little formatting

arbeit/ma.md

@@ -204,12 +204,12 @@ $\|\vec{G}\|_F = 1$, whereby $\|\cdot\|_F$ denotes the Frobenius-Norm.
 
 # 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 $\tau$
-statically as
-$$\tau_i = \nicefrac{i}{n}$$
-whereby $n$ is the number of control-points in that direction.
+statically as $\tau_i = \nicefrac{i}{n}$ 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
 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
 $c_i$
 $$
-v_x = \sum_i N_{i,d,\tau_i}(u) c_i
+v_x \overset{!}{=} \sum_i N_{i,d,\tau_i}(u) c_i
+$$
+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*}
 & \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*}
-
 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
@@ -263,7 +265,7 @@ direction.
 
 Given $n,m,o$ control points in $x,y,z$-direction each Point on the curve is
 defined by
-$$V(u,v,w) = \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot C_{ijk}.$$
+$$V(u,v,w) = \sum_i \sum_j \sum_k N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot C_{ijk}.$$
 
 In this case we have three different B-Splines (one for each dimension) and also
 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
 
-$$Err_x(u,v,w,\vec{p}^{*}) = p^{*}_x - \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} {C_{ijk}}_x N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) $$
+$$Err_x(u,v,w,\vec{p}^{*}) = p^{*}_x - \sum_i \sum_j \sum_k N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot {c_{ijk}}_x $$
 
 To solve this we derive partially, like before:
 
 $$
 \begin{array}{rl}
-    \displaystyle \frac{\partial Err_x}{\partial u} & p^{*}_x - \displaystyle \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} {C_{ijk}}_x N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \\
-  = & \displaystyle - \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} {C_{ijk}}_x N'_i(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w)
+    \displaystyle \frac{\partial Err_x}{\partial u} & p^{*}_x - \displaystyle \sum_i \sum_j \sum_k N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot {c_{ijk}}_x \\
+  = & \displaystyle - \sum_i \sum_j \sum_k N'_i(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot {c_{ijk}}_x
 \end{array}
 $$
 
@@ -303,7 +305,7 @@ $$
 Like leaving out Sums & $i,j,k$-Indices to make obvious what derivative belongs
 where in what case?}
 
-With the Gauss-Newton algorithm we iterate the formula
+With the Gauss-Newton algorithm we iterate via the formula
 $$J(Err(u,v,w)) \cdot \Delta \left( \begin{array}{c} u \\ v \\ w \end{array} \right) = -Err(u,v,w)$$
 and use Cramers rule for inverting the small Jacobian and solving this system of
 linear equations.

arbeit/ma.tex

@@ -237,7 +237,7 @@ evolutionary optimization\cite{Menzel2006}, which we will use later on.
 
 \begin{figure}[!ht]
 \includegraphics[width=\textwidth]{img/hsu_fig7.png}
-\caption{Figure 7 from \cite{hsu1991dmffd}}
+\caption{Figure 7 from \cite{hsu1991dmffd}.}
 \label{fig:hsu_fig7}
 \end{figure}
 
@@ -340,10 +340,11 @@ Frobenius-Norm.
 \chapter{\texorpdfstring{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 \(\tau\)
-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.
 
 \(d\) 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
 \(u\)-value for each vertex. This is achieved by finding out the
 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 \(u\)
-yielding
+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 \\
-& = & 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*}
 
 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
 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)
 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
 
-\[Err_x(u,v,w,\vec{p}^{*}) = p^{*}_x - \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} {C_{ijk}}_x N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \]
+\[Err_x(u,v,w,\vec{p}^{*}) = p^{*}_x - \sum_i \sum_j \sum_k N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot {c_{ijk}}_x \]
 
 To solve this we derive partially, like before:
 
 \[
 \begin{array}{rl}
-    \displaystyle \frac{\partial Err_x}{\partial u} & p^{*}_x - \displaystyle \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} {C_{ijk}}_x N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \\
-  = & \displaystyle - \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} {C_{ijk}}_x N'_i(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w)
+    \displaystyle \frac{\partial Err_x}{\partial u} & p^{*}_x - \displaystyle \sum_i \sum_j \sum_k N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot {c_{ijk}}_x \\
+  = & \displaystyle - \sum_i \sum_j \sum_k N'_i(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot {c_{ijk}}_x
 \end{array}
 \]
 
@@ -444,7 +446,7 @@ J(Err(u,v,w)) =
 Like leaving out Sums & $i,j,k$-Indices to make obvious what derivative belongs
 where in what case?}
 
-With the Gauss-Newton algorithm we iterate the formula
+With the Gauss-Newton algorithm we iterate via the formula
 \[J(Err(u,v,w)) \cdot \Delta \left( \begin{array}{c} u \\ v \\ w \end{array} \right) = -Err(u,v,w)\]
 and use Cramers rule for inverting the small Jacobian and solving this
 system of linear equations.