547 lines
13 KiB
TeX
Executable File
547 lines
13 KiB
TeX
Executable File
\documentclass[xcolor=dvipsnames,notheorem,mathserifs]{beamer}
|
|
\usepackage{amsmath}
|
|
%\documentclass{beamer}
|
|
\usepackage[english]{babel}
|
|
%\usepackage[latin1]{inputenc}
|
|
\usepackage{multicol} % indice en 2 columnas
|
|
\usepackage[utf8]{inputenc}
|
|
|
|
\usepackage{helvet}
|
|
\usefonttheme{serif}
|
|
%\usepackage{ccfonts} % Font family: Concrete Math
|
|
\usepackage[T1]{fontenc}
|
|
|
|
%\usepackage{graphicx}
|
|
%\usepackage{movie15}
|
|
%\usepackage{media9}[2013/11/04]
|
|
\usepackage{xcolor}
|
|
|
|
|
|
\usepackage{graphicx}
|
|
\usepackage{multimedia}
|
|
\usepackage{media9}
|
|
\usepackage{listings,xcolor,caption, mathtools, wrapfig}
|
|
\usepackage{amsfonts}
|
|
\usepackage{amssymb,graphicx,enumerate}
|
|
\usepackage{subcaption}
|
|
\usepackage{hyperref}
|
|
|
|
\usepackage[normalem]{ulem} % for strike out command \sout
|
|
|
|
|
|
|
|
|
|
|
|
%\usetheme{default}
|
|
%\usetheme{AnnArbor}
|
|
%\usetheme{Antibes}
|
|
%\usetheme{Bergen}
|
|
%\usetheme{Berkeley}
|
|
%\usetheme{Berlin}
|
|
%\usetheme{Boadilla}
|
|
%\usetheme{CambridgeUS}
|
|
%\usetheme{Copenhagen}
|
|
%\usetheme{Darmstadt}
|
|
%\usetheme{Dresden}
|
|
%\usetheme{Frankfurt}
|
|
%\usetheme{Goettingen}
|
|
%\usetheme{Hannover}
|
|
%\usetheme{Ilmenau}
|
|
%\usetheme{JuanLesPins}
|
|
%\usetheme{Luebeck}
|
|
%\usetheme{Madrid}
|
|
%\usetheme{Malmoe}
|
|
%\usetheme{Marburg}
|
|
%\usetheme{Montpellier}
|
|
%\usetheme{PaloAlto}
|
|
%\usetheme{Pittsburgh}
|
|
%\usetheme{Rochester}
|
|
%\usetheme{Singapore}
|
|
%\usetheme{Szeged}
|
|
\usetheme{Warsaw}
|
|
|
|
%\usecolortheme{albatross}
|
|
%\usecolortheme{beaver}
|
|
%\usecolortheme{beetle}
|
|
\usecolortheme{crane}
|
|
%\usecolortheme{dolphin}
|
|
%\usecolortheme{dove}
|
|
%\usecolortheme{fly}
|
|
%\usecolortheme{lily}
|
|
%\usecolortheme{orchid}
|
|
%\usecolortheme{rose}
|
|
%\usecolortheme{seagull}
|
|
%\usecolortheme{seahorse}
|
|
%\usecolortheme{whale}
|
|
%\usecolortheme{wolverine}
|
|
|
|
%\useoutertheme{infolines}
|
|
%\useoutertheme{miniframes}
|
|
%\useoutertheme{sidebar}
|
|
\useoutertheme{smoothbars}
|
|
%\useoutertheme{shadow}
|
|
%\useoutertheme{smoothtree}
|
|
%\useoutertheme{split}
|
|
%\useoutertheme{tree}
|
|
|
|
|
|
\usepackage{amssymb,mathrsfs,amsmath,latexsym,amsthm,amsfonts}
|
|
\useinnertheme{rectangles}
|
|
|
|
|
|
\setbeamertemplate{navigation symbols}{} % quitar simbolitos
|
|
|
|
|
|
\setbeamerfont{page number in head/foot}{size=\large}
|
|
%\setbeamertemplate{footline}[frame number] number in footer
|
|
\setbeamertemplate{footline}{}
|
|
|
|
|
|
|
|
\title{Robust parameter estimation in fluid flow models from aliased velocity measurements}
|
|
%\author[Jeremías Garay Labra]
|
|
%{Jeremías Garay Labra}
|
|
\institute[University of Groningen]
|
|
{
|
|
Bernoulli Institute\\
|
|
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} Cristobal Bertoglio.}
|
|
\date{\today}
|
|
|
|
|
|
\begin{document}
|
|
\frame{\titlepage}
|
|
|
|
|
|
% \onslide<1->
|
|
|
|
|
|
|
|
|
|
\begin{frame}
|
|
\frametitle{Index}
|
|
\tableofcontents
|
|
\end{frame}
|
|
|
|
|
|
\section[4D flow MRI]{4D flow MRI}
|
|
\begin{frame}
|
|
\frametitle{4D flow MRI}
|
|
\begin{columns}[c]
|
|
\column{.5\textwidth} % Left column and width
|
|
\footnotesize
|
|
|
|
\begin{itemize}
|
|
\item<2-> Full 3D coverage of the region of interest
|
|
\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
|
|
\onslide<1->
|
|
\begin{figure}[!hbtp]
|
|
\begin{center}
|
|
\includegraphics[height=0.9\textwidth]{images/4dflow.png}
|
|
\caption{\footnotesize 4D flow MRI of a human thorax}
|
|
\end{center}
|
|
\end{figure}
|
|
\end{columns}
|
|
\end{frame}
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\section{The mathematical model}
|
|
|
|
\begin{frame}
|
|
\frametitle{The mathematical model}
|
|
\begin{center}
|
|
The mathematical model
|
|
\end{center}
|
|
\end{frame}
|
|
|
|
|
|
|
|
\begin{frame}
|
|
\frametitle{The mathematical model}
|
|
|
|
\begin{columns}[c]
|
|
\column{.5\textwidth} % Left column and width
|
|
\footnotesize
|
|
|
|
\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 mathematical model}
|
|
|
|
\begin{columns}[c]
|
|
\column{.5\textwidth} % Left column and width
|
|
\footnotesize
|
|
\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}
|
|
\end{itemize}
|
|
|
|
|
|
\column{.54\textwidth} % Right column and width
|
|
\onslide<1->
|
|
\begin{figure}[!hbtp]
|
|
\begin{center}
|
|
\includegraphics[height=0.9\textwidth]{images/windk_model.png}
|
|
\caption{\footnotesize Schematic of the model}
|
|
\end{center}
|
|
\end{figure}
|
|
\end{columns}
|
|
|
|
\end{frame}
|
|
|
|
|
|
\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: $M(\vec{x},t)= M_0(\vec{x}) exp(i\phi_0 + i \frac{\pi}{venc} u(\vec{x},t))$
|
|
\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: $u_z$
|
|
\item<8-> Time interpolation $dt = 1 \ ms \Longrightarrow dt = 30 \ ms$
|
|
\end{itemize}
|
|
\end{columns}
|
|
\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}
|
|
\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{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}
|
|
|
|
|
|
|
|
|
|
\begin{frame}
|
|
\frametitle{The Kalman Filter}
|
|
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 $l=1,...,n_l$
|
|
\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.
|
|
|
|
\end{frame}
|
|
|
|
|
|
|
|
\begin{frame}
|
|
\frametitle{Easy example}
|
|
\footnotesize
|
|
\begin{itemize}
|
|
\item<1-> $\theta_{ref} = (U,\vec{R_d})$ , $U=75$, $\vec{R_d} = (7200,11520,11520)$
|
|
\end{itemize}
|
|
\begin{columns}
|
|
\footnotesize
|
|
\column{.4\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}
|
|
\includegraphics[width=1.2\textwidth]{images/Rd_Pb.png}
|
|
\end{figure}
|
|
\column{.4\textwidth}
|
|
\begin{figure}
|
|
\onslide<2-> \textbf{Test II:} $U_0 = 40$ $\vec{R_{d,0}}= (4000,4000,4000)$
|
|
\onslide<4->
|
|
\includegraphics[width=1.2\textwidth]{images/U_Pc.png}
|
|
\includegraphics[width=1.2\textwidth]{images/Rd_Pc.png}
|
|
\end{figure}
|
|
\end{columns}
|
|
\end{frame}
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\begin{frame}
|
|
\frametitle{Aliased data}
|
|
\begin{center}
|
|
What happend when $venc < u_{max}$ ?
|
|
\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}
|
|
\begin{columns}
|
|
\footnotesize
|
|
\column{.4\textwidth}
|
|
\begin{figure}
|
|
\onslide<1-> \textbf{Test I:} $U_0 = 150$ $\vec{R_{d,0}}= (8760,8760,8760)$
|
|
\onslide<2->
|
|
\includegraphics[width=1.2\textwidth]{images/U_Pb.png}
|
|
\includegraphics[width=1.2\textwidth]{images/Rd_Pb.png}
|
|
\end{figure}
|
|
\column{.4\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}
|
|
\end{figure}
|
|
\end{columns}
|
|
\end{frame}
|
|
|
|
|
|
|
|
|
|
\begin{frame}
|
|
\frametitle{The Kalman Filter: A frequency defined functional}
|
|
\begin{itemize}
|
|
\item<1-> We proposed a change in the cost functional:
|
|
\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}} + \color{red} \sum_{k=1}^N 1- cos \big ( \frac{\pi}{venc} \cdot ( Z_k - \mathbb{H} X_k ) \big )
|
|
\end{equation}
|
|
\end{itemize}
|
|
|
|
\end{frame}
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\section{Numerical Experiments}
|
|
|
|
|
|
\begin{frame}
|
|
\frametitle{Numerical Experiments}
|
|
\begin{center}
|
|
Numerical Experiments
|
|
\end{center}
|
|
\end{frame}
|
|
|
|
\begin{frame}
|
|
\frametitle{Numerical Experiments}
|
|
|
|
\end{frame}
|
|
|
|
|
|
|
|
\section{Conclusions}
|
|
|
|
\begin{frame}
|
|
\begin{center}
|
|
Conclusions
|
|
\end{center}
|
|
\end{frame}
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\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} |