press images

kalman/input_files/aorta.yaml

@@ -8,7 +8,7 @@ fluid:
     implicit_windkessel: True
 
 io:
-    write_path: 'results/Rz_Pc7'
+    write_path: 'results/aorta'
     restart:
         path: ''  # './projects/nse_coa3d/results/test_restart2/' 
         time: 0
@@ -27,8 +27,8 @@ boundary_conditions:
         value: ['0','0','-U*sin(DOLFIN_PI*t/Th)*(t<=Th) + (Th<t)*(U*DOLFIN_PI/Th*(t-Th)*exp(-(t-Th)*beta))']
         parameters:
             #U: 75  #REFERENCE
-            #U: 150 #Pa
-            U: 40 #Pc
+            U: 150 #Pa Pb
+            #U: 40 #Pc
             Th: 0.36
             beta: 70
             t: 0
@@ -57,8 +57,8 @@ boundary_conditions:
             #C: 0.0008   # Pg
             #R_d: 7200  # REFERENCE
             #R_d:  8760  #Pa
-            #R_d: 17520 #Pb x2
-            R_d: 4000  #Pc
+            R_d: 17520 #Pb x2
+            #R_d: 4000  #Pc
             p0: 85
             conv: 1333.223874   
     - 
@@ -73,8 +73,8 @@ boundary_conditions:
             #C: 0.0008 # Pg
             #R_d: 11520 # REFERENCE
             #R_d:  8760  #Pa
-            #R_d: 17520 #Pb x2
-            R_d: 4000 #Pc
+            R_d: 17520 #Pb x2
+            #R_d: 4000 #Pc
             p0: 85
             conv: 1333.223874   
     - 
@@ -88,8 +88,8 @@ boundary_conditions:
             #C: 0.0001 #Pc
             #R_d: 11520 # REFERENCE
             #R_d:  8760 #Pa
-            #R_d: 17520 #Pb x2
-            R_d: 4000 #Pc
+            R_d: 17520 #Pb x2
+            #R_d: 4000 #Pc
             p0: 85
             conv: 1333.223874   
 
@@ -163,17 +163,17 @@ estimation:
             id: 4
             type: 'windkessel'
             mode: 'Rd'
-            initial_stddev: 2
+            initial_stddev: 1
         - 
             id: 5
             type: 'windkessel'
             mode: 'Rd'
-            initial_stddev: 2
+            initial_stddev: 1
         - 
             id: 6
             type: 'windkessel'
             mode: 'Rd'
-            initial_stddev: 2
+            initial_stddev: 1
         - 
             id: 2
             type: 'dirichlet'
@@ -186,13 +186,13 @@ estimation:
             mesh: '/home/yeye/NuMRI/kalman/meshes/coaortaH3_leo2.0.h5'
             #mesh: './meshes/coaortaH1.h5'
             fe_degree: 1
-            xdmf_file: 'measurements/aorta_zdir/Perturbation/Mg15V120/u_all.xdmf'
-            file_root: 'measurements/aorta_zdir/Perturbation/Mg15V120/u{i}.h5'
-            #xdmf_file: 'measurements/aorta_zdir_NV/u_all.xdmf'
-            #file_root: 'measurements/aorta_zdir_NV/u{i}.h5'
+            #xdmf_file: 'measurements/aorta_zdir/Perturbation/Mg15V120/u_all.xdmf'
+            #file_root: 'measurements/aorta_zdir/Perturbation/Mg15V120/u{i}.h5'
+            xdmf_file: 'measurements/aorta/u_all.xdmf'
+            file_root: 'measurements/aorta/u{i}.h5'
             indices: 0 # indices of checkpoints to be processed. 0 == all
-            velocity_direction: [0,0,1]
-            noise_stddev: 25     # standard deviation of Gaussian noise
+            velocity_direction: ~
+            noise_stddev: 0     # standard deviation of Gaussian noise
     
     roukf:
         particles: 'simplex' # unique or simplex

presentations/press_8ecm/press.tex

@@ -98,7 +98,7 @@
 
 
 
-\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}
+\title{Robust parameter estimation in fluid flow models from aliased velocity measurements}
 %\author[Jeremías Garay Labra]
 %{Jeremías Garay Labra}
 \institute[University of Groningen]
@@ -108,7 +108,7 @@ 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]
-  Jeremías Garay Labra \emph{join with} Hernan Mella, Julio Sotelo, Sergio Uribe, Cristobal Bertoglio and Joaquin Mura.}
+  Jeremías Garay Labra \emph{join with} Cristobal Bertoglio.}
 \date{\today}
 
 
@@ -145,8 +145,6 @@ University of Groningen\\[0.5cm]
 \end{itemize}
 
 
-
-
 \column{.54\textwidth} % Right column and width
 \onslide<1-> 
 \begin{figure}[!hbtp]
@@ -159,362 +157,219 @@ University of Groningen\\[0.5cm]
 \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}
+\section{The mathematical model}
 
 \begin{frame}
- \frametitle{The corrector field}
+ \frametitle{The mathematical model}
 \begin{center}
-Methodology
+The mathematical model
 \end{center}
 \end{frame}
 
 
 
 \begin{frame}
- \frametitle{The corrector field}
+ \frametitle{The mathematical model}
+ 
+ \begin{columns}[c] 
+\column{.5\textwidth} % Left column and width
 \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 $(\vec v,q) \in H^1_0(\Omega) \times L^2(\Omega)$.
-
+\column{.54\textwidth} % Right column and width
+\begin{figure}[!hbtp]
+ \begin{center}
+  \includegraphics[height=1.1\textwidth]{images/full_aorta.png} 
+ \end{center}
+ \end{figure}
+\end{columns} 
+ 
 \end{frame}
 
 
 
 \begin{frame}
- \frametitle{The corrector field: Discrete problem}
+ \frametitle{The mathematical model}
+ 
+ \begin{columns}[c] 
+\column{.5\textwidth} % Left column and width
 \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}
+\begin{itemize}
+\item<2-> 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}
+\item<3-> \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}
-
-\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}
- 
+\column{.54\textwidth} % Right column and width
 \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}
+  \includegraphics[height=0.9\textwidth]{images/windk_model.png}
+  \caption{\footnotesize Schematic of the model}
  \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{columns} 
+ 
 \end{frame}
 
 
-
-
 \begin{frame}
- \frametitle{Aliasing and noise}
+ \frametitle{The mathematical model}
+ 
+ \begin{columns}[c] 
+\column{.5\textwidth} % Left column and width
 \footnotesize
+\begin{itemize}
+\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}
+\item \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}
 
-\onslide<1-> For comparison we defined a perfect corrector field as: $\delta \vec u = \vec u_{ref} - \vec u_{meas}$
+
+\column{.54\textwidth} % Right column and width
+\begin{figure}[!hbtp]
+ \begin{center}
+  \includegraphics[height=0.9\textwidth]{images/ref.png}
+  \caption{\footnotesize Schematic of the model}
+ \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: $M(\vec{x},t)= M_0(\vec{x}) exp(i\phi_0 + i \frac{\pi}{venc} u(\vec{x},t))$
+\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:}
+\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: $u_z$
+\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 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} 
+\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:
 
 \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}
+\begin{equation*}
+\hat{\theta} = arg \min_{\theta} J(\theta)
+\end{equation*}
+\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:
+\begin{itemize}
+\item<4-> $Z$ the measurements and $X = (\vec{u} , \pi)$ the state variable
+\item<5-> $\mathbb{H}$ observation operator 
+\item<6-> $\theta_0$ is the initial guess for the parameters
+\item<7-> $P_0$ is the associated covariance matrix
+\item<8-> $W$ is the associated covariance matrix to the meas. noise
+\end{itemize}
 
+\end{itemize}
 
 \end{frame}
 
@@ -522,255 +377,91 @@ For comparison we defined a perfect corrector field as: $\delta \vec u = \vec u_
 
 
 \begin{frame}
- \frametitle{Undersampling}
-\footnotesize
+ \frametitle{The Kalman Filter}
+The parameter vector: 
+ 
+\begin{itemize}
+\item<1->  Plug flow at the inlet: $u_{inlet} = -U f(t) \hat{n}$, with $f(t)$ is the weaveform which simulate a cardiac cycle and $\hat{n}$ is the outward normal vector. 
+\item<2-> Since $R_p << R_d$, we only consider an optimization dependent on $R_d, C$ for every 3D-0D coupled outlet
+\end{itemize}
 
-\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}
+\onslide<3-> $$\theta =  (U,\vec{R_d},\vec{C})$$ \\  with $\vec{R_d} = R_{d,l}$,  $\vec{C} = C_l$ for $l=1,..., n_l$
 
 \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 }
+\section{Numerical Experiments}
 
 
 
 \begin{frame}
- \frametitle{Experiments}
+ \frametitle{Numerical Experiments}
 \begin{center}
-Experiments using real 4D flow data
+Numerical Experiments
 \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}
-%
+ \frametitle{Numerical Experiments}
 
 \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}
 
+
+
+
+
+
+
+%\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}