-\chapter{Continous formulation}
+\chapter{Continuous formulation}
From the introduction we see that there are many different approaches to
the image restoration problem, all with their own strengths and
\section{Anisotropic total variation}
-The method considered will build on the total variation regularisation
-method of Section \ref{sec:total_variation}. From anisotropic diffusion
+The method considered will build on the total variation regularization
+method of Section \ref{sec:total_variation}. From anisotropic diffusion
in Section \ref{sec:anisotropic_diffusion} we borrow the idea of making
-the regularization in each point directionally dependant. We introduce
+the regularization in each point directionally dependent. We introduce
the anisotropic total variation
\begin{equation}
\TVA(u) = \int_\Omega \sqrt{\nabla u(x)^T A(x) \nabla u(x)} \, dx
\nabla \tilde{f}_\sigma(x)$. It obviously contains the same information
as the edge detector itself. Its eigenvalues will be 0 and $\labs{\nabla
\tilde{f}_\sigma(x)}^2$ with corresponding eigenvectors $v_1$ and $v_2$
-perpendicular and parallel to $\nabla \tilde{f}_\sigma(x)$ repectively.
+perpendicular and parallel to $\nabla \tilde{f}_\sigma(x)$ respectively.
Figure \ref{fig:edges} shows that the largest eigenvalue of the
structure tensor is a good edge detector.
Since our space $L^2(\Omega)$ is of infinite dimensions things become a
little bit problematic here. The problem lies in the fact that a
functional which is continuous with respect to sequences is not
-neccessarily continuous with respect to the underlying topology. In
-other words, in these spaces, there is a difference between sequencial
+necessarily continuous with respect to the underlying topology. In
+other words, in these spaces, there is a difference between sequential
continuity and topological continuity. Topological continuity implies
sequential continuity, but not the other way. One way to get around this
would be to consider topological \emph{nets}, an extension of sequences,
but we will stick to proving sequential lower semi-continuity and
-refering to further theory. For further reading on the theory of
+referring to further theory. For further reading on the theory of
sequential versus topological continuity see for example Megginson's
book on Banach space theory \cite{megginson}.
topology, and the same goes for lower semi-continuity.
Before arguing that our own functional is sequentially weakly lower
-semi-continous, we present a much needed result.
+semi-continuous, we present a much needed result.
\begin{lemma}
Assume that the functional $F : L^2(\Omega) \to \mathbb{R}$ is
defined by
\begin{equation}
F = \sup_i F_i
\end{equation}
- where all the $F_i$ are sequentially weakly lower semi-coninuous, then $F$
+ where all the $F_i$ are sequentially weakly lower semi-continuous, then $F$
is sequentially weakly lower semi-continuous, meaning that for any sequence
$u_k \rightharpoonup u$ we have $F(u) \leq \liminf_k F(u_k)$.
\end{lemma}
= \sup \left\{\int_\Omega (u - v) \, \xi \, dx : \xi \in
L^2(\Omega), \norm{\xi}_{L^2} \leq \norm{u-v}_{L^2} \right\}
\end{equation}
-As the map $u \mapsto \int_\Omega (u - v) \xi\, dx$ is continous in the
-weak topology, the fidelity term is then a supremum of weakly continous
+As the map $u \mapsto \int_\Omega (u - v) \xi\, dx$ is continuous in the
+weak topology, the fidelity term is then a supremum of weakly continuous
functionals, and is thus sequentially lower semi-continuous.
For the regularization term the approach is similar. With our extended
This coarea formula is our first step in transforming the anisotropic
total variation into an easily discretizisable expression.
-The anisotropic total variation of the thresholded images occuring in
+The anisotropic total variation of the thresholded images occurring in
the anisotropic coarea formula are very much related to the size of the
boundary of the level set, as the only variation in a characteristic
function, occurs at the boundary of the set. This is why we introduce
In the fields of integral theory and geometric measure theory there are
a number of interesting integral formulas. Several of them fall in a
-category often refered to as \emph{Cauchy--Crofton style formulas}, and
+category often referred to as \emph{Cauchy--Crofton style formulas}, and
give ways to measure geometric objects using the set of all lines in the
plane. The formulas presented here will give a way to measure a curve by
counting the times it intersects line in the set of all lines.
We will now present and prove a Cauchy--Crofton formula in this case
where our domain is equipped with a metric tensor in each point.
\begin{theorem}[The Riemannian Cauchy--Crofton formula]
- Assume that our space $\Omega$ is equipped with a continous metric
+ Assume that our space $\Omega$ is equipped with a continuous metric
tensor $M(x)$, whose eigenvalues are bounded $0 < k \leq
\lambda_2 \leq \lambda_1 \leq K < \infty$ for all $x \in \Omega$.
The Cauchy--Crofton formula for a differentiable curve $C$ of finite
\end{theorem}
Before proving this we present an important result from measure theory
that we will need.
-\begin{theorem}[The Dominated Convergence theorem]
+\begin{theorem}[Lebesgue's Dominated Convergence theorem]
Let $\{ f_n \}$ be a sequence of real-valued measurable functions on
- a measure space $(S, \Sigma, \mu)$. Suppose that the sequence
- converges pointwise to a function $f$ and is dominated by some
- integrable function $g$ in the sense that
- \begin{equation}
- \abs{f_n(x)} \leq g(x)
- \end{equation}
- for all $n$ and all $x \in S$. Then $f$ is integrable and
- \begin{equation}
- \lim_{n \to \infty} \int_S \abs{f_n - f} \, d\mu = 0
- \end{equation}
- which further implies that
+ a space $S$ with measure $d\mu$ which converges almost
+ everywhere to a real-valued measurable function $f$. If there exists
+ an integrable function $g$ such that $\abs{f_n} \leq g$ for all $n$,
+ then $f$ is integrable and
\begin{equation}
\lim_{n \to \infty} \int_S f_n \, d\mu = \int_S f \, d\mu.
\end{equation}
\end{theorem}
For a proof and further background on measure theory and Lebesgue
-integration theory see for example \fixme{ref that is not wikipedia}
+integration theory see for example \cite{bartle1995elements}.
\begin{proof}[Proof of the Riemannian Cauchy--Crofton formula]
Assume first that our space is equipped with at constant metric
tensor $M$. The length of our curve using this tensor can be
\nu\right)^{\sfrac{3}{2}}}
= \frac{\det M}{\left(\nu^T \cdot M \cdot \nu\right)^{\sfrac{3}{2}}}
\end{equation}
- We have now proven that for a constant metric tensor $M$, the length
+ We have now proved that for a constant metric tensor $M$, the length
of the differentiable curve $C$ with regards to this tensor can be
calculated as
\begin{equation}
\label{eq:riemannian_segments}
\end{equation}
As our partition $\pi$ is refined the weight $w_\pi(x)$ converges
- pointwise to the continously varying weight
+ pointwise to the continuously varying weight
\begin{equation}
w(\nu, x) = \frac{\det M(x)}{\left( \nu^T \cdot M(x) \cdot \nu
\right)^{\sfrac{3}{2}}}
The whole transformation from the initial functional in
\eqref{eq:first_anisotropic_functional}
through the coarea formula and the Cauchy--Crofton formula was motivated
-by the discretization wich will be described next. We will see that we
+by the discretization which will be described next. We will see that we
can minimize the energy functional for each level separately, and that
the integral over all lines $\mathcal{L}$ can be approximated by a sum
over some discrete set of lines.
projects / master / commitdiff