1d-scenario description done

arbeit/bibma.bib

@@ -172,4 +172,15 @@
 
   Url                      = {http://jcst.ict.ac.cn:8080/jcst/EN/article/downloadArticleFile.do?attachType=PDF\&id=9543}
 }
-
+@article{giannelli2012thb,
+  title={THB-splines: The truncated basis for hierarchical splines},
+  author={Giannelli, Carlotta and J{\"u}Ttler, Bert and Speleers, Hendrik},
+  journal={Computer Aided Geometric Design},
+  volume={29},
+  number={7},
+  pages={485--498},
+  year={2012},
+  publisher={Elsevier},
+  url={https://pdfs.semanticscholar.org/a858/aa68da617ad9d41de021f6807cc422002258.pdf},
+  doi={10.1016/j.cagd.2012.03.025},
+}

arbeit/ma.md

@@ -424,6 +424,7 @@ formulas for the general case so it can be adapted quite freely.
 
 
 ## Adaption of \ac{FFD}
+\label{sec:ffd:adapt}
 
 As we have established in Chapter \ref{sec:back:ffd} we can define an
 \ac{FFD}--displacement as
@@ -581,22 +582,95 @@ of the control--points.] to simulate different starting-conditions.
 # Scenarios for testing evolvability criteria using \acf{FFD}
 \label{sec:eval}
 
+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
+originally defined in \cite{giannelli2012thb}, where we setup control-points in
+a 2--dimensional manner merely deform in the height--coordinate to get the
+resulting shape.
+
+In the second scenario we increase the degrees of freedom significantly by using
+a 3--dimensional control--grid to deform a sphere into a face. So each control
+point has three degrees of freedom in contrast to first scenario.
+
 ## Test Scenario: 1D Function Approximation
 
-### Optimierungszenario
+\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}
 
-- Ebene -> Template--Fit
+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
+shape is given by the following definition
+$$
+s(x,y) =
+\begin{cases}
+0.5 \cos(4\pi \cdot q^{0.5}) + 0.5 & q(x,y) < \frac{1}{16},\\
+2(y-x) & 0 < y-x < 0.5,\\
+1 & 0.5 < y - x
+\end{cases}
+$$
+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}.
 
-### Matching in 1D
+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.
 
-- Trivial
+Regarding the *fitness--function* $f(\vec{p})$, we use the very simple approach
+of calculating the squared distances for each corresponding vertex
+$$
+\textrm{f(\vec{p})} = \sum_{i=1}^{n} \|(\vec{Up})_i - s_i\|_2^2 = \|\vec{Up} - \vec{s}\|^2 \rightarrow \min
+$$
+where $s_i$ are the respective solution--vertices to the parametrized
+source--vertices^[The parametrization is encoded in $\vec{U}$ and the initial
+position of the control points. See \ref{sec:ffd:adapt}] with the current
+deformation--parameters $\vec{p} = (p_1,\dots, p_m)$. We can do this
+one--to--one--correspondence because we have exactly the same number of
+source and target-vertices do to our setup of just flattening the object.
 
-### Besonderheiten der Auswertung
+This formula is also the least--squares approximation error for which we
+can compute the analytic solution $\vec{p^{*}} = \vec{U^+}\vec{s}$, yielding us
+the correct gradient in which the evolutionary optimizer should move.
 
-- Analytische Lösung einzig beste
-  - Ergebnis auch bei Rauschen konstant?
-  - normierter 1--Vektor auf den Gradienten addieren
-    - Kegel entsteht
+## Procedure: 1D Function Approximation
+
+For our setup we first compute the coefficients of the deformation--matrix and
+use then the formulas for *variability* and *regularity* to get our predictions.
+Afterwards we solve the problem analytically to get the (normalized) correct
+gradient that we use as guess for the *improvement potential*. To check we also
+consider a distorted gradient $\vec{g}_{\textrm{d}}$
+$$
+\vec{g}_{\textrm{d}} = \frac{\vec{g}_{\textrm{c}} + \mathbb{1}}{\|\vec{g}_{\textrm{c}} + \mathbb{1}\|}
+$$
+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}
+\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}.
 
 ## Test Scenario: 3D Function Approximation
 

arbeit/ma.tex

@@ -604,6 +604,8 @@ adapted quite freely.
 \section{\texorpdfstring{Adaption of
 \ac{FFD}}{Adaption of }}\label{adaption-of}
 
+\label{sec:ffd:adapt}
+
 As we have established in Chapter \ref{sec:back:ffd} we can define an
 \ac{FFD}--displacement as
 
@@ -763,43 +765,103 @@ using
 
 \label{sec:eval}
 
+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 originally defined in \cite{giannelli2012thb}, where we
+setup control-points in a 2--dimensional manner merely deform in the
+height--coordinate to get the resulting shape.
+
+In the second scenario we increase the degrees of freedom significantly
+by using a 3--dimensional control--grid to deform a sphere into a face.
+So each control point has three degrees of freedom in contrast to first
+scenario.
+
 \section{Test Scenario: 1D Function
 Approximation}\label{test-scenario-1d-function-approximation}
 
-\subsection{Optimierungszenario}\label{optimierungszenario}
+\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}
 
-\begin{itemize}
-\tightlist
-\item
-  Ebene -\textgreater{} Template--Fit
-\end{itemize}
+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
+shape is given by the following definition \[
+s(x,y) =
+\begin{cases}
+0.5 \cos(4\pi \cdot q^{0.5}) + 0.5 & q(x,y) < \frac{1}{16},\\
+2(y-x) & 0 < y-x < 0.5,\\
+1 & 0.5 < y - x
+\end{cases}
+\] 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}.
 
-\subsection{Matching in 1D}\label{matching-in-1d}
+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.
 
-\begin{itemize}
-\tightlist
-\item
-  Trivial
-\end{itemize}
+Regarding the \emph{fitness--function} \(f(\vec{p})\), we use the very
+simple approach of calculating the squared distances for each
+corresponding vertex \[
+\textrm{f(\vec{p})} = \sum_{i=1}^{n} \|(\vec{Up})_i - s_i\|_2^2 = \|\vec{Up} - \vec{s}\|^2 \rightarrow \min
+\] where \(s_i\) are the respective solution--vertices to the
+parametrized source--vertices\footnote{The parametrization is encoded in
+  \(\vec{U}\) and the initial position of the control points. See
+  \ref{sec:ffd:adapt}} with the current deformation--parameters
+\(\vec{p} = (p_1,\dots, p_m)\). We can do this
+one--to--one--correspondence because we have exactly the same number of
+source and target-vertices do to our setup of just flattening the
+object.
 
-\subsection{Besonderheiten der
-Auswertung}\label{besonderheiten-der-auswertung}
+This formula is also the least--squares approximation error for which we
+can compute the analytic solution \(\vec{p^{*}} = \vec{U^+}\vec{s}\),
+yielding us the correct gradient in which the evolutionary optimizer
+should move.
 
-\begin{itemize}
-\tightlist
-\item
-  Analytische Lösung einzig beste
-\item
-  Ergebnis auch bei Rauschen konstant?
-\item
-  normierter 1--Vektor auf den Gradienten addieren
+\section{Procedure: 1D Function
+Approximation}\label{procedure-1d-function-approximation}
 
-  \begin{itemize}
-  \tightlist
-  \item
-    Kegel entsteht
-  \end{itemize}
-\end{itemize}
+For our setup we first compute the coefficients of the
+deformation--matrix and use then the formulas for \emph{variability} and
+\emph{regularity} to get our predictions. Afterwards we solve the
+problem analytically to get the (normalized) correct gradient that we
+use as guess for the \emph{improvement potential}. To check we also
+consider a distorted gradient \(\vec{g}_{\textrm{d}}\) \[
+\vec{g}_{\textrm{d}} = \frac{\vec{g}_{\textrm{c}} + \mathbb{1}}{\|\vec{g}_{\textrm{c}} + \mathbb{1}\|}
+\] 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 \(r\)\footnote{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}
+\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}

arbeit/settings/commands.tex

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