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25
arbeit/bibma.bib
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@ -69,7 +69,7 @@
Acmid = {2079770},
Doi = {10.1007/11844297_36},
ISBN = {3-540-38990-3, 978-3-540-38990-3},
ISBN = {978-3-540-38990-3},
Location = {Reykjavik, Iceland},
Numpages = {10},
Url = {http://dx.doi.org/10.1007/11844297_36}
@ -86,13 +86,14 @@
Url = {https://www.researchgate.net/profile/Yaneer_Bar-Yam/publication/225104044_Complex_Engineered_Systems_A_New_Paradigm/links/59107f20a6fdccbfd57eb84d/Complex-Engineered-Systems-A-New-Paradigm.pdf}
}
@Article{anrichterEvol,
@InProceedings{anrichterEvol,
Title = {Evolvability as a Quality Criterion for Linear Deformation Representations in Evolutionary Optimization},
Author = {Richter, Andreas and Achenbach, Jascha and Menzel, Stefan and Botsch, Mario},
Author = {Richter, Andreas and Achenbach, Jascha and enzel, Stefan and Botsch, Mario},
Year = {2016},
Note = {\url{http://graphics.uni-bielefeld.de/publications/cec16.pdf}, \url{https://pub.uni-bielefeld.de/publication/2902698}},
Note = {\url{http://graphics.uni-bielefeld.de/publications/disclaimer.php?dlurl=cec16.pdf}, \url{https://pub.uni-bielefeld.de/publication/2902698}},
Booktitle = {IEEE Congress on Evolutionary Computation},
Pages = {901--910},
Location = {Vancouver, Canada},
Publisher = {IEEE}
}
@ -100,12 +101,12 @@
@InProceedings{richter2015evolvability,
Title = {Evolvability of representations in complex system engineering: a survey},
Author = {Richter, Andreas and Botsch, Mario and Menzel, Stefan},
Booktitle = {Evolutionary Computation (CEC), 2015 IEEE Congress on},
Booktitle = {2015 IEEE Congress on Evolutionary Computation (CEC)},
Year = {2015},
Organization = {IEEE},
Pages = {1327--1335},
Url = {http://www.graphics.uni-bielefeld.de/publications/cec15.pdf}
Url = {http://www.graphics.uni-bielefeld.de/publications/disclaimer.php?dlurl=cec15.pdf}
}
@InBook{Rothlauf2006,
@ -168,8 +169,6 @@
Year = {2012},
Number = {5},
Volume = {27},
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},
@ -180,17 +179,17 @@
pages={485--498},
year={2012},
publisher={Elsevier},
url={https://pdfs.semanticscholar.org/a858/aa68da617ad9d41de021f6807cc422002258.pdf},
note={\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},
journal={European Ph. D. in Computer Vision, Universit{\'e} dAuvergne, Cl{\'e}rmont-Ferrand, France, and Technische Universität München, Germany},
year={2010},
url={http://www.brnt.eu/phd/}
}
@article{aschenbach2015,
@InProceedings{aschenbach2015,
author = {Achenbach, Jascha and Zell, Eduard and Botsch, Mario},
booktitle = {Vision, Modeling \& Visualization},
journal = {Proceedings of Vision, Modeling and Visualization},
@ -199,7 +198,7 @@
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},
note = {\url{http://graphics.uni-bielefeld.de/publications/disclaimer.php?dlurl=vmv15.pdf}},
ISBN = {978-3-905674-95-8},
}
@article{hauke2011comparison,
@ -247,7 +246,7 @@
publisher={IEEE},
url={https://www.researchgate.net/profile/Marc_Schoenauer/publication/223460374_Parameter_Control_in_Evolutionary_Algorithms/links/545766440cf26d5090a9b951.pdf},
}
@article{rechenberg1973evolutionsstrategie,
@book{rechenberg1973evolutionsstrategie,
title={Evolutionsstrategie Optimierung technischer Systeme nach Prinzipien der biologischen Evolution},
author={Rechenberg, Ingo},
year={1973},

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@ -44,22 +44,23 @@
\vspace*{\stretch{4}}
\begin{center}
\hspace{0.99cm} {\huge\bfseries Evaluation of the Performance\\[4mm]
\hspace{0.99cm} of Randomized FFD Control Grids}\\[28mm]
\hspace{0.99cm} {\LARGE Master Thesis}\\[3mm]
\hspace{0.99cm} {\Large {\normalsize at the}\\[4mm]
\hspace{0.99cm} AG Computer Graphics}\\[2mm]
\hspace{0.99cm} at the Faculty of Technology\\
\hspace{0.99cm} of Bielefeld University\\[5mm]
\hspace{0.99cm} {\Large by}\\[5mm]
\hspace{0.99cm} {\LARGE Stefan Dresselhaus}\\[8mm]
\hspace{0.99cm} {\large \today}
\hspace{1.5cm} {\huge\bfseries Evaluation of the Performance\\[-3mm]
\hspace{1.5cm} of Randomized\\[4mm]
\hspace{1.5cm} FFD Control Grids}\\[25mm]
\hspace{1.5cm} {\LARGE Master Thesis}\\
\hspace{1.5cm} {\Large {\normalsize at the}\\
\hspace{1.5cm} AG Computer Graphics}\\[2mm]
\hspace{1.5cm} at the Faculty of Technology\\
\hspace{1.5cm} of Bielefeld University\\[3mm]
\hspace{1.5cm} {\Large by}\\[3mm]
\hspace{1.5cm} {\LARGE Stefan Dresselhaus}\\[5mm]
\hspace{1.5cm} {\large \today}
\end{center}
\vspace*{\stretch{2}}
\begin{center}
\begin{tabular}{lrl}
\hspace{0.99cm} Supervisor:~&Prof.~Dr.~&Mario Botsch\\
\hspace{0.99cm} &Dipl.~Math.~&Andreas~Richter
\hspace{1.5cm} Supervisor:~&Prof.~Dr.~&Mario Botsch\\[-5mm]
\hspace{1.5cm} &Dipl.~Math.~&Andreas~Richter
\end{tabular}
\end{center}
\vspace*{\stretch{.2}}

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@ -49,19 +49,19 @@ etc.), the translation of the problem--domain into a simple parametric
representation (the *genome*) can be challenging.
This translation is often necessary as the target of the optimization may have
too many degrees of freedom. In the example of an aerodynamic simulation of drag
onto an object, those object--designs tend to have a high number of vertices to
adhere to various requirements (visual, practical, physical, etc.). A simpler
representation of the same object in only a few parameters that manipulate the
whole in a sensible matter are desirable, as this often decreases the
computation time significantly.
too many degrees of freedom for a reasonable computation. In the example of an
aerodynamic simulation of drag onto an object, those object--designs tend to
have a high number of vertices to adhere to various requirements (visual,
practical, physical, etc.). A simpler representation of the same object in only
a few parameters that manipulate the whole in a sensible matter are desirable,
as this often decreases the computation time significantly.
Additionally one can exploit the fact, that drag in this case is especially
sensitive to non--smooth surfaces, so that a smooth local manipulation of the
surface as a whole is more advantageous than merely random manipulation of the
vertices.
The quality of such a low-dimensional representation in biological evolution is
The quality of such a low--dimensional representation in biological evolution is
strongly tied to the notion of *evolvability*\cite{wagner1996complex}, as the
parametrization of the problem has serious implications on the convergence speed
and the quality of the solution\cite{Rothlauf2006}.
@ -80,14 +80,14 @@ One example of such a general representation of an object is to generate random
points and represent vertices of an object as distances to these points --- for
example via \acf{RBF}. If one (or the algorithm) would move such a point the
object will get deformed only locally (due to the \ac{RBF}). As this results in
a simple mapping from the parameter-space onto the object one can try out
a simple mapping from the parameter--space onto the object one can try out
different representations of the same object and evaluate which criteria may be
suited to describe this notion of *evolvability*. This is exactly what Richter
et al.\cite{anrichterEvol} have done.
As we transfer the results of Richter et al.\cite{anrichterEvol} from using
\acf{RBF} as a representation to manipulate geometric objects to the use of
\acf{FFD} we will use the same definition for evolvability the original author
\acf{FFD} we will use the same definition for *evolvability* the original author
used, namely *regularity*, *variability*, and *improvement potential*. We
introduce these term in detail in Chapter \ref{sec:intro:rvi}. In the original
publication the author could show a correlation between these
@ -95,7 +95,7 @@ evolvability--criteria with the quality and convergence speed of such
optimization.
We will replicate the same setup on the same objects but use \acf{FFD} instead of
\acf{RBF} to create a local deformation near the control points and evaluate if
\acf{RBF} to create a local deformation near the control--points and evaluate if
the evolution--criteria still work as a predictor for *evolvability* of the
representation given the different deformation scheme, as suspected in
\cite{anrichterEvol}.
@ -106,8 +106,8 @@ take an abstract look at the definition of \ac{FFD} for a one--dimensional line
(in \ref{sec:back:ffdgood}).
Then we establish some background--knowledge of evolutionary algorithms (in
\ref{sec:back:evo}) and why this is useful in our domain (in
\ref{sec:back:evogood}) followed by the definition of the different evolvability
criteria established in \cite{anrichterEvol} (in \ref {sec:intro:rvi}).
\ref{sec:back:evogood}) followed by the definition of the different
evolvability--criteria established in \cite{anrichterEvol} (in \ref {sec:intro:rvi}).
In Chapter \ref{sec:impl} we take a look at our implementation of \ac{FFD} and
the adaptation for 3D--meshes that were used. Next, in Chapter \ref{sec:eval},
@ -132,19 +132,20 @@ chapter \ref{3dffd}.
The main idea of \ac{FFD} is to create a function $s : [0,1[^d \mapsto
\mathbb{R}^d$ that spans a certain part of a vector--space and is only linearly
parametrized by some special control points $p_i$ and an constant
parametrized by some special control--points $p_i$ and an constant
attribution--function $a_i(u)$, so
$$
s(\vec{u}) = \sum_i a_i(\vec{u}) \vec{p_i}
$$
can be thought of a representation of the inside of the convex hull generated by
the control points where each point can be accessed by the right $u \in [0,1[^d$.
the control--points where each position inside can be accessed by the right
$u \in [0,1[^d$.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{img/B-Splines.png}
\end{center}
\caption[Example of B-Splines]{Example of a parametrization of a line with
\caption[Example of B--Splines]{Example of a parametrization of a line with
corresponding deformation to generate a deformed objet}
\label{fig:bspline}
\end{figure}
@ -184,7 +185,7 @@ $$\frac{\partial}{\partial u} N_{i,d,r}(u) = \frac{d}{\tau_{i+d} - \tau_i} N_{i,
For a B--Spline
$$s(u) = \sum_{i} N_{i,d,\tau_i}(u) p_i$$
these derivations yield $\frac{\partial^d}{\partial u} s(u) = 0$.
these derivations yield $\left(\frac{\partial}{\partial u}\right)^d s(u) = 0$.
Another interesting property of these recursive polynomials is that they are
continuous (given $d \ge 1$) as every $p_i$ gets blended in between $\tau_i$ and
@ -193,21 +194,21 @@ in every step of the recursion.
This means that all changes are only a local linear combination between the
control--point $p_i$ to $p_{i+d+1}$ and consequently this yields to the
convex--hull--property of B-Splines --- meaning, that no matter how we choose
convex--hull--property of B--Splines --- meaning, that no matter how we choose
our coefficients, the resulting points all have to lie inside convex--hull of
the control--points.
For a given point $v_i$ we can then calculate the contributions
$n_{i,j}~:=~N_{j,d,\tau}$ of each control point $p_j$ to get the
For a given point $s_i$ we can then calculate the contributions
$u_{i,j}~:=~N_{j,d,\tau}$ of each control point $p_j$ to get the
projection from the control--point--space into the object--space:
$$
v_i = \sum_j n_{i,j} \cdot p_j = \vec{n}_i^{T} \vec{p}
s_i = \sum_j u_{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}
\vec{s} = \vec{U} \vec{p}
$$
where $\vec{N}$ is the $n \times m$ transformation--matrix (later on called
where $\vec{U}$ is the $n \times m$ transformation--matrix (later on called
**deformation matrix**) for $n$ object--space--points and $m$ control--points.
\begin{figure}[ht]
@ -220,7 +221,7 @@ 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
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$.}
@ -228,8 +229,8 @@ start at $0$.}
\end{figure}
Furthermore B--Spline--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$
the first and last $d$ control--points\cite{brunet2010contributions}. Therefore
we later on use the border--points $d+1$ times, such that $\sum_j u_{i,j} p_j = p_i$
for these points.
The locality of the influence of each control--point and the partition of unity
@ -240,8 +241,8 @@ was beautifully pictured by Brunet, which we included here as figure
\label{sec:back:ffdgood}
The usage of \ac{FFD} as a tool for manipulating follows directly from the
properties of the polynomials and the correspondence to the control points.
Having only a few control points gives the user a nicer high--level--interface, as
properties of the polynomials and the correspondence to the control--points.
Having only a few control--points gives the user a nicer high--level--interface, as
she only needs to move these points and the model follows in an intuitive
manner. The deformation is smooth as the underlying polygon is smooth as well
and affects as many vertices of the model as needed. Moreover the changes are
@ -317,7 +318,7 @@ the *phenotypes* make certain behaviour observable (algorithmically through our
*fitness--function*, biologically by the ability to survive and produce
offspring). Any individual in our algorithm thus experience a biologically
motivated life cycle of inheriting genes from the parents, modified by mutations
occurring, performing according to a fitness--metric and generating offspring
occurring, performing according to a fitness--metric, and generating offspring
based on this. Therefore each iteration in the while--loop above is also often
named generation.
@ -346,7 +347,7 @@ The main algorithm just repeats the following steps:
- **Selection** takes a selection--function $s : (I^\lambda \cup I^{\mu + \lambda},\Phi) \mapsto I^\mu$ that
selects from the previously generated $I^\lambda$ children and optionally also
the parents (denoted by the set $Q$ in the algorithm) using the
fitness--function $\Phi$. The result of this operation is the next Population
*fitness--function* $\Phi$. The result of this operation is the next Population
of $\mu$ individuals.
All these functions can (and mostly do) have a lot of hidden parameters that
@ -370,7 +371,7 @@ also take ancestry, distance of genes or groups of individuals into account.
\label{sec:back:evogood}
The main advantage of evolutionary algorithms is the ability to find optima of
general functions just with the help of a given fitness--function. Components
general functions just with the help of a given *fitness--function*. Components
and techniques for evolutionary algorithms are specifically known to
help with different problems arising in the domain of
optimization\cite{weise2012evolutionary}. An overview of the typical problems
@ -383,13 +384,13 @@ are shown in figure \ref{fig:probhard}.
\end{figure}
Most of the advantages stem from the fact that a gradient--based procedure has
only one point of observation from where it evaluates the next steps, whereas an
evolutionary strategy starts with a population of guessed solutions. Because an
evolutionary strategy can be modified according to the problem--domain (i.e. by
the ideas given above) it can also approximate very difficult problems in an
efficient manner and even self--tune parameters depending on the ancestry at
runtime^[Some examples of this are explained in detail in
\cite{eiben1999parameter}].
usually only one point of observation from where it evaluates the next steps,
whereas an evolutionary strategy starts with a population of guessed solutions.
Because an evolutionary strategy can be modified according to the
problem--domain (i.e. by the ideas given above) it can also approximate very
difficult problems in an efficient manner and even self--tune parameters
depending on the ancestry at runtime^[Some examples of this are explained in
detail in \cite{eiben1999parameter}].
If an analytic best solution exists and is easily computable (i.e. because the
error--function is convex) an evolutionary algorithm is not the right choice.
@ -421,23 +422,23 @@ coordinates
$$
\Delta \vec{S} = \vec{U} \cdot \Delta \vec{P}
$$
which is isomorphic to the former due to the linear correlation in the
deformation. One can see in this way, that the way the deformation behaves lies
solely in the entries of $\vec{U}$, which is why the three criteria focus on this.
which is isomorphic to the former due to the linearity of the deformation. One
can see in this way, that the way the deformation behaves lies solely in the
entries of $\vec{U}$, which is why the three criteria focus on this.
### Variability
In \cite{anrichterEvol} *variability* is defined as
$$\mathrm{variability}(\vec{U}) := \frac{\mathrm{rank}(\vec{U})}{n},$$
whereby $\vec{U}$ is the $n \times m$ deformation--Matrix used to map the $m$
control points onto the $n$ vertices.
control--points onto the $n$ vertices.
Given $n = m$, an identical number of control--points and vertices, this
quotient will be $=1$ if all control points are independent of each other and
quotient will be $=1$ if all control--points are independent of each other and
the solution is to trivially move every control--point onto a target--point.
In praxis the value of $V(\vec{U})$ is typically $\ll 1$, because as
there are only few control--points for many vertices, so $m \ll n$.
In praxis the value of $V(\vec{U})$ is typically $\ll 1$, because there are only
few control--points for many vertices, so $m \ll n$.
This criterion should correlate to the degrees of freedom the given
parametrization has. This can be seen from the fact, that
@ -459,7 +460,7 @@ value of the deformation--matrix $\vec{U}$.
As we deform the given Object only based on the parameters as $\vec{p} \mapsto
f(\vec{x} + \vec{U}\vec{p})$ this makes sure that $\|\vec{Up}\| \propto
\|\vec{p}\|$ when $\kappa(\vec{U}) \approx 1$. The inversion of $\kappa(\vec{U})$
is only performed to map the criterion--range to $[0..1]$, whereas $1$ is the
is only performed to map the criterion--range to $[0..1]$, where $1$ is the
optimal value and $0$ is the worst value.
On the one hand this criterion should be characteristic for numeric
@ -470,7 +471,7 @@ to the notion of locality\cite{weise2012evolutionary,thorhauer2014locality}.
### Improvement Potential
In contrast to the general nature of *variability* and *regularity*, which are
agnostic of the fitness--function at hand, the third criterion should reflect a
agnostic of the *fitness--function* at hand, the third criterion should reflect a
notion of the potential for optimization, taking a guess into account.
Most of the times some kind of gradient $g$ is available to suggest a
@ -509,7 +510,7 @@ As we have established in Chapter \ref{sec:back:ffd} we can define an
\Delta_x(u) = \sum_i N_{i,d,\tau_i}(u) \Delta_x c_i
\end{equation}
Note that we only sum up the $\Delta$--displacements in the control points $c_i$ to get
Note that we only sum up the $\Delta$--displacements in the control--points $c_i$ to get
the change in position of the point we are interested in.
In this way every deformed vertex is defined by
@ -539,8 +540,8 @@ and do a gradient--descend to approximate the value of $u$ up to an $\epsilon$ o
For this we employ the Gauss--Newton algorithm\cite{gaussNewton}, which
converges into the least--squares solution. An exact solution of this problem is
impossible most of the times, because we usually have way more vertices
than control points ($\#v~\gg~\#c$).
impossible most of the time, because we usually have way more vertices
than control--points ($\#v~\gg~\#c$).
## Adaption of \ac{FFD} for a 3D--Mesh
\label{3dffd}
@ -550,7 +551,7 @@ chapter. But this time things get a bit more complicated. As we have a
3--dimensional grid we may have a different amount of control--points in each
direction.
Given $n,m,o$ control points in $x,y,z$--direction each Point on the curve is
Given $n,m,o$ control--points in $x,y,z$--direction each Point on the curve is
defined by
$$V(u,v,w) = \sum_i \sum_j \sum_k N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot C_{ijk}.$$
@ -624,13 +625,13 @@ beneficial for a good behaviour of the evolutionary algorithm.
As mentioned in chapter \ref{sec:back:evo}, the way of choosing the
representation to map the general problem (mesh--fitting/optimization in our
case) into a parameter-space is very important for the quality and runtime of
case) into a parameter--space is very important for the quality and runtime of
evolutionary algorithms\cite{Rothlauf2006}.
Because our control--points are arranged in a grid, we can accurately represent
each vertex--point inside the grids volume with proper B--Spline--coefficients
between $[0,1[$ and --- as a consequence --- we have to embed our object into it
(or create constant "dummy"-points outside).
(or create constant "dummy"--points outside).
The great advantage of B--Splines is the local, direct impact of each
control point without having a $1:1$--correlation, and a smooth deformation.
@ -651,20 +652,20 @@ control--points.}
One would normally think, that the more control--points you add, the better the
result will be, but this is not the case for our B--Splines. Given any point
$\vec{p}$ only the $2 \cdot (d-1)$ control--points contribute to the parametrization of
that point^[Normally these are $d-1$ to each side, but at the boundaries the
number gets increased to the inside to meet the required smoothness].
This means, that a high resolution can have many control-points that are not
that point^[Normally these are $d-1$ to each side, but at the boundaries border
points get used multiple times to meet the number of points required].
This means, that a high resolution can have many control--points that are not
contributing to any point on the surface and are thus completely irrelevant to
the solution.
We illustrate this phenomenon in figure \ref{fig:enoughCP}, where the four red
We illustrate this phenomenon in figure \ref{fig:enoughCP}, where the red
central points are not relevant for the parametrization of the circle. This
leads to artefacts in the deformation--matrix $\vec{U}$, as the columns
corresponding to those control--points are $0$.
This leads to useless increased complexity, as the parameters corresponding to
those points will never have any effect, but a naive algorithm will still try to
optimize them yielding numeric artefacts in the best and non--terminating or
This also leads to useless increased complexity, as the parameters corresponding
to those points will never have any effect, but a naive algorithm will still try
to optimize them yielding numeric artefacts in the best and non--terminating or
ill--defined solutions^[One example would be, when parts of an algorithm depend
on the inverse of the minimal right singular value leading to a division by $0$.]
at worst.
@ -674,18 +675,18 @@ but this raises the question why they were introduced in the first place. We
will address this in a special scenario in \ref{sec:res:3d:var}.
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
of the control--points.] to simulate different starting-conditions.
of the control--points.] to simulate different starting--conditions.
# Scenarios for testing evolvability criteria using \ac{FFD}
# Scenarios for testing evolvability--criteria using \ac{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 and merely deform in the height--coordinate to get the
resulting shape.
by Richter et al.\cite{anrichterEvol} The first scenario deforms a plane into a shape
originally defined by Giannelli et al.\cite{giannelli2012thb}, where we setup
control--points in a 2--dimensional manner and 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
@ -717,7 +718,7 @@ including a wireframe--overlay of the vertices.}
\label{fig:1dtarget}
\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 already
correct.
@ -728,10 +729,10 @@ of calculating the squared distances for each corresponding vertex
\end{equation}
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
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.
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{t}$, yielding us
@ -762,16 +763,16 @@ these Models can be seen in figure \ref{fig:3dtarget}.
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 and sum up the error from the
respective source and target.
to the closest vertex of the respective other model and sum up the error from
the source and target.
We therefore define the *fitness--function* to be:
\begin{equation}
\mathrm{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)} -
\vec{t_i}\|_2^2}_{\textrm{target-to-source--distance}}
\vec{t_i}\|_2^2}_{\textrm{target--to--source--distance}}
+ \lambda \cdot \textrm{regularization}(\vec{P})
\label{eq:fit3d}
\end{equation}
@ -787,7 +788,7 @@ $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 add a weighted Laplacian of the deformation that has
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
\begin{equation}
@ -812,7 +813,7 @@ ill--defined grids mentioned in section \ref{sec:impl:grid}.
To compare our results to the ones given by Richter et al.\cite{anrichterEvol},
we also use Spearman's rank correlation coefficient. Opposed to other popular
coefficients, like the Pearson correlation coefficient, which measures a linear
relationship between variables, the Spearmans's coefficient assesses \glqq how
relationship between variables, the Spearman's coefficient assesses \glqq how
well an arbitrary monotonic function can describe the relationship between two
variables, without making any assumptions about the frequency distribution of
the variables\grqq\cite{hauke2011comparison}.
@ -846,18 +847,19 @@ well. We leave the parameters at their sensible defaults as further explained in
\label{sec:proc:1d}
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 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}_{\mathrm{d}}$
gradient that we use as guess for the *improvement potential*. To further test
the *improvement potential* we also consider a distorted gradient
$\vec{g}_{\mathrm{d}}$:
$$
\vec{g}_{\mathrm{d}} = \frac{\mu \vec{g}_{\mathrm{c}} + (1-\mu)\mathbb{1}}{\|\mu \vec{g}_{\mathrm{c}} + (1-\mu) \mathbb{1}\|}
$$
where $\mathbb{1}$ is the vector consisting of $1$ in every dimension,
$\vec{g}_\mathrm{c} = \vec{p^{*}} - \vec{p}$ is the calculated correct gradient,
and $\mu$ is used to blend between $\vec{g}_\mathrm{c}$ and $\mathbb{1}$. As
we always start with a gradient of $p = \mathbb{0}$ this means shortens
$\vec{g}_\mathrm{c} = \vec{p^{*}}$.
we always start with a gradient of $p = \mathbb{0}$ this means we can shorten
the definition of $\vec{g}_\mathrm{c}$ to $\vec{g}_\mathrm{c} = \vec{p^{*}}$.
\begin{figure}[ht]
\begin{center}
@ -870,9 +872,9 @@ random distortion to generate a testcase.}
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 self-intersecting grids get tricky to solve
with our implemented newtons--method we avoid the generation of such
self--intersecting grids for our testcases (see section \ref{3dffd}).
randomly inside the x--y--plane. As self--intersecting grids get tricky to solve
with our implemented newtons--method (see section \ref{3dffd}) we avoid the
generation of such self--intersecting grids for our testcases.
To achieve that we generated a gaussian distributed number with $\mu = 0, \sigma=0.25$
and clamped it to the range $[-0.25,0.25]$. We chose such an $r \in [-0.25,0.25]$
@ -899,11 +901,11 @@ In the case of our 1D--Optimization--problem, we have the luxury of knowing the
analytical solution to the given problem--set. We use this to experimentally
evaluate the quality criteria we introduced before. As an evolutional
optimization is partially a random process, we use the analytical solution as a
stopping-criteria. We measure the convergence speed as number of iterations the
stopping--criteria. We measure the convergence speed as number of iterations the
evolutional algorithm needed to get within $1.05 \times$ of the optimal solution.
We used different regular grids that we manipulated as explained in Section
\ref{sec:proc:1d} with a different number of control points. As our grids have
\ref{sec:proc:1d} with a different number of control--points. As our grids have
to be the product of two integers, we compared a $5 \times 5$--grid with $25$
control--points to a $4 \times 7$ and $7 \times 4$--grid with $28$
control--points. This was done to measure the impact an \glqq improper\grqq \
@ -924,7 +926,7 @@ Note that $7 \times 4$ and $4 \times 7$ have the same number of control--points.
\label{fig:1dvar}
\end{figure}
Variability should characterize the potential for design space exploration and
*Variability* should characterize the potential for design space exploration and
is defined in terms of the normalized rank of the deformation matrix $\vec{U}$:
$V(\vec{U}) := \frac{\textrm{rank}(\vec{U})}{n}$, whereby $n$ is the number of
vertices.
@ -933,27 +935,27 @@ grid), we have merely plotted the errors in the box plot in figure
\ref{fig:1dvar}
It is also noticeable, that although the $7 \times 4$ and $4 \times 7$ grids
have a higher variability, they perform not better than the $5 \times 5$ grid.
have a higher *variability*, they perform not better than the $5 \times 5$ grid.
Also the $7 \times 4$ and $4 \times 7$ grids differ distinctly from each other
with a mean$\pm$sigma of $233.09 \pm 12.32$ for the former and $286.32 \pm 22.36$ for the
latter, although they have the same number of control--points. This is an
indication of an impact a proper or improper grid--setup can have. We do not
draw scientific conclusions from these findings, as more research on non-squared
draw scientific conclusions from these findings, as more research on non--squared
grids seem necessary.
Leaving the issue of the grid--layout aside we focused on grids having the same
number of prototypes in every dimension. For the $5 \times 5$, $7 \times 7$ and
$10 \times 10$ grids we found a *very strong* correlation ($-r_S = 0.94, p = 0$)
between the variability and the evolutionary error.
between the *variability* and the evolutionary error.
### Regularity
\begin{figure}[tbh]
\centering
\includegraphics[width=\textwidth]{img/evolution1d/55_to_1010_steps.png}
\caption[Improvement potential and regularity vs. steps]{\newline
Left: Improvement potential against steps until convergence\newline
Right: Regularity against steps until convergence\newline
\caption[Improvement potential and regularity against iterations]{\newline
Left: *Improvement potential* against number of iterations until convergence\newline
Right: *Regularity* against number of iterations until convergence\newline
Coloured by their grid--resolution, both with a linear fit over the whole
dataset.}
\label{fig:1dreg}
@ -966,15 +968,15 @@ $5 \times 5$ & $7 \times 4$ & $4 \times 7$ & $7 \times 7$ & $10 \times 10$\\
\hline
$0.28$ ($0.0045$) & \textcolor{red}{$0.21$} ($0.0396$) & \textcolor{red}{$0.1$} ($0.3019$) & \textcolor{red}{$0.01$} ($0.9216$) & \textcolor{red}{$0.01$} ($0.9185$)
\end{tabular}
\caption[Correlation 1D Regularity/Steps]{Spearman's correlation (and p-values)
between regularity and convergence speed for the 1D function approximation
\caption[Correlation 1D *regularity* against iterations]{Inverted Spearman's correlation (and p--values)
between *regularity* and number of iterations for the 1D function approximation
problem.
\newline Note: Not significant results are marked in \textcolor{red}{red}.
}
\label{tab:1dreg}
\end{table}
Regularity should correspond to the convergence speed (measured in
*Regularity* should correspond to the convergence speed (measured in
iteration--steps of the evolutionary algorithm), and is computed as inverse
condition number $\kappa(\vec{U})$ of the deformation--matrix.
@ -986,9 +988,9 @@ correlation of $- r_S = -0.72, p = 0$, that is opposed to our expectations.
To explain this discrepancy we took a closer look at what caused these high number
of iterations. In figure \ref{fig:1dreg} we also plotted the
improvement-potential against the steps next to the regularity--plot. Our theory
*improvement potential* against the steps next to the *regularity*--plot. Our theory
is that the *very strong* correlation ($-r_S = -0.82, p=0$) between
improvement--potential and number of iterations hints that the employed
*improvement potential* and number of iterations hints that the employed
algorithm simply takes longer to converge on a better solution (as seen in
figure \ref{fig:1dvar} and \ref{fig:1dimp}) offsetting any gain the
regularity--measurement could achieve.
@ -998,14 +1000,14 @@ regularity--measurement could achieve.
\begin{figure}[ht]
\centering
\includegraphics[width=0.8\textwidth]{img/evolution1d/55_to_1010_improvement-vs-evo-error.png}
\caption[Correlation 1D Improvement vs. Error]{Improvement potential plotted
\caption[Correlation 1D Improvement vs. Error]{*Improvement potential* plotted
against the error yielded by the evolutionary optimization for different
grid--resolutions}
\label{fig:1dimp}
\end{figure}
The improvement potential should correlate to the quality of the
fitting--result. We plotted the results for the tested grid-sizes $5 \times 5$,
The *improvement potential* should correlate to the quality of the
fitting--result. We plotted the results for the tested grid--sizes $5 \times 5$,
$7 \times 7$ and $10 \times 10$ in figure \ref{fig:1dimp}. We tested the
$4 \times 7$ and $7 \times 4$ grids as well, but omitted them from the plot.
@ -1035,7 +1037,7 @@ Initially we set up the correspondences $\vec{c_T(\dots)}$ and $\vec{c_S(\dots)}
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.
*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 $\vec{c_S(\dots)}$.
@ -1063,11 +1065,11 @@ iterations until the regularization--effect wears off.
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.
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 in the exact same random manner between
regular grid and move the control--points in the exact same random manner between
their neighbours as described in section \ref{sec:proc:1d}, 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.].
@ -1083,16 +1085,16 @@ Right: A $4 \times 4 \times 7$ grid that we expect to perform worse.}
As is clearly visible from figure \ref{fig:3dgridres}, 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
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.
control--points.
## Results of 3D Function Approximation
In the 3D--Approximation we tried to evaluate further on the impact of the
grid--layout to the overall criteria. As the target--model has many vertices in
concentrated in the facial area we start from a $4 \times 4 \times 4$ grid and
only increase the number of control points in one dimension, yielding a
only increase the number of control--points in one dimension, yielding a
resolution of $7 \times 4 \times 4$ and $4 \times 4 \times 7$ respectively. We
visualized those two grids in figure \ref{fig:3dgridres}.
@ -1121,10 +1123,10 @@ $4 \times 4 \times \mathrm{X}$ & $\mathrm{X} \times 4 \times 4$ & $\mathrm{Y} \t
\hline
0.89 (0) & 0.9 (0) & 0.91 (0) & 0.94 (0)
\end{tabular}
\caption[Correlation between variability and fitting error for 3D]{Correlation
between variability and fitting error for the 3D fitting scenario.\newline
Displayed are the negated Spearman coefficients with the corresponding p-values
in brackets for three cases of increasing variability ($\mathrm{X} \in [4,5,7],
\caption[Correlation between *variability* and fitting error for 3D]{Correlation
between *variability* and fitting error for the 3D fitting scenario.\newline
Displayed are the negated Spearman coefficients with the corresponding p--values
in brackets for three cases of increasing *variability* ($\mathrm{X} \in [4,5,7],
\mathrm{Y} \in [4,5,6]$).
\newline Note: Not significant results are marked in \textcolor{red}{red}.}
\label{tab:3dvar}
@ -1143,37 +1145,37 @@ Interestingly both variants end up closer in terms of fitting error than we
anticipated, which shows that the evolutionary algorithm we employed is capable
of correcting a purposefully created \glqq bad\grqq \ grid. Also this confirms,
that in our cases the number of control--points is more important for quality
than their placement, which is captured by the variability via the rank of the
than their placement, which is captured by the *variability* via the rank of the
deformation--matrix.
Overall the correlation between *variability* and fitness--error were
*significant* and showed a *very strong* correlation in all our tests.
The detailed correlation--coefficients are given in table \ref{tab:3dvar}
alongside their p--values.
As introduces in section \ref{sec:impl:grid} and visualized in figure
\ref{fig:enoughCP}, we know, that not all control--points have to necessarily
contribute to the parametrization of our 3D--model. Because we are starting from
a sphere, some control--points are too far away from the surface to contribute
to the deformation at all.
One can already see in 2D in figure \ref{fig:enoughCP}, that this effect
starts with a regular $9 \times 9$ grid on a perfect circle. To make sure we
observe this, we evaluated the *variability* for 100 randomly moved $10 \times 10 \times 10$
grids on the sphere we start out with.
\begin{figure}[hbt]
\centering
\includegraphics[width=0.8\textwidth]{img/evolution3d/variability2_boxplot.png}
\caption[Histogram of ranks of high--resolution deformation--matrices]{
Histogram of ranks of various $10 \times 10 \times 10$ grids with $1000$
control--points each showing in this case how many control points are actually
control--points each showing in this case how many control--points are actually
used in the calculations.
}
\label{fig:histrank3d}
\end{figure}
Overall the correlation between variability and fitness--error were
*significant* and showed a *very strong* correlation in all our tests.
The detailed correlation--coefficients are given in table \ref{tab:3dvar}
alongside their p--values.
As introduces in section \ref{sec:impl:grid} and visualized in figure
\ref{fig:enoughCP}, we know, that not all control points have to necessarily
contribute to the parametrization of our 3D--model. Because we are starting from
a sphere, some control-points are too far away from the surface to contribute
to the deformation at all.
One can already see in 2D in figure \ref{fig:enoughCP}, that this effect
starts with a regular $9 \times 9$ grid on a perfect circle. To make sure we
observe this, we evaluated the variability for 100 randomly moved $10 \times 10 \times 10$
grids on the sphere we start out with.
As the variability is defined by $\frac{\mathrm{rank}(\vec{U})}{n}$ we can
As the *variability* is defined by $\frac{\mathrm{rank}(\vec{U})}{n}$ we can
easily recover the rank of the deformation--matrix $\vec{U}$. The results are
shown in the histogram in figure \ref{fig:histrank3d}. Especially in the centre
of the sphere and in the corners of our grid we effectively loose
@ -1184,7 +1186,7 @@ to use and one should expect a loss in quality evident by a higher
reconstruction--error opposed to a grid where they are used. Sadly we could not
run a in--depth test on this due to computational limitations.
Nevertheless this hints at the notion, that variability is a good measure for
Nevertheless this hints at the notion, that *variability* is a good measure for
the overall quality of a fit.
### Regularity
@ -1212,19 +1214,19 @@ $4 \times 4 \times 4$ & $4 \times 4 \times 5$ & $4 \times 4 \times 7$ & $4 \time
\cline{2-4}
\multicolumn{3}{c}{} & all: 0.15 (0) \T
\end{tabular}
\caption[Correlation between regularity and iterations for 3D]{Correlation
between regularity and number of iterations for the 3D fitting scenario.
\caption[Correlation between *regularity* and iterations for 3D]{Correlation
between *regularity* and number of iterations for the 3D fitting scenario.
Displayed are the negated Spearman coefficients with the corresponding p--values
in brackets for various given grids ($\mathrm{X} \in [4,5,7], \mathrm{Y} \in [4,5,6]$).
\newline Note: Not significant results are marked in \textcolor{red}{red}.}
\label{tab:3dreg}
\end{table}
Opposed to the predictions of variability our test on regularity gave a mixed
Opposed to the predictions of *variability* our test on *regularity* gave a mixed
result --- similar to the 1D--case.
In roughly half of the scenarios we have a *significant*, but *weak* to *moderate*
correlation between regularity and number of iterations. On the other hand in
correlation between *regularity* and number of iterations. On the other hand in
the scenarios where we increased the number of control--points, namely $125$ for
the $5 \times 5 \times 5$ grid and $216$ for the $6 \times 6 \times 6$ grid we found
a *significant*, but *weak* **anti**--correlation when taking all three tests into
@ -1233,14 +1235,14 @@ findings/trends for the sets with $64$, $80$, and $112$ control--points
(first two rows of table \ref{tab:3dreg}).
Taking all results together we only find a *very weak*, but *significant* link
between regularity and the number of iterations needed for the algorithm to
between *regularity* and the number of iterations needed for the algorithm to
converge.
\begin{figure}[!htb]
\centering
\includegraphics[width=\textwidth]{img/evolution3d/regularity_montage.png}
\caption[Regularity for different 3D--grids]{
Plots of regularity against number of iterations for various scenarios together
Plots of *regularity* against number of iterations for various scenarios together
with a linear fit to indicate trends.}
\label{fig:resreg3d}
\end{figure}
@ -1250,10 +1252,10 @@ the number of control--points helps the convergence--speeds. The
regularity--criterion first behaves as we would like to, but then switches to
behave exactly opposite to our expectations, as can be seen in the first three
plots. While the number of control--points increases from red to green to blue
and the number of iterations decreases, the regularity seems to increase at
and the number of iterations decreases, the *regularity* seems to increase at
first, but then decreases again on higher grid--resolutions.
This can be an artefact of the definition of regularity, as it is defined by the
This can be an artefact of the definition of *regularity*, as it is defined by the
inverse condition--number of the deformation--matrix $\vec{U}$, being the
fraction $\frac{\sigma_{\mathrm{min}}}{\sigma_{\mathrm{max}}}$ between the
least and greatest right singular value.
@ -1264,9 +1266,9 @@ and so a small minimal right singular value occurring on higher
grid--resolutions seems likely the problem.
Adding to this we also noted, that in the case of the $10 \times 10 \times
10$--grid the regularity was always $0$, as a non--contributing control-point
10$--grid the *regularity* was always $0$, as a non--contributing control--point
yields a $0$--column in the deformation--matrix, thus letting
$\sigma_\mathrm{min} = 0$. A better definition for regularity (i.e. using the
$\sigma_\mathrm{min} = 0$. A better definition for *regularity* (i.e. using the
smallest non--zero right singular value) could solve this particular issue, but
not fix the trend we noticed above.
@ -1295,8 +1297,8 @@ $4 \times 4 \times 4$ & $4 \times 4 \times 5$ & $4 \times 4 \times 7$ & $4 \time
\cline{2-4}
\multicolumn{3}{c}{} & all: 0.95 (0) \T
\end{tabular}
\caption[Correlation between improvement--potential and fitting--error for 3D]{Correlation
between improvement--potential and fitting--error for the 3D fitting scenario.
\caption[Correlation between *improvement potential* and fitting--error for 3D]{Correlation
between *improvement potential* and fitting--error for the 3D fitting scenario.
Displayed are the negated Spearman coefficients with the corresponding p--values
in brackets for various given grids ($\mathrm{X} \in [4,5,7], \mathrm{Y} \in [4,5,6]$).
\newline Note: Not significant results are marked in \textcolor{red}{red}.}
@ -1314,20 +1316,20 @@ quality of such gradients anyway.
\centering
\includegraphics[width=\textwidth]{img/evolution3d/improvement_montage.png}
\caption[Improvement potential for different 3D--grids]{
Plots of improvement potential against error given by our fitness--function
Plots of *improvement potential* against error given by our *fitness--function*
after convergence together with a linear fit of each of the plotted data to
indicate trends.}
\label{fig:resimp3d}
\end{figure}
We plotted our findings on the improvement potential in a similar way as we did
before with the regularity. In figure \ref{fig:resimp3d} one can clearly see the
We plotted our findings on the *improvement potential* in a similar way as we did
before with the *regularity*. In figure \ref{fig:resimp3d} one can clearly see the
correlation and the spread within each setup and the behaviour when we increase
the number of control--points.
Along with this we also give the Spearman--coefficients along with their
p--values in table \ref{tab:3dimp}. Within one scenario we only find a *weak* to
*moderate* correlation between the improvement potential and the fitting error,
*moderate* correlation between the *improvement potential* and the fitting error,
but all findings (except for $7 \times 4 \times 4$ and $6 \times 6 \times 6$)
are significant.
@ -1335,14 +1337,14 @@ If we take multiple datasets into account the correlation is *very strong* and
*significant*, which is good, as this functions as a litmus--test, because the
quality is naturally tied to the number of control--points.
All in all the improvement potential seems to be a good and sensible measure of
All in all the *improvement potential* seems to be a good and sensible measure of
quality, even given gradients of varying quality.
Lastly, a small note on the behaviour of improvement potential and convergence
Lastly, a small note on the behaviour of *improvement potential* and convergence
speed, as we used this in the 1D case to argue, why the *regularity* defied our
expectations. As a contrast we wanted to show, that improvement potential cannot
expectations. As a contrast we wanted to show, that *improvement potential* cannot
serve for good predictions of the convergence speed. In figure
\ref{fig:imp1d3d} we show improvement potential against number of iterations
\ref{fig:imp1d3d} we show *improvement potential* against number of iterations
for both scenarios. As one can see, in the 1D scenario we have a *strong*
and *significant* correlation (with $-r_S = -0.72$, $p = 0$), whereas in the 3D
scenario we have the opposite *significant* and *strong* effect (with
@ -1352,11 +1354,11 @@ scenario and are not suited for generalization.
\begin{figure}[hbt]
\centering
\includegraphics[width=\textwidth]{img/imp1d3d.png}
\caption[Improvement potential and convergence speed for 1D and 3D--scenarios]{
\caption[Improvement potential and convergence speed\newline for 1D and 3D--scenarios]{
\newline
Left: Improvement potential against convergence speed for the
Left: *Improvement potential* against convergence speed for the
1D--scenario\newline
Right: Improvement potential against convergence speed for the 3D--scnario
Right: *Improvement potential* against convergence speed for the 3D--scnario
}
\label{fig:imp1d3d}
\end{figure}
@ -1364,23 +1366,23 @@ Right: Improvement potential against convergence speed for the 3D--scnario
# Discussion and outlook
\label{sec:dis}
In this thesis we took a look at the different criteria for evolvability as
In this thesis we took a look at the different criteria for *evolvability* as
introduced by Richter et al.\cite{anrichterEvol}, namely *variability*,
*regularity* and *improvement potential* under different setup--conditions.
Where Richter et al. used \acf{RBF}, we employed \acf{FFD} to set up a
low--complexity parametrization of a more complex vertex--mesh.
In our findings we could show in the 1D--scenario, that there were statistically
significant very strong correlations between *variability and fitting error*
($0.94$) and *improvement--potential and fitting error* ($1.0$) with
*significant* *very strong* correlations between *variability and fitting error*
($0.94$) and *improvement potential and fitting error* ($1.0$) with
comparable results than Richter et al. (with $0.31$ to $0.88$
for the former and $0.75$ to $0.99$ for the latter), whereas we found
only weak correlations for *regularity and convergence--speed* ($0.28$)
only *weak* correlations for *regularity and convergence--speed* ($0.28$)
opposed to Richter et al. with $0.39$ to $0.91$.^[We only took statistically
*significant* results into consideration when compiling these numbers. Details
are given in the respective chapters.]
For the 3D--scenario our results show a very strong, significant correlation
For the 3D--scenario our results show a *very strong*, *significant* correlation
between *variability and fitting error* with $0.89$ to $0.94$, which are pretty
much in line with the findings of Richter et al. ($0.65$ to $0.95$). The
correlation between *improvement potential and fitting error* behave similar,
@ -1410,10 +1412,7 @@ in \cite{anrichterEvol}, whereas we merely used an indirect \ac{FFD}--approach.
As direct manipulations tend to perform better than indirect manipulations, the
usage of \acf{DM--FFD} could also work better with the criteria we examined.
This can also solve the problem of bad singular values for the *regularity* as
the incorporation of the parametrization of the points on the surface, which are
the essential part of a direct--manipulation, could cancel out a bad
the incorporation of the parametrization of the points on the surface --- which
are the essential part of a direct--manipulation --- could cancel out a bad
control--grid as the bad control--points are never or negligibly used to
parametrize those surface--points.
\improvement[inline]{Bibliotheksverzeichnis links anpassen. DOI überschreibt
Direktlinks des Autors.}

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@ -197,20 +197,20 @@ the translation of the problem--domain into a simple parametric
representation (the \emph{genome}) can be challenging.
This translation is often necessary as the target of the optimization
may have too many degrees of freedom. In the example of an aerodynamic
simulation of drag onto an object, those object--designs tend to have a
high number of vertices to adhere to various requirements (visual,
practical, physical, etc.). A simpler representation of the same object
in only a few parameters that manipulate the whole in a sensible matter
are desirable, as this often decreases the computation time
significantly.
may have too many degrees of freedom for a reasonable computation. In
the example of an aerodynamic simulation of drag onto an object, those
object--designs tend to have a high number of vertices to adhere to
various requirements (visual, practical, physical, etc.). A simpler
representation of the same object in only a few parameters that
manipulate the whole in a sensible matter are desirable, as this often
decreases the computation time significantly.
Additionally one can exploit the fact, that drag in this case is
especially sensitive to non--smooth surfaces, so that a smooth local
manipulation of the surface as a whole is more advantageous than merely
random manipulation of the vertices.
The quality of such a low-dimensional representation in biological
The quality of such a low--dimensional representation in biological
evolution is strongly tied to the notion of
\emph{evolvability}\cite{wagner1996complex}, as the parametrization of
the problem has serious implications on the convergence speed and the
@ -230,7 +230,7 @@ One example of such a general representation of an object is to generate
random points and represent vertices of an object as distances to these
points --- for example via \acf{RBF}. If one (or the algorithm) would
move such a point the object will get deformed only locally (due to the
\ac{RBF}). As this results in a simple mapping from the parameter-space
\ac{RBF}). As this results in a simple mapping from the parameter--space
onto the object one can try out different representations of the same
object and evaluate which criteria may be suited to describe this notion
of \emph{evolvability}. This is exactly what Richter et
@ -238,18 +238,19 @@ al.\cite{anrichterEvol} have done.
As we transfer the results of Richter et al.\cite{anrichterEvol} from
using \acf{RBF} as a representation to manipulate geometric objects to
the use of \acf{FFD} we will use the same definition for evolvability
the original author used, namely \emph{regularity}, \emph{variability},
and \emph{improvement potential}. We introduce these term in detail in
Chapter \ref{sec:intro:rvi}. In the original publication the author
could show a correlation between these evolvability--criteria with the
quality and convergence speed of such optimization.
the use of \acf{FFD} we will use the same definition for
\emph{evolvability} the original author used, namely \emph{regularity},
\emph{variability}, and \emph{improvement potential}. We introduce these
term in detail in Chapter \ref{sec:intro:rvi}. In the original
publication the author could show a correlation between these
evolvability--criteria with the quality and convergence speed of such
optimization.
We will replicate the same setup on the same objects but use \acf{FFD}
instead of \acf{RBF} to create a local deformation near the control
points and evaluate if the evolution--criteria still work as a predictor
for \emph{evolvability} of the representation given the different
deformation scheme, as suspected in \cite{anrichterEvol}.
instead of \acf{RBF} to create a local deformation near the
control--points and evaluate if the evolution--criteria still work as a
predictor for \emph{evolvability} of the representation given the
different deformation scheme, as suspected in \cite{anrichterEvol}.
First we introduce different topics in isolation in Chapter
\ref{sec:back}. We take an abstract look at the definition of \ac{FFD}
@ -258,7 +259,7 @@ is a sensible deformation function (in \ref{sec:back:ffdgood}). Then we
establish some background--knowledge of evolutionary algorithms (in
\ref{sec:back:evo}) and why this is useful in our domain (in
\ref{sec:back:evogood}) followed by the definition of the different
evolvability criteria established in \cite{anrichterEvol} (in
evolvability--criteria established in \cite{anrichterEvol} (in
\ref {sec:intro:rvi}).
In Chapter \ref{sec:impl} we take a look at our implementation of
@ -285,18 +286,19 @@ from \cite{spitzmuller1996bezier} here and go into the extension to the
The main idea of \ac{FFD} is to create a function
\(s : [0,1[^d \mapsto \mathbb{R}^d\) that spans a certain part of a
vector--space and is only linearly parametrized by some special control
points \(p_i\) and an constant attribution--function \(a_i(u)\), so \[
vector--space and is only linearly parametrized by some special
control--points \(p_i\) and an constant attribution--function
\(a_i(u)\), so \[
s(\vec{u}) = \sum_i a_i(\vec{u}) \vec{p_i}
\] can be thought of a representation of the inside of the convex hull
generated by the control points where each point can be accessed by the
right \(u \in [0,1[^d\).
generated by the control--points where each position inside can be
accessed by the right \(u \in [0,1[^d\).
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{img/B-Splines.png}
\end{center}
\caption[Example of B-Splines]{Example of a parametrization of a line with
\caption[Example of B--Splines]{Example of a parametrization of a line with
corresponding deformation to generate a deformed objet}
\label{fig:bspline}
\end{figure}
@ -338,7 +340,8 @@ We can even derive this equation straightforward for an arbitrary
\[\frac{\partial}{\partial u} N_{i,d,r}(u) = \frac{d}{\tau_{i+d} - \tau_i} N_{i,d-1,\tau}(u) - \frac{d}{\tau_{i+d+1} - \tau_{i+1}} N_{i+1,d-1,\tau}(u)\]
For a B--Spline \[s(u) = \sum_{i} N_{i,d,\tau_i}(u) p_i\] these
derivations yield \(\frac{\partial^d}{\partial u} s(u) = 0\).
derivations yield
\(\left(\frac{\partial}{\partial u}\right)^d s(u) = 0\).
Another interesting property of these recursive polynomials is that they
are continuous (given \(d \ge 1\)) as every \(p_i\) gets blended in
@ -348,17 +351,17 @@ step of the recursion.
This means that all changes are only a local linear combination between
the control--point \(p_i\) to \(p_{i+d+1}\) and consequently this yields
to the convex--hull--property of B-Splines --- meaning, that no matter
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.
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
For a given point \(s_i\) we can then calculate the contributions
\(u_{i,j}~:=~N_{j,d,\tau}\) of each control point \(p_j\) to get the
projection from the control--point--space into the object--space: \[
v_i = \sum_j n_{i,j} \cdot p_j = \vec{n}_i^{T} \vec{p}
s_i = \sum_j u_{i,j} \cdot p_j = \vec{n}_i^{T} \vec{p}
\] or written for all points at the same time: \[
\vec{v} = \vec{N} \vec{p}
\] where \(\vec{N}\) is the \(n \times m\) transformation--matrix (later
\vec{s} = \vec{U} \vec{p}
\] where \(\vec{U}\) is the \(n \times m\) transformation--matrix (later
on called \textbf{deformation matrix}) for \(n\) object--space--points
and \(m\) control--points.
@ -372,7 +375,7 @@ 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
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 $0$.}
@ -381,8 +384,8 @@ start at $0$.}
Furthermore B--Spline--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 \(\sum_j n_{i,j} p_j = p_i\)
control--points\cite{brunet2010contributions}. Therefore we later on use
the border--points \(d+1\) times, such that \(\sum_j u_{i,j} p_j = p_i\)
for these points.
The locality of the influence of each control--point and the partition
@ -395,13 +398,13 @@ function?}{Why is a good deformation function?}}\label{why-is-a-good-deformatio
\label{sec:back:ffdgood}
The usage of \ac{FFD} as a tool for manipulating follows directly from
the properties of the polynomials and the correspondence to the control
points. Having only a few control points gives the user a nicer
high--level--interface, as she only needs to move these points and the
model follows in an intuitive manner. The deformation is smooth as the
underlying polygon is smooth as well and affects as many vertices of the
model as needed. Moreover the changes are always local so one risks not
any change that a user cannot immediately see.
the properties of the polynomials and the correspondence to the
control--points. Having only a few control--points gives the user a
nicer high--level--interface, as she only needs to move these points and
the model follows in an intuitive manner. The deformation is smooth as
the underlying polygon is smooth as well and affects as many vertices of
the model as needed. Moreover the changes are always local so one risks
not any change that a user cannot immediately see.
But there are also disadvantages of this approach. The user loses the
ability to directly influence vertices and even seemingly simple tasks
@ -479,7 +482,7 @@ through our \emph{fitness--function}, biologically by the ability to
survive and produce offspring). Any individual in our algorithm thus
experience a biologically motivated life cycle of inheriting genes from
the parents, modified by mutations occurring, performing according to a
fitness--metric and generating offspring based on this. Therefore each
fitness--metric, and generating offspring based on this. Therefore each
iteration in the while--loop above is also often named generation.
One should note that there is a subtle difference between
@ -517,8 +520,8 @@ The main algorithm just repeats the following steps:
\(s : (I^\lambda \cup I^{\mu + \lambda},\Phi) \mapsto I^\mu\) that
selects from the previously generated \(I^\lambda\) children and
optionally also the parents (denoted by the set \(Q\) in the
algorithm) using the fitness--function \(\Phi\). The result of this
operation is the next Population of \(\mu\) individuals.
algorithm) using the \emph{fitness--function} \(\Phi\). The result of
this operation is the next Population of \(\mu\) individuals.
\end{itemize}
All these functions can (and mostly do) have a lot of hidden parameters
@ -547,10 +550,10 @@ algorithms}\label{advantages-of-evolutionary-algorithms}
The main advantage of evolutionary algorithms is the ability to find
optima of general functions just with the help of a given
fitness--function. Components and techniques for evolutionary algorithms
are specifically known to help with different problems arising in the
domain of optimization\cite{weise2012evolutionary}. An overview of the
typical problems are shown in figure \ref{fig:probhard}.
\emph{fitness--function}. Components and techniques for evolutionary
algorithms are specifically known to help with different problems
arising in the domain of optimization\cite{weise2012evolutionary}. An
overview of the typical problems are shown in figure \ref{fig:probhard}.
\begin{figure}[!ht]
\includegraphics[width=\textwidth]{img/weise_fig3.png}
@ -559,13 +562,14 @@ typical problems are shown in figure \ref{fig:probhard}.
\end{figure}
Most of the advantages stem from the fact that a gradient--based
procedure has only one point of observation from where it evaluates the
next steps, whereas an evolutionary strategy starts with a population of
guessed solutions. Because an evolutionary strategy can be modified
according to the problem--domain (i.e.~by the ideas given above) it can
also approximate very difficult problems in an efficient manner and even
self--tune parameters depending on the ancestry at runtime\footnote{Some
examples of this are explained in detail in \cite{eiben1999parameter}}.
procedure has usually only one point of observation from where it
evaluates the next steps, whereas an evolutionary strategy starts with a
population of guessed solutions. Because an evolutionary strategy can be
modified according to the problem--domain (i.e.~by the ideas given
above) it can also approximate very difficult problems in an efficient
manner and even self--tune parameters depending on the ancestry at
runtime\footnote{Some examples of this are explained in detail in
\cite{eiben1999parameter}}.
If an analytic best solution exists and is easily computable
(i.e.~because the error--function is convex) an evolutionary algorithm
@ -599,8 +603,8 @@ deformation.
We can also think of the deformation in terms of differences from the
original coordinates \[
\Delta \vec{S} = \vec{U} \cdot \Delta \vec{P}
\] which is isomorphic to the former due to the linear correlation in
the deformation. One can see in this way, that the way the deformation
\] which is isomorphic to the former due to the linearity of the
deformation. One can see in this way, that the way the deformation
behaves lies solely in the entries of \(\vec{U}\), which is why the
three criteria focus on this.
@ -609,14 +613,14 @@ three criteria focus on this.
In \cite{anrichterEvol} \emph{variability} is defined as
\[\mathrm{variability}(\vec{U}) := \frac{\mathrm{rank}(\vec{U})}{n},\]
whereby \(\vec{U}\) is the \(n \times m\) deformation--Matrix used to
map the \(m\) control points onto the \(n\) vertices.
map the \(m\) control--points onto the \(n\) vertices.
Given \(n = m\), an identical number of control--points and vertices,
this quotient will be \(=1\) if all control points are independent of
this quotient will be \(=1\) if all control--points are independent of
each other and the solution is to trivially move every control--point
onto a target--point.
In praxis the value of \(V(\vec{U})\) is typically \(\ll 1\), because as
In praxis the value of \(V(\vec{U})\) is typically \(\ll 1\), because
there are only few control--points for many vertices, so \(m \ll n\).
This criterion should correlate to the degrees of freedom the given
@ -641,7 +645,7 @@ As we deform the given Object only based on the parameters as
\(\vec{p} \mapsto f(\vec{x} + \vec{U}\vec{p})\) this makes sure that
\(\|\vec{Up}\| \propto \|\vec{p}\|\) when \(\kappa(\vec{U}) \approx 1\).
The inversion of \(\kappa(\vec{U})\) is only performed to map the
criterion--range to \([0..1]\), whereas \(1\) is the optimal value and
criterion--range to \([0..1]\), where \(1\) is the optimal value and
\(0\) is the worst value.
On the one hand this criterion should be characteristic for numeric
@ -653,8 +657,8 @@ locality\cite{weise2012evolutionary,thorhauer2014locality}.
\subsection{Improvement Potential}\label{improvement-potential}
In contrast to the general nature of \emph{variability} and
\emph{regularity}, which are agnostic of the fitness--function at hand,
the third criterion should reflect a notion of the potential for
\emph{regularity}, which are agnostic of the \emph{fitness--function} at
hand, the third criterion should reflect a notion of the potential for
optimization, taking a guess into account.
Most of the times some kind of gradient \(g\) is available to suggest a
@ -698,9 +702,9 @@ As we have established in Chapter \ref{sec:back:ffd} we can define an
\Delta_x(u) = \sum_i N_{i,d,\tau_i}(u) \Delta_x c_i
\end{equation}
Note that we only sum up the \(\Delta\)--displacements in the control
points \(c_i\) to get the change in position of the point we are
interested in.
Note that we only sum up the \(\Delta\)--displacements in the
control--points \(c_i\) to get the change in position of the point we
are interested in.
In this way every deformed vertex is defined by \[
\textrm{Deform}(v_x) = v_x + \Delta_x(u)
@ -722,8 +726,8 @@ v_x \overset{!}{=} \sum_i N_{i,d,\tau_i}(u) c_i
For this we employ the Gauss--Newton algorithm\cite{gaussNewton}, which
converges into the least--squares solution. An exact solution of this
problem is impossible most of the times, because we usually have way
more vertices than control points (\(\#v~\gg~\#c\)).
problem is impossible most of the time, because we usually have way more
vertices than control--points (\(\#v~\gg~\#c\)).
\section{\texorpdfstring{Adaption of \ac{FFD} for a
3D--Mesh}{Adaption of for a 3D--Mesh}}\label{adaption-of-for-a-3dmesh}
@ -735,8 +739,8 @@ last chapter. But this time things get a bit more complicated. As we
have a 3--dimensional grid we may have a different amount of
control--points in each direction.
Given \(n,m,o\) control points in \(x,y,z\)--direction each Point on the
curve is defined by
Given \(n,m,o\) control--points in \(x,y,z\)--direction each Point on
the curve is defined by
\[V(u,v,w) = \sum_i \sum_j \sum_k N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot C_{ijk}.\]
In this case we have three different B--Splines (one for each dimension)
@ -814,14 +818,14 @@ behaviour of the evolutionary algorithm.
As mentioned in chapter \ref{sec:back:evo}, the way of choosing the
representation to map the general problem (mesh--fitting/optimization in
our case) into a parameter-space is very important for the quality and
our case) into a parameter--space is very important for the quality and
runtime of evolutionary algorithms\cite{Rothlauf2006}.
Because our control--points are arranged in a grid, we can accurately
represent each vertex--point inside the grids volume with proper
B--Spline--coefficients between \([0,1[\) and --- as a consequence ---
we have to embed our object into it (or create constant ``dummy''-points
outside).
we have to embed our object into it (or create constant
``dummy''--points outside).
The great advantage of B--Splines is the local, direct impact of each
control point without having a \(1:1\)--correlation, and a smooth
@ -844,18 +848,18 @@ One would normally think, that the more control--points you add, the
better the result will be, but this is not the case for our B--Splines.
Given any point \(\vec{p}\) only the \(2 \cdot (d-1)\) control--points
contribute to the parametrization of that point\footnote{Normally these
are \(d-1\) to each side, but at the boundaries the number gets
increased to the inside to meet the required smoothness}. This means,
that a high resolution can have many control-points that are not
are \(d-1\) to each side, but at the boundaries border points get used
multiple times to meet the number of points required}. This means,
that a high resolution can have many control--points that are not
contributing to any point on the surface and are thus completely
irrelevant to the solution.
We illustrate this phenomenon in figure \ref{fig:enoughCP}, where the
four red central points are not relevant for the parametrization of the
red central points are not relevant for the parametrization of the
circle. This leads to artefacts in the deformation--matrix \(\vec{U}\),
as the columns corresponding to those control--points are \(0\).
This leads to useless increased complexity, as the parameters
This also leads to useless increased complexity, as the parameters
corresponding to those points will never have any effect, but a naive
algorithm will still try to optimize them yielding numeric artefacts in
the best and non--terminating or ill--defined solutions\footnote{One
@ -869,22 +873,23 @@ in the first place. We will address this in a special scenario in
\ref{sec:res:3d:var}.
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.
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.
\chapter{\texorpdfstring{Scenarios for testing evolvability criteria
\chapter{\texorpdfstring{Scenarios for testing evolvability--criteria
using
\ac{FFD}}{Scenarios for testing evolvability criteria using }}\label{scenarios-for-testing-evolvability-criteria-using}
\ac{FFD}}{Scenarios for testing evolvability--criteria using }}\label{scenarios-for-testing-evolvabilitycriteria-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 and merely deform in the
height--coordinate to get the resulting shape.
also used by Richter et al.\cite{anrichterEvol} The first scenario
deforms a plane into a shape originally defined by Giannelli et
al.\cite{giannelli2012thb}, where we setup control--points in a
2--dimensional manner and 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,
@ -922,7 +927,7 @@ including a wireframe--overlay of the vertices.}
\label{fig:1dtarget}
\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
already correct.
@ -936,11 +941,11 @@ corresponding vertex
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
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
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
@ -975,16 +980,16 @@ Both of these Models can be seen in figure \ref{fig:3dtarget}.
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 and
sum up the error from the respective source and target.
one vertex is the distance to the closest vertex of the respective other
model and sum up the error from the source and target.
We therefore define the \emph{fitness--function} to be:
\begin{equation}
\mathrm{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)} -
\vec{t_i}\|_2^2}_{\textrm{target-to-source--distance}}
\vec{t_i}\|_2^2}_{\textrm{target--to--source--distance}}
+ \lambda \cdot \textrm{regularization}(\vec{P})
\label{eq:fit3d}
\end{equation}
@ -1002,7 +1007,7 @@ 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 add a weighted Laplacian of the deformation
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
@ -1034,7 +1039,7 @@ To compare our results to the ones given by Richter et
al.\cite{anrichterEvol}, we also use Spearman's rank correlation
coefficient. Opposed to other popular coefficients, like the Pearson
correlation coefficient, which measures a linear relationship between
variables, the Spearmans's coefficient assesses \glqq how well an
variables, the Spearman's coefficient assesses \glqq how well an
arbitrary monotonic function can describe the relationship between two
variables, without making any assumptions about the frequency
distribution of the variables\grqq\cite{hauke2011comparison}.
@ -1072,18 +1077,19 @@ Approximation}\label{procedure-1d-function-approximation}
\label{sec:proc:1d}
For our setup we first compute the coefficients of the
deformation--matrix and use then the formulas for \emph{variability} and
deformation--matrix and use 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 \(\vec{g}_{\mathrm{d}}\) \[
use as guess for the \emph{improvement potential}. To further test the
\emph{improvement potential} we also consider a distorted gradient
\(\vec{g}_{\mathrm{d}}\): \[
\vec{g}_{\mathrm{d}} = \frac{\mu \vec{g}_{\mathrm{c}} + (1-\mu)\mathbb{1}}{\|\mu \vec{g}_{\mathrm{c}} + (1-\mu) \mathbb{1}\|}
\] where \(\mathbb{1}\) is the vector consisting of \(1\) in every
dimension, \(\vec{g}_\mathrm{c} = \vec{p^{*}} - \vec{p}\) is the
calculated correct gradient, and \(\mu\) is used to blend between
\(\vec{g}_\mathrm{c}\) and \(\mathbb{1}\). As we always start with a
gradient of \(p = \mathbb{0}\) this means shortens
\(\vec{g}_\mathrm{c} = \vec{p^{*}}\).
gradient of \(p = \mathbb{0}\) this means we can shorten the definition
of \(\vec{g}_\mathrm{c}\) to \(\vec{g}_\mathrm{c} = \vec{p^{*}}\).
\begin{figure}[ht]
\begin{center}
@ -1096,10 +1102,10 @@ random distortion to generate a testcase.}
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 self-intersecting
grids get tricky to solve with our implemented newtons--method we avoid
the generation of such self--intersecting grids for our testcases (see
section \ref{3dffd}).
grid--vertices randomly inside the x--y--plane. As self--intersecting
grids get tricky to solve with our implemented newtons--method (see
section \ref{3dffd}) we avoid the generation of such self--intersecting
grids for our testcases.
To achieve that we generated a gaussian distributed number with
\(\mu = 0, \sigma=0.25\) and clamped it to the range \([-0.25,0.25]\).
@ -1130,12 +1136,12 @@ In the case of our 1D--Optimization--problem, we have the luxury of
knowing the analytical solution to the given problem--set. We use this
to experimentally evaluate the quality criteria we introduced before. As
an evolutional optimization is partially a random process, we use the
analytical solution as a stopping-criteria. We measure the convergence
analytical solution as a stopping--criteria. We measure the convergence
speed as number of iterations the evolutional algorithm needed to get
within \(1.05 \times\) of the optimal solution.
We used different regular grids that we manipulated as explained in
Section \ref{sec:proc:1d} with a different number of control points. As
Section \ref{sec:proc:1d} with a different number of control--points. As
our grids have to be the product of two integers, we compared a
\(5 \times 5\)--grid with \(25\) control--points to a \(4 \times 7\) and
\(7 \times 4\)--grid with \(28\) control--points. This was done to
@ -1157,7 +1163,7 @@ Note that $7 \times 4$ and $4 \times 7$ have the same number of control--points.
\label{fig:1dvar}
\end{figure}
Variability should characterize the potential for design space
\emph{Variability} should characterize the potential for design space
exploration and is defined in terms of the normalized rank of the
deformation matrix \(\vec{U}\):
\(V(\vec{U}) := \frac{\textrm{rank}(\vec{U})}{n}\), whereby \(n\) is the
@ -1166,29 +1172,30 @@ number of vertices. As all our tested matrices had a constant rank
plotted the errors in the box plot in figure \ref{fig:1dvar}
It is also noticeable, that although the \(7 \times 4\) and
\(4 \times 7\) grids have a higher variability, they perform not better
than the \(5 \times 5\) grid. Also the \(7 \times 4\) and \(4 \times 7\)
grids differ distinctly from each other with a mean\(\pm\)sigma of
\(233.09 \pm 12.32\) for the former and \(286.32 \pm 22.36\) for the
latter, although they have the same number of control--points. This is
an indication of an impact a proper or improper grid--setup can have. We
do not draw scientific conclusions from these findings, as more research
on non-squared grids seem necessary.
\(4 \times 7\) grids have a higher \emph{variability}, they perform not
better than the \(5 \times 5\) grid. Also the \(7 \times 4\) and
\(4 \times 7\) grids differ distinctly from each other with a
mean\(\pm\)sigma of \(233.09 \pm 12.32\) for the former and
\(286.32 \pm 22.36\) for the latter, although they have the same number
of control--points. This is an indication of an impact a proper or
improper grid--setup can have. We do not draw scientific conclusions
from these findings, as more research on non--squared grids seem
necessary.
Leaving the issue of the grid--layout aside we focused on grids having
the same number of prototypes in every dimension. For the
\(5 \times 5\), \(7 \times 7\) and \(10 \times 10\) grids we found a
\emph{very strong} correlation (\(-r_S = 0.94, p = 0\)) between the
variability and the evolutionary error.
\emph{variability} and the evolutionary error.
\subsection{Regularity}\label{regularity-1}
\begin{figure}[tbh]
\centering
\includegraphics[width=\textwidth]{img/evolution1d/55_to_1010_steps.png}
\caption[Improvement potential and regularity vs. steps]{\newline
Left: Improvement potential against steps until convergence\newline
Right: Regularity against steps until convergence\newline
\caption[Improvement potential and regularity against iterations]{\newline
Left: *Improvement potential* against number of iterations until convergence\newline
Right: *Regularity* against number of iterations until convergence\newline
Coloured by their grid--resolution, both with a linear fit over the whole
dataset.}
\label{fig:1dreg}
@ -1201,16 +1208,16 @@ $5 \times 5$ & $7 \times 4$ & $4 \times 7$ & $7 \times 7$ & $10 \times 10$\\
\hline
$0.28$ ($0.0045$) & \textcolor{red}{$0.21$} ($0.0396$) & \textcolor{red}{$0.1$} ($0.3019$) & \textcolor{red}{$0.01$} ($0.9216$) & \textcolor{red}{$0.01$} ($0.9185$)
\end{tabular}
\caption[Correlation 1D Regularity/Steps]{Spearman's correlation (and p-values)
between regularity and convergence speed for the 1D function approximation
\caption[Correlation 1D *regularity* against iterations]{Inverted Spearman's correlation (and p--values)
between *regularity* and number of iterations for the 1D function approximation
problem.
\newline Note: Not significant results are marked in \textcolor{red}{red}.
}
\label{tab:1dreg}
\end{table}
Regularity should correspond to the convergence speed (measured in
iteration--steps of the evolutionary algorithm), and is computed as
\emph{Regularity} should correspond to the convergence speed (measured
in iteration--steps of the evolutionary algorithm), and is computed as
inverse condition number \(\kappa(\vec{U})\) of the deformation--matrix.
As can be seen from table \ref{tab:1dreg}, we could only show a
@ -1222,27 +1229,27 @@ datasets into account we even get a \emph{strong} correlation of
To explain this discrepancy we took a closer look at what caused these
high number of iterations. In figure \ref{fig:1dreg} we also plotted the
improvement-potential against the steps next to the regularity--plot.
Our theory is that the \emph{very strong} correlation
(\(-r_S = -0.82, p=0\)) between improvement--potential and number of
iterations hints that the employed algorithm simply takes longer to
converge on a better solution (as seen in figure \ref{fig:1dvar} and
\ref{fig:1dimp}) offsetting any gain the regularity--measurement could
achieve.
\emph{improvement potential} against the steps next to the
\emph{regularity}--plot. Our theory is that the \emph{very strong}
correlation (\(-r_S = -0.82, p=0\)) between \emph{improvement potential}
and number of iterations hints that the employed algorithm simply takes
longer to converge on a better solution (as seen in figure
\ref{fig:1dvar} and \ref{fig:1dimp}) offsetting any gain the
regularity--measurement could achieve.
\subsection{Improvement Potential}\label{improvement-potential-1}
\begin{figure}[ht]
\centering
\includegraphics[width=0.8\textwidth]{img/evolution1d/55_to_1010_improvement-vs-evo-error.png}
\caption[Correlation 1D Improvement vs. Error]{Improvement potential plotted
\caption[Correlation 1D Improvement vs. Error]{*Improvement potential* plotted
against the error yielded by the evolutionary optimization for different
grid--resolutions}
\label{fig:1dimp}
\end{figure}
The improvement potential should correlate to the quality of the
fitting--result. We plotted the results for the tested grid-sizes
The \emph{improvement potential} should correlate to the quality of the
fitting--result. We plotted the results for the tested grid--sizes
\(5 \times 5\), \(7 \times 7\) and \(10 \times 10\) in figure
\ref{fig:1dimp}. We tested the \(4 \times 7\) and \(7 \times 4\) grids
as well, but omitted them from the plot.
@ -1280,7 +1287,7 @@ Initially we set up the correspondences \(\vec{c_T(\dots)}\) and
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
\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 \(\vec{c_S(\dots)}\).
@ -1310,12 +1317,12 @@ regularization--effect wears off.
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
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 in the
before, we create a regular grid and move the control--points in the
exact same random manner between their neighbours as described in
section \ref{sec:proc:1d}, but in three instead of two
dimensions\footnote{Again, we flip the signs for the edges, if necessary
@ -1332,9 +1339,9 @@ Right: A $4 \times 4 \times 7$ grid that we expect to perform worse.}
As is clearly visible from figure \ref{fig:3dgridres}, 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
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.
predict a suboptimal placement of these control--points.
\section{Results of 3D Function
Approximation}\label{results-of-3d-function-approximation}
@ -1342,8 +1349,8 @@ Approximation}\label{results-of-3d-function-approximation}
In the 3D--Approximation we tried to evaluate further on the impact of
the grid--layout to the overall criteria. As the target--model has many
vertices in concentrated in the facial area we start from a
\(4 \times 4 \times 4\) grid and only increase the number of control
points in one dimension, yielding a resolution of
\(4 \times 4 \times 4\) grid and only increase the number of
control--points in one dimension, yielding a resolution of
\(7 \times 4 \times 4\) and \(4 \times 4 \times 7\) respectively. We
visualized those two grids in figure \ref{fig:3dgridres}.
@ -1374,10 +1381,10 @@ $4 \times 4 \times \mathrm{X}$ & $\mathrm{X} \times 4 \times 4$ & $\mathrm{Y} \t
\hline
0.89 (0) & 0.9 (0) & 0.91 (0) & 0.94 (0)
\end{tabular}
\caption[Correlation between variability and fitting error for 3D]{Correlation
between variability and fitting error for the 3D fitting scenario.\newline
Displayed are the negated Spearman coefficients with the corresponding p-values
in brackets for three cases of increasing variability ($\mathrm{X} \in [4,5,7],
\caption[Correlation between *variability* and fitting error for 3D]{Correlation
between *variability* and fitting error for the 3D fitting scenario.\newline
Displayed are the negated Spearman coefficients with the corresponding p--values
in brackets for three cases of increasing *variability* ($\mathrm{X} \in [4,5,7],
\mathrm{Y} \in [4,5,6]$).
\newline Note: Not significant results are marked in \textcolor{red}{red}.}
\label{tab:3dvar}
@ -1399,40 +1406,42 @@ we anticipated, which shows that the evolutionary algorithm we employed
is capable of correcting a purposefully created \glqq bad\grqq ~grid.
Also this confirms, that in our cases the number of control--points is
more important for quality than their placement, which is captured by
the variability via the rank of the deformation--matrix.
the \emph{variability} via the rank of the deformation--matrix.
Overall the correlation between \emph{variability} and fitness--error
were \emph{significant} and showed a \emph{very strong} correlation in
all our tests. The detailed correlation--coefficients are given in table
\ref{tab:3dvar} alongside their p--values.
As introduces in section \ref{sec:impl:grid} and visualized in figure
\ref{fig:enoughCP}, we know, that not all control--points have to
necessarily contribute to the parametrization of our 3D--model. Because
we are starting from a sphere, some control--points are too far away
from the surface to contribute to the deformation at all.
One can already see in 2D in figure \ref{fig:enoughCP}, that this effect
starts with a regular \(9 \times 9\) grid on a perfect circle. To make
sure we observe this, we evaluated the \emph{variability} for 100
randomly moved \(10 \times 10 \times 10\) grids on the sphere we start
out with.
\begin{figure}[hbt]
\centering
\includegraphics[width=0.8\textwidth]{img/evolution3d/variability2_boxplot.png}
\caption[Histogram of ranks of high--resolution deformation--matrices]{
Histogram of ranks of various $10 \times 10 \times 10$ grids with $1000$
control--points each showing in this case how many control points are actually
control--points each showing in this case how many control--points are actually
used in the calculations.
}
\label{fig:histrank3d}
\end{figure}
Overall the correlation between variability and fitness--error were
\emph{significant} and showed a \emph{very strong} correlation in all
our tests. The detailed correlation--coefficients are given in table
\ref{tab:3dvar} alongside their p--values.
As introduces in section \ref{sec:impl:grid} and visualized in figure
\ref{fig:enoughCP}, we know, that not all control points have to
necessarily contribute to the parametrization of our 3D--model. Because
we are starting from a sphere, some control-points are too far away from
the surface to contribute to the deformation at all.
One can already see in 2D in figure \ref{fig:enoughCP}, that this effect
starts with a regular \(9 \times 9\) grid on a perfect circle. To make
sure we observe this, we evaluated the variability for 100 randomly
moved \(10 \times 10 \times 10\) grids on the sphere we start out with.
As the variability is defined by \(\frac{\mathrm{rank}(\vec{U})}{n}\) we
can easily recover the rank of the deformation--matrix \(\vec{U}\). The
results are shown in the histogram in figure \ref{fig:histrank3d}.
Especially in the centre of the sphere and in the corners of our grid we
effectively loose control--points for our parametrization.
As the \emph{variability} is defined by
\(\frac{\mathrm{rank}(\vec{U})}{n}\) we can easily recover the rank of
the deformation--matrix \(\vec{U}\). The results are shown in the
histogram in figure \ref{fig:histrank3d}. Especially in the centre of
the sphere and in the corners of our grid we effectively loose
control--points for our parametrization.
This of course yields a worse error as when those control--points would
be put to use and one should expect a loss in quality evident by a
@ -1440,7 +1449,7 @@ higher reconstruction--error opposed to a grid where they are used.
Sadly we could not run a in--depth test on this due to computational
limitations.
Nevertheless this hints at the notion, that variability is a good
Nevertheless this hints at the notion, that \emph{variability} is a good
measure for the overall quality of a fit.
\subsection{Regularity}\label{regularity-2}
@ -1468,21 +1477,21 @@ $4 \times 4 \times 4$ & $4 \times 4 \times 5$ & $4 \times 4 \times 7$ & $4 \time
\cline{2-4}
\multicolumn{3}{c}{} & all: 0.15 (0) \T
\end{tabular}
\caption[Correlation between regularity and iterations for 3D]{Correlation
between regularity and number of iterations for the 3D fitting scenario.
\caption[Correlation between *regularity* and iterations for 3D]{Correlation
between *regularity* and number of iterations for the 3D fitting scenario.
Displayed are the negated Spearman coefficients with the corresponding p--values
in brackets for various given grids ($\mathrm{X} \in [4,5,7], \mathrm{Y} \in [4,5,6]$).
\newline Note: Not significant results are marked in \textcolor{red}{red}.}
\label{tab:3dreg}
\end{table}
Opposed to the predictions of variability our test on regularity gave a
mixed result --- similar to the 1D--case.
Opposed to the predictions of \emph{variability} our test on
\emph{regularity} gave a mixed result --- similar to the 1D--case.
In roughly half of the scenarios we have a \emph{significant}, but
\emph{weak} to \emph{moderate} correlation between regularity and number
of iterations. On the other hand in the scenarios where we increased the
number of control--points, namely \(125\) for the
\emph{weak} to \emph{moderate} correlation between \emph{regularity} and
number of iterations. On the other hand in the scenarios where we
increased the number of control--points, namely \(125\) for the
\(5 \times 5 \times 5\) grid and \(216\) for the \(6 \times 6 \times 6\)
grid we found a \emph{significant}, but \emph{weak}
\textbf{anti}--correlation when taking all three tests into
@ -1491,14 +1500,14 @@ contradict the findings/trends for the sets with \(64\), \(80\), and
\(112\) control--points (first two rows of table \ref{tab:3dreg}).
Taking all results together we only find a \emph{very weak}, but
\emph{significant} link between regularity and the number of iterations
needed for the algorithm to converge.
\emph{significant} link between \emph{regularity} and the number of
iterations needed for the algorithm to converge.
\begin{figure}[!htb]
\centering
\includegraphics[width=\textwidth]{img/evolution3d/regularity_montage.png}
\caption[Regularity for different 3D--grids]{
Plots of regularity against number of iterations for various scenarios together
Plots of *regularity* against number of iterations for various scenarios together
with a linear fit to indicate trends.}
\label{fig:resreg3d}
\end{figure}
@ -1509,10 +1518,10 @@ The regularity--criterion first behaves as we would like to, but then
switches to behave exactly opposite to our expectations, as can be seen
in the first three plots. While the number of control--points increases
from red to green to blue and the number of iterations decreases, the
regularity seems to increase at first, but then decreases again on
higher grid--resolutions.
\emph{regularity} seems to increase at first, but then decreases again
on higher grid--resolutions.
This can be an artefact of the definition of regularity, as it is
This can be an artefact of the definition of \emph{regularity}, as it is
defined by the inverse condition--number of the deformation--matrix
\(\vec{U}\), being the fraction
\(\frac{\sigma_{\mathrm{min}}}{\sigma_{\mathrm{max}}}\) between the
@ -1524,12 +1533,12 @@ small minimal right singular value occurring on higher grid--resolutions
seems likely the problem.
Adding to this we also noted, that in the case of the
\(10 \times 10 \times 10\)--grid the regularity was always \(0\), as a
non--contributing control-point yields a \(0\)--column in the
\(10 \times 10 \times 10\)--grid the \emph{regularity} was always \(0\),
as a non--contributing control--point yields a \(0\)--column in the
deformation--matrix, thus letting \(\sigma_\mathrm{min} = 0\). A better
definition for regularity (i.e.~using the smallest non--zero right
singular value) could solve this particular issue, but not fix the trend
we noticed above.
definition for \emph{regularity} (i.e.~using the smallest non--zero
right singular value) could solve this particular issue, but not fix the
trend we noticed above.
\subsection{Improvement Potential}\label{improvement-potential-2}
@ -1556,8 +1565,8 @@ $4 \times 4 \times 4$ & $4 \times 4 \times 5$ & $4 \times 4 \times 7$ & $4 \time
\cline{2-4}
\multicolumn{3}{c}{} & all: 0.95 (0) \T
\end{tabular}
\caption[Correlation between improvement--potential and fitting--error for 3D]{Correlation
between improvement--potential and fitting--error for the 3D fitting scenario.
\caption[Correlation between *improvement potential* and fitting--error for 3D]{Correlation
between *improvement potential* and fitting--error for the 3D fitting scenario.
Displayed are the negated Spearman coefficients with the corresponding p--values
in brackets for various given grids ($\mathrm{X} \in [4,5,7], \mathrm{Y} \in [4,5,6]$).
\newline Note: Not significant results are marked in \textcolor{red}{red}.}
@ -1576,21 +1585,22 @@ gradients anyway.
\centering
\includegraphics[width=\textwidth]{img/evolution3d/improvement_montage.png}
\caption[Improvement potential for different 3D--grids]{
Plots of improvement potential against error given by our fitness--function
Plots of *improvement potential* against error given by our *fitness--function*
after convergence together with a linear fit of each of the plotted data to
indicate trends.}
\label{fig:resimp3d}
\end{figure}
We plotted our findings on the improvement potential in a similar way as
we did before with the regularity. In figure \ref{fig:resimp3d} one can
clearly see the correlation and the spread within each setup and the
behaviour when we increase the number of control--points.
We plotted our findings on the \emph{improvement potential} in a similar
way as we did before with the \emph{regularity}. In figure
\ref{fig:resimp3d} one can clearly see the correlation and the spread
within each setup and the behaviour when we increase the number of
control--points.
Along with this we also give the Spearman--coefficients along with their
p--values in table \ref{tab:3dimp}. Within one scenario we only find a
\emph{weak} to \emph{moderate} correlation between the improvement
potential and the fitting error, but all findings (except for
\emph{weak} to \emph{moderate} correlation between the \emph{improvement
potential} and the fitting error, but all findings (except for
\(7 \times 4 \times 4\) and \(6 \times 6 \times 6\)) are significant.
If we take multiple datasets into account the correlation is \emph{very
@ -1598,30 +1608,30 @@ strong} and \emph{significant}, which is good, as this functions as a
litmus--test, because the quality is naturally tied to the number of
control--points.
All in all the improvement potential seems to be a good and sensible
measure of quality, even given gradients of varying quality.
All in all the \emph{improvement potential} seems to be a good and
sensible measure of quality, even given gradients of varying quality.
Lastly, a small note on the behaviour of improvement potential and
convergence speed, as we used this in the 1D case to argue, why the
Lastly, a small note on the behaviour of \emph{improvement potential}
and convergence speed, as we used this in the 1D case to argue, why the
\emph{regularity} defied our expectations. As a contrast we wanted to
show, that improvement potential cannot serve for good predictions of
the convergence speed. In figure \ref{fig:imp1d3d} we show improvement
potential against number of iterations for both scenarios. As one can
see, in the 1D scenario we have a \emph{strong} and \emph{significant}
correlation (with \(-r_S = -0.72\), \(p = 0\)), whereas in the 3D
scenario we have the opposite \emph{significant} and \emph{strong}
effect (with \(-r_S = 0.69\), \(p=0\)), so these correlations clearly
seem to be dependent on the scenario and are not suited for
generalization.
show, that \emph{improvement potential} cannot serve for good
predictions of the convergence speed. In figure \ref{fig:imp1d3d} we
show \emph{improvement potential} against number of iterations for both
scenarios. As one can see, in the 1D scenario we have a \emph{strong}
and \emph{significant} correlation (with \(-r_S = -0.72\), \(p = 0\)),
whereas in the 3D scenario we have the opposite \emph{significant} and
\emph{strong} effect (with \(-r_S = 0.69\), \(p=0\)), so these
correlations clearly seem to be dependent on the scenario and are not
suited for generalization.
\begin{figure}[hbt]
\centering
\includegraphics[width=\textwidth]{img/imp1d3d.png}
\caption[Improvement potential and convergence speed for 1D and 3D--scenarios]{
\caption[Improvement potential and convergence speed\newline for 1D and 3D--scenarios]{
\newline
Left: Improvement potential against convergence speed for the
Left: *Improvement potential* against convergence speed for the
1D--scenario\newline
Right: Improvement potential against convergence speed for the 3D--scnario
Right: *Improvement potential* against convergence speed for the 3D--scnario
}
\label{fig:imp1d3d}
\end{figure}
@ -1630,36 +1640,36 @@ Right: Improvement potential against convergence speed for the 3D--scnario
\label{sec:dis}
In this thesis we took a look at the different criteria for evolvability
as introduced by Richter et al.\cite{anrichterEvol}, namely
\emph{variability}, \emph{regularity} and \emph{improvement potential}
under different setup--conditions. Where Richter et al. used \acf{RBF},
we employed \acf{FFD} to set up a low--complexity parametrization of a
more complex vertex--mesh.
In this thesis we took a look at the different criteria for
\emph{evolvability} as introduced by Richter et al.\cite{anrichterEvol},
namely \emph{variability}, \emph{regularity} and \emph{improvement
potential} under different setup--conditions. Where Richter et al. used
\acf{RBF}, we employed \acf{FFD} to set up a low--complexity
parametrization of a more complex vertex--mesh.
In our findings we could show in the 1D--scenario, that there were
statistically significant very strong correlations between
\emph{variability and fitting error} (\(0.94\)) and
\emph{improvement--potential and fitting error} (\(1.0\)) with
comparable results than Richter et al. (with \(0.31\) to \(0.88\) for
the former and \(0.75\) to \(0.99\) for the latter), whereas we found
only weak correlations for \emph{regularity and convergence--speed}
(\(0.28\)) opposed to Richter et al. with \(0.39\) to
\(0.91\).\footnote{We only took statistically \emph{significant} results
into consideration when compiling these numbers. Details are given in
the respective chapters.}
statistically \emph{significant} \emph{very strong} correlations between
\emph{variability and fitting error} (\(0.94\)) and \emph{improvement
potential and fitting error} (\(1.0\)) with comparable results than
Richter et al. (with \(0.31\) to \(0.88\) for the former and \(0.75\) to
\(0.99\) for the latter), whereas we found only \emph{weak} correlations
for \emph{regularity and convergence--speed} (\(0.28\)) opposed to
Richter et al. with \(0.39\) to \(0.91\).\footnote{We only took
statistically \emph{significant} results into consideration when
compiling these numbers. Details are given in the respective chapters.}
For the 3D--scenario our results show a very strong, significant
correlation between \emph{variability and fitting error} with \(0.89\)
to \(0.94\), which are pretty much in line with the findings of Richter
et al. (\(0.65\) to \(0.95\)). The correlation between \emph{improvement
potential and fitting error} behave similar, with our findings having a
significant coefficient of \(0.3\) to \(0.95\) depending on the
grid--resolution compared to the \(0.61\) to \(0.93\) from Richter et
al. In the case of the correlation of \emph{regularity and convergence
speed} we found very different (and often not significant) correlations
and anti--correlations ranging from \(-0.25\) to \(0.46\), whereas
Richter et al. reported correlations between \(0.34\) to \(0.87\).
For the 3D--scenario our results show a \emph{very strong},
\emph{significant} correlation between \emph{variability and fitting
error} with \(0.89\) to \(0.94\), which are pretty much in line with the
findings of Richter et al. (\(0.65\) to \(0.95\)). The correlation
between \emph{improvement potential and fitting error} behave similar,
with our findings having a significant coefficient of \(0.3\) to
\(0.95\) depending on the grid--resolution compared to the \(0.61\) to
\(0.93\) from Richter et al. In the case of the correlation of
\emph{regularity and convergence speed} we found very different (and
often not significant) correlations and anti--correlations ranging from
\(-0.25\) to \(0.46\), whereas Richter et al. reported correlations
between \(0.34\) to \(0.87\).
Taking these results into consideration, one can say, that
\emph{variability} and \emph{improvement potential} are very good
@ -1683,14 +1693,11 @@ manipulation in \cite{anrichterEvol}, whereas we merely used an indirect
indirect manipulations, the usage of \acf{DM--FFD} could also work
better with the criteria we examined. This can also solve the problem of
bad singular values for the \emph{regularity} as the incorporation of
the parametrization of the points on the surface, which are the
essential part of a direct--manipulation, could cancel out a bad
the parametrization of the points on the surface --- which are the
essential part of a direct--manipulation --- could cancel out a bad
control--grid as the bad control--points are never or negligibly used to
parametrize those surface--points.
\improvement[inline]{Bibliotheksverzeichnis links anpassen. DOI überschreibt
Direktlinks des Autors.}
% \backmatter
\cleardoublepage
@ -1725,10 +1732,10 @@ Direktlinks des Autors.}
% \addtocounter{chapter}{1}
\newpage
% \listoftables
\listoftodos
% \listoftodos
% \addcontentsline{toc}{chapter}{\protect\numberline{\thechapter}TODOs}
% \addtocounter{chapter}{1}
\newpage
% \newpage
% \printindex
%%%%%%%%%%%%%%% Erklaerung %%%%%%%%%%%%%%%

4
arbeit/template.tex
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@ -175,10 +175,10 @@ $body$
% \addtocounter{chapter}{1}
\newpage
% \listoftables
\listoftodos
% \listoftodos
% \addcontentsline{toc}{chapter}{\protect\numberline{\thechapter}TODOs}
% \addtocounter{chapter}{1}
\newpage
% \newpage
% \printindex
%%%%%%%%%%%%%%% Erklaerung %%%%%%%%%%%%%%%

2
dokumentation/evolution1d/variability.gnuplot
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@ -14,7 +14,7 @@ set xtics norangelimit
set xtics ()
set ytics border in scale 1,0.5 nomirror norotate autojustify
set title "Fitting Errors of 1D Function Approximation for various grids\n"
set ylabel "Squared Error of Vertex-Difference"
set ylabel "Fitting-Error according to fitness-function"
header ="`head -1 errors.csv | sed -s "s/\"//g" | sed -s "s/,/ /g"`"
set for [i=1:words(header)] xtics (word(header,i) i)

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