diff --git a/kalman/input_files/aorta.yaml b/kalman/input_files/aorta.yaml index b7b08e5..f79d1a6 100755 --- a/kalman/input_files/aorta.yaml +++ b/kalman/input_files/aorta.yaml @@ -8,7 +8,7 @@ fluid: implicit_windkessel: True io: - write_path: 'results/Rz_Pc7' + write_path: 'results/aorta' restart: path: '' # './projects/nse_coa3d/results/test_restart2/' time: 0 @@ -27,8 +27,8 @@ boundary_conditions: value: ['0','0','-U*sin(DOLFIN_PI*t/Th)*(t<=Th) + (Th \begin{figure}[!hbtp] @@ -159,362 +157,219 @@ University of Groningen\\[0.5cm] \end{frame} -\begin{frame} - \frametitle{4D flow MRI} -\footnotesize -\onslide<1-> Strategies: -\begin{itemize} -\item<2-> modest spatial resolutions $ \sim (2.5 \times 2.5 \times 2.5 \ mm^3)$ -\item<3-> partial data coverage -\end{itemize} -\begin{columns}[c] -\column{.4\textwidth} % Right column and width -\onslide<4-> -\footnotesize -\begin{figure}[!hbtp] - \begin{center} - \includegraphics[height=0.25\textwidth]{images/channel_noise.png} \\ - (a) Noise - %\caption{Noise} - \end{center} - \end{figure} - \column{.4\textwidth} % Right column and width -\onslide<5-> -\footnotesize -\begin{figure}[!hbtp] - \begin{center} - \includegraphics[height=0.25\textwidth]{images/channel_aliasing.png}\\ - (b) Aliasing - %\caption{Aliasing} - \end{center} - \end{figure} - \column{.4\textwidth} % Right column and width -\onslide<6-> -\footnotesize -\begin{figure}[!hbtp] - \begin{center} - \includegraphics[height=0.25\textwidth]{images/channel_under.png}\\ - (c) Undersampling - %\caption{Aliasing} - \end{center} - \end{figure} -\end{columns} - -\vspace{0.5cm} - -\onslide<7-> Typical quality estimators: SNR, VNR, peak flows/velocities, mass conservation (zero divergence) - -\vspace{0.5cm} - -\onslide<8-> This work $\longrightarrow$ \textbf{conservation of linear momentum} (Navier-Stokes compatibility). - -\end{frame} -\section[]{The corrector field} +\section{The mathematical model} \begin{frame} - \frametitle{The corrector field} + \frametitle{The mathematical model} \begin{center} -Methodology +The mathematical model \end{center} \end{frame} \begin{frame} - \frametitle{The corrector field} + \frametitle{The mathematical model} + + \begin{columns}[c] +\column{.5\textwidth} % Left column and width \footnotesize -\onslide<1-> We assume a perfect physical velocity field $\vec{u}$ -\onslide<2-> \begin{eqnarray*} -\rho \frac{\partial \vec{u}}{\partial t} + \rho \big ( \vec{u} \cdot \nabla \big) \vec{u} - \mu \Delta \vec{u} + \nabla p = 0 \quad \text{in} \quad \Omega \label{eq:NSmom} -\end{eqnarray*} - -\onslide<3-> And a corrector field $\vec{w}$ which satisfies: -\onslide<4-> \begin{align} - \vec{u} & = \vec{u}_{meas} + \vec{w} \quad \text{in} \quad \Omega \label{eq:corrector}\\ -\nabla \cdot \vec w & = 0 \quad \text{in} \quad \Omega \label{eq:correctorDiv} \\ -\vec w & = \vec 0 \quad \text{on} \quad \partial \Omega \label{eq:correctorBC} -\end{align} - -\onslide<5-> The corrector field $\vec{w}$ measures the level of agreedment of the 4D flow measures respect to the Navier-Stokes equations. - -\end{frame} - - -\begin{frame} - \frametitle{The corrector field: Continuum problem} -\footnotesize - -\onslide<1-> Applying the decomposition $\vec{u} = \vec{u}_{meas} + \vec{w}$ into the original equation and writing a variational problem for $\vec w$ we have:\\[0.2cm] -Find $(\vec w(t) ,p(t)) \in H^1_0(\Omega)\times L^2(\Omega)$ such that: -\onslide<2-> \begin{equation*} -\int_{\Omega} \rho \frac{\partial \vec{w}}{\partial t} \cdot \vec{v} + \rho \big ( ( \vec{u}_{meas} + \vec w) \cdot \nabla \big) \vec{w} \cdot \vec{v} + \rho \big ( \vec{w} \cdot \nabla \big) \vec{u}_{meas} \cdot \vec{v} + \mu \nabla \vec{w} : \nabla \vec{v} - p \nabla \cdot \vec{v} + q \nabla \cdot \vec{w} \notag -\end{equation*} -\begin{equation*} -= - \int_{\Omega} \rho \frac{\partial \vec{u}_{meas}}{\partial t} \cdot \vec{v} + \rho \big ( \vec{u}_{meas} \cdot \nabla \big) \vec{u}_{meas} \cdot \vec{v} + \mu \nabla \vec{u}_{meas} : \nabla \vec{v} + q \nabla \cdot \vec{u}_{meas} -\end{equation*} - -\vspace{0.2cm} - -\onslide<3-> or in simple terms: -\onslide<4-> \begin{equation*} -A(\vec w,p;\vec v ,q ) = \mathcal{L} (\vec v) -\end{equation*} - - -for all $(\vec v,q) \in H^1_0(\Omega) \times L^2(\Omega)$. - +\column{.54\textwidth} % Right column and width +\begin{figure}[!hbtp] + \begin{center} + \includegraphics[height=1.1\textwidth]{images/full_aorta.png} + \end{center} + \end{figure} +\end{columns} + \end{frame} \begin{frame} - \frametitle{The corrector field: Discrete problem} + \frametitle{The mathematical model} + + \begin{columns}[c] +\column{.5\textwidth} % Left column and width \footnotesize - -\onslide<1-> In the Discrete, we can write the problem as follows: - -\onslide<2-> \begin{equation} -A_{k}(\vec w,p;\vec v ,q ) + \color{blue}{S^{press}_{k}(\vec w,p;\vec v ,q)} + \color{red}{S^{conv}_{k}(\vec w;\vec v)} \color{black}{ = \mathcal{L}_j (\vec v)} -\label{eq:Corrector_discrete} +\begin{itemize} +\item<2-> Incompressible Navier-Stokes equations: +\begin{equation} +\begin{cases} +\displaystyle \rho \frac{\partial \vec{u}}{\partial t} + \rho \big ( \vec{u} \cdot \nabla \big) \vec{u} - \mu \Delta \vec{u} + \nabla p = 0 \\[0.2cm] +\nabla \cdot \vec{u} = 0 \quad \text{in} \quad \Omega \\[0.2cm] +\vec{u} = \vec{u}_{inlet} \quad \text{on} \quad \Gamma_{in} \\[0.2cm] +\vec{u} = 0 \quad \text{on} \quad \Gamma_{walls} +\end{cases} +\end{equation} +\item<3-> \emph{Three-element} Windkessel coupling at every outlet: +\begin{equation} +\begin{cases} +\displaystyle C_{d,l} \frac{d \pi_l}{dt} + \frac{\pi_l}{R_{d,l}} = Q_l \\[0.2cm] +P_l = R_{p,l} \ Q_l + \pi_l +\end{cases} \end{equation} - -\begin{itemize} -\small -\item<3-> $ -A_{k}(\vec w,p;\vec v ,q ) := \int_{\Omega} \frac{\rho}{\tau} \vec{w} \cdot \vec{v} + \rho \big ( ( \vec{u}_{meas}^k + \vec{w}^{k-1} ) \cdot \nabla \big) \vec{w} \cdot \vec{v} + \rho \big ( \vec{w} \cdot \nabla \big) \vec{u}_{meas}^k \cdot \vec{v} + \mu \nabla \vec{w} : \nabla \vec{v} - p \nabla \cdot \vec{v} + q \nabla \cdot \vec{w} -$ \vspace{0.2cm} -\item<3-> $ \mathcal{L}_j (\vec v) := \int_{\Omega} \frac{\rho}{\tau} \vec{w}^{k-1} \cdot \vec{v} + \mathcal{\ell}_j (\vec v,q) $ - \vspace{0.2cm} -\item<4-> \color{blue}$ -S^{press}_{k}(\vec w,p;\vec v ,q) := \delta \sum_{K \in \Omega}\int_{K} \frac{h_j^2}{\mu} \bigg ( \rho \big ( (\vec u^k_{meas} + \vec w^{k-1}) \cdot \nabla \big) \vec{w} + \rho \big ( \vec{w} \cdot \nabla \big) \vec{u}_{meas}^k + \nabla p \bigg) \cdot \notag \bigg ( \rho \big ( (\vec u^k_{meas} + \vec w^{k-1}) \cdot \nabla \big) \vec{v} + \rho \big ( \vec{v} \cdot \nabla \big) \vec{u}_{meas}^k + \nabla q \bigg ) -$ - \vspace{0.2cm} -\item<5-> \color{red}$ -S^{conv}_{k}(\vec w;\vec v) := \int_{\Omega} \frac{\rho}{2} \ \big( \nabla \cdot (\vec u^k_{meas} + \vec w^{k-1}) \big) \ \vec{w} \cdot \vec{v} -$ \vspace{0.2cm} - \end{itemize} -\end{frame} - - - -\begin{frame} - \frametitle{The corrector field: Well-posedness} -\footnotesize -\onslide<1-> -\begin{theorem} -There exists a unique solution of Problem (\ref{eq:Corrector_discrete}) under the condition: $$\rho/\tau + C_\Omega^{-2} \mu/2 - \rho 3 \| \nabla\vec u_{meas}^k\|_\infty > 0$$ for all $k>0$. -\end{theorem} -\onslide<2-> -We can furthermore prove the following energy balance: -\onslide<3-> -\begin{theorem} For $(\vec w^k ,p^k)$ solution of Problem (\ref{eq:Corrector_discrete}), with $\ell_j(\vec v,q)=0$ it holds -\begin{equation*}\label{eq:energy} - \| \vec w^k \|^2_{L_2(\Omega)} \leq \| \vec w^{k-1} \|^2_{L_2(\Omega)} -\end{equation*} -under the condition -\begin{equation*}\label{eq:condstab} -\mu \geq C_\Omega^2 \rho \| \nabla \vec u_{meas}^k\|_\infty -\end{equation*} -\end{theorem} - - -\end{frame} - - - - - - - - - -\section[Synthetic data]{Experiments using synthetic data } - -\begin{frame} - \frametitle{Experiments} -\begin{center} -Experiments using synthetic data -\end{center} -\end{frame} - - - - - - -\begin{frame} - \frametitle{Numerical tests} - +\column{.54\textwidth} % Right column and width \onslide<1-> -\footnotesize -\begin{columns}[c] -\column{.4\textwidth} % Right column and width -\footnotesize - Simulated channel flow as measurements (Stokes flow) - \column{.5\textwidth} % Right column and width -\footnotesize \begin{figure}[!hbtp] \begin{center} - \includegraphics[height=0.35\textwidth]{images/cilinder_2.png}\\ - (b) Channel mesh - %\caption{Aliasing} + \includegraphics[height=0.9\textwidth]{images/windk_model.png} + \caption{\footnotesize Schematic of the model} \end{center} \end{figure} -\end{columns} - - -\vspace{0.2cm} - -%\onslide<1-> We tested the corrector using CFD simulations as a measurements, in the following testcases: -%\onslide<2-> -%\begin{itemize} -%\item Womersley flow in a cilinder -%\item Navier-Stokes simulations in an aortic mesh -%\end{itemize} -\onslide<2-> Afterwards, perturbations were added: -\begin{itemize} -\item<3-> velocity aliasing (varying the $venc$ parameter) -\item<4-> additive noise (setting SNR in decibels) -\item<5-> simulated k-space undersampling (compressed sensing for the reconstruction) -\end{itemize} -%\onslide<7-> All simulations were done using a stabilized finite element method implemented in FEniCS. Afterwards, all numerical simulations were interpolated into a voxel-type structured mesh -\end{frame} - -% -%\begin{frame} -% \frametitle{Numerical tests: channel} -%\begin{columns}[c] -%\column{.6\textwidth} % Left column and width -%\footnotesize -%\textbf{Channel:} -%\begin{itemize} -%\item Convective term was neglected -%\item Non-slip condition at walls -%\item Oscilatory pressure at $\Gamma_{inlet}$ -%\end{itemize} -%\column{.5\textwidth} % Right column and width -%\footnotesize -%\begin{figure}[!hbtp] -% \begin{center} -% \includegraphics[height=1.0\textwidth]{images/cilinder.png} -% \caption{3D channel mesh} -% \end{center} -% \end{figure} -%\end{columns} -%\end{frame} -% - -\begin{frame} - \frametitle{Numerical tests} -\begin{center} -Results -\end{center} +\end{columns} + \end{frame} - - \begin{frame} - \frametitle{Aliasing and noise} + \frametitle{The mathematical model} + + \begin{columns}[c] +\column{.5\textwidth} % Left column and width \footnotesize +\begin{itemize} +\item Incompressible Navier-Stokes equations: +\begin{equation} +\begin{cases} +\displaystyle \rho \frac{\partial \vec{u}}{\partial t} + \rho \big ( \vec{u} \cdot \nabla \big) \vec{u} - \mu \Delta \vec{u} + \nabla p = 0 \\[0.2cm] +\nabla \cdot \vec{u} = 0 \quad \text{in} \quad \Omega \\[0.2cm] +\vec{u} = \vec{u}_{inlet} \quad \text{on} \quad \Gamma_{in} \\[0.2cm] +\vec{u} = 0 \quad \text{on} \quad \Gamma_{walls} +\end{cases} +\end{equation} +\item \emph{Three-element} Windkessel coupling at every outlet: +\begin{equation} +\begin{cases} +\displaystyle C_{d,l} \frac{d \pi_l}{dt} + \frac{\pi_l}{R_{d,l}} = Q_l \\[0.2cm] +P_l = R_{p,l} \ Q_l + \pi_l +\end{cases} +\end{equation} +\end{itemize} -\onslide<1-> For comparison we defined a perfect corrector field as: $\delta \vec u = \vec u_{ref} - \vec u_{meas}$ + +\column{.54\textwidth} % Right column and width +\begin{figure}[!hbtp] + \begin{center} + \includegraphics[height=0.9\textwidth]{images/ref.png} + \caption{\footnotesize Schematic of the model} + \end{center} + \end{figure} +\end{columns} + +\end{frame} + + + +\begin{frame} + \frametitle{The inverse problem} +\begin{itemize} +\item<1-> Upon this solution $\Longrightarrow$ build a set of measurements +\item<2-> Induce typical arifacts via a Magnetization vector: $M(\vec{x},t)= M_0(\vec{x}) exp(i\phi_0 + i \frac{\pi}{venc} u(\vec{x},t))$ +\end{itemize} +\onslide<3-> + \begin{columns} + \column{.3\textwidth} + \flushleft + \begin{figure} + \includegraphics[width=1.1\textwidth]{images/ref_int.png} + \caption*{(a) Interpolated reference solution} +\end{figure} +\column{.67\textwidth} +\centering +\onslide<4-> \textbf{The measurements:} +\begin{itemize} +\item<5-> Gaussian noise into the magnetization +\item<6-> Different levels of aliasing varying the $venc$ parameter +\item<7-> Only using the dominant component of the velocity: $u_z$ +\end{itemize} +\end{columns} +\end{frame} + + + + + +\begin{frame} + \frametitle{The inverse problem} +\begin{itemize} +\item Upon this solution $\Longrightarrow$ build a set of measurements +\item Induce typical arifacts via a Magnetization vector: $M(\vec{x},t)= M_0(\vec{x}) exp(i\phi_0 + i \frac{\pi}{venc} u(\vec{x},t))$ +\end{itemize} + \begin{columns} + \column{.3\textwidth} + \flushleft + \begin{figure} + \includegraphics[width=1.1\textwidth]{images/ref_int.png} + \caption*{(a) Interpolated reference solution} +\end{figure} +\column{.67\textwidth} +\flushleft + \begin{figure} + \includegraphics[width=1.1\textwidth]{images/supra_venc.png} + \caption*{(b) Aliased measurements with different $vencs = 120,70,30 \%$ of $u_{max}$} + \hfill +\end{figure} +\end{columns} +\end{frame} + + + +\begin{frame} + \frametitle{The inverse problem} +\begin{itemize} +\item Upon this solution $\Longrightarrow$ build a set of measurements +\item Induce typical arifacts via a Magnetization vector: $M(\vec{x},t)= M_0(\vec{x}) exp(i\phi_0 + i \frac{\pi}{venc} u(\vec{x},t))$ +\end{itemize} + \begin{columns} + \column{.3\textwidth} + \flushleft + \begin{figure} + \includegraphics[width=1.1\textwidth]{images/ref_int.png} + \caption*{(a) Interpolated reference solution} +\end{figure} +\column{.67\textwidth} +\flushleft + \begin{figure} + \includegraphics[width=1.1\textwidth]{images/coartation.png} + \caption*{(b) Aliased measurements with different $vencs = 120,70,30 \%$ of $u_{max}$} + \hfill +\end{figure} +\end{columns} +\end{frame} + + + +\begin{frame} + \frametitle{The Kalman Filter} +\begin{itemize} +\item<1-> We use a Reduced Order Unscendent Kalman Filter (ROUKF) to reconstruct the parameter vector $\theta$ solving the next optimization problem: \onslide<2-> -\begin{figure}[!hbtp] - \begin{center} - \includegraphics[height=0.45\textwidth]{images/channel_ppt_1.png} -\caption{\small Fields for the channel: $(SNR,venc) = (\infty,120\%)$. $\vec{w} \times 200$} - \end{center} - \end{figure} - -\end{frame} - -\begin{frame} - \frametitle{Aliasing and noise} -\footnotesize - -For comparison we defined a perfect corrector field as: $\delta \vec u = \vec u_{ref} - \vec u_{meas}$ - - -\begin{figure}[!hbtp] - \begin{center} - \includegraphics[height=0.45\textwidth]{images/channel_ppt_2.png} - \caption{\small Fields for the channel: $(SNR,venc) = (\infty,80\%)$. $\vec{w} \times 4$ } -%\caption{\small Different perturbation scenarios. $(\infty , 120 \%)$: $\vec{w} \times 200$, $(10 \ dB , 120 \%)$: $\delta \vec{u}, \vec{w} \times 4$, rest: $\vec{w} \times 4$ } - \end{center} - \end{figure} - -\end{frame} - - -\begin{frame} - \frametitle{Aliasing and noise} -\footnotesize -For comparison we defined a perfect corrector field as: $\delta \vec u = \vec u_{ref} - \vec u_{meas}$ - -\begin{figure}[!hbtp] - \begin{center} - \includegraphics[height=0.45\textwidth]{images/channel_ppt_3.png} - \caption{\small Fields for the channel: $(SNR,venc) = (10 \ dB,120\%)$. $\delta \vec{u}, \vec{w} \times 4$} -%\caption{\small Different perturbation scenarios. $(\infty , 120 \%)$: $\vec{w} \times 200$, $(10 \ dB , 120 \%)$: $\delta \vec{u}, \vec{w} \times 4$, rest: $\vec{w} \times 4$ } - \end{center} - \end{figure} - -\end{frame} - - -\begin{frame} - \frametitle{Aliasing and noise} -\footnotesize -For comparison we defined a perfect corrector field as: $\delta \vec u = \vec u_{ref} - \vec u_{meas}$ - -\begin{figure}[!hbtp] - \begin{center} - \includegraphics[height=0.45\textwidth]{images/channel_ppt_4.png} - \caption{\small Fields for the channel: $(SNR,venc) = (10 \ dB,80\%)$. $\vec{w} \times 4$} -%\caption{\small Different perturbation scenarios. $(\infty , 120 \%)$: $\vec{w} \times 200$, $(10 \ dB , 120 \%)$: $\delta \vec{u}, \vec{w} \times 4$, rest: $\vec{w} \times 4$ } - \end{center} - \end{figure} - -\end{frame} - - - -\begin{frame} - \frametitle{Aliasing and noise} -\footnotesize - -\begin{figure}[!hbtp] - \begin{center} - \includegraphics[height=0.5\textwidth]{images/channel_curves_SNRinf.png} -\caption{ \footnotesize Evolution of the $L-2$ norms of the components of $\vec w$} - \end{center} - \end{figure} - - -\end{frame} - - -\begin{frame} - \frametitle{Aliasing and noise} -\footnotesize - -\begin{figure}[!hbtp] - \begin{center} - \includegraphics[height=0.5\textwidth]{images/channel_curves_SNR10.png} -\caption{ \footnotesize Evolution of the $L-2$ norms of the components of $\vec w$} - \end{center} - \end{figure} +\begin{equation*} +\hat{\theta} = arg \min_{\theta} J(\theta) +\end{equation*} +\begin{equation} +J(\theta) = \displaystyle \frac{1}{2} || \theta - \theta_0 ||^2_{P_0^{-1}} + \sum_{k=1}^N \frac{1}{2} || Z_k - \mathbb{H} X_k(\theta) ||^2_{W^{-1}} +\end{equation} +\onslide<3-> Where: +\begin{itemize} +\item<4-> $Z$ the measurements and $X = (\vec{u} , \pi)$ the state variable +\item<5-> $\mathbb{H}$ observation operator +\item<6-> $\theta_0$ is the initial guess for the parameters +\item<7-> $P_0$ is the associated covariance matrix +\item<8-> $W$ is the associated covariance matrix to the meas. noise +\end{itemize} +\end{itemize} \end{frame} @@ -522,255 +377,91 @@ For comparison we defined a perfect corrector field as: $\delta \vec u = \vec u_ \begin{frame} - \frametitle{Undersampling} -\footnotesize + \frametitle{The Kalman Filter} +The parameter vector: + +\begin{itemize} +\item<1-> Plug flow at the inlet: $u_{inlet} = -U f(t) \hat{n}$, with $f(t)$ is the weaveform which simulate a cardiac cycle and $\hat{n}$ is the outward normal vector. +\item<2-> Since $R_p << R_d$, we only consider an optimization dependent on $R_d, C$ for every 3D-0D coupled outlet +\end{itemize} -\begin{figure}[!hbtp] - \begin{center} - \includegraphics[height=0.6\textwidth]{images/histo_channel.png} -\caption{ \footnotesize Histograms of different undersampling rates for the channel} - \end{center} - \end{figure} +\onslide<3-> $$\theta = (U,\vec{R_d},\vec{C})$$ \\ with $\vec{R_d} = R_{d,l}$, $\vec{C} = C_l$ for $l=1,..., n_l$ \end{frame} - -%\begin{frame} -% \frametitle{Results for channel: undersampling} -%\footnotesize -% -%\begin{figure}[!hbtp] -% \begin{center} -% \includegraphics[height=0.6\textwidth]{images/undersampling_press.png} -%\caption{ \footnotesize Different undersampling rates for the channel} -% \end{center} -% \end{figure} -% -% -%\end{frame} -% - - - -%\begin{frame} -% \frametitle{Numerical tests: aorta} -% -%\begin{columns}[c] -%\column{.6\textwidth} % Left column and width -%\footnotesize -%\textbf{Aorta} -%\begin{itemize} -%\item a mild coartation was added in the descending aorta -%\item $u_{inlet}$ simulates a cardiac cycle -%\item 3-element Windkessel for the outlets -%\item Non-slip condition at walls -%\end{itemize} - -%\column{.5\textwidth} % Right column and width -%\footnotesize -%\begin{figure}[!hbtp] -% \begin{center} -% \includegraphics[height=1.0\textwidth]{images/aorta_blender.png} -%\caption{Aortic mesh} -% \end{center} -% \end{figure} -%\end{columns} -% -% -%\end{frame} -% -% - - -%\begin{frame} -% \frametitle{Results for aorta: aliasing and noise} -%\footnotesize -% -%\begin{figure}[!hbtp] -% \begin{center} -% \includegraphics[height=0.7\textwidth]{images/aorta_perturbation.png} -%\caption{Different perturbation scenarios for the aortic mesh} -% \end{center} -% \end{figure} -% -%\end{frame} -% -% -%\begin{frame} -% \frametitle{Results for aorta: undersampling} -%\footnotesize -% -%\begin{figure}[!hbtp] -% \begin{center} -% \includegraphics[height=0.6\textwidth]{images/histo_blender.png} -%\caption{ \footnotesize Histograms of different undersampling rates for the aortic mesh} -% \end{center} -% \end{figure} -% -%\end{frame} -% -%\begin{frame} -% \frametitle{Results for aorta: undersampling} -%\footnotesize -% -%\begin{figure}[!hbtp] -% \begin{center} -% \includegraphics[height=0.7\textwidth]{images/undersampling_blender.png} -%\caption{ \footnotesize Different undersampling rates for the aortic mesh} -% \end{center} -% \end{figure} -% -%\end{frame} -% -% - - - -\section[4D flow data]{Experiments using real 4D flow data } +\section{Numerical Experiments} \begin{frame} - \frametitle{Experiments} + \frametitle{Numerical Experiments} \begin{center} -Experiments using real 4D flow data +Numerical Experiments \end{center} \end{frame} - - - \begin{frame} - \frametitle{Experiments} -\footnotesize - -\begin{columns}[c] -\column{.6\textwidth} % Left column and width - -\begin{itemize} -\item<1-> 4D flow measurements were taken from a silicon thoracic aortic phantom made of silicon. -\item<2-> A controled pump (heart rate, peak flow, stroke volume and flow waveform) -\item<3-> A stenosis of $11 \ mm$ of diameter was added in the descending aorta -\item<4-> The phantom was scanned using a clinical $1.5 \ T$ MR scanner (Philips Achieva, Best, The Netherlands) -\end{itemize} - - -\column{.5\textwidth} % Right column and width - -\begin{figure}[!hbtp] - \begin{center} - \footnotesize - \includegraphics[height=\textwidth]{images/phantom.jpg} -\caption{\footnotesize{Experiment done at the Centre of Biomedical Images (CIB) of the Catholic Unversity of Chili (PUC)}} - \end{center} - \end{figure} - -\end{columns} - -%\includemedia[width=0.6\linewidth,height=0.6\linewidth,activate=pageopen, -%passcontext, -%transparent, -%addresource=images/phantom.mp4, -%flashvars={source=images/phantom.mp4} -%]{\includegraphics[width=0.6\linewidth]{images/phantom.jpg}}{VPlayer.swf} -% + \frametitle{Numerical Experiments} \end{frame} - - - -\begin{frame} - \frametitle{Results} -\footnotesize - -\begin{figure} -\begin{subfigure}{.31\textwidth} - \centering - % \includegraphics[trim=100 80 100 150, clip, width=1.0\textwidth]{images/u_15.png} - \caption*{(a) $\vec{u}_{meas}$} -\end{subfigure} -\begin{subfigure}{.01\textwidth} - \hfill -\end{subfigure} -\begin{subfigure}{.31\textwidth} - \centering - %\includegraphics[trim=100 80 100 150, clip, width=1.0\textwidth]{images/w_15.png} - \caption*{(b) $\vec{w}$} -\end{subfigure} -\begin{subfigure}{.01\textwidth} - \hfill -\end{subfigure} -\begin{subfigure}{.31\textwidth} - \centering - %\includegraphics[trim=100 80 100 150, clip, width=1.0\textwidth]{images/uc_15.png} - \caption*{(c) $\vec{u}_{meas}+\vec{w}$} -\end{subfigure} -\caption{Measurements, corrector fields and corrected velocities for all the cases.} -\label{fig:phantom_resolution} -\end{figure} - -\end{frame} - - - - - \section{Conclusions} \begin{frame} - \frametitle{Experiments} \begin{center} Conclusions \end{center} \end{frame} -\begin{frame} - \frametitle{Conclusions and future work} -\footnotesize - -\onslide<1-> Potential of the new quality parameter: - -\begin{itemize} -\item<2-> Vector fields has more details -\item<3-> Artifacts recognition -\end{itemize} - - -\onslide<4-> Future: -\begin{itemize} -\item<5-> The use of the field for create new inverse problems which can be used for further accelerations -\end{itemize} - - - -\end{frame} -\begin{frame} -\begin{center} -\huge{Thank you for your time!} -\end{center} -\end{frame} - - - - - - - -%\includegraphics<1>[height=4.5cm]{images/pat1.png} -%\includegraphics<2>[height=4.5cm]{images/pat2.png} \end{document} + + + + + + +%\begin{frame} +% \frametitle{Results} +%\footnotesize +% +%\begin{figure} +%\begin{subfigure}{.31\textwidth} +% \centering +% % \includegraphics[trim=100 80 100 150, clip, width=1.0\textwidth]{images/u_15.png} +% \caption*{(a) $\vec{u}_{meas}$} +%\end{subfigure} +%\begin{subfigure}{.01\textwidth} +% \hfill +%\end{subfigure} +%\begin{subfigure}{.31\textwidth} +% \centering +% %\includegraphics[trim=100 80 100 150, clip, width=1.0\textwidth]{images/w_15.png} +% \caption*{(b) $\vec{w}$} +%\end{subfigure} +%\begin{subfigure}{.01\textwidth} +% \hfill +%\end{subfigure} +%\begin{subfigure}{.31\textwidth} +% \centering +% %\includegraphics[trim=100 80 100 150, clip, width=1.0\textwidth]{images/uc_15.png} +% \caption*{(c) $\vec{u}_{meas}+\vec{w}$} +%\end{subfigure} +%\caption{Measurements, corrector fields and corrected velocities for all the cases.} +%\label{fig:phantom_resolution} +%\end{figure} +% +%\end{frame} \ No newline at end of file