new version

presentation/pres03.tex

@@ -1,4 +1,5 @@
-\documentclass[xcolor=dvipsnames]{beamer}
+\documentclass[xcolor=dvipsnames,notheorem,mathserifs]{beamer}
+\usepackage{amsmath}
 %\documentclass{beamer}
 \usepackage[english]{babel}
 %\usepackage[latin1]{inputenc}
@@ -13,11 +14,20 @@
 %\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{hyperref}
+
+\usepackage[normalem]{ulem} % for strike out command \sout
+
+
 
 
 
@@ -97,7 +107,7 @@ 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]
-  \texttt{Jeremías Garay Labra \\ \ j.e.garay.labra@rug.nl}
+  \texttt{Jeremías Garay Labra join with Hernan Mella, Julio Sotelo, Sergio Uribe, Cristobal Bertoglio and Joaquin Mura.}
 }
 \date{\today}
 
@@ -124,13 +134,12 @@ University of Groningen\\[0.5cm]
 \column{.55\textwidth} % Left column and width
 \footnotesize
 
-4D flow MRI has been shown potential in the assesment of blood flow dynamics in the heart and also large arteries, allowing wide variety of options for visualization and quantification. 
-
-Some advantages respect others techniques:
+\onslide<1-> 4D flow MRI has been shown potential in the assesment of blood flow dynamics in the heart and also large arteries.\\[0.2cm]
+\onslide<2-> Some advantages:
 \begin{itemize}
-\item Full 3D coverage of the region of interest
-\item Retrospective plane positioning
-\item Rich post-proccesing: derived parameters 
+\item<3-> Full 3D coverage of the region of interest
+\item<4-> Retrospective plane positioning
+\item<5-> Rich post-proccesing: derived parameters 
 \end{itemize}
 
 \column{.5\textwidth} % Right column and width
@@ -142,16 +151,22 @@ Some advantages respect others techniques:
 \begin{frame}
   \frametitle{4D flow MRI}
 \footnotesize
-Main limitation for its clinical applicability is the long scan times involved. Therefore, multiple strategies emerged in order to make acquisition faster, such as:
+\onslide<1->  Main limitation $\longrightarrow$ long scan times involved.\\
+\vspace{0.2cm}
+\onslide<2-> In order to mitigate:
 \begin{itemize}
-\item Navigator gating
-\item modest spatial resolutions $ \sim (2.5 \times 2.5 \times 2.5 \ mm^3)$
-\item partial data coverage  
+\item<3-> Navigator gating
+\item<4-> modest spatial resolutions $ \sim (2.5 \times 2.5 \times 2.5 \ mm^3)$
+\item<5-> partial data coverage  
 \end{itemize}
 
-Typical quality estimators: SNR, VNR, peak flows/velocities, mass conservation (zero divergence) 
+\vspace{0.5cm}
 
-We want to introduce a novel measure for quantify the quality of the 4D flow measurements, using the conservation of momentum of the flow (Navier-Stokes compatibility).
+\onslide<6-> Typical quality estimators: SNR, VNR, peak flows/velocities, mass conservation (zero divergence) 
+
+\vspace{0.5cm}
+
+\onslide<7-> This work $\longrightarrow$  conservation of linear momentum (Navier-Stokes compatibility).
 
 \end{frame}
 
@@ -162,19 +177,19 @@ We want to introduce a novel measure for quantify the quality of the 4D flow mea
  \frametitle{The corrector field}
 \footnotesize
 
-We assume a perfect physical velocity field $\vec{u}$ 
-\begin{eqnarray*}
+\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*}
 
-And a corrector field $\vec{w}$ which satisfies:
-\begin{align}
+\onslide<3-> And a corrector field $\vec{w}$ which satisfies:
+\onslide<4-> \begin{align}
 \vec{u}  & \approx   \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}
 
-The corrector field $\vec{w}$ measures the level of agreedment of the 4D flow measures respect to the Navier-Stokes equations.
+\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}
 
@@ -183,16 +198,18 @@ The corrector field $\vec{w}$ measures the level of agreedment of the 4D flow me
  \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*} 
+\onslide<1-> 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 
+\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*}
 
-or in simple terms:
-\begin{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*}
 
@@ -207,27 +224,26 @@ for all $(\vec v,q) \in H^1_0(\Omega) \times L^2(\Omega)$.
  \frametitle{The corrector field: Discrete problem}
 \footnotesize
 
-In the Discrete, we can write the problem as follows:
+\onslide<1-> 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)
+\onslide<2-> \begin{equation}
+A_{k}(\vec w,p;\vec v ,q ) + \color{red}{S^{conv}_{k}(\vec w;\vec v)} + \color{blue}{S^{press}_{k}(\vec w,p;\vec v ,q)} \color{black}{ =  \mathcal{L}_j (\vec v)}
+\label{eq:Corrector_discrete}
 \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 $
+\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 $
+\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{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}
-\item $
+\item<5-> \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 $ \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}
@@ -235,6 +251,37 @@ $ \vspace{0.2cm}
 
 
 
+\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  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}
@@ -253,27 +300,29 @@ Experiments using synthetic data
  \frametitle{Numerical tests}
 
 \footnotesize
-We tested the corrector using CFD simulations as a measurements, in the following testcases:
 
+\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<3->
 Also perturbations were added into the measurements: 
 \begin{itemize}
-\item  velocity aliasing (varying the $venc$ parameter)
-\item  additive noise (setting SNR in decibels)
-\item  simulated k-space undersampling (compressed sensing for the reconstruction)
+\item<4->  velocity aliasing (varying the $venc$ parameter)
+\item<5->  additive noise (setting SNR in decibels)
+\item<6->  simulated k-space undersampling (compressed sensing for the reconstruction)
 \end{itemize}
 
-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
+%\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: details}
+ \frametitle{Numerical tests: channel}
 \begin{columns}[c] 
 \column{.6\textwidth} % Left column and width
 \footnotesize
@@ -284,16 +333,75 @@ All simulations were done using a stabilized finite element method implemented i
 \item Oscilatory pressure at $\Gamma_{inlet}$
 \end{itemize}
 
-
 \column{.5\textwidth} % Right column and width
 \footnotesize
 \begin{figure}[!hbtp]
  \begin{center}
-  \includegraphics[height=0.3\textwidth]{images/cilinder_2.png}
+  \includegraphics[height=1.0\textwidth]{images/cilinder.png}
+  \caption{3D channel mesh}
  \end{center}
  \end{figure}
 \end{columns}
 
+\end{frame}
+
+
+
+
+
+\begin{frame}
+ \frametitle{Results for channel: 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=0.5\textwidth]{images/perturbation_pres.png}
+\caption{Different perturbation scenarios}
+ \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/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
@@ -311,8 +419,8 @@ All simulations were done using a stabilized finite element method implemented i
 \footnotesize
 \begin{figure}[!hbtp]
  \begin{center}
-  \includegraphics[height=0.7\textwidth]{images/aorta_blender.png}
-\caption{\tiny{Channel mesh}}
+  \includegraphics[height=1.0\textwidth]{images/aorta_blender.png}
+\caption{Aortic mesh}
  \end{center}
  \end{figure}
 \end{columns}
@@ -321,43 +429,10 @@ All simulations were done using a stabilized finite element method implemented i
 \end{frame}
 
 
-\begin{frame}
- \frametitle{Results for channel: aliasing and noise}
-\footnotesize
-
-For comparison we defined a perfect corrector field as: $\delta \vec u = \vec u_{ref} - \vec u_{meas}$
-
-\begin{figure}[!hbtp]
- \begin{center}
-  \includegraphics[height=0.5\textwidth]{images/perturbation_pres.png}
-\caption{Different perturbation scenarios}
- \end{center}
- \end{figure}
 
 
-\end{frame}
 
 
-\begin{frame}
- \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=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}
@@ -374,6 +449,23 @@ other results concerning undersampling....
 \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
@@ -413,10 +505,10 @@ Experiments using real 4D flow data
 \column{.6\textwidth} % Left column and width
 
 \begin{itemize}
-\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)
+\item<1-> 4D flow measurements were taken from a silicon thoracic aortic phantom made of silicon.
+\item<2-> 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<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}
 
 
@@ -426,7 +518,7 @@ Experiments using real 4D flow data
  \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)}
+\caption{\footnotesize{Experiment done at the Centre of Biomedical Images (CIB) of the Catholic Unversity of Chili (PUC)}}
  \end{center}
  \end{figure}
  
@@ -467,14 +559,15 @@ Experiments using real 4D flow data
 
 
 \begin{frame}
- \frametitle{Conclusions and future}
+ \frametitle{Conclusions and future work}
 \footnotesize
 
-potential of the new quality parameter:
+\onslide<1->  Potential of the new quality parameter:
+
 \begin{itemize}
-\item analize real data
-\item use the specificity for label zones with strong disagreedment
-\item  Use the field for create new inverse problems which can be used for further accelerations
+\item<2-> The detect zones with strong disagreedment
+\item<3-> To better recognize common acquisition artifacts
+\item<4-> The use of the field for create new inverse problems which can be used for further accelerations
 \end{itemize}
 
 \end{frame}