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\title [A new mathematical model for verifying the Navier-Stokes compatibility of 4D flow MRI data] { A new mathematical model for verifying the Navier-Stokes compatibility of 4D flow MRI data}
%\author[Jeremías Garay Labra]
%{Jeremías Garay Labra}
\institute [University of Groningen]
{
Bernoulli Institute\\
Faculty of Sciences and Engineering\\
University of Groningen\\ [0.5cm]
%\includegraphics[height=1.5cm]{Imagenes/escudoU2014.pdf}
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\texttt { Jeremías Garay Labra join with Hernan Mella, Julio Sotelo, Sergio Uribe, Cristobal Bertoglio and Joaquin Mura.}
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}
\date { \today }
\begin { document}
\frame { \titlepage }
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% \onslide<1->
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\begin { frame}
\frametitle { Index}
\tableofcontents
\end { frame}
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\section [4D flow MRI] { 4D flow MRI}
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\begin { frame}
\frametitle { 4D flow MRI}
\begin { columns} [c]
\column { .55\textwidth } % Left column and width
\footnotesize
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\onslide <1-> 4D flow MRI has been shown potential in the assesment of blood flow dynamics in the heart and also large arteries.\\ [0.2cm]
\onslide <2-> Some advantages:
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\begin { itemize}
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\item <3-> Full 3D coverage of the region of interest
\item <4-> Retrospective plane positioning
\item <5-> Rich post-proccesing: derived parameters
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\end { itemize}
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\column { .5\textwidth } % Right column and width
\end { columns}
\end { frame}
\begin { frame}
\frametitle { 4D flow MRI}
\footnotesize
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\onslide <1-> Main limitation $ \longrightarrow $ long scan times involved.\\
\vspace { 0.2cm}
\onslide <2-> In order to mitigate:
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\begin { itemize}
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\item <3-> Navigator gating
\item <4-> modest spatial resolutions $ \sim ( 2 . 5 \times 2 . 5 \times 2 . 5 \ mm ^ 3 ) $
\item <5-> partial data coverage
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\end { itemize}
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\vspace { 0.5cm}
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\onslide <6-> Typical quality estimators: SNR, VNR, peak flows/velocities, mass conservation (zero divergence)
\vspace { 0.5cm}
\onslide <7-> This work $ \longrightarrow $ conservation of linear momentum (Navier-Stokes compatibility).
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\end { frame}
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\section [] { The corrector field}
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\begin { frame}
\frametitle { The corrector field}
\footnotesize
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\onslide <1-> We assume a perfect physical velocity field $ \vec { u } $
\onslide <2-> \begin { eqnarray*}
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\rho \frac { \partial \vec { u} } { \partial t} + \rho \big ( \vec { u} \cdot \nabla \big ) \vec { u} - \mu \Delta \vec { u} + \nabla p = 0 \quad \text { in} \quad \Omega \label { eq:NSmom}
\end { eqnarray*}
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\onslide <3-> And a corrector field $ \vec { w } $ which satisfies:
\onslide <4-> \begin { align}
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\vec { u} & \approx \vec { u} _ { meas} + \vec { w} \quad \text { in} \quad \Omega \label { eq:corrector} \\
\nabla \cdot \vec w & = 0 \quad \text { in} \quad \Omega \label { eq:correctorDiv} \\
\vec w & = \vec 0 \quad \text { on} \quad \partial \Omega \label { eq:correctorBC}
\end { align}
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\onslide <5-> The corrector field $ \vec { w } $ measures the level of agreedment of the 4D flow measures respect to the Navier-Stokes equations.
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\end { frame}
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\begin { frame}
\frametitle { The corrector field: Continuum problem}
\footnotesize
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\onslide <1-> Applying the decomposition $ \vec { u } \approx \vec { u } _ { meas } + \vec { w } $ into the original equation and writing a variational problem for $ \vec w $ we have the following: Find $ ( \vec w ( t ) ,p ( t ) ) \in H ^ 1 _ 0 ( \Omega ) \times L ^ 2 ( \Omega ) $ such that
\onslide <2-> \begin { equation*}
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\int _ { \Omega } \rho \frac { \partial \vec { w} } { \partial t} \cdot \vec { v} + \rho \big ( ( \vec { u} _ { meas} + \vec w) \cdot \nabla \big ) \vec { w} \cdot \vec { v} + \rho \big ( \vec { w} \cdot \nabla \big ) \vec { u} _ { meas} \cdot \vec { v} + \mu \nabla \vec { w} : \nabla \vec { v} - p \nabla \cdot \vec { v} + q \nabla \cdot \vec { w} \notag
\end { equation*}
\begin { equation*}
= - \int _ { \Omega } \rho \frac { \partial \vec { u} _ { meas} } { \partial t} \cdot \vec { v} + \rho \big ( \vec { u} _ { meas} \cdot \nabla \big ) \vec { u} _ { meas} \cdot \vec { v} + \mu \nabla \vec { u} _ { meas} : \nabla \vec { v} + q \nabla \cdot \vec { u} _ { meas}
\end { equation*}
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\vspace { 0.2cm}
\onslide <3-> or in simple terms:
\onslide <4-> \begin { equation*}
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A(\vec w,p;\vec v ,q ) = \mathcal { L} (\vec v)
\end { equation*}
for all $ ( \vec v,q ) \in H ^ 1 _ 0 ( \Omega ) \times L ^ 2 ( \Omega ) $ .
\end { frame}
\begin { frame}
\frametitle { The corrector field: Discrete problem}
\footnotesize
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\onslide <1-> In the Discrete, we can write the problem as follows:
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\onslide <2-> \begin { equation}
A_ { k} (\vec w,p;\vec v ,q ) + \color { red} { S^ { conv} _ { k} (\vec w;\vec v)} + \color { blue} { S^ { press} _ { k} (\vec w,p;\vec v ,q)} \color { black} { = \mathcal { L} _ j (\vec v)}
\label { eq:Corrector_ discrete}
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\end { equation}
\begin { itemize}
\small
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\item <3-> $
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A_ { k} (\vec w,p;\vec v ,q ) := \int _ { \Omega } \frac { \rho } { \tau } \vec { w} \cdot \vec { v} + \rho \big ( ( \vec { u} _ { meas} ^ k + \vec { w} ^ { k-1} ) \cdot \nabla \big ) \vec { w} \cdot \vec { v} + \rho \big ( \vec { w} \cdot \nabla \big ) \vec { u} _ { meas} ^ k \cdot \vec { v} + \mu \nabla \vec { w} : \nabla \vec { v} - p \nabla \cdot \vec { v} + q \nabla \cdot \vec { w}
$ \vspace { 0 . 2 cm }
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\item <3-> $ \mathcal { L } _ j ( \vec v ) : = \int _ { \Omega } \frac { \rho } { \tau } \vec { w } ^ { k - 1 } \cdot \vec { v } + \mathcal { \ell } _ j ( \vec v,q ) $
\vspace { 0.2cm}
\item <4-> \color { red} $
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S^ { conv} _ { k} (\vec w;\vec v) := \int _ { \Omega } \frac { \rho } { 2} \ \big ( \nabla \cdot (\vec u^ k_ { meas} + \vec w^ { k-1} ) \big ) \ \vec { w} \cdot \vec { v}
$ \vspace { 0 . 2 cm }
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\item <5-> \color { blue} $
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S^ { press} _ { k} (\vec w,p;\vec v ,q) := \delta \sum _ { K \in \Omega } \int _ { K} \frac { h_ j^ 2} { \mu } \bigg ( \rho \big ( (\vec u^ k_ { meas} + \vec w^ { k-1} ) \cdot \nabla \big ) \vec { w} + \rho \big ( \vec { w} \cdot \nabla \big ) \vec { u} _ { meas} ^ k + \nabla p \bigg ) \cdot \notag \bigg ( \rho \big ( (\vec u^ k_ { meas} + \vec w^ { k-1} ) \cdot \nabla \big ) \vec { v} + \rho \big ( \vec { v} \cdot \nabla \big ) \vec { u} _ { meas} ^ k + \nabla q \bigg )
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$
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\end { itemize}
\end { frame}
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\begin { frame}
\frametitle { The corrector field: Well-posedness}
\footnotesize
\onslide <1->
\begin { theorem}
There exists a unique solution of Problem \ref { eq:Corrector_ discrete} under condition: $$ \rho / \tau + C _ \Omega ^ { - 2 } \mu / 2 - \rho 3 \| \nabla \vec u _ { meas } ^ k \| _ \infty > 0 $$ for all $ k> 0 $ .
\end { theorem}
\onslide <2->
We can furthermore prove the following energy balance:
\onslide <3->
\begin { theorem} For $ ( \vec w ^ k ,p ^ k ) $ solution of Problem \ref { eq:Corrector_ discrete} , with $ \ell _ j ( \vec v,q ) = 0 $ it holds
\begin { equation*} \label { eq:energy}
\| \vec w^ k \| ^ 2_ { L_ 2(\Omega )} \leq \| \vec w^ { k-1} \| ^ 2_ { L_ 2(\Omega )}
\end { equation*}
under the condition
\begin { equation*} \label { eq:condstab}
\mu \geq C_ \Omega ^ 2 \rho \| \nabla \vec u_ { meas} ^ k\| _ \infty
\end { equation*}
\end { theorem}
\end { frame}
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\section [Synthetic data] { Experiments using synthetic data }
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\begin { frame}
\frametitle { Experiments}
\begin { center}
Experiments using synthetic data
\end { center}
\end { frame}
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\begin { frame}
\frametitle { Numerical tests}
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\footnotesize
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\onslide <1-> We tested the corrector using CFD simulations as a measurements, in the following testcases:
\onslide <2->
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\begin { itemize}
\item Womersley flow in a cilinder
\item Navier-Stokes simulations in an aortic mesh
\end { itemize}
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\onslide <3->
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Also perturbations were added into the measurements:
\begin { itemize}
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\item <4-> velocity aliasing (varying the $ venc $ parameter)
\item <5-> additive noise (setting SNR in decibels)
\item <6-> simulated k-space undersampling (compressed sensing for the reconstruction)
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\end { itemize}
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%\onslide<7-> All simulations were done using a stabilized finite element method implemented in FEniCS. Afterwards, all numerical simulations were interpolated into a voxel-type structured mesh
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\end { frame}
\begin { frame}
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\frametitle { Numerical tests: channel}
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\begin { columns} [c]
\column { .6\textwidth } % Left column and width
\footnotesize
\textbf { Channel:}
\begin { itemize}
\item Convective term was neglected
\item Non-slip condition at walls
\item Oscilatory pressure at $ \Gamma _ { inlet } $
\end { itemize}
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\column { .5\textwidth } % Right column and width
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\footnotesize
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\begin { figure} [!hbtp]
\begin { center}
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\includegraphics [height=1.0\textwidth] { images/cilinder.png}
\caption { 3D channel mesh}
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\end { center}
\end { figure}
\end { columns}
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\end { frame}
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\begin { frame}
\frametitle { Results for channel: aliasing and noise}
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\footnotesize
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\onslide <1-> For comparison we defined a perfect corrector field as: $ \delta \vec u = \vec u _ { ref } - \vec u _ { meas } $
\onslide <2->
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\begin { figure} [!hbtp]
\begin { center}
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\includegraphics [height=0.5\textwidth] { images/perturbation_ pres.png}
\caption { Different perturbation scenarios}
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\end { center}
\end { figure}
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\end { frame}
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\begin { frame}
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\frametitle { Results for channel: undersampling}
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\footnotesize
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\begin { figure} [!hbtp]
\begin { center}
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\includegraphics [height=0.6\textwidth] { images/histo_ channel.png}
\caption { \footnotesize Histograms of different undersampling rates for the channel}
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\end { center}
\end { figure}
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\end { frame}
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\begin { frame}
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\frametitle { Results for channel: undersampling}
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\footnotesize
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\begin { figure} [!hbtp]
\begin { center}
\includegraphics [height=0.6\textwidth] { images/undersampling_ press.png}
\caption { \footnotesize Different undersampling rates for the channel}
\end { center}
\end { figure}
\end { frame}
\begin { frame}
\frametitle { Numerical tests: aorta}
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\begin { columns} [c]
\column { .6\textwidth } % Left column and width
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\footnotesize
\textbf { Aorta}
\begin { itemize}
\item a mild coartation was added in the descending aorta
\item $ u _ { inlet } $ simulates a cardiac cycle
\item 3-element Windkessel for the outlets
\item Non-slip condition at walls
\end { itemize}
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\column { .5\textwidth } % Right column and width
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\footnotesize
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\begin { figure} [!hbtp]
\begin { center}
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\includegraphics [height=1.0\textwidth] { images/aorta_ blender.png}
\caption { Aortic mesh}
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\end { center}
\end { figure}
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\end { columns}
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\end { frame}
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\begin { frame}
\frametitle { Results for aorta: aliasing and noise}
\footnotesize
\begin { figure} [!hbtp]
\begin { center}
\includegraphics [height=0.7\textwidth] { images/aorta_ perturbation.png}
\caption { Different perturbation scenarios for the aortic mesh}
\end { center}
\end { figure}
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\end { frame}
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\begin { frame}
\frametitle { Results for aorta: undersampling}
\footnotesize
\begin { figure} [!hbtp]
\begin { center}
\includegraphics [height=0.6\textwidth] { images/histo_ blender.png}
\caption { \footnotesize Histograms of different undersampling rates for the aortic mesh}
\end { center}
\end { figure}
\end { frame}
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\begin { frame}
\frametitle { Results for aorta: undersampling}
\footnotesize
\begin { figure} [!hbtp]
\begin { center}
\includegraphics [height=0.7\textwidth] { images/undersampling_ blender.png}
\caption { \footnotesize Different undersampling rates for the aortic mesh}
\end { center}
\end { figure}
\end { frame}
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\section [4D flow data] { Experiments using real 4D flow data }
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\begin { frame}
\frametitle { Experiments}
\begin { center}
Experiments using real 4D flow data
\end { center}
\end { frame}
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\begin { frame}
\frametitle { Experiments}
\footnotesize
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\begin { columns} [c]
\column { .6\textwidth } % Left column and width
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\begin { itemize}
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\item <1-> 4D flow measurements were taken from a silicon thoracic aortic phantom made of silicon.
\item <2-> A controled pump injects to the system a blood mimicking fluid and allows the control of: heart rate, peak flow, stroke volume and flow waveform
\item <3-> A stenosis of $ 11 \ mm $ of diameter was added in the descending aorta
\item <4-> The phantom was scanned using a clinical $ 1 . 5 \ T $ MR scanner (Philips Achieva, Best, The Netherlands)
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\end { itemize}
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\column { .5\textwidth } % Right column and width
\begin { figure} [!hbtp]
\begin { center}
\footnotesize
\includegraphics [height=\textwidth] { images/phantom.jpg}
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\caption { \footnotesize { Experiment done at the Centre of Biomedical Images (CIB) of the Catholic Unversity of Chili (PUC)} }
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\end { center}
\end { figure}
\end { columns}
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\end { frame}
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\begin { frame}
\frametitle { Results}
\footnotesize
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\begin { figure} [!hbtp]
\begin { center}
\includegraphics [height=0.5\textwidth] { images/phantom_ cib.png}
\caption { At peak systole: a) measurements b) corrector field c) corrected measurements}
\end { center}
\end { figure}
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\end { frame}
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\section { Conclusions}
\begin { frame}
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\frametitle { Conclusions and future work}
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\footnotesize
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\onslide <1-> Potential of the new quality parameter:
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\begin { itemize}
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\item <2-> The detect zones with strong disagreedment
\item <3-> To better recognize common acquisition artifacts
\item <4-> The use of the field for create new inverse problems which can be used for further accelerations
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\end { itemize}
2020-08-11 17:15:00 +02:00
\end { frame}
\begin { frame}
\begin { center}
\huge { Thank you for your time!}
\end { center}
\end { frame}
%\includegraphics<1>[height=4.5cm]{images/pat1.png}
%\includegraphics<2>[height=4.5cm]{images/pat2.png}
\end { document}