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final press 8ecm

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J.E. Garay Labra 2021-06-25 11:07:12 +02:00
parent 1b00a76a17
commit 4311b6f65e
29 changed files with 2868 additions and 162 deletions
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63
kalman/graphics/figure3.py
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@ -146,27 +146,12 @@ def plot_parameters(dat, input_file, deparameterize=False, ref=None):
for i in range(len(ids)):
cur_key = ids[i]
true_level = np.log(true_values[ids[i]]/current_val[i])/np.log(2)
rec_value = np.round(2**theta[-1, idx]*current_val[i],2)
#curve = theta[:,idx] + line_split*idx - true_level
#dash_curve = line_split*idx + t*0
curve = 2**theta[:, idx]*current_val[i]
std_down = 2**(-np.sqrt(P[:, idx, idx]))*curve
std_up = 2**np.sqrt(P[:, idx, idx])*curve
dash_curve = true_values[ids[i]] + t*0
if ids_type[i] == 'dirichlet':
fig3, axes3 = plt.subplots(1,1,figsize=(12,5))
axes3.plot(t, curve , '-', color=col_,label= legends_ + '= ' + str(rec_value) + '/' + str(true_values[cur_key]) + '$', linewidth = 4)
axes3.fill_between(t, std_down, std_up, alpha=0.3, color=col_)
legends_=next(legends)
axes3.plot(t, dash_curve , color=col_,ls='--')
axes3.plot(t, dash_curve , color=col_,ls='--' , linewidth = 3)
axes3.set_ylabel(r'$U$',fontsize=36)
axes3.legend(fontsize=36,loc='upper right')
axes3.set_xlim([-0.01,0.81])
@ -177,28 +162,34 @@ def plot_parameters(dat, input_file, deparameterize=False, ref=None):
plt.savefig('U.png')
plt.close(fig3)
else:
cur_key = ids[i]
rec_value = np.round(2**theta[-1, idx]*current_val[i],2)
curve = 2**theta[:, idx]*current_val[i]
std_down = 2**(-np.sqrt(P[:, idx, idx]))*curve
std_up = 2**np.sqrt(P[:, idx, idx])*curve
dash_curve = true_values[ids[i]] + t*0
axes1.plot(t, curve , '-', color=col_,label= legends_ + '= ' + str(rec_value) + '/' + str(true_values[cur_key]) + '$', linewidth = 3)
axes1.fill_between(t, std_down, std_up, alpha=0.3, color=col_)
axes1.plot(t, dash_curve , color=col_,ls='--')
axes1.plot(t, dash_curve , color=col_,ls='--',linewidth = 3)
legends_=next(legends)
if RC_flag:
if i<len(current_val_C):
rec_value_C = np.round(2**theta[-1, idx+1]*current_val_C[idc],6)
if RC_flag:
if i<len(current_val_C):
true_level_C = np.log(true_values_C[ids[i]]/current_val_C[i])/np.log(2)
rec_value_C = np.round(2**theta[-1, idc]*current_val_C[idc],6)
curve_C = 2**theta[:, idx+1]*current_val_C[idc]
dash_curve_C = true_values_C[ids[i]] + t*0
std_C_down = 2**(-np.sqrt(P[:, idx+1, idx+1]))*curve_C
std_C_up = 2**np.sqrt(P[:, idx+1, idx+1])*curve_C
curve_C = 2**theta[:, idx+1]*current_val_C[idc]
dash_curve_C = true_values_C[ids[i]] + t*0
std_C_down = 2**(-np.sqrt(P[:, idx+1, idx+1]))*curve_C
std_C_up = 2**np.sqrt(P[:, idx+1, idx+1])*curve_C
axes2.plot(t, curve_C , '-', color=col_,label= legends_ + '= ' + str(rec_value_C) + '/' + str(true_values_C[cur_key]) + '$', linewidth = 2)
axes2.fill_between(t, std_C_down, std_C_up, alpha=0.3, color=col_)
axes2.plot(t, dash_curve_C , color=col_,ls='--')
legends_=next(legends)
idx +=1
idc +=1
axes2.plot(t, curve_C , '-', color=col_,label= legends_ + '= ' + str(rec_value_C) + '/' + str(true_values_C[cur_key]) + '$', linewidth = 3)
axes2.fill_between(t, std_C_down, std_C_up, alpha=0.3, color=col_)
axes2.plot(t, dash_curve_C , color=col_,ls='--',linewidth = 3)
legends_=next(legends)
idx +=1
idc +=1
if meas_flag:
@ -221,10 +212,12 @@ def plot_parameters(dat, input_file, deparameterize=False, ref=None):
if RC_flag:
axes2.set_ylabel(r'$C$',fontsize=24)
axes2.legend(fontsize=18,loc='upper right')
axes2.set_ylabel(r'$C$',fontsize=36)
axes2.legend(fontsize=36,loc='upper right')
axes2.set_xlim([-0.01,0.81])
axes2.set_xlabel(r'$t (s)$',fontsize=24)
plt.xticks(fontsize=28)
plt.yticks(fontsize=28)
axes2.set_xlabel(r'$t (s)$',fontsize=36)
fig2.savefig('C.png')
fig1.savefig('Rd.png')

39
kalman/input_files/aorta.yaml
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@ -8,7 +8,7 @@ fluid:
implicit_windkessel: True
io:
write_path: 'results/HRz_Pb_V70'
write_path: 'results/impl_1ms'
restart:
path: '' # './projects/nse_coa3d/results/test_restart2/'
time: 0
@ -27,7 +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 Pb
U: 80
#U: 150 #Pa/Pb
#U: 40 #Pc
Th: 0.36
beta: 70
@ -55,9 +56,9 @@ boundary_conditions:
#C: 0.0010 # Pb
#C: 0.0001 # Pc
#C: 0.0008 # Pg
#R_d: 7200 # REFERENCE
R_d: 7200 # REFERENCE
#R_d: 8760 #Pa
R_d: 17520 #Pb x2
#R_d: 17520 #Pb x2
#R_d: 4000 #Pc
p0: 85
conv: 1333.223874
@ -71,9 +72,9 @@ boundary_conditions:
#C: 0.0010 # Pb
#C: 0.0001 # Pc
#C: 0.0008 # Pg
#R_d: 11520 # REFERENCE
R_d: 11520 # REFERENCE
#R_d: 8760 #Pa
R_d: 17520 #Pb x2
#R_d: 17520 #Pb x2
#R_d: 4000 #Pc
p0: 85
conv: 1333.223874
@ -86,9 +87,9 @@ boundary_conditions:
#C: 0.0005 #Pa
#C: 0.0010 #Pb
#C: 0.0001 #Pc
#R_d: 11520 # REFERENCE
R_d: 11520 # REFERENCE
#R_d: 8760 #Pa
R_d: 17520 #Pb x2
#R_d: 17520 #Pb x2
#R_d: 4000 #Pc
p0: 85
conv: 1333.223874
@ -131,7 +132,7 @@ fem:
convection_skew_symmetric: True # aka Temam term
stabilization:
forced_normal:
enabled: True
enabled: False
boundaries: [3,4,5,6]
gamma: 20
backflow_boundaries: [3,4,5,6]
@ -162,17 +163,17 @@ estimation:
-
id: 4
type: 'windkessel'
mode: 'Rd'
mode: 'RdC'
initial_stddev: 1
-
id: 5
type: 'windkessel'
mode: 'Rd'
mode: 'RdC'
initial_stddev: 1
-
id: 6
type: 'windkessel'
mode: 'Rd'
mode: 'RdC'
initial_stddev: 1
-
id: 2
@ -186,12 +187,12 @@ estimation:
mesh: '/home/yeye/NuMRI/kalman/meshes/coaortaH3_leo2.0.h5'
#mesh: './meshes/coaortaH1.h5'
fe_degree: 1
xdmf_file: 'measurements/aorta_zdir/Perturbation/Mg15V70/u_all.xdmf'
file_root: 'measurements/aorta_zdir/Perturbation/Mg15V70/u{i}.h5'
#xdmf_file: 'measurements/aorta/u_all.xdmf'
#file_root: 'measurements/aorta/u{i}.h5'
#xdmf_file: 'measurements/aorta_dt1ms/Perturbation/Mg15V70/u_all.xdmf'
#file_root: 'measurements/aorta_dt1ms/Perturbation/Mg15V70/u{i}.h5'
xdmf_file: 'measurements/aorta_exp_dt1ms/u_all.xdmf'
file_root: 'measurements/aorta_exp_dt1ms/u{i}.h5'
indices: 0 # indices of checkpoints to be processed. 0 == all
velocity_direction: ~
velocity_direction: [0,0,1]
noise_stddev: 45 # standard deviation of Gaussian noise
roukf:
@ -199,5 +200,5 @@ estimation:
observation_operator: 'postprocessing' #state or postprocessing
reparameterize: True
ODV_functional:
enable: True
VENC: 142
enable: False
VENC: 138

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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

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@ -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}

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\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}