todays work..

arbeit/bibma.bib

@@ -42,7 +42,6 @@
 
   Url                      = {http://graphics.cs.brown.edu/~jfh/papers/Hsu-DMO-1992/paper.pdf}
 }
-
 @Article{gaussNewton,
   Title                    = {An Algorithm for Least-Squares Estimation of Nonlinear Parameters},
   Author                   = {Donald W. Marquardt},
@@ -184,3 +183,10 @@
   url={https://pdfs.semanticscholar.org/a858/aa68da617ad9d41de021f6807cc422002258.pdf},
   doi={10.1016/j.cagd.2012.03.025},
 }
+@article{brunet2010contributions,
+  title={Contributions to parametric image registration and 3d surface reconstruction},
+  author={Brunet, Florent},
+  journal={European Ph. D. in Computer Vision, Universit{\'e} dAuvergne, Cl{\'e}rmont-Ferrand, France, and Technische Universitat Munchen, Germany},
+  year={2010},
+  url={http://www.brnt.eu/phd/}
+}

arbeit/ma.md

@@ -186,12 +186,8 @@ convex--hull--property of B-Splines --- meaning, that no matter how we choose
 our coefficients, the resulting points all have to lie inside convex--hull of
 the control--points.
 
-\improvement[inline]{
-Bilder von Basisfunktionen zur Visualisierung.
-}
-
-For a given number of points $v_1,\dots,v_n$ we can then calculate
-the contributions \linebreak[4]$n_{i,j}~:=~N_{j,d,\tau}$ of each control point $p_j$ to get the
+For a given point $v_i$ we can then calculate the contributions
+$n_{i,j}~:=~N_{j,d,\tau}$ of each control point $p_j$ to get the
 projection from the control--point--space into the object--space:
 $$
 v_i = \sum_j n_{i,j} \cdot p_j = \vec{n}_i^{T} \vec{p}
@@ -203,6 +199,32 @@ $$
 where $\vec{N}$ is the $n \times m$ transformation--matrix (later on called
 **deformation matrix**) for $n$ object--space--points and $m$ control--points.
 
+\begin{figure}[ht]
+\begin{center}
+\includegraphics[width=\textwidth]{img/unity.png}
+\end{center}
+\caption[B--spline--basis--function as partition of unity]{From \cite[Figure 2.13]{brunet2010contributions}:\newline
+\glqq Some interesting properties of the B--splines. On the natural definition domain
+of the B--spline ($[k_0,k_4]$ on this figure), the B--spline basis functions sum
+up to one (partition of unity). In this example, we use B--splines of degree 2.
+The horizontal segment below the abscissa axis represents the domain of
+influence of the B--splines basis function, i.e. the interval on which they are
+not null. At a given point, there are at most $ d+1$ non-zero B--spline basis
+functions (compact support).\grqq \newline
+Note, that Brunet starts his index at $-d$ opposed to our definition, where we
+start at $0$.}
+\label{fig:partition_unity}
+\end{figure}
+
+Furthermore B--splines--basis--functions form a partition of unity for all, but
+the first and last $d$ control-points\cite{brunet2010contributions}. Therefore
+we later on use the border-points $d+1$ times, such that $\sum_j n_{i,j} p_j = p_i$
+for these points.
+
+The locality of the influence of each control--point and the partition of unity
+was beautifully pictured by Brunet, which we included here as figure
+\ref{fig:partition_unity}.
+
 ### Why is \ac{FFD} a good deformation function?
 \label{sec:back:ffdgood}
 
@@ -386,9 +408,9 @@ f(\vec{x} + \vec{U}\vec{p})$ this makes sure that $\|\vec{Up}\| \propto
 is only performed to map the criterion--range to $[0..1]$, whereas $1$ is the
 optimal value and $0$ is the worst value.
 
-This criterion should be characteristic for numeric stability on the on
-hand\cite[chapter 2.7]{golub2012matrix} and for convergence speed of
-evolutionary algorithms on the other hand\cite{anrichterEvol} as it is tied to 
+On the one hand this criterion should be characteristic for numeric
+stability\cite[chapter 2.7]{golub2012matrix} and on the other hand for the
+convergence speed of evolutionary algorithms\cite{anrichterEvol} as it is tied to 
 the notion of locality\cite{weise2012evolutionary,thorhauer2014locality}.
 
 ### Improvement Potential
@@ -581,6 +603,8 @@ of the control--points.] to simulate different starting-conditions.
 
 # Scenarios for testing evolvability criteria using \acf{FFD}
 \label{sec:eval}
+\improvement[inline]{für 1d und 3d entweder konsistent source/target oder
+anders. Weil sonst in 1d $\vec{s}$ das Ziel, in 3d $\vec{t}$ das Ziel.}
 
 In our experiments we use the same two testing--scenarios, that were also used
 by \cite{anrichterEvol}. The first scenario deforms a plane into a shape
@@ -594,15 +618,6 @@ point has three degrees of freedom in contrast to first scenario.
 
 ## Test Scenario: 1D Function Approximation
 
-\begin{figure}[th]
-\begin{center}
-\includegraphics[width=0.7\textwidth]{img/1dtarget.png}
-\end{center}
-\caption{The target--shape for our 1--dimensional optimization--scenario
-including a wireframe--overlay of the vertices.}
-\label{fig:1dtarget}
-\end{figure}
-
 In this scenario we used the shape defined by Giannelli et al.\cite{giannelli2012thb},
 which is also used by Richter et al.\cite{anrichterEvol} using the same
 discretization to $150 \times 150$ points for a total of $n = 22\,500$ vertices. The
@@ -618,6 +633,15 @@ $$
 with $(x,y) \in [0,2] \times [0,1]$ and $q(x,y)=(x-1.5)^2 + (y-0.5)^2$, which we have
 visualized in figure \ref{fig:1dtarget}.
 
+begin{figure}[ht]
+\begin{center}
+\includegraphics[width=0.7\textwidth]{img/1dtarget.png}
+\end{center}
+\caption{The target--shape for our 1--dimensional optimization--scenario
+including a wireframe--overlay of the vertices.}
+\label{fig:1dtarget}
+\end{figure}
+
 As the starting-plane we used the same shape, but set all
 $z$--coordinates to $0$, yielding a flat plane, which is partially already
 correct.
@@ -651,15 +675,6 @@ $$
 where $\mathbb{1}$ is the vector consisting of $1$ in every dimension and
 $\vec{g}_\textrm{c} = \vec{p^{*}}$ the calculated correct gradient.
 
-We then set up a regular 2--dimensional grid around the object with the desired
-grid resolutions. To generate a testcase we then move the grid--vertices
-randomly inside the x--y--plane. As we do not want to generate hard to solve
-grids we avoid the generation of self--intersecting grids.\improvement{besser
-formulieren} To achieve that we select a uniform distributed number
-$r \in [-0.25,0.25]$ per dimension and shrink the distance to the neighbours
-(the smaller neighbour for $r < 0$, the larger for $r > 0$) by the factor
-$r$^[Note: On the Edges this displacement is only applied outwards by flipping the sign of $r$, if appropriate.].
-
 \begin{figure}[ht]
 \begin{center}
 \includegraphics[width=\textwidth]{img/example1d_grid.png}
@@ -669,23 +684,89 @@ random distortion to generate a testcase.}
 \label{fig:example1d_grid}
 \end{figure}
 
+We then set up a regular 2--dimensional grid around the object with the desired
+grid resolutions. To generate a testcase we then move the grid--vertices
+randomly inside the x--y--plane. As we do not want to generate hard to solve
+grids we avoid the generation of self--intersecting grids.\improvement{besser
+formulieren} To achieve that we select a uniform distributed number
+$r \in [-0.25,0.25]$ per dimension and shrink the distance to the neighbours
+(the smaller neighbour for $r < 0$, the larger for $r > 0$) by the factor
+$r$^[Note: On the Edges this displacement is only applied outwards by flipping the sign of $r$, if appropriate.].
+
 An Example of such a testcase can be seen for a $7 \times 4$--grid in figure
 \ref{fig:example1d_grid}.
 
 ## Test Scenario: 3D Function Approximation
 
-### Optimierungsszenario
+Opposed to the 1--dimensional scenario before, the 3--dimensional scenario is
+much more complex --- not only because we have more degrees of freedom on each
+control point, but also because the *fitness--function* we will use has no known
+analytic solution and multiple local minima.
 
-- Ball zu Mario
+\begin{figure}[ht]
+\begin{center}
+\includegraphics[width=0.7\textwidth]{img/3dtarget.png}
+\end{center}
+\caption{\newline
+Left: The sphere we start from with 10\,807 vertices\newline
+Right: The face we want to deform the sphere into with 12\,024 vertices.}
+\label{fig:3dtarget}
+\end{figure}
 
-### Matching in 3D
+First of all we introduce the set up: We have given a triangulated model of a
+sphere consisting of 10\,807 vertices, that we want to deform into a
+the target--model of a face with a total of 12\,024 vertices. Both of
+these Models can be seen in figure \ref{fig:3dtarget}.
 
-- alternierende Optimierung
+Opposed to the 1D--case we cannot map the source and target--vertices in a
+one--to--one--correspondence, which we especially need for the approximation of
+the fitting--error. Hence we state that the error of one vertex is the distance
+to the closest vertex of the other model.
 
-### Besonderheiten der Optimierung
+We therefore define the *fitness--function* to be:
+$$
+f(\vec{P}) = \frac{1}{n} \underbrace{\sum_{i=1}^n \|\vec{c_T(s_i)} -
+\vec{s_i}\|_2^2}_{\textrm{source-to-target--distance}}
++ \frac{1}{m} \underbrace{\sum_{i=1}^m \|\vec{c_S(t_i)} -
+\vec{t_i}\|_2^2}_{\textrm{target-to-source--distance}}
++ \lambda \cdot \textrm{regularization}
+$$
+where $\vec{c_T(s_i)}$ denotes the target--vertex that is corresponding to the
+source--vertex $\vec{s_i}$ and $\vec{c_S(t_i)}$ denotes the source--vertex that
+corresponds to the target--vertex $\vec{t_i}$. Note that the target--vertices
+are given and fixed by the target--model of the face we want to deform into,
+whereas the source--vertices vary depending on the chosen parameters $\vec{P}$,
+as those get calculated by the previously introduces formula $\vec{S} = \vec{UP}$
+with $\vec{S}$ being the $n \times 3$--matrix of source--vertices, $\vec{U}$ the
+$n \times m$--matrix of calculated coefficients for the \ac{FFD} --- analog to
+the 1D case --- and finally $\vec{P}$ being the $m \times 3$--matrix of the
+control--grid defining the whole deformation.
 
-- Analytische Lösung nur bis zur Optimierung der ersten Punkte gültig
-- Kriterien trotzdem gut
+As regularization-term we introduce a weighted decaying
+laplace--coefficient\unsure{heisst der so?}
+that is known to speed up the optimization--process\improvement{cite [34] aus
+ref{anrichterEvol}} and simulates a material that is very stiff in the
+beginning --- to do a coarse deformation --- and gets easier to deform over
+time.
+
+\improvement[inline]{mehr zu regularisierung, Formel etc.}
+
+## Procedure: 3D Function Approximation
+
+Initially we set up the correspondences $\vec{c_T(\dots)}$ and $\vec{c_S(\dots)}$ to be
+the respectively closest vertices of the other model. We then calculate the
+analytical solution given these correspondences via $\vec{P^{*}} = \vec{U^+}\vec{T}$,
+and also use the first solution as guessed gradient for the calculation of the
+*improvement--potential*, as the optimal solution is not known.
+We then let the evolutionary algorithm run up within $1.05$ times the error of 
+this solution and afterwards recalculate the correspondences $\vec{c_T(\dots)}$
+and $\vec{c_S(\dots)}$.
+For the next step we then halve the regularization--impact $\lambda$ and
+calculate the next incremental solution $\vec{P^{*}} = \vec{U^+}\vec{T}$ with
+the updated correspondences to get our next target--error.
+We repeat this process as long as the target--error keeps decreasing.
+
+\improvement[inline]{grid-setup}
 
 # Evaluation of Scenarios
 \label{sec:res}
@@ -716,9 +797,13 @@ An Example of such a testcase can be seen for a $7 \times 4$--grid in figure
 
 
 \begin{figure}[!ht]
-\includegraphics[width=\textwidth]{img/evolution3d/20171007_3dFit_all_append.png}
-\caption{Results 3D}
+\includegraphics[width=\textwidth]{img/evolution3d/4x4xX_montage.png}
+\caption{Results 3D for 4x4xX}
+\end{figure}
 
+\begin{figure}[!ht]
+\includegraphics[width=\textwidth]{img/evolution3d/Xx4x4_montage.png}
+\caption{Results 3D for Xx4x4}
 \end{figure}
 
 <!-- ![Improvement potential vs steps](img/evolution3d/20170926_3dFit_both_improvement-vs-steps.png) -->

arbeit/ma.tex

@@ -339,14 +339,9 @@ to the convex--hull--property of B-Splines --- meaning, that no matter
 how we choose our coefficients, the resulting points all have to lie
 inside convex--hull of the control--points.
 
-\improvement[inline]{
-Bilder von Basisfunktionen zur Visualisierung.
-}
-
-For a given number of points \(v_1,\dots,v_n\) we can then calculate the
-contributions \linebreak[4]\(n_{i,j}~:=~N_{j,d,\tau}\) of each control
-point \(p_j\) to get the projection from the control--point--space into
-the object--space: \[
+For a given point \(v_i\) we can then calculate the contributions
+\(n_{i,j}~:=~N_{j,d,\tau}\) of each control point \(p_j\) to get the
+projection from the control--point--space into the object--space: \[
 v_i = \sum_j n_{i,j} \cdot p_j = \vec{n}_i^{T} \vec{p}
 \] or written for all points at the same time: \[
 \vec{v} = \vec{N} \vec{p}
@@ -354,6 +349,33 @@ v_i = \sum_j n_{i,j} \cdot p_j = \vec{n}_i^{T} \vec{p}
 on called \textbf{deformation matrix}) for \(n\) object--space--points
 and \(m\) control--points.
 
+\begin{figure}[ht]
+\begin{center}
+\includegraphics[width=\textwidth]{img/unity.png}
+\end{center}
+\caption[B--spline--basis--function as partition of unity]{From \cite[Figure 2.13]{brunet2010contributions}:\newline
+\glqq Some interesting properties of the B--splines. On the natural definition domain
+of the B--spline ($[k_0,k_4]$ on this figure), the B--spline basis functions sum
+up to one (partition of unity). In this example, we use B--splines of degree 2.
+The horizontal segment below the abscissa axis represents the domain of
+influence of the B--splines basis function, i.e. the interval on which they are
+not null. At a given point, there are at most $ d+1$ non-zero B--spline basis
+functions (compact support).\grqq \newline
+Note, that Brunet starts his index at $-d$ opposed to our definition, where we
+start at $0$.}
+\label{fig:partition_unity}
+\end{figure}
+
+Furthermore B--splines--basis--functions form a partition of unity for
+all, but the first and last \(d\)
+control-points\cite{brunet2010contributions}. Therefore we later on use
+the border-points \(d+1\) times, such that \(\sum_j n_{i,j} p_j = p_i\)
+for these points.
+
+The locality of the influence of each control--point and the partition
+of unity was beautifully pictured by Brunet, which we included here as
+figure \ref{fig:partition_unity}.
+
 \subsection{\texorpdfstring{Why is \ac{FFD} a good deformation
 function?}{Why is  a good deformation function?}}\label{why-is-a-good-deformation-function}
 
@@ -561,10 +583,10 @@ The inversion of \(\kappa(\vec{U})\) is only performed to map the
 criterion--range to \([0..1]\), whereas \(1\) is the optimal value and
 \(0\) is the worst value.
 
-This criterion should be characteristic for numeric stability on the on
-hand\cite[chapter 2.7]{golub2012matrix} and for convergence speed of
-evolutionary algorithms on the other hand\cite{anrichterEvol} as it is
-tied to the notion of
+On the one hand this criterion should be characteristic for numeric
+stability\cite[chapter 2.7]{golub2012matrix} and on the other hand for
+the convergence speed of evolutionary algorithms\cite{anrichterEvol} as
+it is tied to the notion of
 locality\cite{weise2012evolutionary,thorhauer2014locality}.
 
 \subsection{Improvement Potential}\label{improvement-potential}
@@ -764,6 +786,8 @@ using
 \acf{FFD}}{Scenarios for testing evolvability criteria using }}\label{scenarios-for-testing-evolvability-criteria-using}
 
 \label{sec:eval}
+\improvement[inline]{für 1d und 3d entweder konsistent source/target oder
+anders. Weil sonst in 1d $\vec{s}$ das Ziel, in 3d $\vec{t}$ das Ziel.}
 
 In our experiments we use the same two testing--scenarios, that were
 also used by \cite{anrichterEvol}. The first scenario deforms a plane
@@ -779,15 +803,6 @@ scenario.
 \section{Test Scenario: 1D Function
 Approximation}\label{test-scenario-1d-function-approximation}
 
-\begin{figure}[th]
-\begin{center}
-\includegraphics[width=0.7\textwidth]{img/1dtarget.png}
-\end{center}
-\caption{The target--shape for our 1--dimensional optimization--scenario
-including a wireframe--overlay of the vertices.}
-\label{fig:1dtarget}
-\end{figure}
-
 In this scenario we used the shape defined by Giannelli et
 al.\cite{giannelli2012thb}, which is also used by Richter et
 al.\cite{anrichterEvol} using the same discretization to
@@ -803,6 +818,16 @@ s(x,y) =
 \(q(x,y)=(x-1.5)^2 + (y-0.5)^2\), which we have visualized in figure
 \ref{fig:1dtarget}.
 
+begin\{figure\}{[}ht{]}
+
+\begin{center}
+\includegraphics[width=0.7\textwidth]{img/1dtarget.png}
+\end{center}\caption{The target--shape for our 1--dimensional optimization--scenario
+including a wireframe--overlay of the vertices.}
+\label{fig:1dtarget}
+
+\textbackslash{}end\{figure\}
+
 As the starting-plane we used the same shape, but set all
 \(z\)--coordinates to \(0\), yielding a flat plane, which is partially
 already correct.
@@ -839,6 +864,15 @@ consider a distorted gradient \(\vec{g}_{\textrm{d}}\) \[
 dimension and \(\vec{g}_\textrm{c} = \vec{p^{*}}\) the calculated
 correct gradient.
 
+\begin{figure}[ht]
+\begin{center}
+\includegraphics[width=\textwidth]{img/example1d_grid.png}
+\end{center}
+\caption{\newline Left: A regular $7 \times 4$--grid\newline Right: The same grid after a
+random distortion to generate a testcase.}
+\label{fig:example1d_grid}
+\end{figure}
+
 We then set up a regular 2--dimensional grid around the object with the
 desired grid resolutions. To generate a testcase we then move the
 grid--vertices randomly inside the x--y--plane. As we do not want to
@@ -851,47 +885,84 @@ neighbours (the smaller neighbour for \(r < 0\), the larger for
   displacement is only applied outwards by flipping the sign of \(r\),
   if appropriate.}.
 
-\begin{figure}[ht]
-\begin{center}
-\includegraphics[width=\textwidth]{img/example1d_grid.png}
-\end{center}
-\caption{\newline Left: A regular $7 \times 4$--grid\newline Right: The same grid after a
-random distortion to generate a testcase.}
-\label{fig:example1d_grid}
-\end{figure}
-
 An Example of such a testcase can be seen for a \(7 \times 4\)--grid in
 figure \ref{fig:example1d_grid}.
 
 \section{Test Scenario: 3D Function
 Approximation}\label{test-scenario-3d-function-approximation}
 
-\subsection{Optimierungsszenario}\label{optimierungsszenario}
+Opposed to the 1--dimensional scenario before, the 3--dimensional
+scenario is much more complex --- not only because we have more degrees
+of freedom on each control point, but also because the
+\emph{fitness--function} we will use has no known analytic solution and
+multiple local minima.
 
-\begin{itemize}
-\tightlist
-\item
-  Ball zu Mario
-\end{itemize}
+\begin{figure}[ht]
+\begin{center}
+\includegraphics[width=0.7\textwidth]{img/3dtarget.png}
+\end{center}
+\caption{\newline
+Left: The sphere we start from with 10\,807 vertices\newline
+Right: The face we want to deform the sphere into with 12\,024 vertices.}
+\label{fig:3dtarget}
+\end{figure}
 
-\subsection{Matching in 3D}\label{matching-in-3d}
+First of all we introduce the set up: We have given a triangulated model
+of a sphere consisting of 10,807 vertices, that we want to deform into a
+the target--model of a face with a total of 12,024 vertices. Both of
+these Models can be seen in figure \ref{fig:3dtarget}.
 
-\begin{itemize}
-\tightlist
-\item
-  alternierende Optimierung
-\end{itemize}
+Opposed to the 1D--case we cannot map the source and target--vertices in
+a one--to--one--correspondence, which we especially need for the
+approximation of the fitting--error. Hence we state that the error of
+one vertex is the distance to the closest vertex of the other model.
 
-\subsection{Besonderheiten der
-Optimierung}\label{besonderheiten-der-optimierung}
+We therefore define the \emph{fitness--function} to be: \[
+f(\vec{P}) = \frac{1}{n} \underbrace{\sum_{i=1}^n \|\vec{c_T(s_i)} -
+\vec{s_i}\|_2^2}_{\textrm{source-to-target--distance}}
++ \frac{1}{m} \underbrace{\sum_{i=1}^m \|\vec{c_S(t_i)} -
+\vec{t_i}\|_2^2}_{\textrm{target-to-source--distance}}
++ \lambda \cdot \textrm{regularization}
+\] where \(\vec{c_T(s_i)}\) denotes the target--vertex that is
+corresponding to the source--vertex \(\vec{s_i}\) and \(\vec{c_S(t_i)}\)
+denotes the source--vertex that corresponds to the target--vertex
+\(\vec{t_i}\). Note that the target--vertices are given and fixed by the
+target--model of the face we want to deform into, whereas the
+source--vertices vary depending on the chosen parameters \(\vec{P}\), as
+those get calculated by the previously introduces formula
+\(\vec{S} = \vec{UP}\) with \(\vec{S}\) being the \(n \times 3\)--matrix
+of source--vertices, \(\vec{U}\) the \(n \times m\)--matrix of
+calculated coefficients for the \ac{FFD} --- analog to the 1D case ---
+and finally \(\vec{P}\) being the \(m \times 3\)--matrix of the
+control--grid defining the whole deformation.
 
-\begin{itemize}
-\tightlist
-\item
-  Analytische Lösung nur bis zur Optimierung der ersten Punkte gültig
-\item
-  Kriterien trotzdem gut
-\end{itemize}
+As regularization-term we introduce a weighted decaying
+laplace--coefficient\unsure{heisst der so?} that is known to speed up
+the optimization--process\improvement{cite [34] aus
+ref{anrichterEvol}} and simulates a material that is very stiff in the
+beginning --- to do a coarse deformation --- and gets easier to deform
+over time.
+
+\improvement[inline]{mehr zu regularisierung, Formel etc.}
+
+\section{Procedure: 3D Function
+Approximation}\label{procedure-3d-function-approximation}
+
+Initially we set up the correspondences \(\vec{c_T(\dots)}\) and
+\(\vec{c_S(\dots)}\) to be the respectively closest vertices of the
+other model. We then calculate the analytical solution given these
+correspondences via \(\vec{P^{*}} = \vec{U^+}\vec{T}\), and also use the
+first solution as guessed gradient for the calculation of the
+\emph{improvement--potential}, as the optimal solution is not known. We
+then let the evolutionary algorithm run up within \(1.05\) times the
+error of this solution and afterwards recalculate the correspondences
+\(\vec{c_T(\dots)}\) and \(\vec{c_S(\dots)}\). For the next step we then
+halve the regularization--impact \(\lambda\) and calculate the next
+incremental solution \(\vec{P^{*}} = \vec{U^+}\vec{T}\) with the updated
+correspondences to get our next target--error. We repeat this process as
+long as the target--error keeps decreasing.
+
+\improvement[inline]{grid-setup}
 
 \chapter{Evaluation of Scenarios}\label{evaluation-of-scenarios}
 
@@ -926,9 +997,13 @@ Approximation}\label{results-of-1d-function-approximation}
 Approximation}\label{results-of-3d-function-approximation}
 
 \begin{figure}[!ht]
-\includegraphics[width=\textwidth]{img/evolution3d/20171007_3dFit_all_append.png}
-\caption{Results 3D}
+\includegraphics[width=\textwidth]{img/evolution3d/4x4xX_montage.png}
+\caption{Results 3D for 4x4xX}
+\end{figure}
 
+\begin{figure}[!ht]
+\includegraphics[width=\textwidth]{img/evolution3d/Xx4x4_montage.png}
+\caption{Results 3D for Xx4x4}
 \end{figure}
 
 \chapter{Schluss}\label{schluss}