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@ -10,10 +10,10 @@ chapter.
Unless otherwise noted the following holds:
- lowercase letters $x,y,z$
refer to real variables and represent a point in 3D-Space.
refer to real variables and represent a point in 3D--Space.
- lowercase letters $u,v,w$
refer to real variables between $0$ and $1$ used as coefficients in a 3D
B-Spline grid.
B--Spline grid.
- other lowercase letters
refer to other scalar (real) variables.
- lowercase **bold** letters (e.g. $\vec{x},\vec{y}$)
@ -29,9 +29,9 @@ Many modern industrial design processes require advanced optimization methods
do to the increased complexity. These designs have to adhere to more and more
degrees of freedom as methods refine and/or other methods are used. Examples for
this are physical domains like aerodynamic (i.e. drag), fluid dynamics (i.e.
throughput of liquid) -- where the complexity increases with the temporal and
spatial resolution of the simulation -- or known hard algorithmic problems in
informatics (i.e. layouting of circuit boards or stacking of 3D-objects).
throughput of liquid) --- where the complexity increases with the temporal and
spatial resolution of the simulation --- or known hard algorithmic problems in
informatics (i.e. layouting of circuit boards or stacking of 3D--objects).
Moreover these are typically not static environments but requirements shift over
time or from case to case.
@ -39,8 +39,8 @@ Evolutional algorithms cope especially well with these problem domains while
addressing all the issues at hand\cite{minai2006complex}. One of the main
concerns in these algorithms is the formulation of the problems in terms of a
genome and a fitness function. While one can typically use an arbitrary
cost-function for the fitness-functions (i.e. amount of drag, amount of space,
etc.), the translation of the problem-domain into a simple parametric
cost--function for the fitness--functions (i.e. amount of drag, amount of space,
etc.), the translation of the problem--domain into a simple parametric
representation can be challenging.
The quality of such a representation in biological evolution is called
@ -61,26 +61,26 @@ and potential of such optimization.
We will replicate the same setup on the same meshes 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 given the different deformation
the evolution--criteria still work as a predictor given the different deformation
scheme, as suspected in \cite{anrichterEvol}.
## Outline of this thesis
First we introduce different topics in isolation in Chapter \ref{sec:back}. We
take an abstract look at the definition of \ac{FFD} for a one-dimensional line
take an abstract look at the definition of \ac{FFD} for a one--dimensional line
(in \ref{sec:back:ffd}) and discuss why this is a sensible deformation function
(in \ref{sec:back:ffdgood}).
Then we establish some background-knowledge of evolutional algorithms (in
Then we establish some background--knowledge of evolutional algorithms (in
\ref{sec:back:evo}) and why this is useful in our domain (in
\ref{sec:back:evogood}).
In a third step we take a look at the definition of the different evolvability
criteria established in \cite{anrichterEvol}.
In Chapter \ref{sec:impl} we take a look at our implementation of \ac{FFD} and
the adaptation for 3D-meshes.
the adaptation for 3D--meshes.
Next, in Chapter \ref{sec:eval}, we describe the different scenarios we use to
evaluate the different evolvability-criteria incorporating all aspects
evaluate the different evolvability--criteria incorporating all aspects
introduced in Chapter \ref{sec:back}. Following that, we evaluate the results in
Chapter \ref{sec:res} with further on discussion in Chapter \ref{sec:dis}.
@ -93,7 +93,7 @@ Chapter \ref{sec:res} with further on discussion in Chapter \ref{sec:dis}.
First of all we have to establish how a \ac{FFD} works and why this is a good
tool for deforming meshes in the first place. For simplicity we only summarize
the 1D-case from \cite{spitzmuller1996bezier} here and go into the extension to
the 1D--case from \cite{spitzmuller1996bezier} here and go into the extension to
the 3D case in chapter \ref{3dffd}.
Given an arbitrary number of points $p_i$ alongside a line, we map a scalar
@ -110,20 +110,20 @@ N_{i,d,\tau}(u) = \frac{u-\tau_i}{\tau_{i+d}} N_{i,d-1,\tau}(u) + \frac{\tau_{i+
\end{equation}
If we now multiply every $p_i$ with the corresponding $N_{i,d,\tau_i}(u)$ we get
the contribution of each point $p_i$ to the final curve-point parameterized only
the contribution of each point $p_i$ to the final curve--point parameterized only
by $u \in [0,1[$. As can be seen from \eqref{eqn:ffd1d2} we only access points
$[i..i+d]$ for any given $i$^[one more for each recursive step.], which gives
us, in combination with choosing $p_i$ and $\tau_i$ in order, only a local
interference of $d+1$ points.
We can even derive this equation straightforward for an arbitrary
$N$^[*Warning:* in the case of $d=1$ the recursion-formula yields a $0$
$N$^[*Warning:* in the case of $d=1$ the recursion--formula yields a $0$
denominator, but $N$ is also $0$. The right solution for this case is a
derivative of $0$]:
$$\frac{\partial}{\partial u} N_{i,d,r}(u) = \frac{d}{\tau_{i+d} - \tau_i} N_{i,d-1,\tau}(u) - \frac{d}{\tau_{i+d+1} - \tau_{i+1}} N_{i+1,d-1,\tau}(u)$$
For a B-Spline
For a B--Spline
$$s(u) = \sum_{i} N_{i,d,\tau_i}(u) p_i$$
these derivations yield $\frac{\partial^d}{\partial u} s(u) = 0$.
@ -138,7 +138,7 @@ recursion.
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
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
@ -150,11 +150,11 @@ plateau can be difficult to
achieve\cite[chapter~3.2]{hsu1991dmffd}\cite{hsu1992direct}.
This disadvantages led to the formulation of
\acf{DM-FFD}\cite[chapter~3.3]{hsu1991dmffd} in which the user directly
interacts with the surface-mesh. All interactions will be applied
proportionally to the control-points that make up the parametrization of the
interaction-point itself yielding a smooth deformation of the surface *at* the
surface without seemingly arbitrary scattered control-points. Moreover this
\acf{DM--FFD}\cite[chapter~3.3]{hsu1991dmffd} in which the user directly
interacts with the surface--mesh. All interactions will be applied
proportionally to the control--points that make up the parametrization of the
interaction--point itself yielding a smooth deformation of the surface *at* the
surface without seemingly arbitrary scattered control--points. Moreover this
increases the efficiency of an evolutionary optimization\cite{Menzel2006}, which
we will use later on.
@ -168,7 +168,7 @@ But this approach also has downsides as can be seen in figure
\ref{fig:hsu_fig7}, as the tessellation of the invisible grid has a major impact
on the deformation itself.
All in all \ac{FFD} and \ac{DM-FFD} are still good ways to deform a high-polygon
All in all \ac{FFD} and \ac{DM--FFD} are still good ways to deform a high--polygon
mesh albeit the downsides.
## What is evolutional optimization?
@ -181,12 +181,12 @@ mesh albeit the downsides.
\change[inline]{Needs citations}
The main advantage of evolutional algorithms is the ability to find optima of
general functions just with the help of a given error-function (or
fitness-function in this domain). This avoids the general pitfalls of
gradient-based procedures, which often target the same error-function as an
general functions just with the help of a given error--function (or
fitness--function in this domain). This avoids the general pitfalls of
gradient--based procedures, which often target the same error--function as an
evolutional algorithm, but can get stuck in local optima.
This is mostly due to the fact that a gradient-based procedure has only one
This is mostly due to the fact that a gradient--based procedure has only one
point of observation from where it evaluates the next steps, whereas an
evolutional strategy starts with a population of guessed solutions. Because an
evolutional strategy modifies the solution randomly, keeps the best solutions
@ -194,7 +194,7 @@ and purges the worst, it can also target multiple different hypothesis at the
same time where the local optima die out in the face of other, better
candidates.
If an analytic best solution exists (i.e. because the error-function is convex)
If an analytic best solution exists (i.e. because the error--function is convex)
an evolutional algorithm is not the right choice. Although both converge to the
same solution, the analytic one is usually faster. But in reality many problems
have no analytic solution, because the problem is not convex. Here evolutional
@ -208,31 +208,31 @@ over time.
In \cite{anrichterEvol} *variability* is defined as
$$V(\vec{U}) := \frac{\textrm{rank}(\vec{U})}{n},$$
whereby $\vec{U}$ is the $m \times n$ deformation-Matrix used to map the $m$
whereby $\vec{U}$ is the $m \times n$ deformation--Matrix used to map the $m$
control points onto the $n$ vertices.
Given $n = m$, an identical number of control-points and vertices, this
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
the solution is to trivially move every control-point onto a target-point.
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 $m \ll n$.
there are only few control--points for many vertices, so $m \ll n$.
Additionally in our setup we connect neighbouring control-points in a grid so
Additionally in our setup we connect neighbouring control--points in a grid so
each control point is not independent, but typically depends on $4^d$
control-points for an $d$-dimensional control mesh.
control--points for an $d$--dimensional control mesh.
### Regularity
*Regularity* is defined\cite{anrichterEvol} as
$$R(\vec{U}) := \frac{1}{\kappa(\vec{U})} = \frac{\sigma_{min}}{\sigma_{max}}$$
where $\sigma_{min}$ and $\sigma_{max}$ are the smallest and greatest right singular
value of the deformation-matrix $\vec{U}$.
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 $[0..1]$, whereas $1$ is the
optimal value and $0$ is the worst value.
This criterion should be characteristic for numeric stability on the on
@ -243,7 +243,7 @@ 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 potential.
As during optimization some kind of gradient $g$ is available to suggest a
@ -254,20 +254,20 @@ The definition for an *improvement potential* $P$ is\cite{anrichterEvol}:
$$
P(\vec{U}) := 1 - \|(\vec{1} - \vec{UU}^+)\vec(G)\|^2_F
$$
given some approximate $n \times d$ fitness-gradient $\vec{G}$, normalized to
$\|\vec{G}\|_F = 1$, whereby $\|\cdot\|_F$ denotes the Frobenius-Norm.
given some approximate $n \times d$ fitness--gradient $\vec{G}$, normalized to
$\|\vec{G}\|_F = 1$, whereby $\|\cdot\|_F$ denotes the Frobenius--Norm.
# Implementation of \acf{FFD}
\label{sec:impl}
The general formulation of B-Splines has two free parameters $d$ and $\tau$
The general formulation of B--Splines has two free parameters $d$ and $\tau$
which must be chosen beforehand.
As we usually work with regular grids in our \ac{FFD} we define $\tau$
statically as $\tau_i = \nicefrac{i}{n}$ whereby $n$ is the number of
control-points in that direction.
control--points in that direction.
$d$ defines the *degree* of the B-Spline-Function (the number of times this
$d$ defines the *degree* of the B--Spline--Function (the number of times this
function is differentiable) and for our purposes we fix $d$ to $3$, but give the
formulas for the general case so it can be adapted quite freely.
@ -275,21 +275,21 @@ formulas for the general case so it can be adapted quite freely.
## Adaption of \ac{FFD}
As we have established in Chapter \ref{sec:back:ffd} we can define an
\ac{FFD}-displacement as
\ac{FFD}--displacement as
\begin{equation}
\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
$$
\textrm{Deform}(v_x) = v_x + \Delta_x(u)
$$
with $u \in [0..1[$ being the variable that connects the high-detailed
vertex-mesh to the low-detailed control-grid. To actually calculate the new
position of the vertex we first have to calculate the $u$-value for each
with $u \in [0..1[$ being the variable that connects the high--detailed
vertex--mesh to the low--detailed control--grid. To actually calculate the new
position of the vertex we first have to calculate the $u$--value for each
vertex. This is achieved by finding out the parametrization of $v$ in terms of
$c_i$
$$
@ -299,35 +299,36 @@ so we can minimize the error between those two:
$$
\underset{u}{\argmin}\,Err(u,v_x) = \underset{u}{\argmin}\,2 \cdot \|v_x - \sum_i N_{i,d,\tau_i}(u) c_i\|^2_2
$$
As this error--term is quadratic we just derive by $u$ yielding
$$
\begin{array}{rl}
\frac{\partial}{\partial u} & v_x - \sum_i N_{i,d,\tau_i}(u) c_i \\
= & - \sum_i \left( \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) \right) c_i
\end{array}
$$
and do a gradient--descend to approximate the value of $u$ up to an $\epsilon$ of $0.0001$.
As this error-term is quadratic we just derive by $u$ yielding
\begin{eqnarray*}
& \frac{\partial}{\partial u} & v_x - \sum_i N_{i,d,\tau_i}(u) c_i \\
& = & - \sum_i \left( \frac{d}{\tau_{i+d} - \tau_i} N_{i,d-1,\tau}(u) - \frac{d}{\tau_{i+d+1} - \tau_{i+1}} N_{i+1,d-1,\tau}(u) \right) c_i
\end{eqnarray*}
and do a gradient-descend to approximate the value of $u$ up to an $\epsilon$ of $0.0001$.
For this we use the Gauss-Newton algorithm\cite{gaussNewton} as the solution to
For this we use the Gauss--Newton algorithm\cite{gaussNewton} as the solution to
this problem may not be deterministic, because we usually have way more vertices
than control points ($\#v \gg \#c$).
than control points ($\#v~\gg~\#c$).
## Adaption of \ac{FFD} for a 3D-Mesh
## Adaption of \ac{FFD} for a 3D--Mesh
\label{3dffd}
This is a straightforward extension of the 1D-method presented in the last
This is a straightforward extension of the 1D--method presented in the last
chapter. But this time things get a bit more complicated. As we have a
3-dimensional grid we may have a different amount of control-points in each
3--dimensional grid we may have a different amount of control--points in each
direction.
Given $n,m,o$ control points in $x,y,z$-direction each Point on the curve is
Given $n,m,o$ control points in $x,y,z$--direction each Point on the curve is
defined by
$$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}.$$
In this case we have three different B-Splines (one for each dimension) and also
In this case we have three different B--Splines (one for each dimension) and also
3 variables $u,v,w$ for each vertex we want to approximate.
Given a target vertex $\vec{p}^*$ and an initial guess $\vec{p}=V(u,v,w)$
we define the error-function for the gradient-descent as:
we define the error--function for the gradient--descent as:
$$Err(u,v,w,\vec{p}^{*}) = \vec{p}^{*} - V(u,v,w)$$
@ -368,7 +369,7 @@ $$
\right)
$$
With the Gauss-Newton algorithm we iterate via the formula
With the Gauss--Newton algorithm we iterate via the formula
$$J(Err(u,v,w)) \cdot \Delta \left( \begin{array}{c} u \\ v \\ w \end{array} \right) = -Err(u,v,w)$$
and use Cramers rule for inverting the small Jacobian and solving this system of
linear equations.
@ -378,7 +379,7 @@ linear equations.
- Nachteile von Parametrisierung
- Deformation ist um einen Kontrollpunkt viel direkter zu steuern.
- => DM-FFD?
- => DM--FFD?
# Scenarios for testing evolvability criteria using \acf{FFD}
@ -388,7 +389,7 @@ linear equations.
### Optimierungszenario
- Ebene -> Template-Fit
- Ebene -> Template--Fit
### Matching in 1D
@ -398,7 +399,7 @@ linear equations.
- Analytische Lösung einzig beste
- Ergebnis auch bei Rauschen konstant?
- normierter 1-Vektor auf den Gradienten addieren
- normierter 1--Vektor auf den Gradienten addieren
- Kegel entsteht
## Test Scenario: 3D Function Approximation
@ -419,7 +420,7 @@ linear equations.
# Evaluation of Scenarios
\label{sec:res}
## Spearman/Pearson-Metriken
## Spearman/Pearson--Metriken
- Was ist das?
- Wieso sollte uns das interessieren?
@ -445,7 +446,7 @@ linear equations.
\begin{figure}[!ht]
\includegraphics[width=\textwidth]{img/evolution3d/20170926_3dFit_both_append.png}
\includegraphics[width=\textwidth]{img/evolution3d/20170926_3dFit_all_append.png}
\caption{Results 3D}
\end{figure}

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@ -9,7 +9,11 @@ titlepage,
% pagesize=auto
% openany, % Kapitel koennen auch auf geraden Seiten starten
% draft % schneller compillieren, Bild-dummy
% appendixprefix % Anhang mit Bezeichner
% appendixprefix, % Anhang mit Bezeichner
bibtotocnumbered,
liststotocnumbered,
listof=totocnumbered,
index=totocnumbered,
xcolor=dvipsnames,
]{scrbook}
@ -53,10 +57,10 @@ xcolor=dvipsnames,
\titleformat{name=\chapter,numberless}[hang]{\Huge\bfseries\ }{#1}{20pt}{\Huge\bfseries\ }
\titleformat{\chapter}[hang]{\Huge\bfseries\ }{\color{CadetBlue}\thechapter}{20pt}{\begin{tabular}[t]{@{\color{CadetBlue}\vrule width 2pt}>{\hangindent=20pt\hsp}p{\dimexpr 1\textwidth -44pt}}#1\end{tabular}}
\titleformat{name=\section,numberless}[hang]{\large\bfseries\ }{#1}{32pt}{\large\bfseries\ }
\titleformat{\section}[hang]{\large\bfseries\ }{\color{CadetBlue}\thesection}{32pt}{\begin{tabular}[t]{p{\dimexpr 1\textwidth -44pt}}#1\end{tabular}}
\titleformat{name=\subsection,numberless}[hang]{\bfseries\ }{#1}{27pt}{\bfseries\ }
\titleformat{\subsection}[hang]{\bfseries\ }{\color{CadetBlue}\thesubsection}{27pt}{\begin{tabular}[t]{p{\dimexpr 1\textwidth -44pt}}#1\end{tabular}}
\titleformat{name=\section,numberless}[hang]{\Large\bfseries\ }{#1}{32pt}{\Large\bfseries\ }
\titleformat{\section}[hang]{\Large\bfseries\ }{\color{CadetBlue}\thesection}{32pt}{\begin{tabular}[t]{p{\dimexpr 1\textwidth -44pt}}#1\end{tabular}}
\titleformat{name=\subsection,numberless}[hang]{\large\bfseries\ }{#1}{27pt}{\large\bfseries\ }
\titleformat{\subsection}[hang]{\large\bfseries\ }{\color{CadetBlue}\thesubsection}{27pt}{\begin{tabular}[t]{p{\dimexpr 1\textwidth -44pt}}#1\end{tabular}}
% ### fr 1 seitig
%\usepackage{fancyhdr} %
@ -124,7 +128,8 @@ xcolor=dvipsnames,
%
%%%%%%%%%%%%%%% Verzeichnisse %%%%%%%%%%%%%%%
\frontmatter % Abstrakte Gliederungsebene: Anfang des Buches
\tableofcontents % Rueckseite leer
\renewcommand{\autodot}{}
\tableofcontents % Rueckseite leer
%\lstlistoflistings % fuer listingsverzeichnis mit package listings
%%%%%%%%%%%%%%% Hauptteil %%%%%%%%%%%%%%%
@ -143,11 +148,11 @@ Unless otherwise noted the following holds:
\tightlist
\item
lowercase letters \(x,y,z\)\\
refer to real variables and represent a point in 3D-Space.
refer to real variables and represent a point in 3D--Space.
\item
lowercase letters \(u,v,w\)\\
refer to real variables between \(0\) and \(1\) used as coefficients
in a 3D B-Spline grid.
in a 3D B--Spline grid.
\item
other lowercase letters\\
refer to other scalar (real) variables.
@ -167,10 +172,10 @@ Many modern industrial design processes require advanced optimization
methods do to the increased complexity. These designs have to adhere to
more and more degrees of freedom as methods refine and/or other methods
are used. Examples for this are physical domains like aerodynamic
(i.e.~drag), fluid dynamics (i.e.~throughput of liquid) -- where the
(i.e.~drag), fluid dynamics (i.e.~throughput of liquid) --- where the
complexity increases with the temporal and spatial resolution of the
simulation -- or known hard algorithmic problems in informatics
(i.e.~layouting of circuit boards or stacking of 3D-objects). Moreover
simulation --- or known hard algorithmic problems in informatics
(i.e.~layouting of circuit boards or stacking of 3D--objects). Moreover
these are typically not static environments but requirements shift over
time or from case to case.
@ -178,9 +183,9 @@ Evolutional algorithms cope especially well with these problem domains
while addressing all the issues at hand\cite{minai2006complex}. One of
the main concerns in these algorithms is the formulation of the problems
in terms of a genome and a fitness function. While one can typically use
an arbitrary cost-function for the fitness-functions (i.e.~amount of
drag, amount of space, etc.), the translation of the problem-domain into
a simple parametric representation can be challenging.
an arbitrary cost--function for the fitness--functions (i.e.~amount of
drag, amount of space, etc.), the translation of the problem--domain
into a simple parametric representation can be challenging.
The quality of such a representation in biological evolution is called
\emph{evolvability}\cite{wagner1996complex} and is at the core of this
@ -203,7 +208,7 @@ optimization.
We will replicate the same setup on the same meshes 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
points and evaluate if the evolution--criteria still work as a predictor
given the different deformation scheme, as suspected in
\cite{anrichterEvol}.
@ -211,19 +216,19 @@ given the different deformation scheme, as suspected in
First we introduce different topics in isolation in Chapter
\ref{sec:back}. We take an abstract look at the definition of \ac{FFD}
for a one-dimensional line (in \ref{sec:back:ffd}) and discuss why this
for a one--dimensional line (in \ref{sec:back:ffd}) and discuss why this
is a sensible deformation function (in \ref{sec:back:ffdgood}). Then we
establish some background-knowledge of evolutional algorithms (in
establish some background--knowledge of evolutional algorithms (in
\ref{sec:back:evo}) and why this is useful in our domain (in
\ref{sec:back:evogood}). In a third step we take a look at the
definition of the different evolvability criteria established in
\cite{anrichterEvol}.
In Chapter \ref{sec:impl} we take a look at our implementation of
\ac{FFD} and the adaptation for 3D-meshes.
\ac{FFD} and the adaptation for 3D--meshes.
Next, in Chapter \ref{sec:eval}, we describe the different scenarios we
use to evaluate the different evolvability-criteria incorporating all
use to evaluate the different evolvability--criteria incorporating all
aspects introduced in Chapter \ref{sec:back}. Following that, we
evaluate the results in Chapter \ref{sec:res} with further on discussion
in Chapter \ref{sec:dis}.
@ -238,8 +243,8 @@ in Chapter \ref{sec:dis}.
First of all we have to establish how a \ac{FFD} works and why this is a
good tool for deforming meshes in the first place. For simplicity we
only summarize the 1D-case from \cite{spitzmuller1996bezier} here and go
into the extension to the 3D case in chapter \ref{3dffd}.
only summarize the 1D--case from \cite{spitzmuller1996bezier} here and
go into the extension to the 3D case in chapter \ref{3dffd}.
Given an arbitrary number of points \(p_i\) alongside a line, we map a
scalar value \(\tau_i \in [0,1[\) to each point with
@ -258,7 +263,7 @@ N_{i,d,\tau}(u) = \frac{u-\tau_i}{\tau_{i+d}} N_{i,d-1,\tau}(u) + \frac{\tau_{i+
If we now multiply every \(p_i\) with the corresponding
\(N_{i,d,\tau_i}(u)\) we get the contribution of each point \(p_i\) to
the final curve-point parameterized only by \(u \in [0,1[\). As can be
the final curve--point parameterized only by \(u \in [0,1[\). As can be
seen from \eqref{eqn:ffd1d2} we only access points \([i..i+d]\) for any
given \(i\)\footnote{one more for each recursive step.}, which gives us,
in combination with choosing \(p_i\) and \(\tau_i\) in order, only a
@ -266,12 +271,12 @@ local interference of \(d+1\) points.
We can even derive this equation straightforward for an arbitrary
\(N\)\footnote{\emph{Warning:} in the case of \(d=1\) the
recursion-formula yields a \(0\) denominator, but \(N\) is also \(0\).
The right solution for this case is a derivative of \(0\)}:
recursion--formula yields a \(0\) denominator, but \(N\) is also
\(0\). The right solution for this case is a derivative of \(0\)}:
\[\frac{\partial}{\partial u} N_{i,d,r}(u) = \frac{d}{\tau_{i+d} - \tau_i} N_{i,d-1,\tau}(u) - \frac{d}{\tau_{i+d+1} - \tau_{i+1}} N_{i+1,d-1,\tau}(u)\]
For a B-Spline \[s(u) = \sum_{i} N_{i,d,\tau_i}(u) p_i\] these
For a B--Spline \[s(u) = \sum_{i} N_{i,d,\tau_i}(u) p_i\] these
derivations yield \(\frac{\partial^d}{\partial u} s(u) = 0\).
Another interesting property of these recursive polynomials is that they
@ -288,7 +293,7 @@ function?}{Why is a good deformation function?}}\label{why-is-a-good-deformatio
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
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
@ -300,12 +305,12 @@ as creating a plateau can be difficult to
achieve\cite[chapter~3.2]{hsu1991dmffd}\cite{hsu1992direct}.
This disadvantages led to the formulation of
\acf{DM-FFD}\cite[chapter~3.3]{hsu1991dmffd} in which the user directly
interacts with the surface-mesh. All interactions will be applied
proportionally to the control-points that make up the parametrization of
the interaction-point itself yielding a smooth deformation of the
\acf{DM--FFD}\cite[chapter~3.3]{hsu1991dmffd} in which the user directly
interacts with the surface--mesh. All interactions will be applied
proportionally to the control--points that make up the parametrization
of the interaction--point itself yielding a smooth deformation of the
surface \emph{at} the surface without seemingly arbitrary scattered
control-points. Moreover this increases the efficiency of an
control--points. Moreover this increases the efficiency of an
evolutionary optimization\cite{Menzel2006}, which we will use later on.
\begin{figure}[!ht]
@ -318,8 +323,8 @@ But this approach also has downsides as can be seen in figure
\ref{fig:hsu_fig7}, as the tessellation of the invisible grid has a
major impact on the deformation itself.
All in all \ac{FFD} and \ac{DM-FFD} are still good ways to deform a
high-polygon mesh albeit the downsides.
All in all \ac{FFD} and \ac{DM--FFD} are still good ways to deform a
high--polygon mesh albeit the downsides.
\section{What is evolutional
optimization?}\label{what-is-evolutional-optimization}
@ -335,12 +340,12 @@ algorithms}\label{advantages-of-evolutional-algorithms}
\change[inline]{Needs citations} The main advantage of evolutional
algorithms is the ability to find optima of general functions just with
the help of a given error-function (or fitness-function in this domain).
This avoids the general pitfalls of gradient-based procedures, which
often target the same error-function as an evolutional algorithm, but
can get stuck in local optima.
the help of a given error--function (or fitness--function in this
domain). This avoids the general pitfalls of gradient--based procedures,
which often target the same error--function as an evolutional algorithm,
but can get stuck in local optima.
This is mostly due to the fact that a gradient-based procedure has only
This is mostly due to the fact that a gradient--based procedure has only
one point of observation from where it evaluates the next steps, whereas
an evolutional strategy starts with a population of guessed solutions.
Because an evolutional strategy modifies the solution randomly, keeps
@ -348,7 +353,7 @@ the best solutions and purges the worst, it can also target multiple
different hypothesis at the same time where the local optima die out in
the face of other, better candidates.
If an analytic best solution exists (i.e.~because the error-function is
If an analytic best solution exists (i.e.~because the error--function is
convex) an evolutional algorithm is not the right choice. Although both
converge to the same solution, the analytic one is usually faster. But
in reality many problems have no analytic solution, because the problem
@ -364,33 +369,33 @@ deformations}\label{criteria-for-the-evolvability-of-linear-deformations}
In \cite{anrichterEvol} \emph{variability} is defined as
\[V(\vec{U}) := \frac{\textrm{rank}(\vec{U})}{n},\] whereby \(\vec{U}\)
is the \(m \times n\) deformation-Matrix used to map the \(m\) control
is the \(m \times n\) deformation--Matrix used to map the \(m\) control
points onto the \(n\) vertices.
Given \(n = m\), an identical number of control-points and 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 the solution is to trivially move every control-point
onto a target-point.
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 \(m \ll n\).
there are only few control--points for many vertices, so \(m \ll n\).
Additionally in our setup we connect neighbouring control-points in a
Additionally in our setup we connect neighbouring control--points in a
grid so each control point is not independent, but typically depends on
\(4^d\) control-points for an \(d\)-dimensional control mesh.
\(4^d\) control--points for an \(d\)--dimensional control mesh.
\subsection{Regularity}\label{regularity}
\emph{Regularity} is defined\cite{anrichterEvol} as
\[R(\vec{U}) := \frac{1}{\kappa(\vec{U})} = \frac{\sigma_{min}}{\sigma_{max}}\]
where \(\sigma_{min}\) and \(\sigma_{max}\) are the smallest and
greatest right singular value of the deformation-matrix \(\vec{U}\).
greatest right singular 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 optimal value and
criterion--range to \([0..1]\), whereas \(1\) is the optimal value and
\(0\) is the worst value.
This criterion should be characteristic for numeric stability on the on
@ -402,7 +407,7 @@ 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
\emph{regularity}, which are agnostic of the fitness--function at hand
the third criterion should reflect a notion of potential.
As during optimization some kind of gradient \(g\) is available to
@ -412,87 +417,83 @@ can be achieved in the given direction.
The definition for an \emph{improvement potential} \(P\)
is\cite{anrichterEvol}: \[
P(\vec{U}) := 1 - \|(\vec{1} - \vec{UU}^+)\vec(G)\|^2_F
\] given some approximate \(n \times d\) fitness-gradient \(\vec{G}\),
\] given some approximate \(n \times d\) fitness--gradient \(\vec{G}\),
normalized to \(\|\vec{G}\|_F = 1\), whereby \(\|\cdot\|_F\) denotes the
Frobenius-Norm.
Frobenius--Norm.
\chapter{\texorpdfstring{Implementation of
\acf{FFD}}{Implementation of }}\label{implementation-of}
\label{sec:impl}
The general formulation of B-Splines has two free parameters \(d\) and
The general formulation of B--Splines has two free parameters \(d\) and
\(\tau\) which must be chosen beforehand.
As we usually work with regular grids in our \ac{FFD} we define \(\tau\)
statically as \(\tau_i = \nicefrac{i}{n}\) whereby \(n\) is the number
of control-points in that direction.
of control--points in that direction.
\(d\) defines the \emph{degree} of the B-Spline-Function (the number of
times this function is differentiable) and for our purposes we fix \(d\)
to \(3\), but give the formulas for the general case so it can be
\(d\) defines the \emph{degree} of the B--Spline--Function (the number
of times this function is differentiable) and for our purposes we fix
\(d\) to \(3\), but give the formulas for the general case so it can be
adapted quite freely.
\section{\texorpdfstring{Adaption of
\ac{FFD}}{Adaption of }}\label{adaption-of}
As we have established in Chapter \ref{sec:back:ffd} we can define an
\ac{FFD}-displacement as
\ac{FFD}--displacement as
\begin{equation}
\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
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 \[
\textrm{Deform}(v_x) = v_x + \Delta_x(u)
\] with \(u \in [0..1[\) being the variable that connects the
high-detailed vertex-mesh to the low-detailed control-grid. To actually
calculate the new position of the vertex we first have to calculate the
\(u\)-value for each vertex. This is achieved by finding out the
parametrization of \(v\) in terms of \(c_i\) \[
high--detailed vertex--mesh to the low--detailed control--grid. To
actually calculate the new position of the vertex we first have to
calculate the \(u\)--value for each vertex. This is achieved by finding
out the parametrization of \(v\) in terms of \(c_i\) \[
v_x \overset{!}{=} \sum_i N_{i,d,\tau_i}(u) c_i
\] so we can minimize the error between those two: \[
\underset{u}{\argmin}\,Err(u,v_x) = \underset{u}{\argmin}\,2 \cdot \|v_x - \sum_i N_{i,d,\tau_i}(u) c_i\|^2_2
\]
As this error-term is quadratic we just derive by \(u\) yielding
\begin{eqnarray*}
& \frac{\partial}{\partial u} & v_x - \sum_i N_{i,d,\tau_i}(u) c_i \\
& = & - \sum_i \left( \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) \right) c_i
\end{eqnarray*}
and do a gradient-descend to approximate the value of \(u\) up to an
\] As this error--term is quadratic we just derive by \(u\) yielding \[
\begin{array}{rl}
\frac{\partial}{\partial u} & v_x - \sum_i N_{i,d,\tau_i}(u) c_i \\
= & - \sum_i \left( \frac{d}{\tau_{i+d} - \tau_i} N_{i,d-1,\tau}(u) - \frac{d}{\tau_{i+d+1} - \tau_{i+1}} N_{i+1,d-1,\tau}(u) \right) c_i
\end{array}
\] and do a gradient--descend to approximate the value of \(u\) up to an
\(\epsilon\) of \(0.0001\).
For this we use the Gauss-Newton algorithm\cite{gaussNewton} as the
For this we use the Gauss--Newton algorithm\cite{gaussNewton} as the
solution to this problem may not be deterministic, because we usually
have way more vertices than control points (\(\#v \gg \#c\)).
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-3d-mesh}
3D--Mesh}{Adaption of for a 3D--Mesh}}\label{adaption-of-for-a-3dmesh}
\label{3dffd}
This is a straightforward extension of the 1D-method presented in the
This is a straightforward extension of the 1D--method presented in the
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.
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
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)
In this case we have three different B--Splines (one for each dimension)
and also 3 variables \(u,v,w\) for each vertex we want to approximate.
Given a target vertex \(\vec{p}^*\) and an initial guess
\(\vec{p}=V(u,v,w)\) we define the error-function for the
gradient-descent as:
\(\vec{p}=V(u,v,w)\) we define the error--function for the
gradient--descent as:
\[Err(u,v,w,\vec{p}^{*}) = \vec{p}^{*} - V(u,v,w)\]
@ -533,7 +534,7 @@ J(Err(u,v,w)) =
\right)
\]
With the Gauss-Newton algorithm we iterate via the formula
With the Gauss--Newton algorithm we iterate via the formula
\[J(Err(u,v,w)) \cdot \Delta \left( \begin{array}{c} u \\ v \\ w \end{array} \right) = -Err(u,v,w)\]
and use Cramers rule for inverting the small Jacobian and solving this
system of linear equations.
@ -547,7 +548,7 @@ system of linear equations.
\item
Deformation ist um einen Kontrollpunkt viel direkter zu steuern.
\item
=\textgreater{} DM-FFD?
=\textgreater{} DM--FFD?
\end{itemize}
\chapter{\texorpdfstring{Scenarios for testing evolvability criteria
@ -564,7 +565,7 @@ Approximation}\label{test-scenario-1d-function-approximation}
\begin{itemize}
\tightlist
\item
Ebene -\textgreater{} Template-Fit
Ebene -\textgreater{} Template--Fit
\end{itemize}
\subsection{Matching in 1D}\label{matching-in-1d}
@ -585,7 +586,7 @@ Auswertung}\label{besonderheiten-der-auswertung}
\item
Ergebnis auch bei Rauschen konstant?
\item
normierter 1-Vektor auf den Gradienten addieren
normierter 1--Vektor auf den Gradienten addieren
\begin{itemize}
\tightlist
@ -628,7 +629,7 @@ Optimierung}\label{besonderheiten-der-optimierung}
\label{sec:res}
\section{Spearman/Pearson-Metriken}\label{spearmanpearson-metriken}
\section{Spearman/Pearson--Metriken}\label{spearmanpearsonmetriken}
\begin{itemize}
\tightlist
@ -668,28 +669,28 @@ Approximation}\label{results-of-3d-function-approximation}
HAHA .. als ob -.-
\backmatter
% \backmatter
\cleardoublepage
\renewcommand\thesection{\Roman{section}}
\addtocontents{toc}{\protect\setcounter{tocdepth}{1}}
\setcounter{section}{1} % reset section to 1 so its stars I, II, III,...
\renewcommand\thechapter{\Alph{chapter}}
\chapter*{Appendix}
\addcontentsline{toc}{chapter}{\protect\numberline{}Appendix}
\addtocontents{toc}{\protect\setcounter{tocdepth}{1}}
\setcounter{chapter}{0} % reset section to 1 so its stars I, II, III,...
\pagenumbering{roman}
%%%%%%%%%%%%%%% Literaturverzeichnis %%%%%%%%%%%%%%%
\bibliographystyle{unsrtdin} % \bibliographystyle{natdin}
\bibliography{bibma}
\addcontentsline{toc}{section}{\protect\numberline{\thesection}Bibliography} % Literaturverzeichnis in das Inhaltsverzeichnis aufnehmen
\addtocounter{section}{1}
% \addcontentsline{toc}{chapter}{\protect\numberline{\thechapter}Bibliography} % Literaturverzeichnis in das Inhaltsverzeichnis aufnehmen
% \addtocounter{chapter}{1}
\newpage
%%%%%%%%%%%%%%% Anhang %%%%%%%%%%%%%%%
% \clearpage %spaeter alles wieder rein
% % \input{files/appendix}
\input{settings/abkuerzungen}
\addcontentsline{toc}{section}{\protect\numberline{\thesection}Abbreviations}
\addtocounter{section}{1}
% \addcontentsline{toc}{chapter}{\protect\numberline{\thechapter}Abbreviations}
% \addtocounter{chapter}{1}
\newpage
% \listofalgorithms
@ -698,10 +699,13 @@ HAHA .. als ob -.-
% \newpage
%
\listoffigures
% \addcontentsline{toc}{chapter}{\protect\numberline{\thechapter}List of Figures}
% \addtocounter{chapter}{1}
\newpage
% \listoftables
\listoftodos
\addcontentsline{toc}{section}{\protect\numberline{\thesection}TODOs}
\addtocounter{section}{1}
% \addcontentsline{toc}{chapter}{\protect\numberline{\thechapter}TODOs}
% \addtocounter{chapter}{1}
\newpage
% \printindex

6
arbeit/settings/abkuerzungen.tex
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@ -1,4 +1,4 @@
\chapter*{Abbreviations}
\chapter{Abbreviations}
\label{cha:abbrev}
\begin{acronym}
% Zugriff ueber \ac{BWT} 1te mal Vollreferenz, danach Abk.
@ -10,8 +10,8 @@
%
%\acro{GPL}{GNU General Public License} --
% License for free software, see \url{http://www.gnu.org/copyleft/gpl.html}.
\acro{FFD}{Freeform-Deformation}
\acro{DM-FFD}{Direct Manipulation Freeform-Deformation}
\acro{FFD}{Freeform--Deformation}
\acro{DM--FFD}{Direct Manipulation Freeform--Deformation}
\acro{RBF}{Radial Basis Function}
%

40
arbeit/template.tex
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@ -9,7 +9,11 @@ titlepage,
% pagesize=auto
% openany, % Kapitel koennen auch auf geraden Seiten starten
% draft % schneller compillieren, Bild-dummy
% appendixprefix % Anhang mit Bezeichner
% appendixprefix, % Anhang mit Bezeichner
bibtotocnumbered,
liststotocnumbered,
listof=totocnumbered,
index=totocnumbered,
xcolor=dvipsnames,
]{scrbook}
@ -53,10 +57,10 @@ xcolor=dvipsnames,
\titleformat{name=\chapter,numberless}[hang]{\Huge\bfseries\ }{#1}{20pt}{\Huge\bfseries\ }
\titleformat{\chapter}[hang]{\Huge\bfseries\ }{\color{CadetBlue}\thechapter}{20pt}{\begin{tabular}[t]{@{\color{CadetBlue}\vrule width 2pt}>{\hangindent=20pt\hsp}p{\dimexpr 1\textwidth -44pt}}#1\end{tabular}}
\titleformat{name=\section,numberless}[hang]{\large\bfseries\ }{#1}{32pt}{\large\bfseries\ }
\titleformat{\section}[hang]{\large\bfseries\ }{\color{CadetBlue}\thesection}{32pt}{\begin{tabular}[t]{p{\dimexpr 1\textwidth -44pt}}#1\end{tabular}}
\titleformat{name=\subsection,numberless}[hang]{\bfseries\ }{#1}{27pt}{\bfseries\ }
\titleformat{\subsection}[hang]{\bfseries\ }{\color{CadetBlue}\thesubsection}{27pt}{\begin{tabular}[t]{p{\dimexpr 1\textwidth -44pt}}#1\end{tabular}}
\titleformat{name=\section,numberless}[hang]{\Large\bfseries\ }{#1}{32pt}{\Large\bfseries\ }
\titleformat{\section}[hang]{\Large\bfseries\ }{\color{CadetBlue}\thesection}{32pt}{\begin{tabular}[t]{p{\dimexpr 1\textwidth -44pt}}#1\end{tabular}}
\titleformat{name=\subsection,numberless}[hang]{\large\bfseries\ }{#1}{27pt}{\large\bfseries\ }
\titleformat{\subsection}[hang]{\large\bfseries\ }{\color{CadetBlue}\thesubsection}{27pt}{\begin{tabular}[t]{p{\dimexpr 1\textwidth -44pt}}#1\end{tabular}}
% ### fr 1 seitig
%\usepackage{fancyhdr} %
@ -124,7 +128,8 @@ xcolor=dvipsnames,
%
%%%%%%%%%%%%%%% Verzeichnisse %%%%%%%%%%%%%%%
\frontmatter % Abstrakte Gliederungsebene: Anfang des Buches
\tableofcontents % Rueckseite leer
\renewcommand{\autodot}{}
\tableofcontents % Rueckseite leer
%\lstlistoflistings % fuer listingsverzeichnis mit package listings
%%%%%%%%%%%%%%% Hauptteil %%%%%%%%%%%%%%%
@ -134,28 +139,28 @@ xcolor=dvipsnames,
\pagenumbering{arabic}
$body$
\backmatter
% \backmatter
\cleardoublepage
\renewcommand\thesection{\Roman{section}}
\addtocontents{toc}{\protect\setcounter{tocdepth}{1}}
\setcounter{section}{1} % reset section to 1 so its stars I, II, III,...
\renewcommand\thechapter{\Alph{chapter}}
\chapter*{Appendix}
\addcontentsline{toc}{chapter}{\protect\numberline{}Appendix}
\addtocontents{toc}{\protect\setcounter{tocdepth}{1}}
\setcounter{chapter}{0} % reset section to 1 so its stars I, II, III,...
\pagenumbering{roman}
%%%%%%%%%%%%%%% Literaturverzeichnis %%%%%%%%%%%%%%%
\bibliographystyle{unsrtdin} % \bibliographystyle{natdin}
\bibliography{bibma}
\addcontentsline{toc}{section}{\protect\numberline{\thesection}Bibliography} % Literaturverzeichnis in das Inhaltsverzeichnis aufnehmen
\addtocounter{section}{1}
% \addcontentsline{toc}{chapter}{\protect\numberline{\thechapter}Bibliography} % Literaturverzeichnis in das Inhaltsverzeichnis aufnehmen
% \addtocounter{chapter}{1}
\newpage
%%%%%%%%%%%%%%% Anhang %%%%%%%%%%%%%%%
% \clearpage %spaeter alles wieder rein
% % \input{files/appendix}
\input{settings/abkuerzungen}
\addcontentsline{toc}{section}{\protect\numberline{\thesection}Abbreviations}
\addtocounter{section}{1}
% \addcontentsline{toc}{chapter}{\protect\numberline{\thechapter}Abbreviations}
% \addtocounter{chapter}{1}
\newpage
% \listofalgorithms
@ -164,10 +169,13 @@ $body$
% \newpage
%
\listoffigures
% \addcontentsline{toc}{chapter}{\protect\numberline{\thechapter}List of Figures}
% \addtocounter{chapter}{1}
\newpage
% \listoftables
\listoftodos
\addcontentsline{toc}{section}{\protect\numberline{\thesection}TODOs}
\addtocounter{section}{1}
% \addcontentsline{toc}{chapter}{\protect\numberline{\thechapter}TODOs}
% \addtocounter{chapter}{1}
\newpage
% \printindex

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