J.E. Garay Labra 2 years ago
parent
commit
197e4dcbdc
  1. BIN
      presentation/images/aorta_perturbation.png
  2. BIN
      presentation/images/perturbation_pres.png
  3. BIN
      presentation/images/phantom.jpg
  4. BIN
      presentation/images/phantom.mp4
  5. BIN
      presentation/images/phantom_cib.png
  6. BIN
      presentation/images/undersampling_blender.png
  7. 159
      presentation/pres03.tex

BIN
presentation/images/aorta_perturbation.png

Binary file not shown.

After

Width:  |  Height:  |  Size: 4.3 MiB

BIN
presentation/images/perturbation_pres.png

Binary file not shown.

Before

Width:  |  Height:  |  Size: 4.0 MiB

After

Width:  |  Height:  |  Size: 4.0 MiB

BIN
presentation/images/phantom.jpg

Binary file not shown.

After

Width:  |  Height:  |  Size: 4.5 MiB

BIN
presentation/images/phantom.mp4

Binary file not shown.

BIN
presentation/images/phantom_cib.png

Binary file not shown.

After

Width:  |  Height:  |  Size: 3.0 MiB

BIN
presentation/images/undersampling_blender.png

Binary file not shown.

After

Width:  |  Height:  |  Size: 4.8 MiB

159
presentation/pres03.tex

@ -19,6 +19,8 @@ @@ -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 @@ -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 @@ -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_ @@ -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_ @@ -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}

Loading…
Cancel
Save