From b75e5995e0e27572b3b9d41a15378043a9c95781 Mon Sep 17 00:00:00 2001
From: jeremias <jeremias.garay.l@gmail.com>
Date: Wed, 9 Jun 2021 12:54:31 +0200
Subject: [PATCH] add 8ecm

---
 presentations/press_8ecm/press.tex | 776 +++++++++++++++++++++++++++++
 1 file changed, 776 insertions(+)
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+\documentclass[xcolor=dvipsnames,notheorem,mathserifs]{beamer}
+\usepackage{amsmath}
+%\documentclass{beamer}
+\usepackage[english]{babel}
+%\usepackage[latin1]{inputenc}
+\usepackage{multicol} % indice en 2 columnas
+\usepackage[utf8]{inputenc}
+
+\usepackage{helvet}
+\usefonttheme{serif} 
+%\usepackage{ccfonts}     % Font family: Concrete Math
+\usepackage[T1]{fontenc}
+
+%\usepackage{graphicx}
+%\usepackage{movie15}
+%\usepackage{media9}[2013/11/04]
+\usepackage{xcolor}
+
+
+\usepackage{graphicx}
+\usepackage{multimedia}
+\usepackage{media9}
+\usepackage{listings,xcolor,caption, mathtools, wrapfig}
+\usepackage{amsfonts}
+\usepackage{amssymb,graphicx,enumerate}
+\usepackage{subcaption}
+\usepackage{hyperref}
+
+\usepackage[normalem]{ulem} % for strike out command \sout
+
+
+
+
+
+%\usetheme{default}
+%\usetheme{AnnArbor}
+%\usetheme{Antibes}
+%\usetheme{Bergen}
+%\usetheme{Berkeley}
+%\usetheme{Berlin}
+%\usetheme{Boadilla}
+%\usetheme{CambridgeUS}
+%\usetheme{Copenhagen}
+%\usetheme{Darmstadt}
+%\usetheme{Dresden}
+%\usetheme{Frankfurt}
+%\usetheme{Goettingen}
+%\usetheme{Hannover}
+%\usetheme{Ilmenau}
+%\usetheme{JuanLesPins}
+%\usetheme{Luebeck}
+%\usetheme{Madrid}
+%\usetheme{Malmoe}
+%\usetheme{Marburg}
+%\usetheme{Montpellier}
+%\usetheme{PaloAlto}
+%\usetheme{Pittsburgh}
+%\usetheme{Rochester}
+%\usetheme{Singapore}
+%\usetheme{Szeged}
+\usetheme{Warsaw}
+
+%\usecolortheme{albatross}
+%\usecolortheme{beaver}
+%\usecolortheme{beetle}
+\usecolortheme{crane}
+%\usecolortheme{dolphin}
+%\usecolortheme{dove}
+%\usecolortheme{fly}
+%\usecolortheme{lily}
+%\usecolortheme{orchid}
+%\usecolortheme{rose}
+%\usecolortheme{seagull}
+%\usecolortheme{seahorse}
+%\usecolortheme{whale}
+%\usecolortheme{wolverine}
+
+%\useoutertheme{infolines}
+%\useoutertheme{miniframes}
+%\useoutertheme{sidebar}
+\useoutertheme{smoothbars}
+%\useoutertheme{shadow}
+%\useoutertheme{smoothtree}
+%\useoutertheme{split}
+%\useoutertheme{tree}
+
+
+\usepackage{amssymb,mathrsfs,amsmath,latexsym,amsthm,amsfonts}
+\useinnertheme{rectangles}
+
+
+\setbeamertemplate{navigation symbols}{} % quitar simbolitos
+
+
+\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}
+