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@ -190,3 +190,15 @@
year={2010}, year={2010},
url={http://www.brnt.eu/phd/} url={http://www.brnt.eu/phd/}
} }
@article{aschenbach2015,
author = {Achenbach, Jascha and Zell, Eduard and Botsch, Mario},
booktitle = {Vision, Modeling \& Visualization},
journal = {Proceedings of Vision, Modeling and Visualization},
location = {Aachen, Germany},
pages = {1--8},
publisher = {Eurographics Association},
title = {Accurate Face Reconstruction through Anisotropic Fitting and Eye Correction},
year = {2015},
url = {http://graphics.uni-bielefeld.de/publications/disclaimer.php?dlurl=vmv15.pdf},
ISBN = {978-3-905674-95-8},
}

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@ -548,7 +548,6 @@ $$J(Err(u,v,w)) \cdot \Delta \left( \begin{array}{c} u \\ v \\ w \end{array} \ri
and use Cramers rule for inverting the small Jacobian and solving this system of and use Cramers rule for inverting the small Jacobian and solving this system of
linear equations. linear equations.
## Deformation Grid ## Deformation Grid
As mentioned in chapter \ref{sec:back:evo}, the way of choosing the As mentioned in chapter \ref{sec:back:evo}, the way of choosing the
@ -594,7 +593,7 @@ central points are not relevant for the parametrization of the circle.
\unsure[inline]{erwähnen, dass man aus $\vec{D}$ einfach die Null--Spalten \unsure[inline]{erwähnen, dass man aus $\vec{D}$ einfach die Null--Spalten
entfernen kann?} entfernen kann?}
For our tests we chose different uniformly sized grids and added gaussian noise For our tests we chose different uniformly sized grids and added noise
onto each control-point^[For the special case of the outer layer we only applied onto each control-point^[For the special case of the outer layer we only applied
noise away from the object, so the object is still confined in the convex hull noise away from the object, so the object is still confined in the convex hull
of the control--points.] to simulate different starting-conditions. of the control--points.] to simulate different starting-conditions.
@ -603,8 +602,6 @@ of the control--points.] to simulate different starting-conditions.
# Scenarios for testing evolvability criteria using \acf{FFD} # Scenarios for testing evolvability criteria using \acf{FFD}
\label{sec:eval} \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 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 by \cite{anrichterEvol}. The first scenario deforms a plane into a shape
@ -623,7 +620,7 @@ 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 discretization to $150 \times 150$ points for a total of $n = 22\,500$ vertices. The
shape is given by the following definition shape is given by the following definition
$$ $$
s(x,y) = t(x,y) =
\begin{cases} \begin{cases}
0.5 \cos(4\pi \cdot q^{0.5}) + 0.5 & q(x,y) < \frac{1}{16},\\ 0.5 \cos(4\pi \cdot q^{0.5}) + 0.5 & q(x,y) < \frac{1}{16},\\
2(y-x) & 0 < y-x < 0.5,\\ 2(y-x) & 0 < y-x < 0.5,\\
@ -633,7 +630,7 @@ $$
with $(x,y) \in [0,2] \times [0,1]$ and $q(x,y)=(x-1.5)^2 + (y-0.5)^2$, which we have 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}. visualized in figure \ref{fig:1dtarget}.
begin{figure}[ht] \begin{figure}[ht]
\begin{center} \begin{center}
\includegraphics[width=0.7\textwidth]{img/1dtarget.png} \includegraphics[width=0.7\textwidth]{img/1dtarget.png}
\end{center} \end{center}
@ -649,9 +646,9 @@ correct.
Regarding the *fitness--function* $f(\vec{p})$, we use the very simple approach Regarding the *fitness--function* $f(\vec{p})$, we use the very simple approach
of calculating the squared distances for each corresponding vertex 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 \textrm{f(\vec{p})} = \sum_{i=1}^{n} \|(\vec{Up})_i - t_i\|_2^2 = \|\vec{Up} - \vec{t}\|^2 \rightarrow \min
$$ $$
where $s_i$ are the respective solution--vertices to the parametrized where $t_i$ are the respective target--vertices to the parametrized
source--vertices^[The parametrization is encoded in $\vec{U}$ and the initial 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 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 deformation--parameters $\vec{p} = (p_1,\dots, p_m)$. We can do this
@ -659,11 +656,11 @@ 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. source and target-vertices do to our setup of just flattening the object.
This formula is also the least--squares approximation error for which we 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 can compute the analytic solution $\vec{p^{*}} = \vec{U^+}\vec{t}$, yielding us
the correct gradient in which the evolutionary optimizer should move. the correct gradient in which the evolutionary optimizer should move.
## Procedure: 1D Function Approximation ## Procedure: 1D Function Approximation
For our setup we first compute the coefficients of the deformation--matrix and 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. use then the formulas for *variability* and *regularity* to get our predictions.
Afterwards we solve the problem analytically to get the (normalized) correct Afterwards we solve the problem analytically to get the (normalized) correct
@ -679,19 +676,26 @@ $\vec{g}_\textrm{c} = \vec{p^{*}}$ the calculated correct gradient.
\begin{center} \begin{center}
\includegraphics[width=\textwidth]{img/example1d_grid.png} \includegraphics[width=\textwidth]{img/example1d_grid.png}
\end{center} \end{center}
\caption{\newline Left: A regular $7 \times 4$--grid\newline Right: The same grid after a \caption[Example of a 1D--grid]{\newline Left: A regular $7 \times 4$--grid\newline Right: The same grid after a
random distortion to generate a testcase.} random distortion to generate a testcase.}
\label{fig:example1d_grid} \label{fig:example1d_grid}
\end{figure} \end{figure}
We then set up a regular 2--dimensional grid around the object with the desired 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 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 randomly inside the x--y--plane. As self-intersecting grids get tricky to solve
grids we avoid the generation of self--intersecting grids.\improvement{besser with our implemented newtons--method we avoid the generation of such
formulieren} To achieve that we select a uniform distributed number self--intersecting grids for our testcases.
$r \in [-0.25,0.25]$ per dimension and shrink the distance to the neighbours
(the smaller neighbour for $r < 0$, the larger for $r > 0$) by the factor This is a reasonable thing to do, as self-intersecting grids violate our desired
$r$^[Note: On the Edges this displacement is only applied outwards by flipping the sign of $r$, if appropriate.]. property of locality, as the then farther away control--point has more influence
over some vertices as the next-closer.
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 An Example of such a testcase can be seen for a $7 \times 4$--grid in figure
\ref{fig:example1d_grid}. \ref{fig:example1d_grid}.
@ -729,7 +733,7 @@ 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}} \vec{s_i}\|_2^2}_{\textrm{source-to-target--distance}}
+ \frac{1}{m} \underbrace{\sum_{i=1}^m \|\vec{c_S(t_i)} - + \frac{1}{m} \underbrace{\sum_{i=1}^m \|\vec{c_S(t_i)} -
\vec{t_i}\|_2^2}_{\textrm{target-to-source--distance}} \vec{t_i}\|_2^2}_{\textrm{target-to-source--distance}}
+ \lambda \cdot \textrm{regularization} + \lambda \cdot \textrm{regularization}(\vec{P})
$$ $$
where $\vec{c_T(s_i)}$ denotes the target--vertex that is corresponding to the 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 source--vertex $\vec{s_i}$ and $\vec{c_S(t_i)}$ denotes the source--vertex that
@ -742,14 +746,21 @@ $n \times m$--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 the 1D case --- and finally $\vec{P}$ being the $m \times 3$--matrix of the
control--grid defining the whole deformation. control--grid defining the whole deformation.
As regularization-term we introduce a weighted decaying As regularization-term we add a weighted Laplacian of the deformation that has
laplace--coefficient\unsure{heisst der so?} been used before by Aschenbach et al.\cite[Section 3.2]{aschenbach2015} on
that is known to speed up the optimization--process\improvement{cite [34] aus similar models and was shown to lead to a more precise fit. The Laplacian
ref{anrichterEvol}} and simulates a material that is very stiff in the $$
beginning --- to do a coarse deformation --- and gets easier to deform over \textrm{regularization}(\vec{P}) = \frac{1}{\sum_i A_i} \sum_{i=1}^n A_i \cdot \left( \sum_{\vec{s_j} \in \mathcal{N}(\vec{s_i})} w_j \cdot \|\Delta \vec{s_j} - \Delta \vec{\overline{s}_j}\|^2 \right)
time. $$
is determined by the cotangent weighted displacement $w_j$ of the to $s_i$
connected vertices $\mathcal{N}(s_i)$ and $A_i$ is the Voronoi--area of the corresponding vertex
$\vec{s_i}$. We leave out the $\vec{R}_i$--term from the original paper as our
deformation is merely linear.
\improvement[inline]{mehr zu regularisierung, Formel etc.} This regularization--weight gives us a measure of stiffness for the material
that we will influence via the $\lambda$--coefficient to start out with a stiff
material that will get more flexible per iteration.
\unsure[inline]{Andreas: hast du nen cite, wo gezeigt ist, dass das so sinnvoll ist?}
## Procedure: 3D Function Approximation ## Procedure: 3D Function Approximation
@ -766,7 +777,31 @@ calculate the next incremental solution $\vec{P^{*}} = \vec{U^+}\vec{T}$ with
the updated correspondences to get our next target--error. the updated correspondences to get our next target--error.
We repeat this process as long as the target--error keeps decreasing. We repeat this process as long as the target--error keeps decreasing.
\improvement[inline]{grid-setup} \begin{figure}[ht]
\begin{center}
\includegraphics[width=\textwidth]{img/example3d_grid.png}
\end{center}
\caption[Example of a 3D--grid]{\newline Left: The 3D--setup with a $4\times
4\times 4$--grid.\newline Right: The same grid after added noise to the
control--points.}
\label{fig:setup3d}
\end{figure}
The grid we use for our experiments is just very coarse due to computational
limitations. We are not interested in a good reconstruction, but an estimate if
the mentioned evolvability criteria are good.
In figure \ref{fig:setup3d} we show an example setup of the scene with a
$4\times 4\times 4$--grid. Identical to the 1--dimensional scenario before, we create a
regular grid and move the control-points uniformly random between their
neighbours, but in three instead of two dimensions^[Again, we flip the signs for
the edges, if necessary to have the object still in the convex hull.].
As is clearly visible from figure \ref{fig:3dtarget}, the target--model has many
vertices in the facial area, at the ears and in the neck--region. Therefore we
chose to increase the grid-resolutions for our tests in two different dimensions
and see how well the criteria predict a suboptimal placement of these
control-points.
# Evaluation of Scenarios # Evaluation of Scenarios
\label{sec:res} \label{sec:res}
@ -817,3 +852,6 @@ We repeat this process as long as the target--error keeps decreasing.
\label{sec:dis} \label{sec:dis}
HAHA .. als ob -.- HAHA .. als ob -.-
\improvement[inline]{Bibliotheksverzeichnis links anpassen. DOI überschreibt
Direktlinks des Autors.}

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@ -773,11 +773,11 @@ circle.
\unsure[inline]{erwähnen, dass man aus $\vec{D}$ einfach die Null--Spalten \unsure[inline]{erwähnen, dass man aus $\vec{D}$ einfach die Null--Spalten
entfernen kann?} entfernen kann?}
For our tests we chose different uniformly sized grids and added For our tests we chose different uniformly sized grids and added noise
gaussian noise onto each control-point\footnote{For the special case of onto each control-point\footnote{For the special case of the outer layer
the outer layer we only applied noise away from the object, so the we only applied noise away from the object, so the object is still
object is still confined in the convex hull of the control--points.} confined in the convex hull of the control--points.} to simulate
to simulate different starting-conditions. different starting-conditions.
\unsure[inline]{verweis auf DM--FFD?} \unsure[inline]{verweis auf DM--FFD?}
@ -786,8 +786,6 @@ using
\acf{FFD}}{Scenarios for testing evolvability criteria using }}\label{scenarios-for-testing-evolvability-criteria-using} \acf{FFD}}{Scenarios for testing evolvability criteria using }}\label{scenarios-for-testing-evolvability-criteria-using}
\label{sec:eval} \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 In our experiments we use the same two testing--scenarios, that were
also used by \cite{anrichterEvol}. The first scenario deforms a plane also used by \cite{anrichterEvol}. The first scenario deforms a plane
@ -808,7 +806,7 @@ al.\cite{giannelli2012thb}, which is also used by Richter et
al.\cite{anrichterEvol} using the same discretization to al.\cite{anrichterEvol} using the same discretization to
\(150 \times 150\) points for a total of \(n = 22\,500\) vertices. The \(150 \times 150\) points for a total of \(n = 22\,500\) vertices. The
shape is given by the following definition \[ shape is given by the following definition \[
s(x,y) = t(x,y) =
\begin{cases} \begin{cases}
0.5 \cos(4\pi \cdot q^{0.5}) + 0.5 & q(x,y) < \frac{1}{16},\\ 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,\\ 2(y-x) & 0 < y-x < 0.5,\\
@ -818,15 +816,14 @@ s(x,y) =
\(q(x,y)=(x-1.5)^2 + (y-0.5)^2\), which we have visualized in figure \(q(x,y)=(x-1.5)^2 + (y-0.5)^2\), which we have visualized in figure
\ref{fig:1dtarget}. \ref{fig:1dtarget}.
begin\{figure\}{[}ht{]} \begin{figure}[ht]
\begin{center} \begin{center}
\includegraphics[width=0.7\textwidth]{img/1dtarget.png} \includegraphics[width=0.7\textwidth]{img/1dtarget.png}
\end{center}\caption{The target--shape for our 1--dimensional optimization--scenario \end{center}
\caption{The target--shape for our 1--dimensional optimization--scenario
including a wireframe--overlay of the vertices.} including a wireframe--overlay of the vertices.}
\label{fig:1dtarget} \label{fig:1dtarget}
\end{figure}
\textbackslash{}end\{figure\}
As the starting-plane we used the same shape, but set all As the starting-plane we used the same shape, but set all
\(z\)--coordinates to \(0\), yielding a flat plane, which is partially \(z\)--coordinates to \(0\), yielding a flat plane, which is partially
@ -835,10 +832,10 @@ already correct.
Regarding the \emph{fitness--function} \(f(\vec{p})\), we use the very Regarding the \emph{fitness--function} \(f(\vec{p})\), we use the very
simple approach of calculating the squared distances for each simple approach of calculating the squared distances for each
corresponding vertex \[ corresponding vertex \[
\textrm{f(\vec{p})} = \sum_{i=1}^{n} \|(\vec{Up})_i - s_i\|_2^2 = \|\vec{Up} - \vec{s}\|^2 \rightarrow \min \textrm{f(\vec{p})} = \sum_{i=1}^{n} \|(\vec{Up})_i - t_i\|_2^2 = \|\vec{Up} - \vec{t}\|^2 \rightarrow \min
\] where \(s_i\) are the respective solution--vertices to the \] where \(t_i\) are the respective target--vertices to the parametrized
parametrized source--vertices\footnote{The parametrization is encoded in source--vertices\footnote{The parametrization is encoded in \(\vec{U}\)
\(\vec{U}\) and the initial position of the control points. See and the initial position of the control points. See
\ref{sec:ffd:adapt}} with the current deformation--parameters \ref{sec:ffd:adapt}} with the current deformation--parameters
\(\vec{p} = (p_1,\dots, p_m)\). We can do this \(\vec{p} = (p_1,\dots, p_m)\). We can do this
one--to--one--correspondence because we have exactly the same number of one--to--one--correspondence because we have exactly the same number of
@ -846,7 +843,7 @@ source and target-vertices do to our setup of just flattening the
object. object.
This formula is also the least--squares approximation error for which we This formula is also the least--squares approximation error for which we
can compute the analytic solution \(\vec{p^{*}} = \vec{U^+}\vec{s}\), can compute the analytic solution \(\vec{p^{*}} = \vec{U^+}\vec{t}\),
yielding us the correct gradient in which the evolutionary optimizer yielding us the correct gradient in which the evolutionary optimizer
should move. should move.
@ -868,17 +865,22 @@ correct gradient.
\begin{center} \begin{center}
\includegraphics[width=\textwidth]{img/example1d_grid.png} \includegraphics[width=\textwidth]{img/example1d_grid.png}
\end{center} \end{center}
\caption{\newline Left: A regular $7 \times 4$--grid\newline Right: The same grid after a \caption[Example of a 1D--grid]{\newline Left: A regular $7 \times 4$--grid\newline Right: The same grid after a
random distortion to generate a testcase.} random distortion to generate a testcase.}
\label{fig:example1d_grid} \label{fig:example1d_grid}
\end{figure} \end{figure}
We then set up a regular 2--dimensional grid around the object with the 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 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 grid--vertices randomly inside the x--y--plane. As self-intersecting
generate hard to solve grids we avoid the generation of grids get tricky to solve with our implemented newtons--method we avoid
self--intersecting grids.\improvement{besser the generation of such self--intersecting grids for our testcases.
formulieren} To achieve that we select a uniform distributed number
This is a reasonable thing to do, as self-intersecting grids violate our
desired property of locality, as the then farther away control--point
has more influence over some vertices as the next-closer.
To achieve that we select a uniform distributed number
\(r \in [-0.25,0.25]\) per dimension and shrink the distance to the \(r \in [-0.25,0.25]\) per dimension and shrink the distance to the
neighbours (the smaller neighbour for \(r < 0\), the larger for neighbours (the smaller neighbour for \(r < 0\), the larger for
\(r > 0\)) by the factor \(r\)\footnote{Note: On the Edges this \(r > 0\)) by the factor \(r\)\footnote{Note: On the Edges this
@ -922,7 +924,7 @@ 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}} \vec{s_i}\|_2^2}_{\textrm{source-to-target--distance}}
+ \frac{1}{m} \underbrace{\sum_{i=1}^m \|\vec{c_S(t_i)} - + \frac{1}{m} \underbrace{\sum_{i=1}^m \|\vec{c_S(t_i)} -
\vec{t_i}\|_2^2}_{\textrm{target-to-source--distance}} \vec{t_i}\|_2^2}_{\textrm{target-to-source--distance}}
+ \lambda \cdot \textrm{regularization} + \lambda \cdot \textrm{regularization}(\vec{P})
\] where \(\vec{c_T(s_i)}\) denotes the target--vertex that is \] where \(\vec{c_T(s_i)}\) denotes the target--vertex that is
corresponding to the source--vertex \(\vec{s_i}\) and \(\vec{c_S(t_i)}\) corresponding to the source--vertex \(\vec{s_i}\) and \(\vec{c_S(t_i)}\)
denotes the source--vertex that corresponds to the target--vertex denotes the source--vertex that corresponds to the target--vertex
@ -936,14 +938,22 @@ calculated coefficients for the \ac{FFD} --- analog to the 1D case ---
and finally \(\vec{P}\) being the \(m \times 3\)--matrix of the and finally \(\vec{P}\) being the \(m \times 3\)--matrix of the
control--grid defining the whole deformation. control--grid defining the whole deformation.
As regularization-term we introduce a weighted decaying As regularization-term we add a weighted Laplacian of the deformation
laplace--coefficient\unsure{heisst der so?} that is known to speed up that has been used before by Aschenbach et
the optimization--process\improvement{cite [34] aus al.\cite[Section 3.2]{aschenbach2015} on similar models and was shown to
ref{anrichterEvol}} and simulates a material that is very stiff in the lead to a more precise fit. The Laplacian \[
beginning --- to do a coarse deformation --- and gets easier to deform \textrm{regularization}(\vec{P}) = \frac{1}{\sum_i A_i} \sum_{i=1}^n A_i \cdot \left( \sum_{\vec{s_j} \in \mathcal{N}(\vec{s_i})} w_j \cdot \|\Delta \vec{s_j} - \Delta \vec{\overline{s}_j}\|^2 \right)
over time. \] is determined by the cotangent weighted displacement \(w_j\) of the
to \(s_i\) connected vertices \(\mathcal{N}(s_i)\) and \(A_i\) is the
Voronoi--area of the corresponding vertex \(\vec{s_i}\). We leave out
the \(\vec{R}_i\)--term from the original paper as our deformation is
merely linear.
\improvement[inline]{mehr zu regularisierung, Formel etc.} This regularization--weight gives us a measure of stiffness for the
material that we will influence via the \(\lambda\)--coefficient to
start out with a stiff material that will get more flexible per
iteration.
\unsure[inline]{Andreas: hast du nen cite, wo gezeigt ist, dass das so sinnvoll ist?}
\section{Procedure: 3D Function \section{Procedure: 3D Function
Approximation}\label{procedure-3d-function-approximation} Approximation}\label{procedure-3d-function-approximation}
@ -962,7 +972,33 @@ incremental solution \(\vec{P^{*}} = \vec{U^+}\vec{T}\) with the updated
correspondences to get our next target--error. We repeat this process as correspondences to get our next target--error. We repeat this process as
long as the target--error keeps decreasing. long as the target--error keeps decreasing.
\improvement[inline]{grid-setup} \begin{figure}[ht]
\begin{center}
\includegraphics[width=\textwidth]{img/example3d_grid.png}
\end{center}
\caption[Example of a 3D--grid]{\newline Left: The 3D--setup with a $4\times
4\times 4$--grid.\newline Right: The same grid after added noise to the
control--points.}
\label{fig:setup3d}
\end{figure}
The grid we use for our experiments is just very coarse due to
computational limitations. We are not interested in a good
reconstruction, but an estimate if the mentioned evolvability criteria
are good.
In figure \ref{fig:setup3d} we show an example setup of the scene with a
\(4\times 4\times 4\)--grid. Identical to the 1--dimensional scenario
before, we create a regular grid and move the control-points uniformly
random between their neighbours, but in three instead of two
dimensions\footnote{Again, we flip the signs for the edges, if necessary
to have the object still in the convex hull.}.
As is clearly visible from figure \ref{fig:3dtarget}, the target--model
has many vertices in the facial area, at the ears and in the
neck--region. Therefore we chose to increase the grid-resolutions for
our tests in two different dimensions and see how well the criteria
predict a suboptimal placement of these control-points.
\chapter{Evaluation of Scenarios}\label{evaluation-of-scenarios} \chapter{Evaluation of Scenarios}\label{evaluation-of-scenarios}
@ -1012,6 +1048,9 @@ Approximation}\label{results-of-3d-function-approximation}
HAHA .. als ob -.- HAHA .. als ob -.-
\improvement[inline]{Bibliotheksverzeichnis links anpassen. DOI überschreibt
Direktlinks des Autors.}
% \backmatter % \backmatter
\cleardoublepage \cleardoublepage