--- /dev/null
+\centering
+\begin{tikzpicture}[scale=1.0]
+
+ \node[tiny vertex] (o) at (1, 1) {};
+
+ %\path[dash] (2, 2) -- (5,2);
+ %\path[dash] (2, 2) -- (5,5);
+ %\path[dash] (2, 2) -- (2,5.5);
+
+ \draw (1,1) +(0:1.5cm) arc (0:360:1.5cm);
+ %\path (2,2) ++(22.5:3cm) node {$\phi_{xy}$};
+
+ \draw[dash] (1,1) +(45:0cm) -- +(45:2.6cm);
+ \draw[dash] (1,1) +(63.3:0cm) -- +(63.43:2.6cm);
+
+ \draw (1,1) +(45:1.4cm) -- +(45:1.6cm) +(45:1.9cm)
+ node[font=\small, fill=white, inner sep=0.5pt] {$\phi_k$};
+ \draw (1,1) +(63.43:1.4cm) -- +(63.43:1.6cm) +(67.5:1.9cm)
+ node[font=\small, fill=white, inner sep=0.5pt] {$\phi_{k+1}$};
+ %\path (2,2) ++(67.5:3.8cm) node{$\Delta\phi_{xy}$};
+
+ \draw (1,1) +(45:2.5cm) arc (45:63.43:2.5cm);
+ \path (1,1) ++(54.215:3cm) node {$\Delta \phi_k$};
+
+ \foreach \p in {0,90,180,270} {
+ \draw (1,1) +(\p:1.4cm) -- +(\p:1.6cm);
+ \draw (1,1) +(\p+45:1.4cm) -- +(\p+45:1.6cm);
+ \draw (1,1) +(\p+26.57:1.4cm) -- +(\p+26.57:1.6cm);
+ \draw (1,1) +(\p+90-26.57:1.4cm) -- +(\p+90-26.57:1.6cm);
+ }
+
+\end{tikzpicture}
+\caption{
+ We showed that the angular difference between subsequent angle
+ parameters goes to zero. The discretization in the $\phi$ dimension
+ can be viewed as a rectangle approximation rule of the integral, as
+ the summand is evaluated in one end of the interval $[\phi_k,
+ \phi_{k+1}]$.
+}
+\label{fig:circ_rule}
--- /dev/null
+\centering
+\begin{tikzpicture}[scale=1.2]
+ \foreach \x in {0,...,5} {
+ \foreach \y in {0,...,5} {
+ \node[tiny vertex] (\x\y) at (\x, \y) {};
+ }
+ }
+
+ \clip (0, 0) rectangle (5, 5);
+
+ \path[nedge=4cm] (0,5) -- (2,4);
+ \path[nedge=4cm] (0,4) -- (2,3);
+ \path[nedge=4cm] (0,3) -- (2,2);
+ \path[nedge=4cm] (0,2) -- (2,1);
+ \path[nedge=4cm] (0,1) -- (2,0);
+
+ \path[nedge=4cm] (-1,1) -- (1,0);
+
+ \path[nedge=4cm] (1,5) -- (3,4);
+ \path[nedge=4cm] (1,4) -- (3,3);
+ \path[nedge=4cm] (1,3) -- (3,2);
+ \path[nedge=4cm] (1,2) -- (3,1);
+ \path[nedge=4cm] (1,1) -- (3,0);
+
+ \path[nedge=4cm] (2,5) -- (4,4);
+ \path[nedge=4cm] (3,5) -- (5,4);
+ \path[nedge=4cm] (4,5) -- (6,4);
+
+ \draw [cyan, xshift=4cm] plot [smooth, tension=2] coordinates {
+ (1,1) (2,1) (5,5) (3,3)};
+
+ %\path[dash] (2, 2) -- (5,2);
+ %\path[dash] (2, 2) -- (5,5);
+ %\path[dash] (2, 2) -- (2,5.5);
+
+ %\draw (2,2) +(0:2.5cm) arc (0:45:2.5cm);
+ %\path (2,2) ++(22.5:3cm) node {$\phi_{xy}$};
+
+ %\draw (2,2) +(45:3.3cm) arc (45:90:3.3cm);
+ %\path (2,2) ++(67.5:3.8cm) node{$\Delta\phi_{xy}$};
+
+ %\node[tiny vertex,draw] (a) at (2, 2) {x};
+ %\node[tiny vertex,draw] (b) at (2, 3) {};
+ %\node[tiny vertex,draw] (c) at (2, 1) {};
+ %\node[tiny vertex,draw] (d) at (3, 2) {};
+ %\node[tiny vertex,draw] (e) at (1, 2) {};
+ %\node[tiny vertex,draw] (f) at (3, 3) {y};
+ %\node[tiny vertex,draw] (g) at (1, 1) {};
+ %\node[tiny vertex,draw] (h) at (1, 3) {};
+ %\node[tiny vertex,draw] (i) at (3, 1) {};
+
+ %\path[edge] (a) -- (b);
+ %\path[edge] (a) -- (c);
+ %\path[edge] (a) -- (d);
+ %\path[edge] (a) -- (e);
+ %\path[edge] (a) -- (f);
+ %\path[edge] (a) -- (g);
+ %\path[edge] (a) -- (h);
+ %\path[edge] (a) -- (i);
+
+ \path (4,1) [edge, <->, anchor=center] -- node[anchor=south
+ east,yshift=-3.5pt]
+ {$\Delta \rho$}
+ ($(5,0)!(4,1)!(3,1)$) ;
+
+\end{tikzpicture}
+\caption{One family of lines having the same $\phi$ parameter.}
+\label{fig:curve_edge}
\node[tiny vertex,draw] (f) at (3, 3) {};
\node[tiny vertex,draw] (g) at (3, 1) {};
\node[tiny vertex,draw] (h) at (2, 1) {};
+ \node[tiny vertex,draw] (i) at (2, 0) {};
\path[edge,-] (a) -- (b);
\path[edge,-] (a) -- (c);
\path[edge,-] (a) -- (f);
\path[edge,-] (a) -- (g);
\path[edge,-] (a) -- (h);
+ \path[edge,-] (a) -- (i);
\path[dash] (1, 2) -- +(35.785:4cm);
\path[dash] (1, 2) -- +(54.215:4cm);
--- /dev/null
+\centering
+\begin{tikzpicture}[scale=1.3]
+ \foreach \x in {0,...,3} {
+ \foreach \y in {0,...,2} {
+ \node[tiny vertex] (\x\y) at (\x, \y) {};
+ }
+ }
+
+ \begin{scope}
+ \clip (0, 0) rectangle (3, 2);
+
+ \path[nedge=4cm] (0,5) -- (2,4);
+ \path[nedge=4cm] (0,4) -- (2,3);
+ \path[nedge=4cm] (0,3) -- (2,2);
+ \path[nedge=4cm] (0,2) -- (2,1);
+ \path[nedge=4cm] (0,1) -- (2,0);
+
+ \path[nedge=4cm] (-1,1) -- (1,0);
+
+ \path[nedge=4cm] (1,5) -- (3,4);
+ \path[nedge=4cm] (1,4) -- (3,3);
+ \path[nedge=4cm] (1,3) -- (3,2);
+ \path[nedge=4cm] (1,2) -- (3,1);
+ \path[nedge=4cm] (1,1) -- (3,0);
+
+ \path[nedge=4cm] (2,5) -- (4,4);
+ \path[nedge=4cm] (3,5) -- (5,4);
+ \path[nedge=4cm] (4,5) -- (6,4);
+ \end{scope}
+
+ \path[dash] (2.5,1.5) -- (3.5,1);
+ \path[dash] ($(3.5,0.5)!(2.5,1.5)!(1.5,1.5)$) -- ($(4.5,0.0)!(3.5,1.0)!(2.5,1.0)$);
+
+ \path (3.5,1.0) [edge, <->, anchor=center] -- node[anchor=north
+ west,yshift=+3.0pt,xshift=+1pt, fill=white, font=\small, inner sep=1pt]
+ {$\Delta \rho$}
+ ($(4.5,0.0)!(3.5,1.0)!(2.5,1.0)$) ;
+
+
+ %\path[dash] (2, 2) -- (5,2);
+ %\path[dash] (2, 2) -- (5,5);
+ %\path[dash] (2, 2) -- (2,5.5);
+
+ %\draw (2,2) +(0:2.5cm) arc (0:45:2.5cm);
+ %\path (2,2) ++(22.5:3cm) node {$\phi_{xy}$};
+
+ %\draw (2,2) +(45:3.3cm) arc (45:90:3.3cm);
+ %\path (2,2) ++(67.5:3.8cm) node{$\Delta\phi_{xy}$};
+
+ %\node[tiny vertex,draw] (a) at (2, 2) {x};
+ %\node[tiny vertex,draw] (b) at (2, 3) {};
+ %\node[tiny vertex,draw] (c) at (2, 1) {};
+ %\node[tiny vertex,draw] (d) at (3, 2) {};
+ %\node[tiny vertex,draw] (e) at (1, 2) {};
+ %\node[tiny vertex,draw] (f) at (3, 3) {y};
+ %\node[tiny vertex,draw] (g) at (1, 1) {};
+ %\node[tiny vertex,draw] (h) at (1, 3) {};
+ %\node[tiny vertex,draw] (i) at (3, 1) {};
+
+ %\path[edge] (a) -- (b);
+ %\path[edge] (a) -- (c);
+ %\path[edge] (a) -- (d);
+ %\path[edge] (a) -- (e);
+ %\path[edge] (a) -- (f);
+ %\path[edge] (a) -- (g);
+ %\path[edge] (a) -- (h);
+ %\path[edge] (a) -- (i);
+
+\end{tikzpicture}
+\caption{
+ The discretization in the $\rho$ dimension can be regarded as a
+ midpoint rule approximation of the integral, since the difference
+ $\Delta \rho$ the same for all lines in one line family.
+}
+\label{fig:line_midpoint}
--- /dev/null
+\centering
+\begin{tikzpicture}[scale=0.9]
+ \foreach \x in {0,...,6} {
+ \foreach \y in {0,...,6} {
+ \node[tiny vertex] (\x\y) at (\x, \y) {};
+ }
+ }
+ \node[tiny vertex,draw] (a) at (3, 3) {};
+ \node[tiny vertex,draw] (b) at (3, 4) {};
+ \node[tiny vertex,draw] (c) at (4, 3) {};
+ \node[tiny vertex,draw] (d) at (4, 4) {};
+ \node[tiny vertex,draw] (e) at (4, 5) {};
+ \node[tiny vertex,draw] (f) at (5, 4) {};
+ \node[tiny vertex,draw] (g) at (5, 2) {};
+ \node[tiny vertex,draw] (h) at (4, 2) {};
+ \node[tiny vertex,draw] (i) at (3, 2) {};
+ \node[tiny vertex,draw] (j) at (2, 4) {};
+ \node[tiny vertex,draw] (k) at (2, 3) {};
+ \node[tiny vertex,draw] (l) at (2, 2) {};
+ \node[tiny vertex,draw] (m) at (2, 1) {};
+ \node[tiny vertex,draw] (n) at (1, 2) {};
+ \node[tiny vertex,draw] (o) at (1, 4) {};
+ \node[tiny vertex,draw] (p) at (4, 1) {};
+ \node[tiny vertex,draw] (q) at (2, 5) {};
+
+ \path[edge,-] (a) -- (b);
+ \path[edge,-] (a) -- (c);
+ \path[edge,-] (a) -- (d);
+ \path[edge,-] (a) -- (e);
+ \path[edge,-] (a) -- (f);
+ \path[edge,-] (a) -- (g);
+ \path[edge,-] (a) -- (h);
+ \path[edge,-] (a) -- (i);
+ \path[edge,-] (a) -- (j);
+ \path[edge,-] (a) -- (k);
+ \path[edge,-] (a) -- (l);
+ \path[edge,-] (a) -- (m);
+ \path[edge,-] (a) -- (n);
+ \path[edge,-] (a) -- (o);
+ \path[edge,-] (a) -- (p);
+ \path[edge,-] (a) -- (q);
+
+ \path[dash] (a) -- (5, 5) {};
+ \path[dash] (a) -- (3, 5) {};
+ \path[dash] (a) -- (3, 1) {};
+ \path[dash] (a) -- (1, 3) {};
+ \path[dash] (a) -- (5, 3) {};
+ \path[dash] (a) -- (5, 1) {};
+ \path[dash] (a) -- (1, 5) {};
+ \path[dash] (a) -- (1, 1) {};
+
+ \draw (3,3) +(63.43:1.3cm) arc (63.43:90:1.3cm);
+ \path (3,3) ++(76.715:1.5cm) node {$a$};
+
+ \draw (3,3) +(45:1.7cm) arc (45:63.43:1.7cm);
+ \path (3,3) ++(54.215:1.98cm) node {$b$};
+
+ \draw (0.7, 0.7) rectangle (5.3, 5.3);
+
+ \path[edge,<->] (0.7, 5.6) -- node[fill=white, font=\small, inner
+ sep=1.7pt] {$\sqrt{\delta}$} (5.3, 5.6) {};
+
+\end{tikzpicture}
+\caption{
+ To show that we have a consistent discretization of the
+ Cauchy--Crofton integral formula, we construct a discrete set of
+ lines $\mathcal{L}_D$ such that the length of the edges $\norm{e}$,
+ the angle differences $\Delta \phi$ (here $a$ and $b$) and the
+ distance between lines $\Delta \rho$ goes to zero as $\delta \to 0$.
+}
+\label{fig:square_cons}
approximation is exact for zero or one intersection points, but will, as
we see in Figure \fixme{ref}, not be entirely correct when we have more.
+\begin{figure}
+ \input{fig/curve_edge}
+\end{figure}
+
\fixme{curve or perimeter here, maybe perimeter because then we know it
follows the boundaries of the pixels.}
approximation of the integral.
For the regularization term we will argue that for a differentiable
-curve $C$, the discretization of our domain $\Omega$ and set of lines
-$\mathcal{L}$ gives a discrete Cauchy--Crofton formula that is
+curve $C$, the discretization of our domain $\Omega$ and the set of
+lines $\mathcal{L}$ gives a discrete Cauchy--Crofton formula that is
consistent with the continuous one. We will show that for an
increasingly refined domain $\mathcal{G}$, there exists a choice for
$\mathcal{L}_D$ that leads to a consistent Cauchy--Crofton formula. For
convenience we will use a neighborhood representation of $\mathcal{L}_D$
similar to the one in Figure \ref{fig:line_neigh}.
-Consider a square centered around grid point $x$ with side lengths
+\begin{figure}
+ \input{fig/square_cons}
+\end{figure}
+
+Consider a square centered around a grid point with side lengths
$\sqrt{\delta}$. As $\delta$ goes to zero, the size of this square will
go to zero. Inside this square we can fit a square of $\lfloor 1 /
\sqrt{\delta} \rfloor^2$ grid points. This means that the number of
grid points along the outer edge of this square $\lfloor 1 /
-\sqrt{\delta} \rfloor$ goes to infinity. For each grid point around at
-the boundary of this square, we include a grid point having the same
-angle $\phi$ to the $x$-axis as shown in Figure \fixme{ref}. The maximal
-$\Delta \phi$ will then be between the horizontal or vertical edge and
-its neighbors
+\sqrt{\delta} \rfloor$ goes to infinity. For each grid point along
+the outer edge of this square, we include in our neighborhood a grid point
+having the same angle $\phi$ to the $x$-axis. This means either
+including the actual grid point at the outer edge, or one having the
+same angle, just closer to the center. This construction can be
+seen in Figure \ref{fig:square_cons}.
+The maximal $\Delta \phi$ will then be between the
+horizontal or vertical edge and its neighbors, shown in Figure
+\ref{fig:square_cons} as angle $a$. These angles can be calculated to be
\begin{equation}
\sup \Delta \phi = \arctan \frac{\delta}{\sfrac{\lfloor \sqrt{\delta}
\rfloor}{2}} \leq \arctan \frac{\delta}{\sfrac{\sqrt{\delta}}{2} -
\sup \Delta \rho \leq \frac{\delta^2}{\delta} = \delta \to 0.
\end{equation}
-The edge length is bounded from above by $\sqrt{2} \delta$ and will also
-go to zero. Thus for a given curve $C$, our simplification of only
-counting an intersection between $e$ and $C$ when $C$ crosses $e$ an odd
-number of times becomes increasingly correct.
+Since all edges are inside our square, the edge length is bounded from
+above by $\sqrt{2} \delta$ and will also go to zero. Thus for a given
+curve $C$, our simplification of only counting an intersection between
+$e$ and $C$ when $C$ crosses $e$ an odd number of times becomes
+increasingly correct.
Further it is obvious that our tensor approximation in
\eqref{eq:tensor_approx} converges to the tensor in the intersection
point when the edge length $\norm{e_{ab}}$ goes to zero.
-And thus... the perimeter approximation in \eqref{eq:per_approx1} is
-consistent with the continuous formulation in Theorem
-\ref{thm:riemannian_cauchy_crofton}
+To see that this is enough to give a consistent discretization of our
+integral, we write the approximation as follows
+\begin{equation}
+ \begin{aligned}
+ \abs{C}_M &=
+ \int_\nu \int_\rho \sum_{x \in \ell_{\nu, \rho} \cap C}
+ \, \frac{\det M(x)}{2\left(\nu^T
+ \cdot M(x) \cdot \nu\right)^{\sfrac{3}{2}}} \, d\rho \, d\nu \\
+ &\approx \sum_\nu \sum_\rho \sum_{e_{\nu, \rho} \cap C}
+ \, \frac{\det M(e_{\nu, \rho}) \norm{e_{\nu, \rho}}^3}{2\left(
+ e_{\nu, \rho}^T \cdot M(e_{\nu, \rho}) \cdot e_{\nu, \rho}
+ \right)^{\sfrac{3}{2}}} \, \Delta \rho \, \Delta \nu.
+\end{aligned}
+\end{equation}
+As $\Delta \rho$ is constant for all lines in one family, we can regard
+the discretization in the $\rho$ dimension as a midpoint rule
+approximation, as shown in Figure \ref{fig:line_midpoint}.
+
+\begin{figure}
+ \input{fig/line_midpoint}
+\end{figure}
+
+For the $\phi$ dimension (or $\nu$ equivalently), we chose to show that
+the difference between the angle parameter of one line family and the
+next $\Delta \phi$ goes to zero. As shown in Figure \ref{fig:circ_rule},
+the discretization of $\phi$ can also be regarded as a so-called
+\emph{rectangle method} approximation of the integral, although not the
+midpoint rule. The summand is evaluated in the end point of the angle
+interval $[\phi_k, \phi_{k+1}]$, where $\Delta \phi_k = \phi_{k+1} -
+\phi_k$.
+
+\begin{figure}
+ \input{fig/circ_rule}
+\end{figure}
+
+And thus the approximation has been shown to be equivalent to
+well-known, and consistent integral approximations, so the perimeter
+approximation in \eqref{eq:per_approx1} is consistent with the
+continuous formulation in Theorem \ref{thm:riemannian_cauchy_crofton}.
+
+Note that as we will work with digital images with fixed resolutions, we
+do not really have the chance to refine our discretization. We do
+however have to take these things into account when creating our
+neighborhood stencil, to make sure that we get a reasonable
+approximation of the perimeter lengths.
+
+\fixme{we did not really state that we have arrived at our final
+discretization, and that the rest will be about the solution}
\subsubsection{Stability and convergence}
projects / master / commitdiff