diff --git a/kalman/graphics/figure3.py b/kalman/graphics/figure3.py index 1ff8784..9f76045 100644 --- a/kalman/graphics/figure3.py +++ b/kalman/graphics/figure3.py @@ -146,27 +146,12 @@ def plot_parameters(dat, input_file, deparameterize=False, ref=None): for i in range(len(ids)): - cur_key = ids[i] - - true_level = np.log(true_values[ids[i]]/current_val[i])/np.log(2) - rec_value = np.round(2**theta[-1, idx]*current_val[i],2) - - - #curve = theta[:,idx] + line_split*idx - true_level - #dash_curve = line_split*idx + t*0 - - curve = 2**theta[:, idx]*current_val[i] - std_down = 2**(-np.sqrt(P[:, idx, idx]))*curve - std_up = 2**np.sqrt(P[:, idx, idx])*curve - dash_curve = true_values[ids[i]] + t*0 - - if ids_type[i] == 'dirichlet': fig3, axes3 = plt.subplots(1,1,figsize=(12,5)) axes3.plot(t, curve , '-', color=col_,label= legends_ + '= ' + str(rec_value) + '/' + str(true_values[cur_key]) + '$', linewidth = 4) axes3.fill_between(t, std_down, std_up, alpha=0.3, color=col_) legends_=next(legends) - axes3.plot(t, dash_curve , color=col_,ls='--') + axes3.plot(t, dash_curve , color=col_,ls='--' , linewidth = 3) axes3.set_ylabel(r'$U$',fontsize=36) axes3.legend(fontsize=36,loc='upper right') axes3.set_xlim([-0.01,0.81]) @@ -177,28 +162,34 @@ def plot_parameters(dat, input_file, deparameterize=False, ref=None): plt.savefig('U.png') plt.close(fig3) else: + + cur_key = ids[i] + rec_value = np.round(2**theta[-1, idx]*current_val[i],2) + curve = 2**theta[:, idx]*current_val[i] + std_down = 2**(-np.sqrt(P[:, idx, idx]))*curve + std_up = 2**np.sqrt(P[:, idx, idx])*curve + dash_curve = true_values[ids[i]] + t*0 + axes1.plot(t, curve , '-', color=col_,label= legends_ + '= ' + str(rec_value) + '/' + str(true_values[cur_key]) + '$', linewidth = 3) axes1.fill_between(t, std_down, std_up, alpha=0.3, color=col_) - axes1.plot(t, dash_curve , color=col_,ls='--') + axes1.plot(t, dash_curve , color=col_,ls='--',linewidth = 3) legends_=next(legends) + if RC_flag: + if i -- 0 +< 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defining Unicode char U+017A (decimal 378) + defining Unicode char U+017B (decimal 379) + defining Unicode char U+017C (decimal 380) + defining Unicode char U+017D (decimal 381) + defining Unicode char U+017E (decimal 382) + defining Unicode char U+01CD (decimal 461) + defining Unicode char U+01CE (decimal 462) + defining Unicode char U+01CF (decimal 463) + defining Unicode char U+01D0 (decimal 464) + defining Unicode char U+01D1 (decimal 465) + defining Unicode char U+01D2 (decimal 466) + defining Unicode char U+01D3 (decimal 467) + defining Unicode char U+01D4 (decimal 468) + defining Unicode char U+01E2 (decimal 482) + defining Unicode char U+01E3 (decimal 483) + defining Unicode char U+01E6 (decimal 486) + defining Unicode char U+01E7 (decimal 487) + defining Unicode char U+01E8 (decimal 488) + defining Unicode char U+01E9 (decimal 489) + defining Unicode char U+01EA (decimal 490) + defining Unicode char U+01EB (decimal 491) + defining Unicode char U+01F0 (decimal 496) + defining 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{5}{Conclusions}{56}{Conclusions}{0}} +\headcommand {\slideentry {5}{0}{1}{56/56}{}{0}} +\headcommand {\beamer@framepages {56}{56}} +\headcommand {\slideentry {5}{0}{2}{57/62}{}{0}} +\headcommand {\beamer@framepages {57}{62}} +\headcommand {\slideentry {5}{0}{3}{63/63}{}{0}} +\headcommand {\beamer@framepages {63}{63}} +\headcommand {\beamer@partpages {1}{63}} +\headcommand {\beamer@subsectionpages {56}{63}} +\headcommand {\beamer@sectionpages {56}{63}} +\headcommand {\beamer@documentpages {63}} +\headcommand {\gdef \inserttotalframenumber {25}} diff --git a/presentations/press_8ecm/press.out b/presentations/press_8ecm/press.out new file mode 100644 index 0000000..42323ec --- /dev/null +++ b/presentations/press_8ecm/press.out @@ -0,0 +1,5 @@ +\BOOKMARK [2][]{Outline0.1}{4D flow MRI}{}% 1 +\BOOKMARK [2][]{Outline0.2}{The mathematical model}{}% 2 +\BOOKMARK [2][]{Outline0.3}{The inverse problem}{}% 3 +\BOOKMARK [2][]{Outline0.4}{Numerical Experiments}{}% 4 +\BOOKMARK [2][]{Outline0.5}{Conclusions}{}% 5 diff --git a/presentations/press_8ecm/press.pdf b/presentations/press_8ecm/press.pdf new file mode 100644 index 0000000..afdd7d9 Binary files /dev/null and b/presentations/press_8ecm/press.pdf differ diff --git a/presentations/press_8ecm/press.snm b/presentations/press_8ecm/press.snm new file mode 100644 index 0000000..e69de29 diff --git a/presentations/press_8ecm/press.synctex.gz b/presentations/press_8ecm/press.synctex.gz new file mode 100644 index 0000000..1fbdec6 Binary files /dev/null and b/presentations/press_8ecm/press.synctex.gz differ diff --git a/presentations/press_8ecm/press.tex b/presentations/press_8ecm/press.tex index 4251a0a..61adf95 100755 --- a/presentations/press_8ecm/press.tex +++ b/presentations/press_8ecm/press.tex @@ -135,14 +135,11 @@ University of Groningen\\[0.5cm] \footnotesize \begin{itemize} -\item<2-> Full 3D coverage of the region of interest +\item<2-> Velocities encoded into the magnetization phase \item<3-> Rich post-proccesing: derived parameters \end{itemize} -\onslide<4-> Disadvantages: -\begin{itemize} -\item<5-> Long scan time -\end{itemize} + \column{.54\textwidth} % Right column and width @@ -232,16 +229,26 @@ P_l = R_{p,l} \ Q_l + \pi_l \begin{frame} \frametitle{The mathematical model} - \begin{columns}[c] \column{.5\textwidth} % Left column and width \footnotesize \begin{itemize} -\item<1-> $u_{inlet} = -U f(t) \hat{n}$, with $f(t)$ the weaveform. -\item<2-> Fractional step scheme. -\item<3-> Semi-implicit Windkessel model. -\item<4-> Stabilized $\mathbb{P}1/\mathbb{P}1$ finite elements. -\item<4-> Implemented in FEniCS. +\item Incompressible Navier-Stokes equations: +\begin{equation} +\begin{cases} +\displaystyle \rho \frac{\partial \vec{u}}{\partial t} + \rho \big ( \vec{u} \cdot \nabla \big) \vec{u} - \mu \Delta \vec{u} + \nabla p = 0 \\[0.2cm] +\nabla \cdot \vec{u} = 0 \quad \text{in} \quad \Omega \\[0.2cm] +\vec{u} = \vec{u}_{inlet} \quad \text{on} \quad \Gamma_{in} \\[0.2cm] +\vec{u} = 0 \quad \text{on} \quad \Gamma_{walls} +\end{cases} +\end{equation} + \emph{Three-element} Windkessel coupling at every outlet: +\begin{equation} +\begin{cases} +\displaystyle C_{d,l} \frac{d \pi_l}{dt} + \frac{\pi_l}{R_{d,l}} = Q_l \\[0.2cm] +P_l = R_{p,l} \ Q_l + \pi_l +\end{cases} +\end{equation} \end{itemize} @@ -258,90 +265,105 @@ P_l = R_{p,l} \ Q_l + \pi_l +%\begin{frame} +% \frametitle{The mathematical model} +% +% \begin{columns}[c] +%\column{.5\textwidth} % Left column and width +%\footnotesize +%\begin{itemize} +%\item<1-> $u_{inlet} = -U f(t) \hat{n}$, with $f(t)$ the weaveform. +%\item<2-> Fractional step scheme. +%\item<3-> Semi-implicit Windkessel model. +%\item<4-> Stabilized $\mathbb{P}1/\mathbb{P}1$ finite elements. +%\item<4-> Implemented in FEniCS. +%\end{itemize} +% +% +%\column{.54\textwidth} % Right column and width +%\begin{figure}[!hbtp] +% \begin{center} +% \includegraphics[height=0.9\textwidth]{images/ref.png} +% \caption{\footnotesize Reference solution at peak systole} +% \end{center} +% \end{figure} +%\end{columns} +% +%\end{frame} + + + \begin{frame} \frametitle{The inverse problem} \begin{itemize} \item<1-> Upon this solution $\Longrightarrow$ build a set of measurements -\item<2-> Induce typical arifacts via a Magnetization vector: $M(\vec{x},t)= M_0(\vec{x}) exp(i\phi_0 + i \frac{\pi}{venc} u(\vec{x},t))$ +\item<2-> $M(\vec{x},t)= M_0(\vec{x}) exp(i\phi_0 + i \frac{\pi}{venc} u(\vec{x},t))$ +\item<3-> reconstructed velocity: $u \in \big ( -venc, + venc \big )$ +\item<4-> $VNR \sim 1/venc$ \end{itemize} -\onslide<3-> - \begin{columns} - \column{.3\textwidth} - \flushleft - \begin{figure} - \includegraphics[width=1.1\textwidth]{images/ref_int.png} - \caption*{(a) Interpolated reference solution} -\end{figure} -\column{.67\textwidth} \centering -\onslide<4-> \textbf{The measurements:} +\onslide<5-> \textbf{The measurements:} \begin{itemize} -\item<5-> Gaussian noise into the magnetization -\item<6-> Different levels of aliasing varying the $venc$ parameter -\item<7-> Only using the dominant component of the velocity: $u_z$ -\item<8-> Time interpolation $dt = 1 \ ms \Longrightarrow dt = 30 \ ms$ +\item<6-> Gaussian noise into the magnetization +\item<7-> Spatial and temporal interpolation +\item<8-> Only using the dominant component of the velocity: $u_z$ +\item<9-> Different levels of aliasing varying the $venc$ parameter \end{itemize} -\end{columns} \end{frame} + +\begin{frame} + \frametitle{The inverse problem} +\begin{itemize} +\item Upon this solution $\Longrightarrow$ build a set of measurements +\item $M(\vec{x},t)= M_0(\vec{x}) exp(i\phi_0 + i \frac{\pi}{venc} u(\vec{x},t))$ +\item reconstructed velocity: $u \in \big ( -venc, + venc \big )$ +\item $VNR \sim 1/venc$ +\end{itemize} + \begin{figure} + \includegraphics[width=0.7\textwidth]{images/supra_venc.png} + \caption*{Aliased measurements with different $vencs = 120,70,30 \%$ of $u_{max}$} + \hfill +\end{figure} +\end{frame} + + + +\begin{frame} + \frametitle{The inverse problem} +\begin{itemize} +\item Upon this solution $\Longrightarrow$ build a set of measurements +\item $M(\vec{x},t)= M_0(\vec{x}) exp(i\phi_0 + i \frac{\pi}{venc} u(\vec{x},t))$ +\item reconstructed velocity: $u \in \big ( -venc, + venc \big )$ +\item $VNR \sim 1/venc$ +\end{itemize} + \begin{figure} + \includegraphics[width=0.7\textwidth]{images/coartation.png} + \caption*{ Aliased measurements with different $vencs = 120,70,30 \%$ of $u_{max}$} + \hfill +\end{figure} +\end{frame} + + \section{The inverse problem} -\begin{frame} - \frametitle{The inverse problem} -\begin{itemize} -\item Upon this solution $\Longrightarrow$ build a set of measurements -\item Induce typical arifacts via a Magnetization vector: $M(\vec{x},t)= M_0(\vec{x}) exp(i\phi_0 + i \frac{\pi}{venc} u(\vec{x},t))$ -\end{itemize} - \begin{columns} - \column{.3\textwidth} - \flushleft - \begin{figure} - \includegraphics[width=1.1\textwidth]{images/ref_int.png} - \caption*{(a) Interpolated reference solution} -\end{figure} -\column{.67\textwidth} -\flushleft - \begin{figure} - \includegraphics[width=1.1\textwidth]{images/supra_venc.png} - \caption*{(b) Aliased measurements with different $vencs = 120,70,30 \%$ of $u_{max}$} - \hfill -\end{figure} -\end{columns} -\end{frame} - - \begin{frame} \frametitle{The inverse problem} -\begin{itemize} -\item Upon this solution $\Longrightarrow$ build a set of measurements -\item Induce typical arifacts via a Magnetization vector: $M(\vec{x},t)= M_0(\vec{x}) exp(i\phi_0 + i \frac{\pi}{venc} u(\vec{x},t))$ -\end{itemize} - \begin{columns} - \column{.3\textwidth} - \flushleft - \begin{figure} - \includegraphics[width=1.1\textwidth]{images/ref_int.png} - \caption*{(a) Interpolated reference solution} -\end{figure} -\column{.67\textwidth} -\flushleft - \begin{figure} - \includegraphics[width=1.1\textwidth]{images/coartation.png} - \caption*{(b) Aliased measurements with different $vencs = 120,70,30 \%$ of $u_{max}$} - \hfill -\end{figure} -\end{columns} +\begin{center} +Parameter optimization +\end{center} \end{frame} + \begin{frame} \frametitle{The Kalman Filter} \begin{itemize} -\item<1-> We use a Reduced Order Unscendent Kalman Filter (ROUKF) to reconstruct the parameter vector $\theta$ solving the next optimization problem: +\item<1-> We use a Reduced Order Unscendent Kalman Filter (ROUKF) to reconstruct the parameter vector $\theta$: \onslide<2-> \begin{equation*} @@ -350,7 +372,7 @@ P_l = R_{p,l} \ Q_l + \pi_l \begin{equation} J(\theta) = \displaystyle \frac{1}{2} || \theta - \theta_0 ||^2_{P_0^{-1}} + \sum_{k=1}^N \frac{1}{2} || Z_k - \mathbb{H} X_k(\theta) ||^2_{W^{-1}} \end{equation} -\onslide<3-> Where: +\onslide<4-> Where: \begin{itemize} \item<4-> $Z$ the measurements and $X = (\vec{u} , \pi)$ the state variable \item<5-> $\mathbb{H}$ observation operator @@ -372,32 +394,41 @@ The parameter vector: \begin{itemize} \item<1-> Amplitude of the inlet velocity: $U$ -\item<2-> Since $R_p << R_d$, we only consider an optimization dependent on $\big ( R_{d,l}, C_l \big )$ for $l=1,...,n_l$ +\item<2-> Only the higher resistence: $R_d$ \end{itemize} -\onslide<3-> $$\theta = (U,\vec{R_d},\vec{C})$$ \\ with $\vec{R_d} = R_{d,l}$, $\vec{C} = C_l$ for $l=1,..., \color{red} n_{l-1}$ \\[0.3cm] -\onslide<4-> \color{red} Not all the resistences can be recovered at once $\Longrightarrow$ desc. aorta fixed. +\onslide<4-> $$\theta = (U,\vec{R_d})$$ \\ with $\vec{R_d} = R_{d,l}$ for $l=1,..., \color{red} n_{l-1}$ \\[0.3cm] +\onslide<5-> \color{red} Not all the resistences can be recovered at once $\Longrightarrow$ desc. aorta fixed. \end{frame} + +\section{Numerical Experiments} + + \begin{frame} - \frametitle{Easy example} + \frametitle{Numerical Experiments} +\begin{center} +Numerical Experiments +\end{center} +\end{frame} + + +\begin{frame} + \frametitle{Numerical Experiments} \footnotesize -\begin{itemize} -\item<1-> $\theta_{ref} = (U,\vec{R_d})$ , $U=75$, $\vec{R_d} = (7200,11520,11520)$ -\end{itemize} +\onslide<1-> $\theta_{ref} = (U,\vec{R_d})$ , $U=75$, $\vec{R_d} = (7200,11520,11520)$ \begin{columns} \footnotesize - \column{.4\textwidth} + \column{.45\textwidth} \begin{figure} - \onslide<2-> \textbf{Test I:} $U_0 = 150$ $\vec{R_{d,0}}= (8760,8760,8760)$ - \onslide<3-> - \includegraphics[width=1.2\textwidth]{images/U_Pb.png} + \onslide<2-> \textbf{Test I:} $U_0 = 150$ $\vec{R_{d,0}}= (17520,17520,17520)$ + \onslide<3-> \includegraphics[width=1.2\textwidth]{images/U_Pb.png} \includegraphics[width=1.2\textwidth]{images/Rd_Pb.png} \end{figure} -\column{.4\textwidth} +\column{.45\textwidth} \begin{figure} \onslide<2-> \textbf{Test II:} $U_0 = 40$ $\vec{R_{d,0}}= (4000,4000,4000)$ \onslide<4-> @@ -415,32 +446,48 @@ The parameter vector: \begin{frame} \frametitle{Aliased data} \begin{center} -What happend when $venc < u_{max}$ ? +\onslide<1-> What happend when $venc < u_{max}$ ? +\begin{figure} + \onslide<2-> \includegraphics[width=0.45\textwidth]{images/v120.png} + \caption{Measurement set with $venc = 120 \% u_{max}$} +\end{figure} \end{center} \end{frame} +\begin{frame} + \frametitle{Aliased data} +\begin{center} +What happend when $venc < u_{max}$ ? +\begin{figure} + \includegraphics[width=0.45\textwidth]{images/v70.png} + \caption{Measurement set with $venc = 70 \% u_{max}$} +\end{figure} +\end{center} +\end{frame} + + + + \begin{frame} \frametitle{Easy example: with $venc = 70 \% u_{max}$} \footnotesize -\begin{itemize} -\item $\theta_{ref} = (U,\vec{R_d})$ , $U=75$, $\vec{R_d} = (7200,11520,11520)$ -\end{itemize} +$\theta_{ref} = (U,\vec{R_d})$ , $U=75$, $\vec{R_d} = (7200,11520,11520)$ \begin{columns} \footnotesize - \column{.4\textwidth} + \column{.45\textwidth} \begin{figure} - \onslide<1-> \textbf{Test I:} $U_0 = 150$ $\vec{R_{d,0}}= (8760,8760,8760)$ + \onslide<1-> \textbf{Test I:} $U_0 = 150$ $\vec{R_{d,0}}= (17520,17520,17520)$ \onslide<2-> - \includegraphics[width=1.2\textwidth]{images/U_Pb.png} - \includegraphics[width=1.2\textwidth]{images/Rd_Pb.png} + \includegraphics[width=1.2\textwidth]{images/U_Pb_V70.png} + \includegraphics[width=1.2\textwidth]{images/Rd_Pb_V70.png} \end{figure} -\column{.4\textwidth} +\column{.45\textwidth} \begin{figure} \onslide<1-> \textbf{Test II:} $U_0 = 40$ $\vec{R_{d,0}}= (4000,4000,4000)$ \onslide<2-> - \includegraphics[width=1.2\textwidth]{images/U_Pc.png} - \includegraphics[width=1.2\textwidth]{images/Rd_Pc.png} + \includegraphics[width=1.2\textwidth]{images/U_Pc_V70.png} + \includegraphics[width=1.2\textwidth]{images/Rd_Pc_V70.png} \end{figure} \end{columns} \end{frame} @@ -465,30 +512,70 @@ J(\theta) = \displaystyle \frac{1}{2} || \theta - \theta_0 ||^2_{P_0^{-1}} + \ - - - - - - - -\section{Numerical Experiments} +\begin{frame} + \frametitle{Easy example: with $venc = 70 \% u_{max}$} + \footnotesize +$\theta_{ref} = (U,\vec{R_d})$ , $U=75$, $\vec{R_d} = (7200,11520,11520)$ + \begin{columns} + \footnotesize + \column{.45\textwidth} + \begin{figure} + \onslide<1-> \textbf{Test I:} $U_0 = 150$ $\vec{R_{d,0}}= (17520,17520,17520)$ + \onslide<2-> + \includegraphics[width=1.2\textwidth]{images/HU_Pb_V70.png} + \includegraphics[width=1.2\textwidth]{images/HRd_Pb_V70.png} +\end{figure} +\column{.45\textwidth} +\begin{figure} + \onslide<1-> \textbf{Test II:} $U_0 = 40$ $\vec{R_{d,0}}= (4000,4000,4000)$ + \onslide<3-> + \includegraphics[width=1.2\textwidth]{images/HU_Pc_V70.png} + \includegraphics[width=1.2\textwidth]{images/HRd_Pc_V70.png} +\end{figure} +\end{columns} +\end{frame} \begin{frame} - \frametitle{Numerical Experiments} + \frametitle{Aliased data} \begin{center} -Numerical Experiments +Or even higher aliasing... +\begin{figure} + \includegraphics[width=0.45\textwidth]{images/v30.png} + \caption{Measurement set with $venc = 30 \% u_{max}$} +\end{figure} \end{center} \end{frame} -\begin{frame} - \frametitle{Numerical Experiments} + + +\begin{frame} + \frametitle{Easy example: with $venc = 30 \% u_{max}$} + \footnotesize +$\theta_{ref} = (U,\vec{R_d})$ , $U=75$, $\vec{R_d} = (7200,11520,11520)$ + \begin{columns} + \footnotesize + \column{.45\textwidth} + \begin{figure} + \onslide<1-> \textbf{Test I:} $U_0 = 150$ $\vec{R_{d,0}}= (17520,17520,17520)$ + \onslide<2-> + \includegraphics[width=1.2\textwidth]{images/HU_Pb_V30.png} + \includegraphics[width=1.2\textwidth]{images/HRd_Pb_V30.png} +\end{figure} +\column{.45\textwidth} +\begin{figure} + \onslide<1-> \textbf{Test II:} $U_0 = 40$ $\vec{R_{d,0}}= (4000,4000,4000)$ + \onslide<3-> + \includegraphics[width=1.2\textwidth]{images/HU_Pc_V30.png} + \includegraphics[width=1.2\textwidth]{images/HRd_Pc_V30.png} +\end{figure} +\end{columns} \end{frame} + \section{Conclusions} \begin{frame} @@ -499,8 +586,32 @@ Conclusions +\begin{frame} +\frametitle{Conclusions} + \footnotesize +\begin{itemize} +\item<1-> 4D Flow measurements are promising for extracting data via inverse problems +\item<2-> Noise and aliasing are the typical artifacts involved. +\item<3-> Using a suitable Kalman filter, have shown to bypass aliasing defining the funcional in terms of the frequencies. +\end{itemize} + +\onslide<4-> Future Work + +\begin{itemize} +\item<5-> To include the capacitancies in the inverse problem (adding some pressure meas.) +\item<6-> Real data! +\end{itemize} + +\end{frame} +\begin{frame} +\begin{center} +Thank you for your time! +\end{center} + + +\end{frame} diff --git a/presentations/press_8ecm/press.toc b/presentations/press_8ecm/press.toc new file mode 100644 index 0000000..0943b01 --- /dev/null +++ b/presentations/press_8ecm/press.toc @@ -0,0 +1,6 @@ +\babel@toc {english}{} +\beamer@sectionintoc {1}{4D flow MRI}{3}{0}{1} +\beamer@sectionintoc {2}{The mathematical model}{6}{0}{2} +\beamer@sectionintoc {3}{The inverse problem}{23}{0}{3} +\beamer@sectionintoc {4}{Numerical Experiments}{37}{0}{4} +\beamer@sectionintoc {5}{Conclusions}{56}{0}{5}