282 lines
11 KiB
Markdown
282 lines
11 KiB
Markdown
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fontsize: 11pt
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---
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\chapter*{How to read this Thesis}
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As a guide through the nomenclature used in the formulas we prepend this
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chapter.
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Unless otherwise noted the following holds:
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- lowercase letters $x,y,z$
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refer to real variables and represent a point in 3D-Space.
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- lowercase letters $u,v,w$
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refer to real variables between $0$ and $1$ used as coefficients in a 3D
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B-Spline grid.
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- other lowercase letters
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refer to other scalar (real) variables.
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- lowercase **bold** letters (e.g. $\vec{x},\vec{y}$)
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refer to 3D coordinates
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- uppercase **BOLD** letters (e.g. $\vec{D}, \vec{M}$)
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refer to Matrices
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# Introduction
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In this Master Thesis we try to extend a previously proposed concept of
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predicting the evolvability of \acf{FFD} given a
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Deformation-Matrix\cite{anrichterEvol}. In the original publication the author
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used random sampled points weighted with \acf{RBF} to deform the mesh and
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defined three different criteria that can be calculated prior to using an
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evolutional optimization algorithm to asses the quality and potential of such
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optimization.
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We will replicate the same setup on the same meshes but use \acf{FFD} instead of
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\acf{RBF} to create a deformation and evaluate if the evolution-criteria still
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work as a predictor given the different deformation scheme.
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## What is \acf{FFD}?
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First of all we have to establish how a \ac{FFD} works and why this is a good
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tool for deforming meshes in the first place. For simplicity we only summarize
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the 1D-case from \cite{spitzmuller1996bezier} here and go into the extension to
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the 3D case in chapter \ref{3dffd}.
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Given an arbitrary number of points $p_i$ alongside a line, we map a scalar
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value $\tau_i \in [0,1[$ to each point with $\tau_i < \tau_{i+1} \forall i$.
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Given a degree of the target polynomial $d$ we define the curve
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$N_{i,d,\tau_i}(u)$ as follows:
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\begin{equation} \label{eqn:ffd1d1}
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N_{i,0,\tau}(u) = \begin{cases} 1, & u \in [\tau_i, \tau_{i+1}[ \\ 0, & \mbox{otherwise} \end{cases}
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\end{equation}
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and
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\begin{equation} \label{eqn:ffd1d2}
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N_{i,d,\tau}(u) = \frac{u-\tau_i}{\tau_{i+d}} N_{i,d-1,\tau}(u) + \frac{\tau_{i+d+1} - u}{\tau_{i+d+1}-\tau_{i+1}} N_{i+1,d-1,\tau}(u)
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\end{equation}
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If we now multiply every $p_i$ with the corresponding $N_{i,d,\tau_i}(u)$ we get
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the contribution of each point $p_i$ to the final curve-point parameterized only
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by $u \in [0,1[$. As can be seen from \eqref{eqn:ffd1d2} we only access points
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$[i..i+d]$ for any given $i$^[one more for each recursive step.], which gives
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us, in combination with choosing $p_i$ and $\tau_i$ in order, only a local
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interference of $d+1$ points.
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We can even derive this equation straightforward for an arbitrary
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$N$^[*Warning:* in the case of $d=1$ the recursion-formula yields a $0$
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denominator, but $N$ is also $0$. The right solution for this case is a
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derivative of $0$]:
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$$\frac{\partial}{\partial u} N_{i,d,r}(u) = \frac{d}{\tau_{i+d} - \tau_i} N_{i,d-1,\tau}(u) - \frac{d}{\tau_{i+d+1} - \tau_{i+1}} N_{i+1,d-1,\tau}(u)$$
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For a B-Spline
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$$s(u) = \sum_{i} N_{i,d,\tau_i}(u) p_i$$
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these derivations yield $\frac{\partial^d}{\partial u} s(u) = 0$.
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Another interesting property of these recursive polynomials is that they are
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continuous (given $d \ge 1$) as every $p_i$ gets blended in linearly between
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$\tau_i$ and $\tau_{i+d}$ and out linearly between $\tau_{i+1}$ and
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$\tau_{i+d+1}$ as can bee seen from the two coefficients in every step of the
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recursion.
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### Why is \ac{FFD} a good deformation function?
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The usage of \ac{FFD} as a tool for manipulating follows directly from the
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properties of the polynomials and the correspondence to the control points.
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Having only a few control points gives the user a nicer high-level-interface, as
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she only needs to move these points and the model follows in an intuitive
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manner. The deformation is smooth as the underlying polygon is smooth as well
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and affects as many vertices of the model as needed. Moreover the changes are
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always local so one risks not any change that a user cannot immediately see.
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But there are also disadvantages of this approach. The user loses the ability to
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directly influence vertices and even seemingly simple tasks as creating a
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plateau can be difficult to
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achieve\cite[chapter~3.2]{hsu1991dmffd}\cite{hsu1992direct}.
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This disadvantages led to the formulation of
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\acf{DM-FFD}\cite[chapter~3.3]{hsu1991dmffd} in which the user directly
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interacts with the surface-mesh. All interactions will be applied
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proportionally to the control-points that make up the parametrization of the
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interaction-point itself yielding a smooth deformation of the surface *at* the
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surface without seemingly arbitrary scattered control-points. Moreover this
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increases the efficiency of an evolutionary optimization\cite{Menzel2006}, which
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we will use later on.
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But this approach also has downsides as can be seen in
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\cite[figure~7]{hsu1991dmffd}\unsure{figure hier einfügen?}, as the tessellation
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of the invisible grid has a major impact on the deformation itself.
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All in all \ac{FFD} and \ac{DM-FFD} are still good ways to deform a high-polygon
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mesh albeit the downsides.
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## What is evolutional optimization?
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## Advantages of evolutional algorithms
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\improvement[inline]{Needs citations}
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The main advantage of evolutional algorithms is the ability to find optima of
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general functions just with the help of a given error-function (or
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fitness-function in this domain). This avoids the general pitfalls of
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gradient-based procedures, which often target the same error-function as an
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evolutional algorithm, but can get stuck in local optima.
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This is mostly due to the fact that a gradient-based procedure has only one
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point of observation from where it evaluates the next steps, whereas an
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evolutional strategy starts with a population of guessed solutions. Because an
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evolutional strategy modifies the solution randomly, keeps the best solutions
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and purges the worst, it can also target multiple different hypothesis at the
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same time where the local optima die out in the face of other, better
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candidates.
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If an analytic best solution exists (i.e. because the error-function is convex)
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an evolutional algorithm is not the right choice. Although both converge to the
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same solution, the analytic one is usually faster. But in reality many problems
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have no analytic solution, because the problem is not convex. Here evolutional
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optimization has one more advantage as you get bad solutions fast, which refine
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over time.
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## Criteria for the evolvability of linear deformations
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### Variability
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In \cite{anrichterEvol} variability is defined as
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$$V(\vec{U}) := \frac{\textrm{rank}(\vec{U})}{n},$$
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whereby $\vec{U}$ is the $m \times n$ deformation-Matrix used to map the $m$
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control points onto the $n$ vertices.
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Given $n = m$, an identical number of control-points and vertices, this
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quotient will be $=1$ if all control points are independent of each other and
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the solution is to trivially move every control-point onto a target-point.
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In praxis the value of $V(\vec{U})$ is typically $\ll 1$, because as
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there are only few control-points for many vertices, so $m \ll n$.
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Additionally in our setup we connect neighbouring control-points in a grid so
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each control point is not independent, but typically depends on $4^d$
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control-points for an $d$-dimensional control mesh.
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### Regularity
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Regularity is defined\cite{anrichterEvol} as
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$$R(\vec{U}) := \frac{1}{\kappa(\vec{U})} = \frac{\sigma_{min}}{\sigma_{max}}$$
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where $\sigma_{min}$ and $\sigma_{max}$ are the smallest and greatest right singular
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value of the deformation-matrix $\vec{U}$.
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As we deform the given Object only based on the parameters as $\vec{p} \mapsto
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f(\vec{x} + \vec{U}\vec{p})$ this makes sure that $\|\vec{Up}\| \propto
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\|\vec{p}\|$ when $\kappa(\vec{U}) \approx 1$. The inversion of $\kappa(\vec{U})$
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is only performed to map the criterion-range to $[0..1]$, whereas $1$ is the
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optimal value and $0$ is the worst value.
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This criterion should be characteristic for numeric stability on the on
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hand\cite[chapter 2.7]{golub2012matrix} and for convergence speed of evolutional
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algorithms on the other hand\cite{anrichterEvol} as it is tied to the notion of
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locality\cite{weise2012evolutionary,thorhauer2014locality}.
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### Improvement Potential
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In contrast to the general nature of variability and regularity, which are
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agnostic of the fitness-function at hand the third criterion should reflect a
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notion of potential.
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As during optimization some kind of gradient $g$ is available to suggest a
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direction worth pursuing we use this to guess how much change can be achieved in
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the given direction.
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The definition for an improvement potential $P$ is\cite{anrichterEvol}:
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$$
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P(\vec{U}) := 1 - \|(\vec{1} - \vec{UU}^+)\vec(G)\|^2_F
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$$
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given some approximate $n \times d$ fitness-gradient $\vec{G}$, normalized to
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$\|\vec{G}\|_F = 1$, whereby $\|\cdot\|_F$ denotes the Frobenius-Norm.
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# Implementation of \acf{FFD}
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## Was ist FFD?
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\label{3dffd}
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- Definition
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- Wieso Newton-Optimierung?
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- Was folgt daraus?
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## Test Scenario: 1D Function Approximation
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### Optimierungszenario
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- Ebene -> Template-Fit
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### Matching in 1D
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- Trivial
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### Besonderheiten der Auswertung
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- Analytische Lösung einzig beste
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- Ergebnis auch bei Rauschen konstant?
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- normierter 1-Vektor auf den Gradienten addieren
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- Kegel entsteht
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## Test Scenario: 3D Function Approximation
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### Optimierungsszenario
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- Ball zu Mario
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### Matching in 3D
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- alternierende Optimierung
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### Besonderheiten der Optimierung
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- Analytische Lösung nur bis zur Optimierung der ersten Punkte gültig
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- Kriterien trotzdem gut
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# Evaluation of Scenarios
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## Spearman/Pearson-Metriken
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- Was ist das?
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- Wieso sollte uns das interessieren?
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- Wieso reicht Monotonie?
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- Haben wir das gezeigt?
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- Statistik, Bilder, blah!
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## Results of 1D Function Approximation
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\begin{figure}[!ht]
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\includegraphics[width=\textwidth]{img/evolution1d/20170830-evolution1D_5x5_100Times-all_appended.png}
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\caption{Results 1D}
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\end{figure}
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<!--  -->
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<!-- -->
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<!--  -->
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<!-- -->
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<!--  -->
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## Results of 3D Function Approximation
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\begin{figure}[!ht]
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\includegraphics[width=\textwidth]{img/evolution3d/20170926_3dFit_both_append.png}
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\caption{Results 3D}
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\end{figure}
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<!--  -->
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<!-- -->
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<!--  -->
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<!-- -->
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<!--  -->
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# Schluss
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HAHA .. als ob -.-
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