version viernes

presentation/pres03.tex

@@ -97,7 +97,7 @@
 
 
 
-\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 data}
+\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]
@@ -107,8 +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 join with Hernan Mella, Julio Sotelo, Sergio Uribe, Cristobal Bertoglio and Joaquin Mura.}
-}
+  Jeremías Garay Labra \emph{join with} Hernan Mella, Julio Sotelo, Sergio Uribe, Cristobal Bertoglio and Joaquin Mura.}
 \date{\today}
 
 
@@ -134,15 +133,13 @@ University of Groningen\\[0.5cm]
 \column{.5\textwidth} % Left column and width
 \footnotesize
 
-\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<3-> Full 3D coverage of the region of interest
-\item<4-> Retrospective plane positioning
-\item<5-> Rich post-proccesing: derived parameters 
+\item<2-> Full 3D coverage of the region of interest
+\item<3-> Rich post-proccesing: derived parameters 
 \end{itemize}
 
 \column{.54\textwidth} % Right column and width
+\onslide<1-> 
 \begin{figure}[!hbtp]
  \begin{center}
   \includegraphics[height=0.9\textwidth]{images/4dflow.png}
@@ -156,28 +153,69 @@ University of Groningen\\[0.5cm]
 \begin{frame}
   \frametitle{4D flow MRI}
 \footnotesize
-\onslide<1->  Main limitation $\longrightarrow$ long scan times involved.\\
-\vspace{0.2cm}
-\onslide<2-> In order to mitigate:
+\onslide<1-> Disadvantages:
 \begin{itemize}
-\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  
+\item<2-> Long scan time
+\item<3-> modest spatial resolutions $ \sim (2.5 \times 2.5 \times 2.5 \ mm^3)$
+\item<4-> partial data coverage  
 \end{itemize}
 
-\vspace{0.5cm}
 
-\onslide<6-> Typical quality estimators: SNR, VNR, peak flows/velocities, mass conservation (zero divergence) 
+\begin{columns}[c] 
+\column{.4\textwidth} % Right column and width
+\onslide<5->
+\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<6->
+\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<7->
+\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{0.3cm}
+
+\onslide<8-> 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).
+\onslide<9-> This work $\longrightarrow$  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
@@ -189,7 +227,7 @@ University of Groningen\\[0.5cm]
 
 \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} \\
+ \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}
@@ -203,8 +241,8 @@ University of Groningen\\[0.5cm]
  \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 the following:\\ 
-Find $(\vec w(t) ,p(t)) \in H^1_0(\Omega)\times L^2(\Omega)$ such that 
+\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*}
@@ -233,7 +271,7 @@ for all $(\vec v,q) \in H^1_0(\Omega) \times L^2(\Omega)$.
 \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{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)}
+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}
 
@@ -244,12 +282,14 @@ A_{k}(\vec w,p;\vec v ,q ) := \int_{\Omega} \frac{\rho}{\tau} \vec{w} \cdot \vec
 $ \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{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<5-> \color{blue}$
+\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}
@@ -304,55 +344,65 @@ Experiments using synthetic data
 
 \begin{frame}
  \frametitle{Numerical tests}
-
+ 
+\onslide<1-> 
 \footnotesize
-
-\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<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}
-
-%\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
+\column{.4\textwidth} % Right 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
+ Simulated channel flow as measurements (Stokes flow)
+ \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}
+  \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}
@@ -364,8 +414,72 @@ Also perturbations were added into the measurements:
 \onslide<2->
 \begin{figure}[!hbtp]
  \begin{center}
-  \includegraphics[height=0.5\textwidth]{images/perturbation_pres.png}
-\caption{\small Different perturbation scenarios. $(\infty , 120 \%)$: $\vec{w} \times 200$, $(10 \ dB , 120 \%)$: $\delta \vec{u}, \vec{w} \times 4$, rest: $\vec{w} \times 4$ }
+  \includegraphics[height=0.45\textwidth]{images/channel_ppt_1.png}
+\caption{\small Fields for the channel in terms of (SNR,$venc$)}
+ \end{center}
+ \end{figure}
+
+\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=0.45\textwidth]{images/channel_ppt_2.png}
+  \caption{\small Fields for the channel in terms of (SNR,$venc$)}
+%\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{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=0.45\textwidth]{images/channel_ppt_3.png}
+  \caption{\small Fields for the channel in terms of (SNR,$venc$)}
+%\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{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=0.45\textwidth]{images/channel_ppt_4.png}
+  \caption{\small Fields for the channel in terms of (SNR,$venc$)}
+%\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{Results for channel: 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}
 
@@ -373,6 +487,23 @@ Also perturbations were added into the measurements:
 \end{frame}
 
 
+\begin{frame}
+ \frametitle{Results for channel: 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{Results for channel: undersampling}
 \footnotesize
@@ -406,86 +537,77 @@ Also perturbations were added into the measurements:
 
 
 
-\begin{frame}
- \frametitle{Numerical tests: aorta}
+%\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}
 
-\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}
+%\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}
+%
+%
 
 
 
@@ -512,7 +634,7 @@ Experiments using real 4D flow data
 
 \begin{itemize}
 \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<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}
@@ -551,7 +673,7 @@ Experiments using real 4D flow data
 \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}
+\caption{At peak systole: a) measurements b) corrector field c) corrected measurements: $\vec u_{meas} + \vec w$}
  \end{center}
  \end{figure}
 
@@ -578,11 +700,18 @@ Conclusions
 \onslide<1->  Potential of the new quality parameter:
 
 \begin{itemize}
-\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
+\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}