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presentations/press_4dflow/pres03.tex

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+\usepackage{amsmath}
+%\documentclass{beamer}
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+\usepackage{multicol} % indice en 2 columnas
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+
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+
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+\usepackage{hyperref}
+
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+%\usecolortheme{beetle}
+\usecolortheme{crane}
+%\usecolortheme{dolphin}
+%\usecolortheme{dove}
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+%\usecolortheme{lily}
+%\usecolortheme{orchid}
+%\usecolortheme{rose}
+%\usecolortheme{seagull}
+%\usecolortheme{seahorse}
+%\usecolortheme{whale}
+%\usecolortheme{wolverine}
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+\setbeamertemplate{navigation symbols}{} % quitar simbolitos
+
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+\setbeamerfont{page number in head/foot}{size=\large}
+%\setbeamertemplate{footline}[frame number] number in footer
+\setbeamertemplate{footline}{}
+
+
+
+\title[A new mathematical model for verifying the Navier-Stokes compatibility of 4D flow MRI data]{ A new mathematical model for verifying the Navier-Stokes compatibility of 4D flow MRI}
+%\author[Jeremías Garay Labra]
+%{Jeremías Garay Labra}
+\institute[University of Groningen]
+{
+Bernoulli Institute\\
+Faculty of Sciences and Engineering\\
+University of Groningen\\[0.5cm]
+ %\includegraphics[height=1.5cm]{Imagenes/escudoU2014.pdf}   
+  %  \includegraphics[height=1cm]{Imagenes/fcfm.png} \\[0.5cm]
+  Jeremías Garay Labra \emph{join with} Hernan Mella, Julio Sotelo, Sergio Uribe, Cristobal Bertoglio and Joaquin Mura.}
+\date{\today}
+
+
+\begin{document}
+\frame{\titlepage}
+
+
+% \onslide<1->
+
+
+
+
+\begin{frame}
+  \frametitle{Index}
+  \tableofcontents
+\end{frame}
+
+
+\section[4D flow MRI]{4D flow MRI}
+\begin{frame}
+  \frametitle{4D flow MRI}  
+ \begin{columns}[c] 
+\column{.5\textwidth} % Left column and width
+\footnotesize
+
+\begin{itemize}
+\item<2-> Full 3D coverage of the region of interest
+\item<3-> Rich post-proccesing: derived parameters 
+\end{itemize}
+
+\onslide<4-> Disadvantages:
+\begin{itemize}
+\item<5-> Long scan time
+\end{itemize}
+
+
+
+
+\column{.54\textwidth} % Right column and width
+\onslide<1-> 
+\begin{figure}[!hbtp]
+ \begin{center}
+  \includegraphics[height=0.9\textwidth]{images/4dflow.png}
+  \caption{\footnotesize 4D flow MRI of a human thorax}
+ \end{center}
+ \end{figure}
+\end{columns}
+\end{frame}
+
+
+\begin{frame}
+  \frametitle{4D flow MRI}
+\footnotesize
+\onslide<1-> Strategies:
+\begin{itemize}
+\item<2-> modest spatial resolutions $ \sim (2.5 \times 2.5 \times 2.5 \ mm^3)$
+\item<3-> partial data coverage  
+\end{itemize}
+
+
+\begin{columns}[c] 
+\column{.4\textwidth} % Right column and width
+\onslide<4->
+\footnotesize
+\begin{figure}[!hbtp]
+ \begin{center}
+  \includegraphics[height=0.25\textwidth]{images/channel_noise.png} \\
+  (a) Noise
+  %\caption{Noise}
+ \end{center}
+ \end{figure}
+ \column{.4\textwidth} % Right column and width
+\onslide<5->
+\footnotesize
+\begin{figure}[!hbtp]
+ \begin{center}
+  \includegraphics[height=0.25\textwidth]{images/channel_aliasing.png}\\
+  (b) Aliasing
+  %\caption{Aliasing}
+ \end{center}
+ \end{figure}
+  \column{.4\textwidth} % Right column and width
+\onslide<6->
+\footnotesize
+\begin{figure}[!hbtp]
+ \begin{center}
+  \includegraphics[height=0.25\textwidth]{images/channel_under.png}\\
+  (c) Undersampling
+  %\caption{Aliasing}
+ \end{center}
+ \end{figure}
+\end{columns}
+
+\vspace{0.5cm}
+
+\onslide<7-> Typical quality estimators: SNR, VNR, peak flows/velocities, mass conservation (zero divergence) 
+
+\vspace{0.5cm}
+
+\onslide<8-> This work $\longrightarrow$  \textbf{conservation of linear momentum} (Navier-Stokes compatibility).
+
+\end{frame}
+
+
+\section[]{The corrector field}
+
+\begin{frame}
+ \frametitle{The corrector field}
+\begin{center}
+Methodology
+\end{center}
+\end{frame}
+
+
+
+\begin{frame}
+ \frametitle{The corrector field}
+\footnotesize
+
+\onslide<1-> We assume a perfect physical velocity field $\vec{u}$ 
+\onslide<2-> \begin{eqnarray*}
+\rho \frac{\partial \vec{u}}{\partial t} + \rho \big ( \vec{u}  \cdot \nabla \big) \vec{u} - \mu \Delta \vec{u} + \nabla p =  0 \quad \text{in} \quad \Omega \label{eq:NSmom}
+\end{eqnarray*}
+
+\onslide<3-> And a corrector field $\vec{w}$ which satisfies:
+\onslide<4-> \begin{align}
+ \vec{u}  & =   \vec{u}_{meas} + \vec{w} \quad \text{in} \quad \Omega \label{eq:corrector}\\
+\nabla \cdot \vec w  & =   0  \quad \text{in}  \quad \Omega \label{eq:correctorDiv} \\
+\vec w & =   \vec 0  \quad \text{on} \quad \partial \Omega \label{eq:correctorBC}
+\end{align}
+
+\onslide<5-> The corrector field $\vec{w}$ measures the level of agreedment of the 4D flow measures respect to the Navier-Stokes equations.
+
+\end{frame}
+
+
+\begin{frame}
+ \frametitle{The corrector field: Continuum problem}
+\footnotesize
+
+\onslide<1-> Applying the decomposition $\vec{u}   =  \vec{u}_{meas} + \vec{w}$ into the original equation and writing a variational problem for $\vec w$ we have:\\[0.2cm]
+Find $(\vec w(t) ,p(t)) \in H^1_0(\Omega)\times L^2(\Omega)$ such that: 
+\onslide<2-> \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*}
+
+\vspace{0.2cm}
+
+\onslide<3-> or in simple terms:
+\onslide<4-> \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
+
+\onslide<1-> In the Discrete, we can write the problem as follows:
+
+\onslide<2-> \begin{equation}
+A_{k}(\vec w,p;\vec v ,q ) + \color{blue}{S^{press}_{k}(\vec w,p;\vec v ,q)} + \color{red}{S^{conv}_{k}(\vec w;\vec v)} \color{black}{ =  \mathcal{L}_j (\vec v)}
+\label{eq:Corrector_discrete}
+\end{equation}
+
+\begin{itemize} 
+\small
+\item<3-> $
+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<3-> $ \mathcal{L}_j (\vec v) := \int_{\Omega} \frac{\rho}{\tau} \vec{w}^{k-1} \cdot \vec{v} + \mathcal{\ell}_j (\vec v,q)   $
+ \vspace{0.2cm}
+\item<4-> \color{blue}$
+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<5-> \color{red}$
+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}
+
+\end{itemize}
+
+\end{frame}
+
+
+
+
+\begin{frame}
+ \frametitle{The corrector field: Well-posedness}
+\footnotesize
+\onslide<1->
+\begin{theorem} 
+There exists a unique solution of Problem (\ref{eq:Corrector_discrete}) under the condition: $$\rho/\tau + C_\Omega^{-2} \mu/2 - \rho 3 \| \nabla\vec u_{meas}^k\|_\infty > 0$$ for all $k>0$.
+\end{theorem}
+\onslide<2->
+We can furthermore prove the following energy balance:
+\onslide<3->
+\begin{theorem} For $(\vec w^k ,p^k)$ solution of Problem (\ref{eq:Corrector_discrete}),  with $\ell_j(\vec v,q)=0$ it holds
+\begin{equation*}\label{eq:energy}
+   \| \vec w^k \|^2_{L_2(\Omega)} \leq \| \vec w^{k-1} \|^2_{L_2(\Omega)} 
+\end{equation*}
+under the condition
+\begin{equation*}\label{eq:condstab}
+\mu \geq C_\Omega^2 \rho \| \nabla \vec u_{meas}^k\|_\infty
+\end{equation*}
+\end{theorem}
+
+
+\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}
+ 
+\onslide<1-> 
+\footnotesize
+\begin{columns}[c] 
+\column{.4\textwidth} % Right column and width
+\footnotesize
+ Simulated channel flow as measurements (Stokes flow)
+ \column{.5\textwidth} % Right column and width
+\footnotesize
+\begin{figure}[!hbtp]
+ \begin{center}
+  \includegraphics[height=0.35\textwidth]{images/cilinder_2.png}\\
+  (b) Channel mesh
+  %\caption{Aliasing}
+ \end{center}
+ \end{figure}
+\end{columns}
+
+
+\vspace{0.2cm}
+
+%\onslide<1-> We tested the corrector using CFD simulations as a measurements, in the following testcases:
+%\onslide<2->
+%\begin{itemize}
+%\item Womersley flow in a cilinder
+%\item Navier-Stokes simulations in an aortic mesh
+%\end{itemize}
+\onslide<2-> Afterwards, perturbations were added: 
+\begin{itemize}
+\item<3->  velocity aliasing (varying the $venc$ parameter)
+\item<4->  additive noise (setting SNR in decibels)
+\item<5->  simulated k-space undersampling (compressed sensing for the reconstruction)
+\end{itemize}
+%\onslide<7-> All simulations were done using a stabilized finite element method implemented in FEniCS. Afterwards, all numerical simulations were interpolated into a voxel-type structured mesh
+\end{frame}
+
+%
+%\begin{frame}
+% \frametitle{Numerical tests: channel}
+%\begin{columns}[c] 
+%\column{.6\textwidth} % Left column and width
+%\footnotesize
+%\textbf{Channel:}
+%\begin{itemize}
+%\item Convective term was neglected
+%\item Non-slip condition at walls
+%\item Oscilatory pressure at $\Gamma_{inlet}$
+%\end{itemize}
+%\column{.5\textwidth} % Right column and width
+%\footnotesize
+%\begin{figure}[!hbtp]
+% \begin{center}
+%  \includegraphics[height=1.0\textwidth]{images/cilinder.png}
+%  \caption{3D channel mesh}
+% \end{center}
+% \end{figure}
+%\end{columns}
+%\end{frame}
+%
+
+\begin{frame}
+ \frametitle{Numerical tests}
+\begin{center}
+Results
+\end{center}
+\end{frame}
+
+
+
+
+\begin{frame}
+ \frametitle{Aliasing and noise}
+\footnotesize
+
+\onslide<1-> For comparison we defined a perfect corrector field as: $\delta \vec u = \vec u_{ref} - \vec u_{meas}$
+
+\onslide<2->
+\begin{figure}[!hbtp]
+ \begin{center}
+  \includegraphics[height=0.45\textwidth]{images/channel_ppt_1.png}
+\caption{\small Fields for the channel: $(SNR,venc) = (\infty,120\%)$. $\vec{w} \times 200$}
+ \end{center}
+ \end{figure}
+
+\end{frame}
+
+\begin{frame}
+ \frametitle{Aliasing and noise}
+\footnotesize
+
+For comparison we defined a perfect corrector field as: $\delta \vec u = \vec u_{ref} - \vec u_{meas}$
+
+
+\begin{figure}[!hbtp]
+ \begin{center}
+  \includegraphics[height=0.45\textwidth]{images/channel_ppt_2.png}
+  \caption{\small Fields for the channel: $(SNR,venc) = (\infty,80\%)$. $\vec{w} \times 4$ }
+%\caption{\small Different perturbation scenarios. $(\infty , 120 \%)$: $\vec{w} \times 200$, $(10 \ dB , 120 \%)$: $\delta \vec{u}, \vec{w} \times 4$, rest: $\vec{w} \times 4$ }
+ \end{center}
+ \end{figure}
+
+\end{frame}
+
+
+\begin{frame}
+ \frametitle{Aliasing and noise}
+\footnotesize
+For comparison we defined a perfect corrector field as: $\delta \vec u = \vec u_{ref} - \vec u_{meas}$
+
+\begin{figure}[!hbtp]
+ \begin{center}
+  \includegraphics[height=0.45\textwidth]{images/channel_ppt_3.png}
+  \caption{\small Fields for the channel: $(SNR,venc) = (10 \ dB,120\%)$. $\delta \vec{u}, \vec{w} \times 4$}
+%\caption{\small Different perturbation scenarios. $(\infty , 120 \%)$: $\vec{w} \times 200$, $(10 \ dB , 120 \%)$: $\delta \vec{u}, \vec{w} \times 4$, rest: $\vec{w} \times 4$ }
+ \end{center}
+ \end{figure}
+
+\end{frame}
+
+
+\begin{frame}
+ \frametitle{Aliasing and noise}
+\footnotesize
+For comparison we defined a perfect corrector field as: $\delta \vec u = \vec u_{ref} - \vec u_{meas}$
+
+\begin{figure}[!hbtp]
+ \begin{center}
+  \includegraphics[height=0.45\textwidth]{images/channel_ppt_4.png}
+  \caption{\small Fields for the channel: $(SNR,venc) = (10 \ dB,80\%)$.  $\vec{w} \times 4$}
+%\caption{\small Different perturbation scenarios. $(\infty , 120 \%)$: $\vec{w} \times 200$, $(10 \ dB , 120 \%)$: $\delta \vec{u}, \vec{w} \times 4$, rest: $\vec{w} \times 4$ }
+ \end{center}
+ \end{figure}
+
+\end{frame}
+
+
+
+\begin{frame}
+ \frametitle{Aliasing and noise}
+\footnotesize
+
+\begin{figure}[!hbtp]
+ \begin{center}
+  \includegraphics[height=0.5\textwidth]{images/channel_curves_SNRinf.png}
+\caption{ \footnotesize Evolution of the $L-2$ norms of the components of $\vec w$}
+ \end{center}
+ \end{figure}
+
+
+\end{frame}
+
+
+\begin{frame}
+ \frametitle{Aliasing and noise}
+\footnotesize
+
+\begin{figure}[!hbtp]
+ \begin{center}
+  \includegraphics[height=0.5\textwidth]{images/channel_curves_SNR10.png}
+\caption{ \footnotesize Evolution of the $L-2$ norms of the components of $\vec w$}
+ \end{center}
+ \end{figure}
+
+
+\end{frame}
+
+
+
+
+\begin{frame}
+ \frametitle{Undersampling}
+\footnotesize
+
+\begin{figure}[!hbtp]
+ \begin{center}
+  \includegraphics[height=0.6\textwidth]{images/histo_channel.png}
+\caption{ \footnotesize Histograms of different undersampling rates for the channel}
+ \end{center}
+ \end{figure}
+
+\end{frame}
+
+
+
+
+%\begin{frame}
+% \frametitle{Results for channel: undersampling}
+%\footnotesize
+%
+%\begin{figure}[!hbtp]
+% \begin{center}
+%  \includegraphics[height=0.6\textwidth]{images/undersampling_press.png}
+%\caption{ \footnotesize Different undersampling rates for the channel}
+% \end{center}
+% \end{figure}
+%
+%
+%\end{frame}
+%
+
+
+
+%\begin{frame}
+% \frametitle{Numerical tests: aorta}
+%
+%\begin{columns}[c] 
+%\column{.6\textwidth} % Left column and width
+%\footnotesize
+%\textbf{Aorta}
+%\begin{itemize}
+%\item a mild coartation was added in the descending aorta
+%\item $u_{inlet}$ simulates a cardiac cycle
+%\item 3-element Windkessel for the outlets
+%\item Non-slip condition at walls
+%\end{itemize}
+
+%\column{.5\textwidth} % Right column and width
+%\footnotesize
+%\begin{figure}[!hbtp]
+% \begin{center}
+%  \includegraphics[height=1.0\textwidth]{images/aorta_blender.png}
+%\caption{Aortic mesh}
+% \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=0.6\textwidth]{images/histo_blender.png}
+%\caption{ \footnotesize Histograms of different undersampling rates 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<1-> 4D flow measurements were taken from a silicon thoracic aortic phantom made of silicon.
+\item<2-> A controled pump (heart rate, peak flow, stroke volume and flow waveform)
+\item<3-> A stenosis of $11 \ mm$ of diameter was added in the descending aorta 
+\item<4->  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}
+
+
+
+
+
+
+\begin{frame}
+ \frametitle{Results}
+\footnotesize
+
+\begin{figure}
+\begin{subfigure}{.31\textwidth}
+  \centering
+        \includegraphics[trim=100 80 100 150, clip, width=1.0\textwidth]{images/u_15.png}
+  \caption*{(a) $\vec{u}_{meas}$}
+\end{subfigure}
+\begin{subfigure}{.01\textwidth}
+ \hfill
+\end{subfigure}
+\begin{subfigure}{.31\textwidth}
+  \centering
+        \includegraphics[trim=100 80 100 150, clip, width=1.0\textwidth]{images/w_15.png}
+  \caption*{(b) $\vec{w}$}
+\end{subfigure}
+\begin{subfigure}{.01\textwidth}
+ \hfill
+\end{subfigure}
+\begin{subfigure}{.31\textwidth}
+  \centering
+        \includegraphics[trim=100 80 100 150, clip, width=1.0\textwidth]{images/uc_15.png}
+  \caption*{(c) $\vec{u}_{meas}+\vec{w}$}
+\end{subfigure}
+\caption{Measurements, corrector fields and corrected velocities for all the cases.}
+\label{fig:phantom_resolution}
+\end{figure}
+
+\end{frame}
+
+
+
+
+
+\section{Conclusions}
+
+\begin{frame}
+ \frametitle{Experiments}
+\begin{center}
+Conclusions
+\end{center}
+\end{frame}
+
+
+\begin{frame}
+ \frametitle{Conclusions and future work}
+\footnotesize
+
+\onslide<1->  Potential of the new quality parameter:
+
+\begin{itemize}
+\item<2-> Vector fields has more details
+\item<3-> Artifacts recognition
+\end{itemize}
+
+
+\onslide<4->  Future:
+\begin{itemize}
+\item<5-> The use of the field for create new inverse problems which can be used for further accelerations
+\end{itemize}
+
+
+
+\end{frame}
+
+
+
+
+
+\begin{frame}
+\begin{center}
+\huge{Thank you for your time!}
+\end{center}
+\end{frame}
+
+
+
+
+
+
+
+%\includegraphics<1>[height=4.5cm]{images/pat1.png}
+%\includegraphics<2>[height=4.5cm]{images/pat2.png}
+
+
+ 
+\end{document}
+

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+\BOOKMARK [2][]{Outline0.1}{Introduction}{}% 1
+\BOOKMARK [2][]{Outline0.2}{Application: Parameter recovery}{}% 2
+\BOOKMARK [2][]{Outline0.3}{Summary}{}% 3

presentations/press_kalman/press.tex

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+
+\documentclass{beamer}
+
+
+\usetheme{Boadilla}
+\usefonttheme[onlylarge]{serif}
+\usebeamercolor{dolphin}
+\setbeamerfont*{frametitle}{size=\normalsize,series=\bfseries}
+\setbeamertemplate{navigation symbols}{}
+
+
+% Standard packages
+
+\usepackage[english]{babel}
+\usepackage[latin1]{inputenc}
+\usepackage{times}
+\usepackage[T1]{fontenc}
+
+
+% Setup TikZ
+
+\usepackage{tikz}
+\usetikzlibrary{arrows}
+\tikzstyle{block}=[draw opacity=0.7,line width=1.4cm]
+
+
+% Author, Title, etc.
+
+\title[] 
+{%
+	Data assimilation on the Kalman filter
+}
+
+\author[Garay]
+{
+  Jeremias Garay %\inst{1}
+  %\textcolor{green!50!black}{Till~Tantau}\inst{5}
+}
+
+%\institute[University of Groningen]
+%{
+%  \inst{1}%
+%  University of Groningen, The Netherlands
+%  \and
+%  \vskip-2mm
+%}
+
+\date
+
+
+
+% The main document
+
+\begin{document}
+
+\begin{frame}
+  \titlepage
+\end{frame}
+
+%\begin{frame}{Outline}
+%  \tableofcontents
+%\end{frame}
+
+
+
+
+\section{Introduction}
+
+\begin{frame}
+\begin{center}
+\large{Introduction}
+\end{center}
+\end{frame}
+
+
+
+
+\begin{frame}{Stationary Case: Least square estimation}
+
+\onslide<1->
+\textit{Assume we want to find an estimator $\hat{X}$ of a unknown vector $X$, with a certain guess available $\hat{X}^-$, associated with a confidence matrix $(P^-)^{-1}$. Assume also that we have partial observation $Z$, satisfying $Z = HX + \zeta^Z$, associated with a confidence matrix $W^{-1}$.}  \\[0.5cm]
+
+\onslide<2->
+A quantity taking care of $\hat{X}^{-}$ and $Z$ can be obtained minimizing the cuadratic cost functional:
+
+\onslide<3->
+\begin{equation}
+J(\hat{X}) = \frac{1}{2} (\hat{X} - \hat{X}^-) (P^-)^{-1} (\hat{X} - \hat{X}^-)   +    \frac{1}{2} (Z -H\hat{X}) W^{-1} (Z - H\hat{X})
+\end{equation}
+
+\end{frame}
+
+
+
+
+\begin{frame}{Stationary Case: Least square estimation}
+
+\onslide<1->
+Find the optimal state imposing: $\frac{dJ}{d\hat{X}}(\hat{X}^+) = 0$:
+
+\onslide<2->
+\begin{eqnarray*}
+-H^T W^{-1} Z + H^T W^{-1} H \hat{X} - (P^-)^{-1} \hat{X}^- + (P^-)^{-1} \hat{X} \equiv 0
+\end{eqnarray*}
+
+\onslide<3->
+or reordering terms:
+
+\onslide<4->
+\begin{equation*}
+\hat{X}^+ = \hat{X}^- + K (Z-H\hat{X}^-)
+\end{equation*}
+\vspace{0.4cm}
+
+With $K = P^+ H^T W^{-1}$ the Kalman matrix and $P^+ = ((P^-)^{-1} + H^T W^{-1} H)^{-1}$.
+
+\end{frame}
+
+
+
+\begin{frame}{Time dependent problems}
+
+\onslide<1->
+The method could be easily expanded into time-dependent systems ($\dot{X} = AX + F$):
+
+
+\begin{itemize}
+\item[1.]<2-> Assume that $\hat{X}^{+}_{n-1}$ is known with a covariance $P^+_{n-1}$
+
+    \begin{exampleblock}{Prediction}
+	$$\hat{X}^{-}_n = A_n  \hat{X}^{+}_{n-1}  + F_n$$
+	by linearity of $A_n$, the covariance of $\hat{X}^-_n$ is equal to $A_n P_{n-1}^+ A_n^T$
+    \end{exampleblock}
+
+\item[2. ]<3-> Afterwards
+
+    \begin{exampleblock}{Correction}
+	$$\hat{X}^{+}_n = \hat{X}^{-}_n + K_n (Z_n - H_n \hat{X}_n^-) $$
+
+    \end{exampleblock}
+
+\end{itemize}
+\end{frame}
+
+
+\begin{frame}{Non-linear problems}
+
+\begin{itemize}
+\item[1.]<1-> \emph{Extended Kalman Filter (EKF)}
+
+\begin{itemize}
+\item[a.]<2-> Taylor's expansion on the non-linear operator (tangent operators)
+\item[b.]<3-> High cost if the Jacobian can be found numerically
+\item[c.]<4-> Not optimal when the system is highly non-linear
+\end{itemize}
+
+\item[2.]<5-> \emph{Unscented Kalman Filter (UKF)}
+
+\begin{itemize}
+\item[a.]<6-> Approximate propagation of vectors by propagating suitable particles
+\item[b.]<7-> Could be shown that by computing mean and covariance of the particles, a better approx could be reached. 
+\end{itemize}
+
+\item[3.]<8-> \emph{Reduced Order Unscented Kalman Filter (ROUKF)}
+
+\begin{itemize}
+\item[a.]<9-> LU factorization could be performed on the covariance matrix $P_n^-$
+\end{itemize}
+
+
+\end{itemize}
+
+
+\end{frame}
+
+
+
+
+
+
+\begin{frame}{Graphical Picture: Initial State}
+\begin{figure}
+\includegraphics[width=\textwidth]{pictures/kalman1.png}
+\end{figure}
+\end{frame}
+
+\begin{frame}{Graphical Picture: Initial State}
+\begin{figure}
+\includegraphics[width=\textwidth]{pictures/kalman2.png}
+\end{figure}
+\end{frame}
+
+\begin{frame}{Graphical Picture: Prediction}
+\begin{figure}
+\includegraphics[width=\textwidth]{pictures/kalman3.png}
+\end{figure}
+\end{frame}
+
+
+\begin{frame}{Graphical Picture: Updating Measurements}
+\begin{figure}
+\includegraphics[width=\textwidth]{pictures/kalman4.png}
+\end{figure}
+\end{frame}
+
+\begin{frame}{Graphical Picture: Correction}
+\begin{figure}
+\includegraphics[width=\textwidth]{pictures/kalman5.png}
+\end{figure}
+\end{frame}
+
+
+
+\section{Application: Parameter recovery}
+
+\begin{frame}
+\begin{center}
+\large{Application: Parameter recovery}
+\end{center}
+\end{frame}
+
+
+
+
+\begin{frame}{Application: Parameter recovery}
+\onslide<1-> Consider a Poiseuille flow in a cylinder coming from a simulation. Assume we have:
+
+  \begin{columns}[t]
+    \column{.4\textwidth}
+    
+
+\vspace{0.3cm}
+
+\column{.4\textwidth}
+
+\onslide<2->
+
+\begin{figure}
+\includegraphics[width=1.3\textwidth]{pictures/u_ref.png}
+\vspace{1.5cm}
+\end{figure}
+
+  \end{columns}
+\end{frame}
+
+
+
+
+\begin{frame}{Application: Parameter recovery}
+ Consider a Poiseuille flow in a cylinder coming from a simulation. Assume we have:
+
+  \begin{columns}[t]
+    \column{.4\textwidth}
+    
+\begin{itemize}
+\item[1.]The velocity measurements with the addition of some noise
+\item[2.]<2-> The measurement's mesh
+\end{itemize} 
+\vspace{0.3cm}
+
+\onslide<3-> We want to estimate the amplitude of the inlet flow
+
+\onslide<4-> $$ u_{inlet} = \alert{U} \ (R^2-r^2) \ sin(\pi t / T) $$
+
+\column{.4\textwidth}
+\onslide<1->
+\begin{figure}
+\includegraphics[width=1.3\textwidth]{pictures/u_noi.png}
+\vspace{1.5cm}
+\end{figure}
+
+  \end{columns}
+\end{frame}
+
+
+
+
+\begin{frame}{Application: Parameter recovery}
+
+\begin{figure}
+\includegraphics[width=0.8\textwidth]{pictures/channel_inlet.png}
+\end{figure}
+
+\begin{itemize}
+\item[] Reparametrized value: $\theta_0 \cdot 2^\theta$
+\end{itemize}
+
+\end{frame}
+
+
+
+
+\begin{frame}{Application: More complex scenario}
+ 
+ \onslide<1-> Aortic velocity data with reduced order boundary condition:
+
+  \begin{columns}[t]
+    \column{.55\textwidth}
+    
+\begin{itemize}
+\item[1.]<2-> Navier-Stokes simulation with a \emph{plug-flow} at the intlet:
+\[   
+u_{inlet} = 
+     \begin{cases}
+        U sin(\pi t/T) & \text{if} \ t<T^* \\
+        \alpha U sin( \pi t/T')e^{- \gamma t} & \text{if} \ t \geq T^*\\
+     \end{cases}
+\]
+\item[2.]<3-> A 1-element Windkessel boundary condition is defined in every inlet.
+
+\end{itemize} 
+
+\vspace{0.3cm}
+
+\onslide<4-> We want to recover the proximal resistances $R_i$, $i=1,2,3,4$ and the amplitude $U$ from noisy velocity measurements.
+
+
+
+\column{.4\textwidth}
+
+\onslide<1->
+\begin{figure}
+\includegraphics[width=1.0\textwidth]{pictures/windk_model.png}
+\end{figure}
+  \end{columns}
+
+\end{frame}
+
+
+\begin{frame}{Application: Parameter recovery $\theta_0 \ 2^\theta$}
+
+\begin{figure}
+\includegraphics[width=0.9\textwidth]{pictures/windk_res.png}
+\end{figure}
+\end{frame}
+
+
+
+\begin{frame}{Application: Parameter recovery}
+
+\begin{tabular}{ l c r }
+      & \emph{true} & \emph{recovered} \\[0.1cm]
+      \hline
+  $R_1 \ (dyn\cdot s \cdot cm^{-5})$ & $250$ & $242.14$ \\
+  $R_2 \ (dyn\cdot s \cdot cm^{-5})$ & $250$ & $249.16$ \\
+  $R_3 \ (dyn\cdot s \cdot cm^{-5})$ & $250$ & $246.03$ \\
+  $R_4 \ (dyn\cdot s \cdot cm^{-5})$ & $10$ & $9.87$ \\
+  $U \ (cm/s)$ & $30$ & $29.94$ \\
+    
+\end{tabular}
+
+\end{frame}
+
+
+\begin{frame}{Application: Parameter recovery (\alert{only using 1 vel. component})}
+
+\onslide<2->
+\begin{figure}
+\includegraphics[width=0.9\textwidth]{pictures/windk_res2.png}
+\end{figure}
+\end{frame}
+
+
+
+\begin{frame}{Application: Parameter recovery}
+
+\begin{tabular}{ l c c c }
+      & \emph{true} & \emph{recovered} & \emph{recovered with reduced vel}\\[0.1cm]
+      \hline
+  $R_1 \ (dyn\cdot s \cdot cm^{-5})$ & $250$ &  $242.14$ & $247.31$ \\
+  $R_2 \ (dyn\cdot s \cdot cm^{-5})$ & $250$ &  $249.16$ & $255.56$ \\
+  $R_3 \ (dyn\cdot s \cdot cm^{-5})$ & $250$ & $246.03$ & $277.37$ \\
+  $R_4 \ (dyn\cdot s \cdot cm^{-5})$ & $10$ & $9.87$ & $8.03$ \\
+  $U \ (cm/s)$ & $30$ & $29.94$ & $29.80$ \\
+    
+\end{tabular}
+
+\end{frame}
+
+
+
+
+
+
+
+
+
+
+
+
+\section{Summary}
+
+
+
+\begin{frame}
+\begin{center}
+\large{Summary}
+\end{center}
+\end{frame}
+
+
+\begin{frame}
+  \frametitle<presentation>{Summary}
+
+  \begin{itemize}
+  \item<1->
+    Kalman's filter uses a series of measurements and produce an estimate in two steps: Prediction and Correction
+   \item<2-> 
+   The Reduced Order Kalman Filter (ROUKF) its a simplification for non-linear problems which generally run faster than others methods. (no derivatives are need it)
+   \item<3-> Parameter recovery its a straightforward application.
+
+  \end{itemize}
+\end{frame}
+
+
+
+\end{document}
+

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