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J.E. Garay Labra 1 year ago
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  1. 36
      kalman/input_files/aorta.yaml
  2. BIN
      presentations/press_8ecm/images/4dflow.png
  3. BIN
      presentations/press_8ecm/images/coartation.png
  4. BIN
      presentations/press_8ecm/images/full_aorta.png
  5. BIN
      presentations/press_8ecm/images/ref.png
  6. BIN
      presentations/press_8ecm/images/ref_int.png
  7. BIN
      presentations/press_8ecm/images/supra_venc.png
  8. BIN
      presentations/press_8ecm/images/windk_model.png
  9. 703
      presentations/press_8ecm/press.tex

36
kalman/input_files/aorta.yaml

@ -8,7 +8,7 @@ fluid: @@ -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: @@ -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: @@ -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: @@ -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: @@ -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: @@ -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: @@ -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

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703
presentations/press_8ecm/press.tex

@ -98,7 +98,7 @@ @@ -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\\ @@ -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] @@ -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] @@ -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}
\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}
\frametitle{The mathematical model}
\begin{columns}[c]
\column{.5\textwidth} % Left column and width
\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: $\delta \vec u = \vec u_{ref} - \vec u_{meas}$
\onslide<2->
\column{.54\textwidth} % Right column and width
\begin{figure}[!hbtp]
\begin{center}
\includegraphics[height=0.45\textwidth]{images/channel_ppt_1.png}
\caption{\small Fields for the channel: $(SNR,venc) = (\infty,120\%)$. $\vec{w} \times 200$}
\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{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}
\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{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}
\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{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}
\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{Aliasing and noise}
\footnotesize
\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:
\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}
\onslide<2->
\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_ @@ -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}
\end{document}
%\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}
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