@ -98,7 +98,7 @@
\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}
\title { Robust parameter estimation in fluid flow models from aliased velocity measurements}
% \author [Jeremías Garay Labra]
% { Jeremías Garay Labra}
\institute [University of Groningen]
@ -108,7 +108,7 @@ Faculty of Sciences and Engineering\\
University of Groningen\\ [0.5cm]
% \includegraphics [height=1.5cm] { Imagenes/escudoU2014.pdf}
% \includegraphics [height=1cm] { Imagenes/fcfm.png} \\ [0.5cm]
Jeremías Garay Labra \emph { join with} Hernan Mella, Julio Sotelo, Sergio Uribe, Cristobal Bertoglio and Joaquin Mura .}
Jeremías Garay Labra \emph { join with} Cristobal Bertoglio.}
\date { \today }
@ -145,8 +145,6 @@ University of Groningen\\[0.5cm]
\end { itemize}
\column { .54\textwidth } % Right column and width
\onslide <1->
\begin { figure} [!hbtp]
@ -159,362 +157,219 @@ University of Groningen\\[0.5cm]
\end { frame}
\begin { frame}
\frametitle { 4D flow MRI}
\footnotesize
\onslide <1-> Strategies:
\begin { itemize}
\item <2-> modest spatial resolutions $ \sim ( 2 . 5 \times 2 . 5 \times 2 . 5 \ mm ^ 3 ) $
\item <3-> partial data coverage
\end { itemize}
\begin { columns} [c]
\column { .4\textwidth } % Right column and width
\onslide <4->
\footnotesize
\begin { figure} [!hbtp]
\begin { center}
\includegraphics [height=0.25\textwidth] { images/channel_ noise.png} \\
(a) Noise
% \caption { Noise}
\end { center}
\end { figure}
\column { .4\textwidth } % Right column and width
\onslide <5->
\footnotesize
\begin { figure} [!hbtp]
\begin { center}
\includegraphics [height=0.25\textwidth] { images/channel_ aliasing.png} \\
(b) Aliasing
% \caption { Aliasing}
\end { center}
\end { figure}
\column { .4\textwidth } % Right column and width
\onslide <6->
\footnotesize
\begin { figure} [!hbtp]
\begin { center}
\includegraphics [height=0.25\textwidth] { images/channel_ under.png} \\
(c) Undersampling
% \caption { Aliasing}
\end { center}
\end { figure}
\end { columns}
\vspace { 0.5cm}
\onslide <7-> Typical quality estimators: SNR, VNR, peak flows/velocities, mass conservation (zero divergence)
\vspace { 0.5cm}
\onslide <8-> This work $ \longrightarrow $ \textbf { conservation of linear momentum} (Navier-Stokes compatibility).
\end { frame}
\section [] { The corrector field}
\section { The mathematical model}
\begin { frame}
\frametitle { The corrector field }
\frametitle { The mathematical model}
\begin { center}
Methodology
The mathematical model
\end { center}
\end { frame}
\begin { frame}
\frametitle { The corrector field}
\footnotesize
\onslide <1-> We assume a perfect physical velocity field $ \vec { u } $
\onslide <2-> \begin { eqnarray*}
\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*}
\onslide <3-> And a corrector field $ \vec { w } $ which satisfies:
\onslide <4-> \begin { align}
\vec { u} & = \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}
\onslide <5-> The corrector field $ \vec { w } $ measures the level of agreedment of the 4D flow measures respect to the Navier-Stokes equations.
\end { frame}
\begin { frame}
\frametitle { The corrector field: Continuum problem}
\frametitle { The mathematical model}
\begin { columns} [c]
\column { .5\textwidth } % Left column and width
\footnotesize
\onslide <1-> Applying the decomposition $ \vec { u } = \vec { u } _ { meas } + \vec { w } $ into the original equation and writing a variational problem for $ \vec w $ we have:\\ [0.2cm]
Find $ ( \vec w ( t ) ,p ( t ) ) \in H ^ 1 _ 0 ( \Omega ) \times L ^ 2 ( \Omega ) $ such that:
\onslide <2-> \begin { equation*}
\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*}
\vspace { 0.2cm}
\onslide <3-> or in simple terms:
\onslide <4-> \begin { equation*}
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 ) $ .
\column { .54\textwidth } % Right column and width
\begin { figure} [!hbtp]
\begin { center}
\includegraphics [height=1.1\textwidth] { images/full_ aorta.png}
\end { center}
\end { figure}
\end { columns}
\end { frame}
\begin { frame}
\frametitle { The corrector field: Discrete problem}
\frametitle { The mathematical model}
\begin { columns} [c]
\column { .5\textwidth } % Left column and width
\footnotesize
\onslide <1-> In the Discrete, we can write the problem as follows:
\onslide <2-> \begin { equation}
A_ { k} (\vec w,p;\vec v ,q ) + \color { blue} { S^ { press} _ { k} (\vec w,p;\vec v ,q)} + \color { red} { S^ { conv} _ { k} (\vec w;\vec v)} \color { black} { = \mathcal { L} _ j (\vec v)}
\label { eq:Corrector_ discrete}
\begin { itemize}
\item <2-> Incompressible Navier-Stokes equations:
\begin { equation}
\begin { cases}
\displaystyle \rho \frac { \partial \vec { u} } { \partial t} + \rho \big ( \vec { u} \cdot \nabla \big ) \vec { u} - \mu \Delta \vec { u} + \nabla p = 0 \\ [0.2cm]
\nabla \cdot \vec { u} = 0 \quad \text { in} \quad \Omega \\ [0.2cm]
\vec { u} = \vec { u} _ { inlet} \quad \text { on} \quad \Gamma _ { in} \\ [0.2cm]
\vec { u} = 0 \quad \text { on} \quad \Gamma _ { walls}
\end { cases}
\end { equation}
\item <3-> \emph { Three-element} Windkessel coupling at every outlet:
\begin { equation}
\begin { cases}
\displaystyle C_ { d,l} \frac { d \pi _ l} { dt} + \frac { \pi _ l} { R_ { d,l} } = Q_ l \\ [0.2cm]
P_ l = R_ { p,l} \ Q_ l + \pi _ l
\end { cases}
\end { equation}
\begin { itemize}
\small
\item <3-> $
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 }
\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 { blue} $
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 )
$
\vspace { 0.2cm}
\item <5-> \color { red} $
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 }
\end { itemize}
\end { frame}
\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 the 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}
\section [Synthetic data] { Experiments using synthetic data }
\begin { frame}
\frametitle { Experiments}
\begin { center}
Experiments using synthetic data
\end { center}
\end { frame}
\begin { frame}
\frametitle { Numerical tests}
\column { .54\textwidth } % Right column and width
\onslide <1->
\footnotesize
\begin { columns} [c]
\column { .4\textwidth } % Right column and width
\footnotesize
Simulated channel flow as measurements (Stokes flow)
\column { .5\textwidth } % Right column and width
\footnotesize
\begin { figure} [!hbtp]
\begin { center}
\includegraphics [height=0.35\textwidth] { images/cilinder_ 2.png} \\
(b) Channel mesh
% \caption { Aliasing}
\includegraphics [height=0.9\textwidth] { images/windk_ model.png}
\caption { \footnotesize Schematic of the model}
\end { center}
\end { figure}
\end { columns}
\vspace { 0.2cm}
% \onslide <1-> We tested the corrector using CFD simulations as a measurements, in the following testcases:
% \onslide <2->
% \begin { itemize}
% \item Womersley flow in a cilinder
% \item Navier-Stokes simulations in an aortic mesh
% \end { itemize}
\onslide <2-> Afterwards, perturbations were added:
\begin { itemize}
\item <3-> velocity aliasing (varying the $ venc $ parameter)
\item <4-> additive noise (setting SNR in decibels)
\item <5-> simulated k-space undersampling (compressed sensing for the reconstruction)
\end { itemize}
% \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
\end { frame}
%
% \begin { frame}
% \frametitle { Numerical tests: channel}
% \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}
% \column { .5\textwidth } % Right column and width
% \footnotesize
% \begin { figure} [!hbtp]
% \begin { center}
% \includegraphics [height=1.0\textwidth] { images/cilinder.png}
% \caption { 3D channel mesh}
% \end { center}
% \end { figure}
% \end { columns}
% \end { frame}
%
\begin { frame}
\frametitle { Numerical tests}
\begin { center}
Results
\end { center}
\end { columns}
\end { frame}
\begin { frame}
\frametitle { Aliasing and noise}
\frametitle { The mathematical model}
\begin { columns} [c]
\column { .5\textwidth } % Left column and width
\footnotesize
\begin { itemize}
\item Incompressible Navier-Stokes equations:
\begin { equation}
\begin { cases}
\displaystyle \rho \frac { \partial \vec { u} } { \partial t} + \rho \big ( \vec { u} \cdot \nabla \big ) \vec { u} - \mu \Delta \vec { u} + \nabla p = 0 \\ [0.2cm]
\nabla \cdot \vec { u} = 0 \quad \text { in} \quad \Omega \\ [0.2cm]
\vec { u} = \vec { u} _ { inlet} \quad \text { on} \quad \Gamma _ { in} \\ [0.2cm]
\vec { u} = 0 \quad \text { on} \quad \Gamma _ { walls}
\end { cases}
\end { equation}
\item \emph { Three-element} Windkessel coupling at every outlet:
\begin { equation}
\begin { cases}
\displaystyle C_ { d,l} \frac { d \pi _ l} { dt} + \frac { \pi _ l} { R_ { d,l} } = Q_ l \\ [0.2cm]
P_ l = R_ { p,l} \ Q_ l + \pi _ l
\end { cases}
\end { equation}
\end { itemize}
\onslide <1-> For comparison we defined a perfect corrector field as: $ \delta \vec u = \vec u _ { ref } - \vec u _ { meas } $
\onslide <2->
\column { .54\textwidth } % Right column and width
\begin { figure} [!hbtp]
\begin { center}
\includegraphics [height=0.45\textwidth] { images/channel_ ppt_ 1.png}
\caption { \small Fields for the channel: $ ( SNR,venc ) = ( \infty , 120 \% ) $ . $ \vec { w } \times 200 $ }
\includegraphics [height=0.9\textwidth] { images/ref.png}
\caption { \footnotesize Schematic of the model}
\end { center}
\end { figure}
\end { columns}
\end { frame}
\begin { frame}
\frametitle { Aliasing and noise}
\footnotesize
For comparison we defined a perfect corrector field as: $ \delta \vec u = \vec u _ { ref } - \vec u _ { meas } $
\begin { figure} [!hbtp]
\begin { center}
\includegraphics [height=0.45\textwidth] { images/channel_ ppt_ 2.png}
\caption { \small Fields for the channel: $ ( SNR,venc ) = ( \infty , 80 \% ) $ . $ \vec { w } \times 4 $ }
% \caption { \small Different perturbation scenarios. $ ( \infty , 120 \% ) $ : $ \vec { w } \times 200 $ , $ ( 10 \ dB , 120 \% ) $ : $ \delta \vec { u } , \vec { w } \times 4 $ , rest: $ \vec { w } \times 4 $ }
\end { center}
\end { figure}
\begin { frame}
\frametitle { The inverse problem}
\begin { itemize}
\item <1-> Upon this solution $ \Longrightarrow $ build a set of measurements
\item <2-> Induce typical arifacts via a Magnetization vector: $ M ( \vec { x } ,t ) = M _ 0 ( \vec { x } ) exp ( i \phi _ 0 + i \frac { \pi } { venc } u ( \vec { x } ,t ) ) $
\end { itemize}
\onslide <3->
\begin { columns}
\column { .3\textwidth }
\flushleft
\begin { figure}
\includegraphics [width=1.1\textwidth] { images/ref_ int.png}
\caption * { (a) Interpolated reference solution}
\end { figure}
\column { .67\textwidth }
\centering
\onslide <4-> \textbf { The measurements:}
\begin { itemize}
\item <5-> Gaussian noise into the magnetization
\item <6-> Different levels of aliasing varying the $ venc $ parameter
\item <7-> Only using the dominant component of the velocity: $ u _ z $
\end { itemize}
\end { columns}
\end { frame}
\begin { frame}
\frametitle { Aliasing and noise}
\footnotesize
For comparison we defined a perfect corrector field as: $ \delta \vec u = \vec u _ { ref } - \vec u _ { meas } $
\begin { figure} [!hbtp]
\begin { center}
\includegraphics [height=0.45\textwidth] { images/channel_ ppt_ 3.png}
\caption { \small Fields for the channel: $ ( SNR,venc ) = ( 10 \ dB, 120 \% ) $ . $ \delta \vec { u } , \vec { w } \times 4 $ }
% \caption { \small Different perturbation scenarios. $ ( \infty , 120 \% ) $ : $ \vec { w } \times 200 $ , $ ( 10 \ dB , 120 \% ) $ : $ \delta \vec { u } , \vec { w } \times 4 $ , rest: $ \vec { w } \times 4 $ }
\end { center}
\end { figure}
\end { frame}
\begin { frame}
\frametitle { Aliasing and noise}
\footnotesize
For comparison we defined a perfect corrector field as: $ \delta \vec u = \vec u _ { ref } - \vec u _ { meas } $
\begin { figure} [!hbtp]
\begin { center}
\includegraphics [height=0.45\textwidth] { images/channel_ ppt_ 4.png}
\caption { \small Fields for the channel: $ ( SNR,venc ) = ( 10 \ dB, 80 \% ) $ . $ \vec { w } \times 4 $ }
% \caption { \small Different perturbation scenarios. $ ( \infty , 120 \% ) $ : $ \vec { w } \times 200 $ , $ ( 10 \ dB , 120 \% ) $ : $ \delta \vec { u } , \vec { w } \times 4 $ , rest: $ \vec { w } \times 4 $ }
\end { center}
\end { figure}
\frametitle { The inverse problem}
\begin { itemize}
\item Upon this solution $ \Longrightarrow $ build a set of measurements
\item Induce typical arifacts via a Magnetization vector: $ M ( \vec { x } ,t ) = M _ 0 ( \vec { x } ) exp ( i \phi _ 0 + i \frac { \pi } { venc } u ( \vec { x } ,t ) ) $
\end { itemize}
\begin { columns}
\column { .3\textwidth }
\flushleft
\begin { figure}
\includegraphics [width=1.1\textwidth] { images/ref_ int.png}
\caption * { (a) Interpolated reference solution}
\end { figure}
\column { .67\textwidth }
\flushleft
\begin { figure}
\includegraphics [width=1.1\textwidth] { images/supra_ venc.png}
\caption * { (b) Aliased measurements with different $ vencs = 120 , 70 , 30 \% $ of $ u _ { max } $ }
\hfill
\end { figure}
\end { columns}
\end { frame}
\begin { frame}
\frametitle { Aliasing and noise}
\footnotesize
\begin { figure} [!hbtp]
\begin { center}
\includegraphics [height=0.5\textwidth] { images/channel_ curves_ SNRinf.png}
\caption { \footnotesize Evolution of the $ L - 2 $ norms of the components of $ \vec w $ }
\end { center}
\end { figure}
\frametitle { The inverse problem}
\begin { itemize}
\item Upon this solution $ \Longrightarrow $ build a set of measurements
\item Induce typical arifacts via a Magnetization vector: $ M ( \vec { x } ,t ) = M _ 0 ( \vec { x } ) exp ( i \phi _ 0 + i \frac { \pi } { venc } u ( \vec { x } ,t ) ) $
\end { itemize}
\begin { columns}
\column { .3\textwidth }
\flushleft
\begin { figure}
\includegraphics [width=1.1\textwidth] { images/ref_ int.png}
\caption * { (a) Interpolated reference solution}
\end { figure}
\column { .67\textwidth }
\flushleft
\begin { figure}
\includegraphics [width=1.1\textwidth] { images/coartation.png}
\caption * { (b) Aliased measurements with different $ vencs = 120 , 70 , 30 \% $ of $ u _ { max } $ }
\hfill
\end { figure}
\end { columns}
\end { frame}
\begin { frame}
\frametitle { Aliasing and noise}
\footnotesize
\frametitle { The Kalman Filter}
\begin { itemize}
\item <1-> We use a Reduced Order Unscendent Kalman Filter (ROUKF) to reconstruct the parameter vector $ \theta $ solving the next optimization problem:
\begin { figure} [!hbtp]
\begin { center}
\includegraphics [height=0.5\textwidth] { images/channel_ curves_ SNR10.png}
\caption { \footnotesize Evolution of the $ L - 2 $ norms of the components of $ \vec w $ }
\end { center}
\end { figure}
\onslide <2->
\begin { equation*}
\hat { \theta } = arg \min _ { \theta } J(\theta )
\end { equation*}
\begin { equation}
J(\theta ) = \displaystyle \frac { 1} { 2} || \theta - \theta _ 0 ||^ 2_ { P_ 0^ { -1} } + \sum _ { k=1} ^ N \frac { 1} { 2} || Z_ k - \mathbb { H} X_ k(\theta ) ||^ 2_ { W^ { -1} }
\end { equation}
\onslide <3-> Where:
\begin { itemize}
\item <4-> $ Z $ the measurements and $ X = ( \vec { u } , \pi ) $ the state variable
\item <5-> $ \mathbb { H } $ observation operator
\item <6-> $ \theta _ 0 $ is the initial guess for the parameters
\item <7-> $ P _ 0 $ is the associated covariance matrix
\item <8-> $ W $ is the associated covariance matrix to the meas. noise
\end { itemize}
\end { itemize}
\end { frame}
@ -522,255 +377,91 @@ For comparison we defined a perfect corrector field as: $\delta \vec u = \vec u_
\begin { frame}
\frametitle { Undersampling}
\footnotesize
\frametitle { The Kalman Filter}
The parameter vector:
\begin { itemize}
\item <1-> Plug flow at the inlet: $ u _ { inlet } = - U f ( t ) \hat { n } $ , with $ f ( t ) $ is the weaveform which simulate a cardiac cycle and $ \hat { n } $ is the outward normal vector.
\item <2-> Since $ R _ p << R _ d $ , we only consider an optimization dependent on $ R _ d, C $ for every 3D-0D coupled outlet
\end { itemize}
\begin { figure} [!hbtp]
\begin { center}
\includegraphics [height=0.6\textwidth] { images/histo_ channel.png}
\caption { \footnotesize Histograms of different undersampling rates for the channel}
\end { center}
\end { figure}
\onslide <3-> $$ \theta = ( U, \vec { R _ d } , \vec { C } ) $$ \\ with $ \vec { R _ d } = R _ { d,l } $ , $ \vec { C } = C _ l $ for $ l = 1 ,..., n _ l $
\end { frame}
% \begin { frame}
% \frametitle { Results for channel: undersampling}
% \footnotesize
%
% \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}
%
% \begin { columns} [c]
% \column { .6\textwidth } % Left column and width
% \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}
% \column { .5\textwidth } % Right column and width
% \footnotesize
% \begin { figure} [!hbtp]
% \begin { center}
% \includegraphics [height=1.0\textwidth] { images/aorta_ blender.png}
% \caption { Aortic mesh}
% \end { center}
% \end { figure}
% \end { columns}
%
%
% \end { frame}
%
%
% \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}
%
% \end { frame}
%
%
% \begin { frame}
% \frametitle { Results for aorta: undersampling}
% \footnotesize
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% \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}
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% \end { frame}
%
% \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}
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% \end { frame}
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%
\section [4D flow data] { Experiments using real 4D flow data }
\section { Numerical Experiments}
\begin { frame}
\frametitle { Experiments}
\frametitle { Numerical Experiments}
\begin { center}
Experiments using real 4D flow data
Numerical Experiments
\end { center}
\end { frame}
\begin { frame}
\frametitle { Experiments}
\footnotesize
\begin { columns} [c]
\column { .6\textwidth } % Left column and width
\begin { itemize}
\item <1-> 4D flow measurements were taken from a silicon thoracic aortic phantom made of silicon.
\item <2-> A controled pump (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)
\end { itemize}
\column { .5\textwidth } % Right column and width
\begin { figure} [!hbtp]
\begin { center}
\footnotesize
\includegraphics [height=\textwidth] { images/phantom.jpg}
\caption { \footnotesize { Experiment done at the Centre of Biomedical Images (CIB) of the Catholic Unversity of Chili (PUC)} }
\end { center}
\end { figure}
\end { columns}
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% ]{ \includegraphics [width=0.6\linewidth] { images/phantom.jpg} } { VPlayer.swf}
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\frametitle { Numerical Experiments}
\end { frame}
\begin { frame}
\frametitle { Results}
\footnotesize
\begin { figure}
\begin { subfigure} { .31\textwidth }
\centering
% \includegraphics [trim=100 80 100 150, clip, width=1.0\textwidth] { images/u_ 15.png}
\caption * { (a) $ \vec { u } _ { meas } $ }
\end { subfigure}
\begin { subfigure} { .01\textwidth }
\hfill
\end { subfigure}
\begin { subfigure} { .31\textwidth }
\centering
% \includegraphics [trim=100 80 100 150, clip, width=1.0\textwidth] { images/w_ 15.png}
\caption * { (b) $ \vec { w } $ }
\end { subfigure}
\begin { subfigure} { .01\textwidth }
\hfill
\end { subfigure}
\begin { subfigure} { .31\textwidth }
\centering
% \includegraphics [trim=100 80 100 150, clip, width=1.0\textwidth] { images/uc_ 15.png}
\caption * { (c) $ \vec { u } _ { meas } + \vec { w } $ }
\end { subfigure}
\caption { Measurements, corrector fields and corrected velocities for all the cases.}
\label { fig:phantom_ resolution}
\end { figure}
\end { frame}
\section { Conclusions}
\begin { frame}
\frametitle { Experiments}
\begin { center}
Conclusions
\end { center}
\end { frame}
\begin { frame}
\frametitle { Conclusions and future work}
\footnotesize
\onslide <1-> Potential of the new quality parameter:
\begin { itemize}
\item <2-> Vector fields has more details
\item <3-> Artifacts recognition
\end { itemize}
\onslide <4-> Future:
\begin { itemize}
\item <5-> The use of the field for create new inverse problems which can be used for further accelerations
\end { itemize}
\end { frame}
\begin { frame}
\begin { center}
\huge { Thank you for your time!}
\end { center}
\end { frame}
\end { document}
% \includegraphics <1>[height=4.5cm]{ images/pat1.png}
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\end { document}
% \begin { frame}
% \frametitle { Results}
% \footnotesize
%
% \begin { figure}
% \begin { subfigure} { .31\textwidth }
% \centering
% % \includegraphics [trim=100 80 100 150, clip, width=1.0\textwidth] { images/u_ 15.png}
% \caption * { (a) $ \vec { u } _ { meas } $ }
% \end { subfigure}
% \begin { subfigure} { .01\textwidth }
% \hfill
% \end { subfigure}
% \begin { subfigure} { .31\textwidth }
% \centering
% % \includegraphics [trim=100 80 100 150, clip, width=1.0\textwidth] { images/w_ 15.png}
% \caption * { (b) $ \vec { w } $ }
% \end { subfigure}
% \begin { subfigure} { .01\textwidth }
% \hfill
% \end { subfigure}
% \begin { subfigure} { .31\textwidth }
% \centering
% % \includegraphics [trim=100 80 100 150, clip, width=1.0\textwidth] { images/uc_ 15.png}
% \caption * { (c) $ \vec { u } _ { meas } + \vec { w } $ }
% \end { subfigure}
% \caption { Measurements, corrector fields and corrected velocities for all the cases.}
% \label { fig:phantom_ resolution}
% \end { figure}
%
% \end { frame}
