press images
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@ -8,7 +8,7 @@ fluid:
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implicit_windkessel: True
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io:
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write_path: 'results/Rz_Pc7'
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write_path: 'results/aorta'
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restart:
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path: '' # './projects/nse_coa3d/results/test_restart2/'
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time: 0
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@ -27,8 +27,8 @@ boundary_conditions:
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value: ['0','0','-U*sin(DOLFIN_PI*t/Th)*(t<=Th) + (Th<t)*(U*DOLFIN_PI/Th*(t-Th)*exp(-(t-Th)*beta))']
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parameters:
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#U: 75 #REFERENCE
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#U: 150 #Pa
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U: 40 #Pc
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U: 150 #Pa Pb
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#U: 40 #Pc
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Th: 0.36
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beta: 70
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t: 0
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@ -57,8 +57,8 @@ boundary_conditions:
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#C: 0.0008 # Pg
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#R_d: 7200 # REFERENCE
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#R_d: 8760 #Pa
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#R_d: 17520 #Pb x2
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R_d: 4000 #Pc
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R_d: 17520 #Pb x2
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#R_d: 4000 #Pc
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p0: 85
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conv: 1333.223874
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-
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@ -73,8 +73,8 @@ boundary_conditions:
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#C: 0.0008 # Pg
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#R_d: 11520 # REFERENCE
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#R_d: 8760 #Pa
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#R_d: 17520 #Pb x2
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R_d: 4000 #Pc
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R_d: 17520 #Pb x2
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#R_d: 4000 #Pc
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p0: 85
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conv: 1333.223874
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-
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@ -88,8 +88,8 @@ boundary_conditions:
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#C: 0.0001 #Pc
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#R_d: 11520 # REFERENCE
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#R_d: 8760 #Pa
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#R_d: 17520 #Pb x2
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R_d: 4000 #Pc
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R_d: 17520 #Pb x2
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#R_d: 4000 #Pc
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p0: 85
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conv: 1333.223874
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@ -163,17 +163,17 @@ estimation:
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id: 4
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type: 'windkessel'
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mode: 'Rd'
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initial_stddev: 2
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initial_stddev: 1
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-
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id: 5
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type: 'windkessel'
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mode: 'Rd'
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initial_stddev: 2
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initial_stddev: 1
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-
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id: 6
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type: 'windkessel'
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mode: 'Rd'
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initial_stddev: 2
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initial_stddev: 1
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-
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id: 2
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type: 'dirichlet'
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@ -186,13 +186,13 @@ estimation:
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mesh: '/home/yeye/NuMRI/kalman/meshes/coaortaH3_leo2.0.h5'
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#mesh: './meshes/coaortaH1.h5'
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fe_degree: 1
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xdmf_file: 'measurements/aorta_zdir/Perturbation/Mg15V120/u_all.xdmf'
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file_root: 'measurements/aorta_zdir/Perturbation/Mg15V120/u{i}.h5'
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#xdmf_file: 'measurements/aorta_zdir_NV/u_all.xdmf'
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#file_root: 'measurements/aorta_zdir_NV/u{i}.h5'
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#xdmf_file: 'measurements/aorta_zdir/Perturbation/Mg15V120/u_all.xdmf'
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#file_root: 'measurements/aorta_zdir/Perturbation/Mg15V120/u{i}.h5'
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xdmf_file: 'measurements/aorta/u_all.xdmf'
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file_root: 'measurements/aorta/u{i}.h5'
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indices: 0 # indices of checkpoints to be processed. 0 == all
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velocity_direction: [0,0,1]
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noise_stddev: 25 # standard deviation of Gaussian noise
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velocity_direction: ~
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noise_stddev: 0 # standard deviation of Gaussian noise
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roukf:
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particles: 'simplex' # unique or simplex
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@ -98,7 +98,7 @@
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\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}
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\title{Robust parameter estimation in fluid flow models from aliased velocity measurements}
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%\author[Jeremías Garay Labra]
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%{Jeremías Garay Labra}
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\institute[University of Groningen]
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@ -108,7 +108,7 @@ Faculty of Sciences and Engineering\\
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University of Groningen\\[0.5cm]
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%\includegraphics[height=1.5cm]{Imagenes/escudoU2014.pdf}
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% \includegraphics[height=1cm]{Imagenes/fcfm.png} \\[0.5cm]
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Jeremías Garay Labra \emph{join with} Hernan Mella, Julio Sotelo, Sergio Uribe, Cristobal Bertoglio and Joaquin Mura.}
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Jeremías Garay Labra \emph{join with} Cristobal Bertoglio.}
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\date{\today}
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@ -145,8 +145,6 @@ University of Groningen\\[0.5cm]
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\end{itemize}
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\column{.54\textwidth} % Right column and width
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\onslide<1->
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\begin{figure}[!hbtp]
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@ -159,362 +157,219 @@ University of Groningen\\[0.5cm]
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\end{frame}
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\begin{frame}
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\frametitle{4D flow MRI}
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\footnotesize
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\onslide<1-> Strategies:
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\begin{itemize}
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\item<2-> modest spatial resolutions $ \sim (2.5 \times 2.5 \times 2.5 \ mm^3)$
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\item<3-> partial data coverage
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\end{itemize}
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\begin{columns}[c]
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\column{.4\textwidth} % Right column and width
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\onslide<4->
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\footnotesize
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\begin{figure}[!hbtp]
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\begin{center}
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\includegraphics[height=0.25\textwidth]{images/channel_noise.png} \\
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(a) Noise
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%\caption{Noise}
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\end{center}
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\end{figure}
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\column{.4\textwidth} % Right column and width
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\onslide<5->
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\footnotesize
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\begin{figure}[!hbtp]
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\begin{center}
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\includegraphics[height=0.25\textwidth]{images/channel_aliasing.png}\\
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(b) Aliasing
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%\caption{Aliasing}
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\end{center}
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\end{figure}
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\column{.4\textwidth} % Right column and width
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\onslide<6->
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\footnotesize
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\begin{figure}[!hbtp]
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\begin{center}
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\includegraphics[height=0.25\textwidth]{images/channel_under.png}\\
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(c) Undersampling
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%\caption{Aliasing}
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\end{center}
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\end{figure}
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\end{columns}
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\vspace{0.5cm}
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\onslide<7-> Typical quality estimators: SNR, VNR, peak flows/velocities, mass conservation (zero divergence)
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\vspace{0.5cm}
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\onslide<8-> This work $\longrightarrow$ \textbf{conservation of linear momentum} (Navier-Stokes compatibility).
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\end{frame}
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\section[]{The corrector field}
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\section{The mathematical model}
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\begin{frame}
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\frametitle{The corrector field}
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\frametitle{The mathematical model}
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\begin{center}
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Methodology
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The mathematical model
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\end{center}
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\end{frame}
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\begin{frame}
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\frametitle{The corrector field}
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\frametitle{The mathematical model}
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\begin{columns}[c]
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\column{.5\textwidth} % Left column and width
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\footnotesize
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\onslide<1-> We assume a perfect physical velocity field $\vec{u}$
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\onslide<2-> \begin{eqnarray*}
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\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}
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\end{eqnarray*}
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\onslide<3-> And a corrector field $\vec{w}$ which satisfies:
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\onslide<4-> \begin{align}
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\vec{u} & = \vec{u}_{meas} + \vec{w} \quad \text{in} \quad \Omega \label{eq:corrector}\\
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\nabla \cdot \vec w & = 0 \quad \text{in} \quad \Omega \label{eq:correctorDiv} \\
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\vec w & = \vec 0 \quad \text{on} \quad \partial \Omega \label{eq:correctorBC}
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\end{align}
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\onslide<5-> The corrector field $\vec{w}$ measures the level of agreedment of the 4D flow measures respect to the Navier-Stokes equations.
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\end{frame}
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\begin{frame}
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\frametitle{The corrector field: Continuum problem}
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\footnotesize
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\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]
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Find $(\vec w(t) ,p(t)) \in H^1_0(\Omega)\times L^2(\Omega)$ such that:
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\onslide<2-> \begin{equation*}
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\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
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\end{equation*}
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\begin{equation*}
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= - \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}
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\end{equation*}
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\vspace{0.2cm}
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\onslide<3-> or in simple terms:
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\onslide<4-> \begin{equation*}
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A(\vec w,p;\vec v ,q ) = \mathcal{L} (\vec v)
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\end{equation*}
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for all $(\vec v,q) \in H^1_0(\Omega) \times L^2(\Omega)$.
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\column{.54\textwidth} % Right column and width
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\begin{figure}[!hbtp]
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\begin{center}
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\includegraphics[height=1.1\textwidth]{images/full_aorta.png}
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\end{center}
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\end{figure}
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\end{columns}
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\end{frame}
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\begin{frame}
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\frametitle{The corrector field: Discrete problem}
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\frametitle{The mathematical model}
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\begin{columns}[c]
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\column{.5\textwidth} % Left column and width
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\footnotesize
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\onslide<1-> In the Discrete, we can write the problem as follows:
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\onslide<2-> \begin{equation}
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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)}
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\label{eq:Corrector_discrete}
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\begin{itemize}
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\item<2-> Incompressible Navier-Stokes equations:
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\begin{equation}
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\begin{cases}
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\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]
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\nabla \cdot \vec{u} = 0 \quad \text{in} \quad \Omega \\[0.2cm]
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\vec{u} = \vec{u}_{inlet} \quad \text{on} \quad \Gamma_{in} \\[0.2cm]
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\vec{u} = 0 \quad \text{on} \quad \Gamma_{walls}
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\end{cases}
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\end{equation}
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\item<3-> \emph{Three-element} Windkessel coupling at every outlet:
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\begin{equation}
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\begin{cases}
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\displaystyle C_{d,l} \frac{d \pi_l}{dt} + \frac{\pi_l}{R_{d,l}} = Q_l \\[0.2cm]
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P_l = R_{p,l} \ Q_l + \pi_l
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\end{cases}
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\end{equation}
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\begin{itemize}
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\small
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\item<3-> $
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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}
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$ \vspace{0.2cm}
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\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) $
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\vspace{0.2cm}
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\item<4-> \color{blue}$
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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 )
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$
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\vspace{0.2cm}
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\item<5-> \color{red}$
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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}
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$ \vspace{0.2cm}
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{The corrector field: Well-posedness}
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\footnotesize
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\onslide<1->
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\begin{theorem}
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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 $k>0$.
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\end{theorem}
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\onslide<2->
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We can furthermore prove the following energy balance:
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\onslide<3->
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\begin{theorem} For $(\vec w^k ,p^k)$ solution of Problem (\ref{eq:Corrector_discrete}), with $\ell_j(\vec v,q)=0$ it holds
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\begin{equation*}\label{eq:energy}
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\| \vec w^k \|^2_{L_2(\Omega)} \leq \| \vec w^{k-1} \|^2_{L_2(\Omega)}
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\end{equation*}
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under the condition
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\begin{equation*}\label{eq:condstab}
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\mu \geq C_\Omega^2 \rho \| \nabla \vec u_{meas}^k\|_\infty
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\end{equation*}
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\end{theorem}
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\end{frame}
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\section[Synthetic data]{Experiments using synthetic data }
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\begin{frame}
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\frametitle{Experiments}
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\begin{center}
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Experiments using synthetic data
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\end{center}
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\end{frame}
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\begin{frame}
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\frametitle{Numerical tests}
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\column{.54\textwidth} % Right column and width
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\onslide<1->
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\footnotesize
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\begin{columns}[c]
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\column{.4\textwidth} % Right column and width
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\footnotesize
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Simulated channel flow as measurements (Stokes flow)
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\column{.5\textwidth} % Right column and width
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\footnotesize
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\begin{figure}[!hbtp]
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\begin{center}
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\includegraphics[height=0.35\textwidth]{images/cilinder_2.png}\\
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(b) Channel mesh
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%\caption{Aliasing}
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\includegraphics[height=0.9\textwidth]{images/windk_model.png}
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\caption{\footnotesize Schematic of the model}
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\end{center}
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\end{figure}
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\end{columns}
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\vspace{0.2cm}
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%\onslide<1-> We tested the corrector using CFD simulations as a measurements, in the following testcases:
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%\onslide<2->
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%\begin{itemize}
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%\item Womersley flow in a cilinder
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%\item Navier-Stokes simulations in an aortic mesh
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%\end{itemize}
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\onslide<2-> Afterwards, perturbations were added:
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\begin{itemize}
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\item<3-> velocity aliasing (varying the $venc$ parameter)
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\item<4-> additive noise (setting SNR in decibels)
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\item<5-> simulated k-space undersampling (compressed sensing for the reconstruction)
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\end{itemize}
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%\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
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\end{frame}
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%
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%\begin{frame}
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% \frametitle{Numerical tests: channel}
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%\begin{columns}[c]
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%\column{.6\textwidth} % Left column and width
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%\footnotesize
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%\textbf{Channel:}
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%\begin{itemize}
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%\item Convective term was neglected
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%\item Non-slip condition at walls
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||||
%\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}
|
||||
Loading…
Reference in New Issue