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\author{\ \\ Jan Hendrik Dithmar \\ \texttt{jadi5003@stud.uni-saarland.de}}
\title{\color{miablue}{Euler-Lagrange Formalism}}

\begin{document}
\maketitle
\slidetitle{\normalsize\bfseries%
    Mathematical Image Analysis Group, Saarland University\\
    Seminar "Optimisation for Visual Computing"
%    \href{http://www.mia.uni-saarland.de}{http://www.mia.uni-saarland.de}
}

\begin{center}
July 18th, 2009
\end{center}

%\subhead{Outline:}
%\begin{enumerate} 
%\end{enumerate}

% -----------------------------------------------------------------------
\foilhead{Motivation}

\stitle{Motivation}
\newline
\begin{itemize}
	\item Euler-Lagrange equations appear in
	\begin{itemize}
		\item (classical) mechanics,
		\item functional minimisation,
		\item field theory,
%		\item diffusion processes,
		\item ...
	\end{itemize}
%	\item ..., but also in applications like landmark matching for cardiac anatomy.
	\item Connections to iterative algorithms (Gradient Descent, Newton's method) to solve optimisation problems.
\end{itemize}

% -----------------------------------------------------------------------
\foilhead{Outline}

\stitle{Outline}
\begin{center}
	\includegraphics[width=0.62\textwidth]{Pic/Outline.pdf}
\end{center}

% -----------------------------------------------------------------------
\foilhead{Basic Observations}

\stitle{Basic observations}
\newline \newline
{\color{miablue}\textbf{Standard calculus}}
\begin{itemize}
	\item Considers real-valued functions $f: \mathbb{R}^N \to \mathbb{R}$, $(\xi_1, ..., \xi_N) \in \mathbb{R}^N$.
	\item If $f$ has minimum in $\xi$, then $f'(\xi) = 0$.
	\item If $f$ strictly convex and $f'(\xi) = 0$, then $\xi$ is unique minimum of $f$.
\end{itemize}
\ \\ \ \\
{\color{miablue}\textbf{Calculus of variations}}
\begin{itemize}
	\item Considers real-valued functionals $\mathcal{E}: \mathcal{F} \to \mathbb{R}$, where $\mathcal{F}$ is a function space.
	\item If $\mathcal{E}$ minimised by function $u$, then $u$ has to satisfy so-called \important{Euler-Lagrange equations}.
	\item If $\mathcal{E}$ strictly convex and satisfies Euler-Lagrange equations, then $u$ is unique minimiser of $\mathcal{E}$.
\end{itemize}

\newpage

{\color{miablue}{\textbf{Functionals}}}
\begin{itemize}
	\item Structure (with regularised smoothness term)
		$$\mathcal{E}(u) = \int_a^b \underbrace{(u-f)^2}_{\mbox{data term}} + \alpha \underbrace{\Psi_S(|\nabla u|^2)}_{\mbox{smoothness term}} \; \mathrm d \textbf{x}$$
		\begin{itemize}
			\item $u$: (unknown) function
%			\item $(u-f)^2$: function $u$ should be close to "measured" function $f$
			\item $\alpha$: smoothness weight
			\item $\Psi_S$: regulariser, penalises deviations from smoothness
		\end{itemize}
	\item \textsl{Remark:} Data term can be robustified by using a penaliser $\Psi_D$. \\[0.5cm]
		($\rightarrow$ Talk by Laurent Hoeltgen about "Approximation and Fitting")
\end{itemize}

% -----------------------------------------------------------------------
\foilhead{Euler-Lagrange Equations}

\stitle{Euler-Lagrange equations}
\begin{center}
	\includegraphics[width=0.4\textwidth]{Pic/euler-ready.pdf} \includegraphics[width=0.4\textwidth]{Pic/lagrange-ready.pdf} \\
	{\small \textbf{Left:} Leonhard Euler (1707--1783). \textbf{Right:} Joseph-Louis Lagrange (1736--1813). \textbf{Source:} Wikipedia.}
\end{center}

\newpage

{\color{miablue}\textbf{Theorem (1-D case)}} \\ \ \\
A \important{smooth function} $u(x)$ with $x \in [a,b]$ that minimises the 1-D energy functional
$$\mathcal{E}(u) = \int_a^b \mathcal{L}(x, u, u') \; \mathrm dx$$
satisfies necessarily the \important{Euler-Lagrange equation}
$$\mathcal{L}_{u} - \frac{\mathrm d}{\mathrm dx} \mathcal{L}_{u'} = 0$$
and the so-called \important{natural boundary conditions}
$$\mathcal{L}_{u'} = 0$$
for the boundary ($x=a$ and $x=b$).

\newpage

{\color{miablue}\textbf{Theorem (2-D case)}} \\ \ \\
A minimiser of the 2-D energy functional
$$\mathcal{E}(u) = \int_\Omega \mathcal{L}(x_1, x_2, u, u_{x_1}, u_{x_2}) \; \mathrm d\textbf{x}$$
satisfies necessarily the \important{Euler-Lagrange equation}
$$\mathcal{L}_{u} - \frac{\partial}{\partial x_1} \mathcal{L}_{u_{x_1}} - \frac{\partial}{\partial x_2} \mathcal{L}_{u_{x_2}} = 0$$
with the \important{natural boundary conditions}
$$\textbf{n}^\top \left(\begin{array}{c}\mathcal{L}_{u_{x_1}} \\ \mathcal{L}_{u_{x_2}}\end{array}\right) = 0$$
at the image boundaries $\partial \Omega$ with the normal vector $\textbf{n}$.

% -----------------------------------------------------------------------
\foilhead{Example}

\stitle{Example for a Variational Problem}
\newline \newline
\important{Problem:} Find the shortest plane curve joining two points $A$ and $B$. \pause%\newline \newline
This means: Find the curve $u(x)$ for which the functional
$$\mathcal{E}(u(x)) = \int_a^b \sqrt{1 + \left(\frac{\mathrm d}{\mathrm dx}u(x)\right)^2} \; \mathrm dx$$
achieves its minimum where $a \leq x \leq b$. \newline \newline \newline
\important{Solution:} Straight line segment joining $A$ and $B$. Why?
\pause
The Euler-Lagrange equation yields the differential equation
$$\frac{\mathrm d}{\mathrm du} \sqrt{1 + \left(\frac{\mathrm d}{\mathrm dx}u(x)\right)^2} = 0 \quad \Longrightarrow \quad \frac{\mathrm d}{\mathrm dx}u(x) = C,$$
in other words a straight line.

% -----------------------------------------------------------------------
\foilhead{Variations of an Energy Functional}

\stitle{Variations of an Energy Functional}
\newline \newline
{\color{miablue}{\textbf{Notation}}}
\begin{itemize}
	\item Continuous
		\begin{itemize}
			\item $u$ is a function
			\item $\mathcal{E}$ is a functional (takes functions as arguments)
		\end{itemize}
	\item Discrete
		\begin{itemize}
			\item $\textbf{u}$ is a vector representing function values
			\item $E$ is a function (takes vectors as arguments)
		\end{itemize}
\end{itemize}
{\color{miablue}{\textbf{What is it about?}}}
\begin{itemize}
	\item \textsl{Problem:} How to compute a derivative w.r.t.\ a function?
	\item \textsl{Remedy:} Construct \textsl{something} that we can differentiate.
\end{itemize}

\newpage
{\color{miablue}{\textbf{Variations of an Energy Functional}}} \\ \ \\
\begin{itemize}
	\item \textsl{Problem:} $\mathcal{E}: {\color{red}{\mathcal{F}}} \to \mathbb{R}$.
	\item \textsl{Remedy:} $\mathcal{P}: \mathbb{R} \to \mathbb{R}$.
	\pause
	\item \textsl{We do:} Construct $\mathcal{P}$, s.t.\ $\mathcal{P}: \mathbb{R} \to \mathbb{R}$
		\begin{itemize}
			\item $\mathcal{S}: \mathbb{R} \to \mathcal{F}; \; \varepsilon \mapsto u + \varepsilon v$ brings us to function space $\mathcal{F}$.
			\item $\mathcal{E}: \mathcal{F} \to \mathbb{R}$ brings us back to $\mathbb{R}$.
			\item[$\Rightarrow$] $\mathcal{P}: \mathbb{R} \to \mathbb{R}; \; \varepsilon \mapsto \mathcal{E}(u + \varepsilon v)$.
		\end{itemize} \ \\[1cm]
	\item \textsl{Remark:} If $\mathcal{F} = \mathbb{R}^N$, then $\textbf{u} = (u_1,...,u_N)$ and $\textbf{v} = (v_1,...,v_N)$. \\[1cm]
	\item first variation $\approx$ first derivative
	\item second variation $\approx$ second derivative
\end{itemize}

\newpage
{\color{miablue}{\textbf{Overview}}}
\newline \newline
\begin{center}
\begin{tabular}{p{10cm}|p{10cm}}
	\textsl{Continuous} & \textsl{Discrete} \\
	{\color{miablue}{First Variation}} & {\color{miablue}{?}}\\ \\
	$\displaystyle \frac{\partial}{\partial \varepsilon} \mathcal{E}(u + \varepsilon v) \Big|_{\varepsilon = 0}$ & $\displaystyle \frac{\partial}{\partial \varepsilon} E(\textbf{u} + \varepsilon \textbf{v}) \Big|_{\varepsilon = 0} = \quad ?$ \\[2cm]
	\hline
	{\color{miablue}{Second Variation}} & {\color{miablue}{?}} \\ \\
	$\displaystyle \frac{\partial^2}{\partial \varepsilon^2} \mathcal{E} (u + \varepsilon v) \Big|_{\varepsilon = 0}$ & $\displaystyle \frac{\partial^2}{\partial \varepsilon^2} E(\textbf{u} + \varepsilon \textbf{v}) \Big|_{\varepsilon = 0} = \quad ?$ \\[2cm]
\end{tabular}
\end{center}

\newpage

{\color{miablue}{\textbf{First Derivative: Gradient}}} \\ \ \\
\begin{itemize}
	\item First derivative in $\textbf{u} \in \mathbb{R}^N$ in direction $\textbf{v}$ is given by
		\begin{eqnarray*}
			\textbf{D}E(\textbf{u})\textbf{v} & = & \frac{\partial}{\partial \varepsilon} E(\textbf{u} + \varepsilon \textbf{v}) \Big|_{\varepsilon = 0} \\
%			& = & \left. \left(\frac{\partial}{\partial u_1} E(\textbf{u} + \varepsilon \textbf{v}), ..., \frac{\partial}{\partial u_N} E(\textbf{u} + \varepsilon \textbf{v})\right) \right|_{\varepsilon = 0} \cdot \textbf{v} \\
%			& = & \left(\frac{\partial}{\partial u_1} E(\textbf{u}), ..., \frac{\partial}{\partial u_N} E(\textbf{u})\right) \cdot \textbf{v} \\
			& = & \nabla E(\textbf{u}) \cdot \textbf{v} \\
%			& = & \sum_{j=1}^N \left[\frac{\partial}{\partial u_j} E(\textbf{u})\right] v_j
		\end{eqnarray*}
\end{itemize}

\newpage

{\color{miablue}{\textbf{Second Derivative: Hessian}}} \\ \ \\
\begin{itemize}
	\item Second derivative in $\textbf{u} \in \mathbb{R}^N$ in direction $\textbf{v}$ is given by
		\begin{eqnarray*}
			\textbf{D}^2E(\textbf{u})\textbf{v} & = & \left. \frac{\partial^2}{\partial \varepsilon^2} E(\textbf{u} + \varepsilon \textbf{v}) \right|_{\varepsilon = 0} \\
			& = & \left. \frac{\partial}{\partial \varepsilon} (\nabla E(\textbf{u} + \varepsilon \textbf{v}) \cdot \textbf{v}) \right|_{\varepsilon = 0} \\
			& = & \left. \frac{\partial}{\partial \varepsilon} \left(\left(\frac{\partial}{\partial u_1} E(\textbf{u} + \varepsilon \textbf{v}), ..., \frac{\partial}{\partial u_N} E(\textbf{u} + \varepsilon \textbf{v})\right) \cdot \textbf{v}\right) \right|_{\varepsilon = 0} \\
			& = & \left(\left(\left(\frac{\partial^2}{\partial u_1 \partial u_i} E(\textbf{u})\right), ..., \left(\frac{\partial^2}{\partial u_N \partial u_i} E(\textbf{u})\right)\right)_{i=1,...,N} \cdot \textbf{v}\right) \cdot \textbf{v} \\
			& = & \left(\left(\underbrace{\left(\frac{\partial^2}{\partial u_j \partial u_i} E(\textbf{u})\right)}_{\mathcal{H} E(\textbf{u})}\right)_{i,j=1,..,N} \cdot \textbf{v}\right) \cdot \textbf{v} \\
			& = & \textbf{v}^\top \mathcal{H} E(\textbf{u}) \textbf{v}
		\end{eqnarray*}
\end{itemize}

\newpage
{\color{miablue}{\textbf{Overview}}}
\newline \newline
\begin{center}
\begin{tabular}{p{10cm}|p{10cm}}
	\textsl{Continuous} & \textsl{Discrete} \\
	{\color{miablue}{First Variation}} & {\color{miablue}{Gradient}} \\ \\
	$\displaystyle \frac{\partial}{\partial \varepsilon} \mathcal{E}(u + \varepsilon v) \Big|_{\varepsilon = 0}$ & $\displaystyle \frac{\partial}{\partial \varepsilon} E(\textbf{u} + \varepsilon \textbf{v}) \Big|_{\varepsilon = 0} = \nabla E(\textbf{u}) \cdot \textbf{v}$ \\[2cm]
	\hline
	{\color{miablue}{Second Variation}} & {\color{miablue}{Hessian Matrix}} \\ \\
	$\displaystyle \frac{\partial^2}{\partial \varepsilon^2} \mathcal{E} (u + \varepsilon v) \Big|_{\varepsilon = 0}$ & $\displaystyle \frac{\partial^2}{\partial \varepsilon^2} E(\textbf{u} + \varepsilon \textbf{v}) \Big|_{\varepsilon = 0} = \textbf{v}^\top \mathcal{H} E(\textbf{u}) \textbf{v}$ \\[2cm]
\end{tabular}
\end{center}

% -----------------------------------------------------------------------
\foilhead{Appearance}

\stitle{Appearance of the concept}
\newline
\begin{itemize}
	\item \textsl{Assume:} energy functional = cost function (discrete!)
	\item \textsl{Idea:} Minimise the cost
	\item \textsl{Ingredients:}
		\begin{itemize}
			\item $\nabla E(\textbf{u})$: gradient of the energy functional
				$$\nabla E(\textbf{u}) = \left(\frac{\partial}{\partial u_j} \left(E(\textbf{u})\right)\right)_{j = 1,...,N}$$
			\item $\mathcal{H} E(\textbf{u})$: Hessian matrix of the energy functional
				$$\mathcal{H} E(\textbf{u}) = \left(\frac{\partial^2}{\partial u_j \partial u_i} \left(E(\textbf{u})\right)\right)_{i,j = 1,...,N}$$
		\end{itemize}
\end{itemize}

\newpage

\stitle{Gradient Descent Algorithm}
\newline
\begin{itemize}
	\item give a starting point $x$
	\item repeat
		\begin{enumerate}
			\item $\textbf{d} = -\nabla E(\textbf{u})$
			\item Choose step size $\gamma$ (e.g. by backtracking line search)
			\item $\textbf{x} := \textbf{x} + \gamma \cdot \textbf{d}$
		\end{enumerate}
		until stopping criterion is satisfied.
\end{itemize}
\begin{center}
	\includegraphics[width=0.25\textwidth]{Pic/Gradient_descent.png} \\
	{\small Illustration of gradient descent. \textbf{Source:} Wikipedia.}
\end{center}

\newpage

\stitle{Newton's Method}
\newline
\begin{itemize}
	\item give a starting point $x$
	\item repeat
		\begin{enumerate}
			\item $\textbf{d} := - {\color{red}\big[\mathcal{H}E(\textbf{u})\big]^{-1}} \nabla E(\textbf{u})$
			\item $\textbf{x} := \textbf{x} + \textbf{d}$
		\end{enumerate}
		until stopping criterion is satisfied.
\end{itemize}
\textsl{Need:} strict convexity $\Longleftrightarrow$ Hessian positive definite \\[0.5cm]
\textsl{Given:} Hessian $\mathcal{H} E(u)$ not (always) positive definite \\[0.5cm]
\textsl{Solution:} Perturb it to be positive definite! $\rightarrow$ Trust-Region Method

\newpage

\stitle{Connection to the Euler-Lagrange Formalism}
\newline
\begin{itemize}
	\item Assume the energy functional
		$$\mathcal{E}(u) = \int_a^b (u-f)^2 + \alpha \Psi_S\big((u')^2\big) \; \mathrm dx$$
	\item Two approaches:
		\begin{itemize}
			\item First employ necessary optimality condition, then discretise. (OD)
			\item First discretise, then employ necessary optimality condition. (DO)
		\end{itemize}
	\item What's better? OD or DO?
\end{itemize}

\newpage
{\color{miablue}{\textbf{Necessary optimality condition $\rightarrow$ Discretise: OD}}}
\newline\newline
The Euler-Lagrange equation
\begin{eqnarray*}
	0 & = & 2 (u-f) - \frac{\mathrm d}{\mathrm dx} \Big[2 \alpha \Psi_S'\big((u')^2\big) \cdot u'\Big] \\
	& = & u-f - \alpha \frac{\mathrm d}{\mathrm dx} \Big[\Psi_S'\big((u')^2\big) \cdot u'\Big]
\end{eqnarray*}
After discretisation
\vfill
\begin{eqnarray*}
	0 & = & u_j - f_j - \alpha \frac{\mathrm d}{\mathrm dx} \Big[\Psi_S'\big((u')^2\big) \cdot u'\Big]\Big|_{x=x_j} \\
	& = & u_j - f_j - \alpha \Psi_S'\left(\frac{(u_{j+1} - u_j)^2}{\Delta x^2}\right) \cdot \frac{u_{j+1} - u_j}{\Delta x^2} \\
	& & \quad \quad \quad \; \: + \alpha \Psi_S'\left(\frac{(u_j - u_{j-1})^2}{\Delta x^2}\right) \cdot \frac{u_j - u_{j-1}}{\Delta x^2} \quad , j = 1,...,N.
\end{eqnarray*}

\newpage
{\color{miablue}{\textbf{Discretise $\rightarrow$ Necessary optimality condition: DO}}}
\newline\newline
%Employ "mid-point rule"
%$$\int_{x_{j-\frac{1}{2}}}^{x_{j+\frac{1}{2}}} \varphi(x) \; \mathrm dx \approx \underbrace{\left(x_{j+\frac{1}{2}} - x_{j-\frac{1}{2}}\right)}_{\Delta x} \cdot \varphi(x_j)$$
Discretised energy functional
$$E(\textbf{u}) = \sum_{j=1}^N (u_j - f_j)^2 + \alpha \Psi_S\Big((\textbf{u}')^2\Big)\Big|_{x=x_j}.$$
Necessary optimality condition
\vfill
\begin{eqnarray*}
	0 & = & \frac{\partial}{\partial u_j} \left(E(\textbf{u})\right) \\
	& = & u_j - f_j - \alpha \Psi_S'\left(\frac{(u_{j+1} - u_j)^2}{\Delta x^2}\right) \cdot \frac{u_{j+1} - u_j}{\Delta x^2} \\
	& & \quad \quad \quad \; \: + \alpha \Psi_S'\left(\frac{(u_j - u_{j-1})^2}{\Delta x^2}\right) \cdot \frac{u_j - u_{j-1}}{\Delta x^2} \quad , j = 1,...,N.
\end{eqnarray*}

% -----------------------------------------------------------------------
\foilhead{Examples in Image Processing: Variational Image Restoration}

\stitle{Example in Image Processing: Gaussian Noise}
\newline \newline
Consider
$$\mathcal{E}(u) = \int_a^b (u-f)^2 + \alpha \Psi_S(|\nabla u|^2) \; \mathrm dx$$
with Charbonnier regulariser
$$\Psi_S(|\nabla u|^2) = 2(\lambda^2 \sqrt{1 + |\nabla u|^2/\lambda^2} - \lambda^2)$$
\begin{center}
	\includegraphics[width=0.25\textwidth]{Pic/couple.pdf} \quad
	\includegraphics[width=0.25\textwidth]{Pic/couple40.pdf} \\
	{\small \textbf{Left:} Original image. \textbf{Right:} Noisy version with Gaussian noise ($\sigma = 40$).}
\end{center}

\newpage

{\color{miablue}{\textbf{Denoised version after 20 iterations}}}
\begin{center}
	\includegraphics[width=0.25\textwidth]{Pic/couple.pdf} \quad
	\includegraphics[width=0.25\textwidth]{Pic/couple40.pdf} \\[0.3cm]
	\includegraphics[width=0.25\textwidth]{Pic/couple40_1_20.pdf} \quad
	\includegraphics[width=0.25\textwidth]{Pic/couple40_2_20.pdf} \quad
	\includegraphics[width=0.25\textwidth]{Pic/couple40_5_20.pdf}
\end{center}
{\small \textbf{Top Left:} Original image. \textbf{Top Right:} Noisy version with Gaussian noise ($\sigma = 40$). \textbf{Bottom row:} Denoised version with $\lambda = 1,2,5$.}

\newpage

\stitle{Example in Image Processing: Salt-and-Pepper Noise}
\newline
\begin{center}
	\includegraphics[width=0.45\textwidth]{Pic/example_ip_noisy.pdf} \quad \quad
	\includegraphics[width=0.45\textwidth]{Pic/example_ip_noisefree.pdf}
\end{center}
{\small \textbf{Left:} Noisy version with salt-and-pepper noise. \textbf{Right:} Denoised version using the regularised data term $\Psi_D(s^2) = 2 (\sqrt{s^2 + \varepsilon^2} - \varepsilon)$ and a Charbonnier regulariser as smoothness term $\Psi_S(s^2) = 2 (\lambda^2 \sqrt{1 + s^2 / \lambda^2} - \lambda^2)$. \textbf{Author:} J. Weickert.}


% -----------------------------------------------------------------------
\foilhead{Summary}

\stitle{Summary}
\newline
\begin{itemize}
	\item Euler-Lagrange equation: necessary condition for a minimiser of an energy functional.
	\item Several approaches use the Euler-Lagrange theory. Good for solving optimisation problems.
	\item Gradient Descent Method and Newton's Method are algorithms to find a local extremum. The Euler-Lagrange formalism is related to these methods:
		\begin{center}
			\includegraphics[width=0.4\textwidth]{Pic/E-L.pdf}
		\end{center}
	\item OD and DO lead to the same result.
	\item Energy functional minimisation is used for image denoising.
\end{itemize}


% -----------------------------------------------------------------------
\foilhead{References}

\stitle{References}
\newline
{\small
\begin{itemize}
	\item Pablo Pedregal. Introduction to Optimization. Springer New York. 2004. \\
		\important{(Chapter 5 deals with variational problems and the Euler-Lagrange equations.)}
	\item I.M. Gelfand, S.V. Fomin. Calculus of Variations. Dover New York. 2000. \\
		\important{(Section 4 gives an introduction to the Euler-Lagrange equations. Sections 16 ff. go into detail.)}
	\item Euler-Lagrange equations (Wikipedia). \\
		\texttt{http://en.wikipedia.org/wiki/Euler-Lagrange\_equation}
%	\item Fundamental Lemma of Calculus of Variations (Wikipedia). \\
%		\texttt{http://en.wikipedia.org/wiki/Fundamental\_lemma\_of\_calculus\_of\_variations}
	\item Lecture "Differential Equations in Image Processing and Computer Vision". SS 2008. \\
		\important{(Lecture 10 gives an introduction to Euler-Lagrange equations.)}
	\item Lecture "Numerical Algorithms for Visual Computing III". SS 2009. \\
		\important{(Covers all the material presented here in a more detailed way.)}
	\item L. Bar, G. Sapiro. Generalized Newton-Type Methods for Energy Formulations in Image Processing. SIAM Journal on Imaging Sciences, 2009. \\
		\texttt{http://www.ece.umn.edu/users/barxx002/papers/Siam09.pdf}
%	\item M.F. Beg et al.The Euler-Lagrange Equation for Interpolating Sequence of Landmark Datasets: Applications to Cardiac Anatomy. \\
%		\texttt{http://cis.jhu.edu/posters/Beg.MICCAIposter4cis.Jan04.pdf}
\end{itemize}
}


% -----------------------------------------------------------------------
\foilhead{Thank you}

\ \\ \ \\ \ \\ \ \\ \ \\ \ \\ \ \\ \ \\
\begin{center}
{\Huge \color{miablue}{Thank you for your attention!}}
\end{center}
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