From 8ff1869027416edb54c11018ef2c1607180b885c Mon Sep 17 00:00:00 2001 From: =?utf8?q?Bj=C3=B8rn=20Rustad?= Date: Wed, 26 Nov 2014 19:18:10 +0100 Subject: [PATCH] MOEAR --- bib.bib | 7 ++++ fig/edges.pgm | Bin 0 -> 65551 bytes fig/edges.png | Bin 0 -> 58319 bytes introduction.tex | 2 ++ theory.tex | 92 +++++++++++++++++++++++++++++++++-------------- 5 files changed, 75 insertions(+), 26 deletions(-) create mode 100644 fig/edges.pgm create mode 100644 fig/edges.png diff --git a/bib.bib b/bib.bib index 009fec9..f3efebf 100644 --- a/bib.bib +++ b/bib.bib @@ -99,6 +99,13 @@ publisher={Elsevier} } +@article{project, + title={Total variation based image restoration using graph cuts}, + author={Rustad, Bjørn}, + year={2014}, + publisher={NTNU} +} + @article{darbon2006image, AUTHOR = {Darbon, J{\'e}r{\^o}me and Sigelle, Marc}, TITLE = {Image restoration with discrete constrained total variation. diff --git a/fig/edges.pgm b/fig/edges.pgm new file mode 100644 index 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P A(x) P^T$ and curve length +calculation in +\eqref{eq:per_to_length2} fulfill all the assumptions of Theorem \ref{thm:riemannian_cauchy_crofton}. \fixme{did we argue for -continuous?} Inserting the Cauchy--Crofton formula, we obtain the final -expression for the continuous anisotropic total variation +continuous?} Thus we can apply the Cauchy--Crofton formula of +\eqref{eq:riemannian_cauchy_crofton} to calculate the curve length in +\eqref{eq:per_to_length2} and through the anisotropic coarea formula in +Theorem \ref{thm:anisotropic_coarea} obtain our final expression for the +continuous anisotropic total variation \begin{equation} \TVA(u) = \int_{-\infty}^{\infty} \int_\mathcal{L} \sum_{x \in \ell_{\nu, \rho} \cap \gamma_s } \frac{\det M(x)} @@ -1009,7 +1047,8 @@ $\det A = \det PAP^T = \det M$, and from our eigendecomposition in amounts to switching the two eigenvalues $\sigma_1$ and $\sigma_2$ in $\Sigma$. -The whole transformation from the initial formulation in \fixme{ref} +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 can minimize the energy functional for each level separately, and that @@ -1036,7 +1075,8 @@ energyfunction as a sum over all the levels $\mathcal{L}$. \subsection{Fidelity term} -For some pixel position $x \in \mathcal{G}$ and some level value $k \in +The fidelity term can be discretize as in my project work +\cite{project}. For some pixel position $x \in \mathcal{G}$ and some level value $k \in \mathcal{L}$, we define the following function \begin{equation} N_x(k) = \abs{k - v_x}^p -- 2.47.3