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@ -204,12 +204,12 @@ $\|\vec{G}\|_F = 1$, whereby $\|\cdot\|_F$ denotes the Frobenius-Norm.
# Implementation of \acf{FFD} # Implementation of \acf{FFD}
As general B-Splines have a 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 $\tau$ As we usually work with regular grids in our \ac{FFD} we define $\tau$
statically as statically as $\tau_i = \nicefrac{i}{n}$ whereby $n$ is the number of
$$\tau_i = \nicefrac{i}{n}$$ control-points in that direction.
whereby $n$ is the number of 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 function is differentiable) and for our purposes we fix $d$ to $3$, but give the
@ -237,16 +237,18 @@ 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 vertex. This is achieved by finding out the parametrization of $v$ in terms of
$c_i$ $c_i$
$$ $$
v_x = \sum_i N_{i,d,\tau_i}(u) 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 the B-Spline-functions are smooth and convex we just derive by $u$ yielding As this error-term is quadratic we just derive by $u$ yielding
\begin{eqnarray*} \begin{eqnarray*}
& \frac{\partial}{\partial u} & v_x - \sum_i N_{i,d,\tau_i}(u) c_i \\ & \frac{\partial}{\partial u} & v_x - \sum_i N_{i,d,\tau_i}(u) c_i \\
& = & v_x - \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 & = & - \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*} \end{eqnarray*}
and do a gradient-descend to approximate the value of $u$ up to an $\epsilon$ of $0.0001$. 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
@ -263,7 +265,7 @@ 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 defined by
$$V(u,v,w) = \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot C_{ijk}.$$ $$V(u,v,w) = \sum_i \sum_j \sum_k N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot C_{ijk}.$$
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. 3 variables $u,v,w$ for each vertex we want to approximate.
@ -275,14 +277,14 @@ $$Err(u,v,w,\vec{p}^{*}) = \vec{p}^{*} - V(u,v,w)$$
And the partial version for just one direction as And the partial version for just one direction as
$$Err_x(u,v,w,\vec{p}^{*}) = p^{*}_x - \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} {C_{ijk}}_x N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) $$ $$Err_x(u,v,w,\vec{p}^{*}) = p^{*}_x - \sum_i \sum_j \sum_k N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot {c_{ijk}}_x $$
To solve this we derive partially, like before: To solve this we derive partially, like before:
$$ $$
\begin{array}{rl} \begin{array}{rl}
\displaystyle \frac{\partial Err_x}{\partial u} & p^{*}_x - \displaystyle \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} {C_{ijk}}_x N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \\ \displaystyle \frac{\partial Err_x}{\partial u} & p^{*}_x - \displaystyle \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}}_x \\
= & \displaystyle - \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} {C_{ijk}}_x N'_i(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) = & \displaystyle - \sum_i \sum_j \sum_k N'_i(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot {c_{ijk}}_x
\end{array} \end{array}
$$ $$
@ -303,7 +305,7 @@ $$
Like leaving out Sums & $i,j,k$-Indices to make obvious what derivative belongs Like leaving out Sums & $i,j,k$-Indices to make obvious what derivative belongs
where in what case?} where in what case?}
With the Gauss-Newton algorithm we iterate the formula With the Gauss-Newton algorithm we iterate via the formula
$$J(Err(u,v,w)) \cdot \Delta \left( \begin{array}{c} u \\ v \\ w \end{array} \right) = -Err(u,v,w)$$ $$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 and use Cramers rule for inverting the small Jacobian and solving this system of
linear equations. linear equations.

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arbeit/ma.tex
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@ -237,7 +237,7 @@ evolutionary optimization\cite{Menzel2006}, which we will use later on.
\begin{figure}[!ht] \begin{figure}[!ht]
\includegraphics[width=\textwidth]{img/hsu_fig7.png} \includegraphics[width=\textwidth]{img/hsu_fig7.png}
\caption{Figure 7 from \cite{hsu1991dmffd}} \caption{Figure 7 from \cite{hsu1991dmffd}.}
\label{fig:hsu_fig7} \label{fig:hsu_fig7}
\end{figure} \end{figure}
@ -340,10 +340,11 @@ Frobenius-Norm.
\chapter{\texorpdfstring{Implementation of \chapter{\texorpdfstring{Implementation of
\acf{FFD}}{Implementation of }}\label{implementation-of} \acf{FFD}}{Implementation of }}\label{implementation-of}
As general B-Splines have a 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 \(\tau\) 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 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 \(d\) defines the \emph{degree} of the B-Spline-Function (the number of
@ -372,15 +373,16 @@ 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 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 \(u\)-value for each vertex. This is achieved by finding out the
parametrization of \(v\) in terms of \(c_i\) \[ parametrization of \(v\) in terms of \(c_i\) \[
v_x = \sum_i N_{i,d,\tau_i}(u) c_i v_x \overset{!}{=} \sum_i N_{i,d,\tau_i}(u) c_i
\] 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 the B-Spline-functions are smooth and convex we just derive by \(u\) As this error-term is quadratic we just derive by \(u\) yielding
yielding
\begin{eqnarray*} \begin{eqnarray*}
& \frac{\partial}{\partial u} & v_x - \sum_i N_{i,d,\tau_i}(u) c_i \\ & \frac{\partial}{\partial u} & v_x - \sum_i N_{i,d,\tau_i}(u) c_i \\
& = & v_x - \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 & = & - \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*} \end{eqnarray*}
and do a gradient-descend to approximate the value of \(u\) up to an and do a gradient-descend to approximate the value of \(u\) up to an
@ -402,7 +404,7 @@ 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 curve is defined by
\[V(u,v,w) = \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot C_{ijk}.\] \[V(u,v,w) = \sum_i \sum_j \sum_k N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot C_{ijk}.\]
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. and also 3 variables \(u,v,w\) for each vertex we want to approximate.
@ -415,14 +417,14 @@ gradient-descent as:
And the partial version for just one direction as And the partial version for just one direction as
\[Err_x(u,v,w,\vec{p}^{*}) = p^{*}_x - \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} {C_{ijk}}_x N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \] \[Err_x(u,v,w,\vec{p}^{*}) = p^{*}_x - \sum_i \sum_j \sum_k N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot {c_{ijk}}_x \]
To solve this we derive partially, like before: To solve this we derive partially, like before:
\[ \[
\begin{array}{rl} \begin{array}{rl}
\displaystyle \frac{\partial Err_x}{\partial u} & p^{*}_x - \displaystyle \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} {C_{ijk}}_x N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \\ \displaystyle \frac{\partial Err_x}{\partial u} & p^{*}_x - \displaystyle \sum_i \sum_j \sum_k N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot {c_{ijk}}_x \\
= & \displaystyle - \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} {C_{ijk}}_x N'_i(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) = & \displaystyle - \sum_i \sum_j \sum_k N'_i(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot {c_{ijk}}_x
\end{array} \end{array}
\] \]
@ -444,7 +446,7 @@ J(Err(u,v,w)) =
Like leaving out Sums & $i,j,k$-Indices to make obvious what derivative belongs Like leaving out Sums & $i,j,k$-Indices to make obvious what derivative belongs
where in what case?} where in what case?}
With the Gauss-Newton algorithm we iterate the formula With the Gauss-Newton algorithm we iterate via the formula
\[J(Err(u,v,w)) \cdot \Delta \left( \begin{array}{c} u \\ v \\ w \end{array} \right) = -Err(u,v,w)\] \[J(Err(u,v,w)) \cdot \Delta \left( \begin{array}{c} u \\ v \\ w \end{array} \right) = -Err(u,v,w)\]
and use Cramers rule for inverting the small Jacobian and solving this and use Cramers rule for inverting the small Jacobian and solving this
system of linear equations. system of linear equations.