asd

presentation/pres03.tex

@@ -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{0.2cm}
+\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{0.2cm}
+\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{0.2cm}
+\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}
+
+%\includemedia[width=0.6\linewidth,height=0.6\linewidth,activate=pageopen,
+%passcontext,
+%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}