medskip
\quotes_language english
\papercolumns 1
\papersides 2
\paperpagestyle plain
\tracking_changes false
\output_changes false
\author "" 
\author "" 
\end_header

\begin_body

\begin_layout Title
Spin propagator and diffusion crossings
\end_layout

\begin_layout Section
Overview
\end_layout

\begin_layout Standard
The idea that only the orientation information from the diffusion signal
 is adequate for accurate tractography generation is under question.
 We propose a new method based on the work of Yeh.
 et.
 al 
\begin_inset CommandInset citation
LatexCommand cite
key "Yeh2010"

\end_inset

 who extends the current derivations for diffusion signal reconstruction
 and provides both scalar and vector metrics that facilitate the understanding
 of the underlying signal and its use for fast and accurate tractography.
 The key issue is the reentrance of the spin density as an important part
 of the diffusion voxel reconstruction.
 The spin density together with the diffusion propagator give rise to the
 spin propagator.
 From the spin propagator both old metrics like FA and MD can be calculated
 non-parametricly along with new metrics as QA and XA (FA for X) which give
 higher insight on the underlying signal in a model free fashion.
\end_layout

\begin_layout Section
From k-space to q-space
\end_layout

\begin_layout Standard
The starting point is the classical formulation seen in eq.
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:kq"

\end_inset

 of joint k-space and q-space imaging (Calaghan 8.3.1 p.
 438)
\begin_inset CommandInset citation
LatexCommand cite
key "Callaghan1991OUP"

\end_inset

 using the narrow pulse gradient spin echo (PGSE) sequence of Tanner and
 Stejskal
\end_layout

\begin_layout Standard
\align center
\begin_inset Formula \begin{eqnarray}
\mathit{\mathcal{RF}}(\mathbf{k},\mathbf{q}) & = & \int\rho(\mathbf{r})\exp(j2\pi\mathbf{k}\cdot\mathbf{r})\int p_{\Delta}(\mathbf{r},\mathbf{R})\exp(j2\pi\mathbf{q}\cdot\mathbf{R})d\mathbf{R}d\mathbf{r}\label{eq:kq}\end{eqnarray}

\end_inset


\end_layout

\begin_layout Standard
Here 
\begin_inset Formula $RF$
\end_inset

 is the complex RF signal measured at spatial wave number 
\begin_inset Formula $\mathbf{k}$
\end_inset

 and magnetic gradient wave number 
\begin_inset Formula $\mathbf{q}$
\end_inset

, 
\begin_inset Formula $\rho$
\end_inset

 is the local spin density (number of protons per unit volume contributing
 to the RF signal), 
\begin_inset Formula $\Delta$
\end_inset

 is the diffusion time scale of the sequence, 
\begin_inset Formula $p_{\Delta}$
\end_inset

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

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


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

 is the diffusion displacement.
 
\end_layout

\begin_layout Standard
The k-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

 
\begin_inset CommandInset citation
LatexCommand cite
key "Wedeen"

\end_inset

 of each voxel [
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:W"

\end_inset

]
\end_layout

\begin_layout Standard
\align center
\begin_inset Formula \begin{eqnarray}
S(\mathbf{r},\mathbf{q}) & = & \int\rho(\mathbf{r})p_{\Delta}(\mathbf{r},\mathbf{R})\exp(j2\pi\mathbf{q}\cdot\mathbf{R})d\mathbf{R}\label{eq:W}\end{eqnarray}

\end_inset


\end_layout

\begin_layout Standard
In order to represent the average propagator in the scale of spin quantity
 Yeh et.al.
 introduced a 
\emph on
spin density function 
\emph default
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 $Q(\mathbf{r},\mathbf{R})=\rho(\mathbf{r})p_{\Delta}(\mathbf{r},\mathbf{R})$
\end_inset

.
 
\end_layout

\begin_layout Standard
\align center

\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
\lang british
\begin_inset Formula \begin{eqnarray}
S(\mathbf{r},\mathbf{q}) & = & \int Q(\mathbf{r},\mathbf{R})\exp(j2\pi\mathbf{q}\cdot\mathbf{R})d\mathbf{R}\label{eq:W2Q}\end{eqnarray}

\end_inset


\end_layout

\begin_layout Standard
We can apply the inverse fourier transform in eq.
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:W2Q"

\end_inset

 and obtain
\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray}
Q(\mathbf{r},\mathbf{R}) & = & \int S(\mathbf{r},\mathbf{q})exp(-j2\pi\mathbf{q}\cdot\mathbf{R})d\mathbf{q}\label{eq:Q2S_complex}\end{eqnarray}

\end_inset


\end_layout

\begin_layout Standard
Because
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
\lang british
 
\begin_inset Formula $Q(\mathbf{r},\mathbf{R})$
\end_inset


\family default
\series default
\shape default
\size default
\emph default
\bar default
\noun default
\color inherit
\lang english
 is real, 
\begin_inset Formula $S(\mathbf{r},\mathbf{q})$
\end_inset

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

 we can use directly the fourier cosine transform 
\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray}
Q(\mathbf{r},\mathbf{R}) & = & \int S(\mathbf{r},\mathbf{q})cos(2\pi\mathbf{q}\cdot\mathbf{R})d\mathbf{q}\label{eq:cosine_transform}\end{eqnarray}

\end_inset


\end_layout

\begin_layout Standard
and obtain the spin distribution function 
\begin_inset Formula $\psi$
\end_inset

 from an unweighted trancated radial projection 
\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray}
\psi_{Q}(\mathbf{r},\mathbf{\hat{u}}) & = & \intop_{0}^{L_{\Delta}}Q(\mathbf{r},L\mathbf{\hat{u}})dL\label{eq:SDF}\end{eqnarray}

\end_inset


\begin_inset Note Note
status open

\begin_layout Plain Layout
!!! add here equations 5 & 6 from Frank's paper and their matrix form
\end_layout

\end_inset


\end_layout

\begin_layout Standard
where 
\begin_inset Formula $L_{\Delta}$
\end_inset

is the diffusion sampling length.
 This parameter acts like a smoothing factor.
 The highest the 
\begin_inset Formula $L_{\Delta}$
\end_inset

more detailed the SDF will be but also more noisy.
 Typical values for the diffusion sampling length are 
\begin_inset Formula $[1.1-1.3]$
\end_inset

.
 Usually 
\begin_inset Formula $L_{\Delta}=1.2$
\end_inset

 is a good ballance between noisy reconstruction and over-smoothing.
\end_layout

\begin_layout Standard
Assuming that the diffusion voxel can be represented by a single tensor
 model eq.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:W"

\end_inset

 can be written in the form
\end_layout

\begin_layout Standard
\align center
\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 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 could try to 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 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


\end_layout

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

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


\family default
\series default
\shape default
\size default
\emph default
\bar default
\noun default
\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 CommandInset ref
LatexCommand ref
reference "eq:TensorQ2"

\end_inset

 illustrates the remarkable result that we can calculate analyticaly 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/sampled spin odf
 that is derived in 
\begin_inset CommandInset citation
LatexCommand cite
key "Yeh2010"

\end_inset

 with eq.
 
\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 CommandInset ref
LatexCommand ref
reference "eq:TensorQ2"

\end_inset

.
 Because calculating a spherical intergral from eq.
\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
Lets 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 surface integral needed in eq.
 
\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 hypergeometric 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 $A$
\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 Section
From the spin density ODF to FA
\end_layout

\begin_layout Standard
As 
\begin_inset Formula $FA(\lambda_{1},\lambda_{2},\lambda_{3})$
\end_inset

 is a function of the eigen-values of the diffusion tensor model we could
 try to approximate directly FA from the sODF by finding 
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none

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

 through the global maximum , equatorial maximum and inferior maximum of
 the sODF.
\end_layout

\begin_layout Standard

\lang british
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename Ldelta1_Ldelta_10.png
	scale 80
	rotateOrigin center

\end_inset


\end_layout

\begin_layout Plain Layout

\lang british
\begin_inset Caption

\begin_layout Plain Layout
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout
import scipy.stats.distributions as ssd 
\begin_inset Newline newline
\end_inset

used ssd.norm.cdf
\end_layout

\end_inset


\begin_inset Newline newline
\end_inset

if k=0 then cdf = NaN but 
\begin_inset Formula $\lim_{k\rightarrow0}\backsimeq0.398$
\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Section
What is QA then?
\end_layout

\begin_layout Standard
QA is defined by the sODF value at the resolved fiber orientation 
\begin_inset Formula $\hat{u}$
\end_inset

 (local maximum) minus the isotropic component (global minimum) for the
 voxel.
 QA can be also very easily normalized by the maximum sODF value of all
 voxels.
\end_layout

\begin_layout Section
Appendix
\end_layout

\begin_layout Subsection
The cosine transform
\end_layout

\begin_layout Standard
\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)+isin(its)+cos(its)-isin(its)]dt\\
 & = & 2\int_{0}^{\infty}f(t)cos(st)dt\end{eqnarray*}

\end_inset

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

 
\end_layout

\begin_layout Standard
If you want to compute the integral in the hole space because of symmetry
 you have 
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none

\begin_inset Formula $F(s)=\int_{-\infty}^{\infty}f(t)cos(st)dt$
\end_inset


\end_layout

\begin_layout Standard

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

\end_inset


\end_layout

\end_body
\end_document