From 0537618e169be7f63ea23a646e5ea95c44399712 Mon Sep 17 00:00:00 2001 From: =?utf8?q?Bj=C3=B8rn=20Rustad?= Date: Sun, 7 Dec 2014 18:45:49 +0100 Subject: [PATCH] Dominated conv thm and spelling --- bib.bib | 7 +++++++ theory.tex | 55 ++++++++++++++++++++++++------------------------------ 2 files changed, 31 insertions(+), 31 deletions(-) diff --git a/bib.bib b/bib.bib index f3efebf..879eba1 100644 --- a/bib.bib +++ b/bib.bib @@ -1,5 +1,12 @@ +@book{bartle1995elements, + title={The {E}lements of {I}ntegration and {L}ebesgue {M}easure}, + author={Bartle, Robert G}, + year={1995}, + publisher={John Wiley \& Sons} +} + @article{goldberg1988new, AUTHOR = {Goldberg, Andrew V. and Tarjan, Robert E.}, TITLE = {A new approach to the maximum-flow problem}, diff --git a/theory.tex b/theory.tex index cf51eb0..b6e408d 100644 --- a/theory.tex +++ b/theory.tex @@ -1,4 +1,4 @@ -\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 @@ -12,10 +12,10 @@ important theorems, transform it into a form that is easy to discretize. \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 @@ -136,7 +136,7 @@ First consider the tensor $S_0(x) = \nabla \tilde{f}_\sigma(x) \otimes \nabla \tilde{f}_\sigma(x)$. 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. @@ -297,13 +297,13 @@ in Figure \ref{fig:lower_semicont}. 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}. @@ -323,14 +323,14 @@ weaker version of continuity, but rather continuity in the weak 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} @@ -361,8 +361,8 @@ consider the fidelity term = \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 @@ -551,7 +551,7 @@ given in \cite{olsson2009extending}. 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 @@ -616,7 +616,7 @@ definition of the TV...} 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. @@ -664,7 +664,7 @@ a curve $\gamma$ parametrized by some parameter $t$ becomes 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 @@ -680,25 +680,18 @@ where our domain is equipped with a metric tensor in each point. \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 @@ -816,7 +809,7 @@ integration theory see for example \fixme{ref that is not wikipedia} \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} @@ -867,7 +860,7 @@ integration theory see for example \fixme{ref that is not wikipedia} \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}}} @@ -1050,7 +1043,7 @@ $\Sigma$. 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. -- 2.47.3