final press 8ecm

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presentations/press_8ecm/press.out

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+\BOOKMARK [2][]{Outline0.1}{4D flow MRI}{}% 1
+\BOOKMARK [2][]{Outline0.2}{The mathematical model}{}% 2
+\BOOKMARK [2][]{Outline0.3}{The inverse problem}{}% 3
+\BOOKMARK [2][]{Outline0.4}{Numerical Experiments}{}% 4
+\BOOKMARK [2][]{Outline0.5}{Conclusions}{}% 5

presentations/press_8ecm/press.tex

@@ -135,14 +135,11 @@ University of Groningen\\[0.5cm]
 \footnotesize
 
 \begin{itemize}
-\item<2-> Full 3D coverage of the region of interest
+\item<2-> Velocities encoded into the magnetization phase
 \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
@@ -232,16 +229,26 @@ P_l = R_{p,l} \ Q_l + \pi_l
 
 \begin{frame}
  \frametitle{The mathematical model}
- 
  \begin{columns}[c] 
 \column{.5\textwidth} % Left column and width
 \footnotesize
 \begin{itemize}
-\item<1-> $u_{inlet} = -U f(t) \hat{n}$, with $f(t)$ the weaveform.
-\item<2-> Fractional step scheme.
-\item<3-> Semi-implicit Windkessel model.
-\item<4-> Stabilized $\mathbb{P}1/\mathbb{P}1$ finite elements.
-\item<4-> Implemented in FEniCS.
+\item Incompressible Navier-Stokes equations:
+\begin{equation}
+\begin{cases}
+\displaystyle \rho \frac{\partial \vec{u}}{\partial t} + \rho \big ( \vec{u}  \cdot \nabla \big) \vec{u} - \mu \Delta \vec{u} + \nabla p  =  0  \\[0.2cm]
+\nabla \cdot \vec{u}  =  0 \quad \text{in} \quad \Omega \\[0.2cm]
+\vec{u}  =  \vec{u}_{inlet}  \quad \text{on} \quad \Gamma_{in} \\[0.2cm]
+\vec{u} = 0 \quad \text{on} \quad \Gamma_{walls}
+\end{cases}
+\end{equation}
+ \emph{Three-element} Windkessel coupling at every outlet:
+\begin{equation}
+\begin{cases}
+\displaystyle C_{d,l} \frac{d \pi_l}{dt} + \frac{\pi_l}{R_{d,l}} = Q_l \\[0.2cm]
+P_l = R_{p,l} \ Q_l + \pi_l
+\end{cases}
+\end{equation}
 \end{itemize}
 
 
@@ -258,90 +265,105 @@ P_l = R_{p,l} \ Q_l + \pi_l
 
 
 
+%\begin{frame}
+% \frametitle{The mathematical model}
+% 
+% \begin{columns}[c] 
+%\column{.5\textwidth} % Left column and width
+%\footnotesize
+%\begin{itemize}
+%\item<1-> $u_{inlet} = -U f(t) \hat{n}$, with $f(t)$ the weaveform.
+%\item<2-> Fractional step scheme.
+%\item<3-> Semi-implicit Windkessel model.
+%\item<4-> Stabilized $\mathbb{P}1/\mathbb{P}1$ finite elements.
+%\item<4-> Implemented in FEniCS.
+%\end{itemize}
+%
+%
+%\column{.54\textwidth} % Right column and width
+%\begin{figure}[!hbtp]
+% \begin{center}
+%  \includegraphics[height=0.9\textwidth]{images/ref.png}
+%  \caption{\footnotesize Reference solution at peak systole}
+% \end{center}
+% \end{figure}
+%\end{columns} 
+% 
+%\end{frame}
+
+
+
 \begin{frame}
  \frametitle{The inverse problem}
 \begin{itemize}
 \item<1-> Upon this solution $\Longrightarrow$ build a set of measurements
-\item<2-> Induce typical arifacts via a Magnetization vector: $M(\vec{x},t)= M_0(\vec{x}) exp(i\phi_0 + i \frac{\pi}{venc} u(\vec{x},t))$
+\item<2->  $M(\vec{x},t)= M_0(\vec{x}) exp(i\phi_0 + i \frac{\pi}{venc} u(\vec{x},t))$
+\item<3-> reconstructed velocity: $u \in \big ( -venc, + venc  \big )$
+\item<4-> $VNR \sim 1/venc$
 \end{itemize}
-\onslide<3-> 
- \begin{columns}
- \column{.3\textwidth}
- \flushleft
- \begin{figure}
-        \includegraphics[width=1.1\textwidth]{images/ref_int.png}
-  \caption*{(a) Interpolated reference solution}
-\end{figure}
-\column{.67\textwidth}
 \centering
-\onslide<4-> \textbf{The measurements:}
+\onslide<5-> \textbf{The measurements:}
 \begin{itemize}
-\item<5-> Gaussian noise into the magnetization
-\item<6-> Different levels of aliasing varying the $venc$ parameter
-\item<7-> Only using the dominant component of the velocity: $u_z$
-\item<8-> Time interpolation $dt = 1 \ ms  \Longrightarrow dt = 30 \ ms$ 
+\item<6-> Gaussian noise into the magnetization
+\item<7-> Spatial and temporal interpolation
+\item<8-> Only using the dominant component of the velocity: $u_z$
+\item<9-> Different levels of aliasing varying the $venc$ parameter
 \end{itemize}
-\end{columns} 
 \end{frame}
 
 
 
+
+\begin{frame}
+ \frametitle{The inverse problem}
+\begin{itemize}
+\item Upon this solution $\Longrightarrow$ build a set of measurements
+\item $M(\vec{x},t)= M_0(\vec{x}) exp(i\phi_0 + i \frac{\pi}{venc} u(\vec{x},t))$
+\item reconstructed velocity: $u \in \big ( -venc, + venc  \big )$
+\item $VNR \sim 1/venc$
+\end{itemize}
+ \begin{figure}
+        \includegraphics[width=0.7\textwidth]{images/supra_venc.png}
+  \caption*{Aliased measurements with different $vencs = 120,70,30 \%$ of $u_{max}$}
+  \hfill
+\end{figure}
+\end{frame}
+
+
+
+\begin{frame}
+ \frametitle{The inverse problem}
+\begin{itemize}
+\item Upon this solution $\Longrightarrow$ build a set of measurements
+\item $M(\vec{x},t)= M_0(\vec{x}) exp(i\phi_0 + i \frac{\pi}{venc} u(\vec{x},t))$
+\item reconstructed velocity: $u \in \big ( -venc, + venc  \big )$
+\item $VNR \sim 1/venc$
+\end{itemize}
+ \begin{figure}
+        \includegraphics[width=0.7\textwidth]{images/coartation.png}
+  \caption*{ Aliased measurements with different $vencs = 120,70,30 \%$ of $u_{max}$}
+  \hfill
+\end{figure}
+\end{frame}
+
+
 \section{The inverse problem}
 
-\begin{frame}
- \frametitle{The inverse problem}
-\begin{itemize}
-\item Upon this solution $\Longrightarrow$ build a set of measurements
-\item Induce typical arifacts via a Magnetization vector: $M(\vec{x},t)= M_0(\vec{x}) exp(i\phi_0 + i \frac{\pi}{venc} u(\vec{x},t))$
-\end{itemize}
- \begin{columns}
- \column{.3\textwidth}
- \flushleft
- \begin{figure}
-        \includegraphics[width=1.1\textwidth]{images/ref_int.png}
-  \caption*{(a) Interpolated reference solution}
-\end{figure}
-\column{.67\textwidth}
-\flushleft
- \begin{figure}
-        \includegraphics[width=1.1\textwidth]{images/supra_venc.png}
-  \caption*{(b) Aliased measurements with different $vencs = 120,70,30 \%$ of $u_{max}$}
-  \hfill
-\end{figure}
-\end{columns} 
-\end{frame}
-
-
 
 \begin{frame}
  \frametitle{The inverse problem}
-\begin{itemize}
-\item Upon this solution $\Longrightarrow$ build a set of measurements
-\item Induce typical arifacts via a Magnetization vector: $M(\vec{x},t)= M_0(\vec{x}) exp(i\phi_0 + i \frac{\pi}{venc} u(\vec{x},t))$
-\end{itemize}
- \begin{columns}
- \column{.3\textwidth}
- \flushleft
- \begin{figure}
-        \includegraphics[width=1.1\textwidth]{images/ref_int.png}
-  \caption*{(a) Interpolated reference solution}
-\end{figure}
-\column{.67\textwidth}
-\flushleft
- \begin{figure}
-        \includegraphics[width=1.1\textwidth]{images/coartation.png}
-  \caption*{(b) Aliased measurements with different $vencs = 120,70,30 \%$ of $u_{max}$}
-  \hfill
-\end{figure}
-\end{columns} 
+\begin{center}
+Parameter optimization
+\end{center}
 \end{frame}
 
 
 
+
 \begin{frame}
  \frametitle{The Kalman Filter}
 \begin{itemize}
-\item<1-> We use a Reduced Order Unscendent Kalman Filter (ROUKF) to reconstruct the parameter vector $\theta$ solving the next optimization problem:
+\item<1-> We use a Reduced Order Unscendent Kalman Filter (ROUKF) to reconstruct the parameter vector $\theta$:
 
 \onslide<2->
 \begin{equation*}
@@ -350,7 +372,7 @@ P_l = R_{p,l} \ Q_l + \pi_l
 \begin{equation}
 J(\theta) = \displaystyle \frac{1}{2} || \theta - \theta_0  ||^2_{P_0^{-1}}  + \sum_{k=1}^N \frac{1}{2}  || Z_k - \mathbb{H} X_k(\theta)   ||^2_{W^{-1}}
 \end{equation}
-\onslide<3-> Where:
+\onslide<4-> Where:
 \begin{itemize}
 \item<4-> $Z$ the measurements and $X = (\vec{u} , \pi)$ the state variable
 \item<5-> $\mathbb{H}$ observation operator 
@@ -372,32 +394,41 @@ The parameter vector:
  
 \begin{itemize}
 \item<1->  Amplitude of the inlet velocity: $U$ 
-\item<2-> Since $R_p << R_d$, we only consider an optimization dependent on $\big ( R_{d,l}, C_l \big )$ for $l=1,...,n_l$
+\item<2-> Only the higher resistence: $R_d$
 \end{itemize}
 
-\onslide<3-> $$\theta =  (U,\vec{R_d},\vec{C})$$ \\  with $\vec{R_d} = R_{d,l}$,  $\vec{C} = C_l$ for $l=1,..., \color{red} n_{l-1}$ \\[0.3cm]
-\onslide<4-> \color{red} Not all the resistences can be recovered at once $\Longrightarrow$ desc. aorta fixed.
+\onslide<4-> $$\theta =  (U,\vec{R_d})$$ \\  with $\vec{R_d} = R_{d,l}$ for $l=1,..., \color{red} n_{l-1}$ \\[0.3cm]
+\onslide<5-> \color{red} Not all the resistences can be recovered at once $\Longrightarrow$ desc. aorta fixed.
 
 \end{frame}
 
 
 
+
+\section{Numerical Experiments}
+
+
 \begin{frame}
- \frametitle{Easy example}
+ \frametitle{Numerical Experiments}
+\begin{center}
+Numerical Experiments
+\end{center}
+\end{frame}
+
+
+\begin{frame}
+ \frametitle{Numerical Experiments}
  \footnotesize
-\begin{itemize}
-\item<1-> $\theta_{ref} = (U,\vec{R_d})$ , $U=75$, $\vec{R_d} = (7200,11520,11520)$
-\end{itemize}
+\onslide<1-> $\theta_{ref} = (U,\vec{R_d})$ , $U=75$, $\vec{R_d} = (7200,11520,11520)$
  \begin{columns}
   \footnotesize
- \column{.4\textwidth}
+ \column{.45\textwidth}
  \begin{figure}
- \onslide<2-> \textbf{Test I:} $U_0 = 150$  $\vec{R_{d,0}}= (8760,8760,8760)$
-  \onslide<3->
-        \includegraphics[width=1.2\textwidth]{images/U_Pb.png}
+ \onslide<2-> \textbf{Test I:} $U_0 = 150$  $\vec{R_{d,0}}= (17520,17520,17520)$
+  \onslide<3-> \includegraphics[width=1.2\textwidth]{images/U_Pb.png}
          \includegraphics[width=1.2\textwidth]{images/Rd_Pb.png}
 \end{figure}
-\column{.4\textwidth}
+\column{.45\textwidth}
 \begin{figure}
  \onslide<2-> \textbf{Test II:} $U_0 = 40$  $\vec{R_{d,0}}= (4000,4000,4000)$
  \onslide<4->
@@ -415,32 +446,48 @@ The parameter vector:
 \begin{frame}
  \frametitle{Aliased data}
 \begin{center}
-What happend when $venc < u_{max}$ ?
+\onslide<1-> What happend when $venc < u_{max}$ ?
+\begin{figure}
+ \onslide<2-> \includegraphics[width=0.45\textwidth]{images/v120.png}
+ \caption{Measurement set with $venc = 120 \% u_{max}$}
+\end{figure}
 \end{center}
 \end{frame}
 
 
+\begin{frame}
+ \frametitle{Aliased data}
+\begin{center}
+What happend when $venc < u_{max}$ ?
+\begin{figure}
+ \includegraphics[width=0.45\textwidth]{images/v70.png}
+ \caption{Measurement set with $venc = 70 \% u_{max}$}
+\end{figure}
+\end{center}
+\end{frame}
+
+
+
+
 \begin{frame}
  \frametitle{Easy example: with $venc = 70 \% u_{max}$}
  \footnotesize
-\begin{itemize}
-\item $\theta_{ref} = (U,\vec{R_d})$ , $U=75$, $\vec{R_d} = (7200,11520,11520)$
-\end{itemize}
+$\theta_{ref} = (U,\vec{R_d})$ , $U=75$, $\vec{R_d} = (7200,11520,11520)$
  \begin{columns}
   \footnotesize
- \column{.4\textwidth}
+ \column{.45\textwidth}
  \begin{figure}
-  \onslide<1-> \textbf{Test I:} $U_0 = 150$  $\vec{R_{d,0}}= (8760,8760,8760)$
+  \onslide<1-> \textbf{Test I:} $U_0 = 150$  $\vec{R_{d,0}}= (17520,17520,17520)$
   \onslide<2->
-        \includegraphics[width=1.2\textwidth]{images/U_Pb.png}
-         \includegraphics[width=1.2\textwidth]{images/Rd_Pb.png}
+        \includegraphics[width=1.2\textwidth]{images/U_Pb_V70.png}
+         \includegraphics[width=1.2\textwidth]{images/Rd_Pb_V70.png}
 \end{figure}
-\column{.4\textwidth}
+\column{.45\textwidth}
 \begin{figure}
   \onslide<1->  \textbf{Test II:} $U_0 = 40$  $\vec{R_{d,0}}= (4000,4000,4000)$
  \onslide<2->
-        \includegraphics[width=1.2\textwidth]{images/U_Pc.png}
-         \includegraphics[width=1.2\textwidth]{images/Rd_Pc.png}
+        \includegraphics[width=1.2\textwidth]{images/U_Pc_V70.png}
+         \includegraphics[width=1.2\textwidth]{images/Rd_Pc_V70.png}
 \end{figure}
 \end{columns} 
 \end{frame}
@@ -465,30 +512,70 @@ J(\theta) = \displaystyle \frac{1}{2} || \theta - \theta_0  ||^2_{P_0^{-1}}  + \
 
 
 
-
-
-
-
-
-
-
-\section{Numerical Experiments}
+\begin{frame}
+ \frametitle{Easy example: with $venc = 70 \% u_{max}$}
+ \footnotesize
+$\theta_{ref} = (U,\vec{R_d})$ , $U=75$, $\vec{R_d} = (7200,11520,11520)$
+ \begin{columns}
+  \footnotesize
+ \column{.45\textwidth}
+ \begin{figure}
+  \onslide<1-> \textbf{Test I:} $U_0 = 150$  $\vec{R_{d,0}}= (17520,17520,17520)$
+  \onslide<2->
+        \includegraphics[width=1.2\textwidth]{images/HU_Pb_V70.png}
+         \includegraphics[width=1.2\textwidth]{images/HRd_Pb_V70.png}
+\end{figure}
+\column{.45\textwidth}
+\begin{figure}
+  \onslide<1->  \textbf{Test II:} $U_0 = 40$  $\vec{R_{d,0}}= (4000,4000,4000)$
+ \onslide<3->
+        \includegraphics[width=1.2\textwidth]{images/HU_Pc_V70.png}
+         \includegraphics[width=1.2\textwidth]{images/HRd_Pc_V70.png}
+\end{figure}
+\end{columns} 
+\end{frame}
 
 
 \begin{frame}
- \frametitle{Numerical Experiments}
+ \frametitle{Aliased data}
 \begin{center}
-Numerical Experiments
+Or even higher aliasing...
+\begin{figure}
+ \includegraphics[width=0.45\textwidth]{images/v30.png}
+ \caption{Measurement set with $venc = 30 \% u_{max}$}
+\end{figure}
 \end{center}
 \end{frame}
 
-\begin{frame}
- \frametitle{Numerical Experiments}
 
+
+
+\begin{frame}
+ \frametitle{Easy example: with $venc = 30 \% u_{max}$}
+ \footnotesize
+$\theta_{ref} = (U,\vec{R_d})$ , $U=75$, $\vec{R_d} = (7200,11520,11520)$
+ \begin{columns}
+  \footnotesize
+ \column{.45\textwidth}
+ \begin{figure}
+  \onslide<1-> \textbf{Test I:} $U_0 = 150$  $\vec{R_{d,0}}= (17520,17520,17520)$
+  \onslide<2->
+        \includegraphics[width=1.2\textwidth]{images/HU_Pb_V30.png}
+         \includegraphics[width=1.2\textwidth]{images/HRd_Pb_V30.png}
+\end{figure}
+\column{.45\textwidth}
+\begin{figure}
+  \onslide<1->  \textbf{Test II:} $U_0 = 40$  $\vec{R_{d,0}}= (4000,4000,4000)$
+ \onslide<3->
+        \includegraphics[width=1.2\textwidth]{images/HU_Pc_V30.png}
+         \includegraphics[width=1.2\textwidth]{images/HRd_Pc_V30.png}
+\end{figure}
+\end{columns} 
 \end{frame}
 
 
 
+
 \section{Conclusions}
 
 \begin{frame}
@@ -499,8 +586,32 @@ Conclusions
 
 
 
+\begin{frame}
+\frametitle{Conclusions}
+ \footnotesize
+\begin{itemize}
+\item<1-> 4D Flow measurements are promising for extracting data via inverse problems
+\item<2-> Noise and aliasing are the typical artifacts involved.
+\item<3-> Using a suitable Kalman filter, have shown to bypass aliasing defining the funcional in terms of the frequencies.
+\end{itemize}
+
+\onslide<4-> Future Work
+
+\begin{itemize}
+\item<5-> To include the capacitancies in the inverse problem (adding some pressure meas.)
+\item<6-> Real data!
+\end{itemize}
+
+\end{frame}
 
 
+\begin{frame}
+\begin{center}
+Thank you for your time!
+\end{center}
+
+
+\end{frame}
 
 
 

presentations/press_8ecm/press.toc

@@ -0,0 +1,6 @@
+\babel@toc {english}{}
+\beamer@sectionintoc {1}{4D flow MRI}{3}{0}{1}
+\beamer@sectionintoc {2}{The mathematical model}{6}{0}{2}
+\beamer@sectionintoc {3}{The inverse problem}{23}{0}{3}
+\beamer@sectionintoc {4}{Numerical Experiments}{37}{0}{4}
+\beamer@sectionintoc {5}{Conclusions}{56}{0}{5}