### asd

This commit is contained in:
parent c4efe84537
commit 197e4dcbdc
7 changed files with 152 additions and 7 deletions

#### BIN presentation/images/aorta_perturbation.png Normal file View File

Binary file not shown.
 After Width:  |  Height:  |  Size: 4.3 MiB

#### BIN presentation/images/perturbation_pres.png View File

Binary file not shown.
 Before Width:  |  Height:  |  Size: 4.0 MiB After Width:  |  Height:  |  Size: 4.0 MiB

#### BIN presentation/images/phantom.jpg Normal file View File

Binary file not shown.
 After Width:  |  Height:  |  Size: 4.5 MiB

#### BIN presentation/images/phantom.mp4 Normal file View File

Binary file not shown.

#### BIN presentation/images/phantom_cib.png Normal file View File

Binary file not shown.
 After Width:  |  Height:  |  Size: 3.0 MiB

#### BIN presentation/images/undersampling_blender.png Normal file View File

Binary file not shown.
 After Width:  |  Height:  |  Size: 4.8 MiB

#### 159 presentation/pres03.tex View File

 @ -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}