From 922d6518ac11aa64515e9ef34c497edee7cc3f54 Mon Sep 17 00:00:00 2001 From: =?utf8?q?Bj=C3=B8rn=20Rustad?= Date: Fri, 6 Feb 2015 20:44:37 +0100 Subject: [PATCH] Nomencl work --- continuous.tex | 22 +++++++++++----------- discrete.tex | 12 ++++++------ maxflow.tex | 4 ++-- methods.tex | 41 ++++++++++++++++++++--------------------- 4 files changed, 39 insertions(+), 40 deletions(-) diff --git a/continuous.tex b/continuous.tex index 820f738..77713a4 100644 --- a/continuous.tex +++ b/continuous.tex @@ -73,7 +73,7 @@ $\lnorm{\eta}_A^* = \sup_x (\eta^T A^{-1} \eta)^{\sfrac{1}{2}}$, and with that w \nomenclature{$\lnorm{\xi}_A$}{The norm $\sup_x (\xi^T A \xi)^{\sfrac{1}{2}}$}% \nomenclature{$\lnorm{\eta}_A^*$}{The norm $\sup_x (\eta^T A^{-1} -\eta)^{\sfrac{1}{2}}$.}% +\eta)^{\sfrac{1}{2}}$}% present the formal definition of the anisotropic total variation. \begin{definition}[Anisotropic total variation] For a function $u \in L^2(\Omega)$ and a continuous symmetric @@ -88,7 +88,7 @@ present the formal definition of the anisotropic total variation. \label{def:extended_tv} \end{definition} \nomenclature{$\TVA(u)$}{Anisotropic total variation of image $u$ given -the anisotropy tensor $A$.}% +the anisotropy tensor $A$}% With this extended definition, we have arrived at a minimization problem where we seek to find a minimizer of the functional @@ -227,7 +227,7 @@ and for $\sigma_1$ and $\sigma_2$ we choose \end{aligned} \label{eq:sigma_construction} \end{equation} -\nomenclature{$\omega$}{Anisotropy parameter.} +\nomenclature{$\omega$}{Anisotropy parameter} Thus the eigenvectors of $A(x)$ and $S_\rho(x)$ are equal, while the eigenvalues are different. A visualization of the two tensors can be seen in Figure~\ref{fig:tensor_viz} where the two tensors are shown at @@ -776,7 +776,7 @@ the following definition of the anisotropic set perimeter. \end{equation} \end{definition} \nomenclature{$\PerA(U;\Omega)$}{Anisotropic perimeter of set $U$ using -anisotropy tensor $A$.}% +anisotropy tensor $A$}% The anisotropic set perimeter is not like the regular set perimeter and does not measure the length of the boundary of the set, but it can for sufficiently nice level sets be calculated in the following way @@ -844,11 +844,11 @@ measure on this set $d\mathcal{L} = \dpdr$ we are ready to introduce the Cauchy--Crofton formula. Note that the measure $d\mathcal{L}$ is invariant under rotations. \nomenclature{$\mathcal{L}$}{The set of all straight lines in the -plane.}% +plane}% \nomenclature{$\ell_{\phi, \rho}$}{A line given by the angle of the -normal $\phi$ and the distance to origin $\rho$.}% +normal $\phi$ and the distance to origin $\rho$}% \nomenclature{$\ell_{\nu, \rho}$}{A line given by a tangent vector $\nu$ -and the distance to origin $\rho$.}% +and the distance to origin $\rho$}% \begin{theorem}[The Euclidean Cauchy--Crofton formula] Given a differentiable curve $C$ in $\mathbb{R}^2$, the length of this curve $\abs{C}$ is related to the set of lines $\mathcal{L}$ as @@ -868,10 +868,10 @@ If our space is equipped with a metric tensor $M(x)$ such that the inner product of two vectors $a$ and $b$ in a point $x$ is calculated as $\langle a, b\rangle_M = \langle a, M(x) b \rangle$, then the length of a curve $\gamma$ parametrized by some parameter $t$ becomes -\nomenclature{$M(x)$}{A metric tensor.}% -\nomenclature{$\abs{C}$}{The length of the curve $C$.}% +\nomenclature{$M(x)$}{A metric tensor}% +\nomenclature{$\abs{C}$}{The length of the curve $C$}% \nomenclature{$\abs{C}_M$}{The length of the curve $C$ calculated using -the metric tensor $M$.}% +the metric tensor $M$}% \begin{equation} \abs{\gamma}_M = \int_\gamma \sqrt{\langle \dot{\gamma}, M\big(\gamma(t)\big) \, \dot{\gamma} \rangle} \, dt. @@ -919,7 +919,7 @@ by a sum over all these lines. transformation $F : \mathcal{L} \to \mathcal{L}$, which maps $\ell_{\phi, \rho} \mapsto \Mhalf \ell_{\phi, \rho}$. \nomenclature{$J_M(\ell_{\phi, \rho})$}{Jacobian of the coordinate - transformation induced by the metric tensor $M$.} + transformation induced by the metric tensor $M$} We will now compute the Jacobian $J_M(\ell_{\phi, \rho})$. As $M\in \mathbb{R}^{2\times2}$ is symmetric, so is $M^{\sfrac{1}{2}}$, and diff --git a/discrete.tex b/discrete.tex index a63c617..0cd9f22 100644 --- a/discrete.tex +++ b/discrete.tex @@ -36,7 +36,7 @@ level value $k \in \begin{equation} N_x(k) = \abs{k - f_x}^2 \end{equation} -\nomenclature{$N_x(k)$}{Helper function $N_x(k) = \abs{k-f_x}^2$.}% +\nomenclature{$N_x(k)$}{Helper function $N_x(k) = \abs{k-f_x}^2$}% which is the value of the fidelity term if we were to give $u_x$ a value of $k$. This allows us write \begin{equation} @@ -145,7 +145,7 @@ discretization $\mathcal{L}_D$, then $a \in \mathcal{N}(b)$. Extending the edges Figure~\ref{fig:line_neigh} gives all lines going through the point considered. Figure~\ref{fig:line_family} shows all lines of a given family, i.e.\ lines having the same angle parameter $\phi$. -\nomenclature{$\mathcal{N}(x)$}{Neighborhood of pixel $x$.}% +\nomenclature{$\mathcal{N}(x)$}{Neighborhood of pixel $x$}% Thus not only have we discretized the set of lines, but each line is made up of edges going from one grid point to the next. We will @@ -471,8 +471,8 @@ with capacity function $c$, one might call it a capacitated graph or a flow network, but as all our graphs will be capacitated from this point, we will just call them graphs and we write $G = (V, E, c)$. \nomenclature{$G = (V,E,c)$}{A graph given by the set of vertices $V$, -the set of edges $E$, and the capacity function $c$.}% -\nomenclature{$c(u,v)$}{The capacity function $c : V \times V \to \left[0, \infty \right)$.}% +the set of edges $E$, and the capacity function $c$}% +\nomenclature{$c(u,v)$}{The capacity function $c : V \times V \to \left[0, \infty \right)$}% There are two special vertices in the graph, the source $s$ and the sink $t$. Contrary to other vertices, which can neither produce nor @@ -480,8 +480,8 @@ receive excess flow, the source can produce and the sink can receive an unlimited amount of flow. The most basic problem in graph flow theory is the question of how much flow it is possible to send through the graph from the source to the sink. -\nomenclature{$s$}{The source vertex.}% -\nomenclature{$t$}{The sink vertex.}% +\nomenclature{$s$}{The source vertex}% +\nomenclature{$t$}{The sink vertex}% What we seek in our final graph is a minimum $s$-$t$-cut, a ``minimal'' line through the graph that cuts a set of edges and diff --git a/maxflow.tex b/maxflow.tex index 196de92..8fc5b1d 100644 --- a/maxflow.tex +++ b/maxflow.tex @@ -257,7 +257,7 @@ As in most of the cited push-relabel literature, we define $N = which represents the amount of flow which disappears in vertex $u$. Equivalent to the preflow conservation constraint is stating that $e(u) \geq 0$ for all vertices $u \in V - \{s,t\}$. -\nomenclature{$e(u)$}{Excess in vertex $u$.}% +\nomenclature{$e(u)$}{Excess in vertex $u$}% %The idea of the algorithm is to maintain a height map of the %vertices in the graph where connected vertices can not have a large @@ -274,7 +274,7 @@ d(v) + 1$. For all vertices $u$, the label $d(u)$ will be a lower bound on the length from $u$ to $t$ in $G_f$ which is why it is also often called a distance labeling. \nomenclature{$d(u)$}{Height map or distance labeling $d : V \to -\mathbb{N}$.}% +\mathbb{N}$}% A vertex $u$ is \emph{active} if $u \in V - \{s,t\}$, it has positive excess ($e(u) > 0$) and $d(u) < N$. These are the vertices we want to diff --git a/methods.tex b/methods.tex index 4e92d36..823f7df 100644 --- a/methods.tex +++ b/methods.tex @@ -10,8 +10,8 @@ background on image processing in general. In this chapter, and also in the rest of the thesis we will assume that we are given an image $f : \Omega \to \mathbb{R}$ where $\Omega$ is a rectangular, open domain. Because of limitations in the numerical method -\nomenclature{$f$}{Original, noisy image.}% -\nomenclature{$\Omega$}{Rectangular, open domain.}% +\nomenclature{$f$}{Original, noisy image}% +\nomenclature{$\Omega$}{Rectangular, open domain}% used, the codomain is $\mathbb{R}$ and we are thus restricted to monochrome, or grayscale images. Such images are produced in large numbers by for example ultrasound, X-ray and MRI machines. @@ -24,9 +24,9 @@ types of noise. We will assume that the given image $f$ is a combination of an underlying, actual image $u^*$, and some noise $\delta$. The simplest -\nomenclature{$\delta$}{Noise.}% +\nomenclature{$\delta$}{Noise}% model is additive noise where the assumption is that $f = u^* + \delta$. -\nomenclature{$u^*$}{Usually unknown, actual image without noise.}% +\nomenclature{$u^*$}{Usually unknown, actual image without noise}% There is also multiplicative noise where $f = u^* \cdot \delta$. An other much seen noise type is salt and pepper noise, which is when black and white pixels randomly appear in the image. @@ -53,7 +53,7 @@ Gaussian function y^2}{2\sigma^2} \right). \label{eq:gaussian_function} \end{equation} -\nomenclature{$K_\sigma(x,y)$}{The Gaussian kernel.}% +\nomenclature{$K_\sigma(x,y)$}{The Gaussian kernel}% In the discrete setting where the image consists of a grid of pixels, the Gaussian blur amounts to calculating each pixel in the output image as a weighted average of its neighboring pixels in the input image. @@ -88,7 +88,7 @@ diffusivity} $\alpha$ such that the equation becomes \end{aligned} \right. \end{equation} -\nomenclature{$\alpha(\nabla u)$}{Scalar thermal diffusivity.}% +\nomenclature{$\alpha(\nabla u)$}{Scalar thermal diffusivity}% The thermal diffusivity $\alpha(\nabla u) = \alpha(x, \nabla u)$ is material dependent, and can also vary throughout the object. It specifies how well heat travels through the specific point in the object. We can make use of this in the image @@ -155,19 +155,19 @@ problem becomes \label{eq:aniso_diff} \end{equation} \nomenclature{$A(u)$}{Thermal diffusivity tensor, or anisotropy -tensor.}% +tensor}% where $\nu$ is the outer normal of $\Omega$. The tensor $A(u)$ is constructed such as to diminish the effect of $\nabla u$ across what we believe to be edges in the image. This way, there will also be less diffusion through these edges. Weickert \cite{weickert1998anisotropic} suggests constructing $A(u)$ based on the edge estimator $\nabla u_\sigma$ where -\nomenclature{$\nabla u_\sigma$}{Edge estimator.}% +\nomenclature{$\nabla u_\sigma$}{Edge estimator}% \begin{equation} u_\sigma := K_\sigma * \tilde{u} \end{equation} \nomenclature{$u_\sigma$}{Edge estimator, image $u$ smoothed with a Gaussian of -parameter $\sigma$.}% +parameter $\sigma$}% and $\tilde{u}$ is an extension of $u$ from $\Omega$ to $\mathbb{R}^2$ \nomenclature{$\tilde{u}$}{Symmetric extension of $u$ from $\Omega$ to $\mathbb{R}^2$}% @@ -182,7 +182,7 @@ scale, the \emph{structure tensor} is introduced \begin{equation} S_\rho(x) := K_\rho * (\nabla u_\sigma \otimes \nabla u_\sigma), \end{equation} -\nomenclature{$S_\rho(x)$}{Structure tensor.}% +\nomenclature{$S_\rho(x)$}{Structure tensor}% where the convolution with the Gaussian function $K_\rho$ is done component-wise. The anisotropy tensor $A(u)$ can then be constructed based on the eigenvectors and eigenvalues of $S_\rho(x)$. @@ -239,10 +239,9 @@ is usually formulated as a minimization problem \label{eq:first_min_presentation} \end{gathered} \end{equation} -\nomenclature{$F(u)$}{Variational functional on $u$.}% \nomenclature{$L^p(\Omega)$}{Real functions $f$ on $\Omega$ for which -$\int_\Omega \labs{f}^p < \infty$.}% -\nomenclature{$\beta$}{Regularization parameter.}% +$\int_\Omega \labs{f}^p < \infty$}% +\nomenclature{$\beta$}{Regularization parameter}% where $p$ is normally taken to be 1 or 2. The fidelity term penalizes images $u$ that are far from the original image $f$. The @@ -278,11 +277,11 @@ variation using the distributional derivative. $\Omega$. \label{def:tv} \end{definition} -\nomenclature{$\TV(u)$}{Total variation of the image $u$.}% +\nomenclature{$\TV(u)$}{Total variation of the image $u$}% \nomenclature{$C^\infty_c\left(\Omega, \mathbb{R}^2\right)$}{The space of smooth functions from $\Omega$ to $\mathbb{R}^2$ with compact support in -$\Omega$.}% -\nomenclature{$\varphi$}{Test function.}% +$\Omega$}% +\nomenclature{$\varphi$}{Test function}% Note that since $\Omega$ is open and bounded, the test functions $\varphi$ vanish on the boundary of $\Omega$. Thus no variation is @@ -300,7 +299,7 @@ to introduce the space of functions of bounded variation. \end{equation} \end{definition} \nomenclature{$\BV(\Omega)$}{Functions of bounded variation in -$\Omega$.}% +$\Omega$}% Our optimization problem has thus become \begin{equation} \min_{u \in \BV(\Omega)} \int_\Omega \abs{u - v}^p \, dx + \beta \, @@ -396,8 +395,8 @@ the functional. In the discrete setting our image consists of pixels, and is represented by a function $u : \mathcal{G} \to \mathcal{P}$ where $\mathcal{G}$ is a -\nomenclature{$\mathcal{G}$}{Regular grid of pixels over $\Omega$.}% -\nomenclature{$\mathcal{P}$}{Discrete set of pixel values, or levels.}% +\nomenclature{$\mathcal{G}$}{Regular grid of pixels over $\Omega$}% +\nomenclature{$\mathcal{P}$}{Discrete set of pixel values, or levels}% regular grid over $\Omega$, and $\mathcal{P} = \{0, \hdots, L-1\}$ is the discrete set of pixel values, or \emph{levels}. We denote the value in pixel $x$ as $u(x) = u_x$. @@ -406,12 +405,12 @@ For an image $u$ and a level $\lambda$ we denote the \emph{level set} by $\{ u > \lambda\}$, defined as the set $\{ x \in \Omega : u_x > \lambda \}$. The thresholded image $u^\lambda$, an indicator function, is then defined as -\nomenclature{$u^\lambda$}{Thresholded image at level $\lambda$.}% +\nomenclature{$u^\lambda$}{Thresholded image at level $\lambda$}% \begin{equation} u^\lambda = \idfun_{u > \lambda}. \end{equation} Here, $\idfun_E$ signifies the characteristic function of the set $E$, -\nomenclature{$\idfun_E$}{Characteristic function of the set $E$.}% +\nomenclature{$\idfun_E$}{Characteristic function of the set $E$}% the function which is equal to one in every point in $E$, and zero elsewhere. -- 2.47.3