\begin{aligned}
F(k) &= \sum_{\lambda = 0}^{k-1} \big( F(\lambda + 1) -
F(\lambda) \big) + F(0) \\
- &= \sum_{\lambda = 0}^{L-2} \big( F(\lambda - 1) - F(\lambda) \big)
+ &= \sum_{\lambda = 0}^{L-2} \big( F(\lambda + 1) - F(\lambda) \big)
I( \lambda < k) + F(0),
\end{aligned}
\end{equation}
rewrite \eqref{eq:fidelity_approx_1} and obtain
\begin{equation}
\sum_{x \in \mathcal{G}} \abs{u_x - v_x}^p =
- \sum_{x \in \mathcal{G}} N_x(u_x) = \sum_{\lambda = 0}^{L-2} \sum_x
+ \sum_{x \in \mathcal{G}} N_x(u_x) = \sum_{\lambda = 0}^{L-2} \sum_{x
+ \in \mathcal{G}}
\big( N_x(\lambda + 1) - N_x(\lambda) \big) \, u_x^\lambda + N_x(0).
\end{equation}
-\fixme{Why stop at $L-2$}
+Note that since our image takes values in $\mathcal{L} = \{0, \hdots,
+L-1\}$, the thresholded image $u^{L-1}$ is equal to zero everywhere.
\subsection{Regularization term}
\approx \sum_{\lambda = 0}^{L-2} \PerA( \{ u > \lambda \}; \Omega)
\, \Delta \lambda.
\end{equation}
-Note that we do not include any $\Delta \lambda$ difference, as we can
+Note that we will later ignore the $\Delta \lambda$ difference, as we can
just absorb this into the $\beta$ parameter. The perimeter is then
calculated using a discretized version of the Cauchy--Crofton formula
-introduced in \fixme{ref}.
-\fixme{Why stop at $L-2$}
+introduced in Theorem \ref{thm:riemannian_cauchy_crofton}. Again, we
+stop the sum at $L-2$ since the level set $\{ u > L - 1\}$ is empty and
+has zero perimeter.
\subsubsection{Discrete Riemannian Cauchy--Crofton formula}
\end{figure}
By approximating the integral in \fixme{REF} by a discrete sum we obtain
the approximation
-\begin{align}
- \abs{C}_M &= \int_\mathcal{L} \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\mathcal{L}(\ell_{\nu, \rho}) \\
- &\approx \sum_{\ell_{\nu, \rho} \in \mathcal{L}_D}
- \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}}} \, \Delta\rho \,
- \Delta\phi \\
- &\approx \sum_\nu \sum_\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}}} \, \Delta\rho \,
- \Delta\phi.
- \label{eq:cauchy_crofton_approx1}
-\end{align}
+\begin{equation}
+ \begin{aligned}
+ \abs{C}_M &= \int_\mathcal{L} \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\mathcal{L}(\ell_{\nu, \rho}) \\
+ &\approx \sum_{\ell_{\nu, \rho} \in \mathcal{L}_D}
+ \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}}} \, \Delta\rho \,
+ \Delta\phi \\
+ &= \sum_\nu \sum_\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}}} \, \Delta\rho \,
+ \Delta\phi.
+ \label{eq:cauchy_crofton_approx1}
+ \end{aligned}
+\end{equation}
The set of lines $\mathcal{L}$ has been discretized to the lines
$\mathcal{L}_D$. Note that we are approximating the length of the
\emph{differentiable} curve $C$ in $\Omega$. Further we need to
$x$ somewhere on the edge $e_{ab}$, we approximate the metric tensor by
\begin{equation}
M(x) \approx M(e_{ab}) = \frac{M(a) + M(b)}{2},
+ \label{eq:tensor_approx}
\end{equation}
the component-wise average of the tensors in the two end points of the
edge. \fixme{really? componentwise? will that not mess up the
\, \delta^2 \, \Delta\phi}{2 \left(e^T \cdot M(e) \cdot
e\right)^{\sfrac{3}{2}}}.
\end{equation}
-Going back to the perimeter of the level set $\{ u > \lambda \}$ we get
+Going back to the perimeter of the level set $\{ u > \lambda \}$ we can
+rewrite the sum to be a sum over all edges that goes from one side of
+the set to the other such that
\begin{equation}
\PerA(\{u > \lambda\}; \mathcal{G}) = \sum_{e_{ab}} \abs{
- u^\lambda(a) - u^\lambda(b)} \frac{\det M(e_{ab}) \norm{e_{ab}}^2
+ u^\lambda_a - u^\lambda_b} \frac{\det M(e_{ab}) \norm{e_{ab}}^2
\, \delta^2 \, \Delta\phi}{2 \left(e_{ab}^T \cdot M(e_{ab}) \cdot
e_{ab}\right)^{\sfrac{3}{2}}}.
+ \label{eq:per_approx1}
\end{equation}
-The absolute value $\abs{u^\lambda(a) - u^\lambda(b)}$ is one if one of
+The absolute value $\abs{u^\lambda_a - u^\lambda_b}$ is one if one of
$a$ and $b$ lie inside the level set and the other lies outside, and
-zero otherwise. In other words if the absolute value is one if $e_{ab}$
+zero otherwise. In other words the absolute value is one if $e_{ab}$
crosses the perimeter of $\{ u > \lambda\}$ an odd number of times, and
zero otherwise.
\fixme{We now have the problem that $C$ is a curve, but later we want it
to be a cut...}
-\subsubsection{Consistency, stability and convergence}
+\fixme{edges here are a bit different from edges later}
+
+\subsubsection{Consistency}
As for any discretization these are important properties to consider.
Consistency relates to how well a solution to the continuous problem
-fits in the discretized equation. This would optimally be controlled by
-the resolution of the discretization in such a way that if we refine the
-discretization, the
-
-Consider a square centered around grid point $p$ with side lengths
-$\sqrt{\delta}$. The size of this square will go to zero, while the
-number of points along the outer edge $\lfloor 1 / \delta \rfloor$ of
-the square, goes to infinity. Also, the number of points in the square
-(and thus the possible number of edges) goes to infinity so the
-discretization is finer and finer. So, as $\delta \to 0$, the length of
-the edges are bounded by $\sqrt{\delta} \to 0$, and $\Delta \phi \to 0$
-because the number of points around the edge of the square goes to
-infinity! \fixme{waving with hands} Also, since the curve is somewhat
-regular, when $\norm{e}$ goes to zero, the number of times the curve can
-cross $e$ is maximum $1$ so that's great. Also, $M(e) \to M(p)$
-obviously. \fixme{Some figure here aswell. And maybe some consideration
-of $\sup \Delta \phi$ etc.}
-
-\subsubsection{Back to the energy...}
-
-\fixme{We talk about edges, but it's not really a graph yet...}
-The curve $C$ that we are measuring is the perimeter of a level set
-$X = \{ u > \lambda \}$. This means that the edges crossing the curve --
-$e \cap C$ -- are exactly those edges with one terminal inside $X$ and
-the other terminal outside $X$. \fixme{Terminal?}. This means we can
-write the discrete regularization energy term as
+fits in the discretized equation, in other words, how well the
+discretized equation approximates the continuous one.
+
+It is obvious that the discretization of the fidelity term in
+\eqref{eq:fidelity_approx_1} is consistent. We have left out the pixel
+size $\Delta x$ in the sum, and absorbed it into the regularization
+parameter $\beta$, but apart from that, the sum is the midpoint rule
+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
+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
+$\sqrt{\delta}$. 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
+\begin{equation}
+ \sup \Delta \phi = \arctan \frac{\delta}{\sfrac{\lfloor \sqrt{\delta}
+ \rfloor}{2}} \leq \arctan \frac{\delta}{\sfrac{\sqrt{\delta}}{2} -
+ \delta} = \arctan \frac{1}{\frac{1}{2\sqrt{\delta}} - 1} \to 0.
+\end{equation}
+
+Further we know that for each line family
\begin{equation}
- \sum_{\lambda = 0}^{L-2} \PerA( \{ u > \lambda \}; \Omega)
- \approx \sum_{\lambda = 0}^{L-2} \sum_{e} w_e
- \abs{u_{\delta_1(e)}^\lambda - u_{\delta_2(e)}^\lambda}
+ \delta^2 = \Delta \rho \norm{e},
\end{equation}
-where the weight $w_e$ comes directly from the Riemannian
-Cauchy--Crofton formula
+and since the edge length $\norm{e}$ is bounded from below by $\delta$,
+we have
\begin{equation}
- w_e = \frac{\det M(e) \norm{e}^2 \delta^2 \Delta \phi}{2 (e^T \cdot
- M(e) \cdot e)^{\sfrac{3}{2}}}.
+ \sup \Delta \rho \leq \frac{\delta^2}{\delta} = \delta \to 0.
\end{equation}
-Now we just have to make sure that this fits well with the
-graph-representable function definition and all will be good.
+
+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}
+
+\subsubsection{Stability and convergence}
+
+For a discretization of a problem to be stable, the solution needs to
+depend continuously on the input data. So in our case the output image
+$u$ should depend continuously on the given input image $f$. We will not
+prove stability here, but intuitively there is no reason to believe the
+method is unstable though, as small changes in $f$ only gives small
+changes to the functional $F$. \fixme{well this is not nearly enough I
+guess} \fixme{ref?}
+
+Convergence is in some nice cases, but not nearly all the time,
+equivalent to consistency and stability. It answers the question of
+whether a the solutions of the discretized problem converges to the
+solution of the continuous problem as the discretization is refined.
+This will also not be considered here, but we refer to \fixme{ref}. And
+note that even though convergence is important, in practice, we have no
+possibility of refining the discretization, as the grid structure is
+defined by the digital image which is already composed of discrete
+pixels.
\subsection{Total energy}
projects / master / commitdiff