@ -19,6 +19,8 @@
\usepackage { multimedia}
\usepackage { media9}
% \usetheme { default}
% \usetheme { AnnArbor}
% \usetheme { Antibes}
@ -177,9 +179,75 @@ The corrector field $\vec{w}$ measures the level of agreedment of the 4D flow me
\end { frame}
\begin { frame}
\frametitle { The corrector field: Continuum problem}
\footnotesize
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
\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*}
or in simple terms:
\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 ) $ .
\end { frame}
\begin { frame}
\frametitle { The corrector field: Discrete problem}
\footnotesize
In the Discrete, we can write the problem as follows:
\begin { equation}
A_ { k} (\vec w,p;\vec v ,q ) + S^ { conv} _ { k} (\vec w;\vec v) + S^ { press} _ { k} (\vec w,p;\vec v ,q) = \mathcal { L} _ j (\vec v)
\end { equation}
With $ S ^ { conv } _ { k } ( \vec w; \vec v ) $ and $ S ^ { press } _ { k } ( \vec w,p; \vec v ,q ) $ terms for the stabilization of the convection and pressure respectively.
\begin { itemize}
\small
\item $
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 $
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 }
\item $
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 . 2 cm }
\item $ \mathcal { L } _ j ( \vec v ) : = \int _ { \Omega } \frac { \rho } { \tau } \vec { w } ^ { k - 1 } \cdot \vec { v } + \mathcal { \ell } _ j ( \vec v,q ) $
\end { itemize}
\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}
@ -254,7 +322,7 @@ All simulations were done using a stabilized finite element method implemented i
\begin { frame}
\frametitle { Results: aliasing and noise}
\frametitle { Results for channel : aliasing and noise}
\footnotesize
For comparison we defined a perfect corrector field as: $ \delta \vec u = \vec u _ { ref } - \vec u _ { meas } $
@ -271,35 +339,107 @@ For comparison we defined a perfect corrector field as: $\delta \vec u = \vec u_
\begin { frame}
\frametitle { Results: undersampling}
\frametitle { Results for channel : undersampling}
\footnotesize
\begin { columns} [c]
\column { .6\textwidth } % Left column and width
other results concerning undersampling....
\column { .5\textwidth } % Right column and width
\begin { figure} [!hbtp]
\begin { center}
\includegraphics [height= 0.6 \textwidth]{ images/undersampling_ final.png}
\caption { Different perturbation scenarios }
\includegraphics [height= 1.2 \textwidth]{ images/undersampling_ final.png}
\caption { \footnotesize Different undersampling rates for the channel }
\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
\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}
\section [4D flow data] { Experiments using real 4D flow data }
\begin { frame}
\frametitle { Experiments}
\begin { center}
Experiments using real 4D flow data
\end { center}
\end { frame}
\begin { frame}
\frametitle { Experiments}
\footnotesize
\begin { columns} [c]
\column { .6\textwidth } % Left column and width
\begin { itemize}
\item We performed 4D flow measurements in a silicon aortic phantom
\item 4 healthy volunteers were scanned using a clinical standard 4D flow protocol.
\item 4D flow measurements were taken from a silicon thoracic aortic phantom made of silicon.
\item 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 A stenosis of $ 11 \ mm $ of diameter was added in the descending aorta
\item 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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% transparent,
% addresource=images/phantom.mp4,
% flashvars={ source=images/phantom.mp4}
% ]{ \includegraphics [width=0.6\linewidth] { images/phantom.jpg} } { VPlayer.swf}
%
\end { frame}
@ -310,7 +450,12 @@ For comparison we defined a perfect corrector field as: $\delta \vec u = \vec u_
\frametitle { Results}
\footnotesize
results for experimental phantom
\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}
\end { frame}