chapter 1-4 complete. 2 left.

arbeit/bibma.bib

@@ -190,3 +190,15 @@
   year={2010},
   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},
+}

arbeit/ma.md

@@ -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
 linear equations.
 
-
 ## Deformation Grid
 
 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
 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
 noise away from the object, so the object is still confined in the convex hull
 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}
 \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
@@ -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
 shape is given by the following definition
 $$
-s(x,y) =
+t(x,y) =
 \begin{cases}
 0.5 \cos(4\pi \cdot q^{0.5}) + 0.5 & q(x,y) < \frac{1}{16},\\
 2(y-x) & 0 < y-x < 0.5,\\
@@ -633,7 +630,7 @@ $$
 with $(x,y) \in [0,2] \times [0,1]$ and $q(x,y)=(x-1.5)^2 + (y-0.5)^2$, which we have
 visualized in figure \ref{fig:1dtarget}.
 
-begin{figure}[ht]
+\begin{figure}[ht]
 \begin{center}
 \includegraphics[width=0.7\textwidth]{img/1dtarget.png}
 \end{center}
@@ -649,9 +646,9 @@ correct.
 Regarding the *fitness--function* $f(\vec{p})$, we use the very simple approach
 of calculating the squared distances for each corresponding vertex
 $$
-\textrm{f(\vec{p})} = \sum_{i=1}^{n} \|(\vec{Up})_i - s_i\|_2^2 = \|\vec{Up} - \vec{s}\|^2 \rightarrow \min
+\textrm{f(\vec{p})} = \sum_{i=1}^{n} \|(\vec{Up})_i - t_i\|_2^2 = \|\vec{Up} - \vec{t}\|^2 \rightarrow \min
 $$
-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
 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
@@ -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.
 
 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.
 
 ## 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
@@ -679,19 +676,26 @@ $\vec{g}_\textrm{c} = \vec{p^{*}}$ the calculated correct gradient.
 \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
+\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.}
 \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.].
+randomly inside the x--y--plane. As self-intersecting grids get tricky to solve
+with our implemented newtons--method we avoid the generation of such
+self--intersecting grids for our testcases.
+
+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 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}.
@@ -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}}
 + \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}
++ \lambda \cdot \textrm{regularization}(\vec{P})
 $$
 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
@@ -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
 control--grid defining the whole deformation.
 
-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.
+As regularization-term we add a weighted Laplacian of the deformation that has
+been used before by Aschenbach et al.\cite[Section 3.2]{aschenbach2015} on
+similar models and was shown to lead to a more precise fit. The Laplacian
+$$
+\textrm{regularization}(\vec{P}) = \frac{1}{\sum_i A_i} \sum_{i=1}^n A_i \cdot \left( \sum_{\vec{s_j} \in \mathcal{N}(\vec{s_i})} w_j \cdot \|\Delta \vec{s_j} - \Delta \vec{\overline{s}_j}\|^2 \right)
+$$
+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
 
@@ -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.
 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
 \label{sec:res}
@@ -817,3 +852,6 @@ We repeat this process as long as the target--error keeps decreasing.
 \label{sec:dis}
 
 HAHA .. als ob -.-
+
+\improvement[inline]{Bibliotheksverzeichnis links anpassen. DOI überschreibt
+Direktlinks des Autors.}

arbeit/ma.tex

@@ -773,11 +773,11 @@ circle.
 \unsure[inline]{erwähnen, dass man aus $\vec{D}$ einfach die Null--Spalten
 entfernen kann?}
 
-For our tests we chose different uniformly sized grids and added
-gaussian noise onto each control-point\footnote{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 of the control--points.}
-to simulate different starting-conditions.
+For our tests we chose different uniformly sized grids and added noise
+onto each control-point\footnote{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 of the control--points.} to simulate
+different starting-conditions.
 
 \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}
 
 \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
@@ -808,7 +806,7 @@ 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) =
+t(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,\\
@@ -818,15 +816,14 @@ 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{figure}[ht]
 \begin{center}
 \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.}
 \label{fig:1dtarget}
-
-\textbackslash{}end\{figure\}
+\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
@@ -835,10 +832,10 @@ already correct.
 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
+\textrm{f(\vec{p})} = \sum_{i=1}^{n} \|(\vec{Up})_i - t_i\|_2^2 = \|\vec{Up} - \vec{t}\|^2 \rightarrow \min
+\] where \(t_i\) are the respective target--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
@@ -846,7 +843,7 @@ 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
-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
 should move.
 
@@ -868,17 +865,22 @@ correct gradient.
 \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
+\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.}
 \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
+grid--vertices randomly inside the x--y--plane. As self-intersecting
+grids get tricky to solve with our implemented newtons--method we avoid
+the generation of such self--intersecting grids for our testcases.
+
+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
 neighbours (the smaller neighbour for \(r < 0\), the larger for
 \(r > 0\)) 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}}
 + \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}
++ \lambda \cdot \textrm{regularization}(\vec{P})
 \] 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
@@ -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
 control--grid defining the whole deformation.
 
-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.
+As regularization-term we add a weighted Laplacian of the deformation
+that has been used before by Aschenbach et
+al.\cite[Section 3.2]{aschenbach2015} on similar models and was shown to
+lead to a more precise fit. The Laplacian \[
+\textrm{regularization}(\vec{P}) = \frac{1}{\sum_i A_i} \sum_{i=1}^n A_i \cdot \left( \sum_{\vec{s_j} \in \mathcal{N}(\vec{s_i})} w_j \cdot \|\Delta \vec{s_j} - \Delta \vec{\overline{s}_j}\|^2 \right)
+\] 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
 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
 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}
 
@@ -1012,6 +1048,9 @@ Approximation}\label{results-of-3d-function-approximation}
 
 HAHA .. als ob -.-
 
+\improvement[inline]{Bibliotheksverzeichnis links anpassen. DOI überschreibt
+Direktlinks des Autors.}
+
 % \backmatter
 \cleardoublepage