diff --git a/presentation/images/aorta_perturbation.png b/presentation/images/aorta_perturbation.png new file mode 100644 index 0000000..a44450d Binary files /dev/null and b/presentation/images/aorta_perturbation.png differ diff --git a/presentation/images/perturbation_pres.png b/presentation/images/perturbation_pres.png index 0334924..f1ff03b 100644 Binary files a/presentation/images/perturbation_pres.png and b/presentation/images/perturbation_pres.png differ diff --git a/presentation/images/phantom.jpg b/presentation/images/phantom.jpg new file mode 100644 index 0000000..0dbcc58 Binary files /dev/null and b/presentation/images/phantom.jpg differ diff --git a/presentation/images/phantom.mp4 b/presentation/images/phantom.mp4 new file mode 100644 index 0000000..884e0d5 Binary files /dev/null and b/presentation/images/phantom.mp4 differ diff --git a/presentation/images/phantom_cib.png b/presentation/images/phantom_cib.png new file mode 100644 index 0000000..c69cbdb Binary files /dev/null and b/presentation/images/phantom_cib.png differ diff --git a/presentation/images/undersampling_blender.png b/presentation/images/undersampling_blender.png new file mode 100644 index 0000000..0d2eef6 Binary files /dev/null and b/presentation/images/undersampling_blender.png differ diff --git a/presentation/pres03.tex b/presentation/pres03.tex index 0b380f2..54747a4 100755 --- a/presentation/pres03.tex +++ b/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}