From 4a8369bad59fce1f4abede62a406b559ea6b984d Mon Sep 17 00:00:00 2001 From: =?utf8?q?Bj=C3=B8rn=20Rustad?= Date: Thu, 5 Feb 2015 19:35:40 +0100 Subject: [PATCH] Fixes and stuff --- bib.bib | 8 ++ conclusion.tex | 29 +++--- continuous.tex | 15 +-- discrete.tex | 12 +-- fig/factory/{n_q200.pgm => nfinger.pgm} | Bin fig/factory/nlena.pgm | Bin 0 -> 262159 bytes fig/factory/pmake.py | 18 +++- introduction.tex | 47 +++++----- main.tex | 31 ++++--- maxflow.tex | 12 ++- methods.tex | 1 + results.tex | 118 ++++++++++++------------ 12 files changed, 165 insertions(+), 126 deletions(-) rename fig/factory/{n_q200.pgm => nfinger.pgm} (100%) create mode 100644 fig/factory/nlena.pgm diff --git a/bib.bib b/bib.bib index 253f7f0..a8a8c1c 100644 --- a/bib.bib +++ b/bib.bib @@ -19,6 +19,14 @@ URL = {http://dx.doi.org/10.1007/978-1-4612-1015-3}, } +@PhdThesis{kolmogorovphd, + TITLE={{G}raph {B}ased {A}lgorithms for {S}cene {R}econstruction from {T}wo or {M}ore {V}iews}, + AUTHOR={Kolmogorov, Vladimir}, + SCHOOL={Cornell University}, + MONTH={September}, + YEAR={2003} +} + @misc{github, AUTHOR={Bjørn Rustad}, TITLE={\cpp{} implementation}, diff --git a/conclusion.tex b/conclusion.tex index 228b169..15ad845 100644 --- a/conclusion.tex +++ b/conclusion.tex @@ -1,14 +1,16 @@ \chapter{Discussion and conclusion} +\label{chap:conclusion} In this thesis we have seen how the total variation restoration method can be extended using an anisotropy tensor, and how this fits into the discretization and graph cut framework used in my project work \cite{project}. The anisotropy tensor was introduced in hopes of -reducing the amount of regularization across edges in the image. It was -constructed based on the structure tensor with three parameters exposed -for controlling the method. +reducing the amount of regularization applied across edges in the image. It was +constructed based on the structure tensor which contains information +about edge direction and steepness, and three parameters were exposed +for controlling the anisotropy. -The continuous functional we initally wanted to minimize was then +The continuous functional we initially wanted to minimize was then transformed using an anisotropic coarea formula and an anisotropic Cauchy--Crofton formula, both described in detail. The functional was discretized, and the discretization was shown---under some @@ -16,10 +18,11 @@ restrictions---to be consistent with the continuous functional. As in my project work~\cite{project} the discrete functional was minimized using maximum flow algorithms to obtain successive minimum -graph cuts. The description of the push-relabel algorithm was included from -my project work for completeness, and a description of the -Boykov--Kolmogorov algorithm was also given, as it is taylored for these -kinds of graph cut applications. +graph cuts. A description of the push-relabel algorithm was given as it +is known to have +good performance for general graphs. Further the +Boykov--Kolmogorov algorithm was described, as it is tailored for the +kinds of graphs appearing in imaging applications. Effort was put into giving a complete overview of the method, describing each part needed to go from the initial continuous problem, to the @@ -28,13 +31,14 @@ inner workings of the method, and also easily be able to implement it. Further work is possible in the study of the continuous problem, its well-foundedness, and also the anisotropic coarea and Cauchy--Crofton -formulas, which can be studied on a measure-theoretic foundation. -\fixme{tensor construction?} +formulas, which can be studied on a measure-theoretic foundation. The +construction of the tensor offers choices that can be explored further. + Regarding the discretization, the choice of neighborhood stencil also allows for further discussion, as approximation error can be traded for algorithm performance. Opening for non-uniform stencils, where the stencil choice depends on the level of detail in the neighborhood is -also a possiblity. +a possibility. In the end we looked at how the different parameters affect the performance of the algorithm. We will not give any unified conclusion as @@ -42,7 +46,8 @@ to whether this method is ``better'' or ``worse'' than the regular total variation method it is based on, or other methods. In different applications the input images have different properties, and one might also have different hopes for, and restrictions on the output image. But -we have seen that the anisotropy can help reduce the contrast loss. +we have seen how the anisotropy tensor affects the restoration, and that +it can have positive effects on contrast loss as well as method noise. %Because we have spread our attention across all parts of the restoration %method, it is also possible to go deeper into the theory behind all of diff --git a/continuous.tex b/continuous.tex index 8c3e80e..8d2f390 100644 --- a/continuous.tex +++ b/continuous.tex @@ -1,4 +1,5 @@ \chapter{Continuous formulation} +\label{chap:continuous} In the previous chapter we saw that there are many different approaches to the image restoration problem, all with their own strengths and @@ -37,7 +38,7 @@ below and above. If $A(x)$ is the identity matrix we get the regular total variation found in \eqref{eq:first_min_presentation}. When reducing the regular total variation, we will also try to reduce the variation over known edges in the image. This can lead to unwanted -contrast loss, expecially in fine details. +contrast loss, especially in fine details. By controlling $A(x)$ such that the contribution of $\nabla u(x)$ is reduced across known edges, we hope to retain the regularization properties of the original method while reducing some of the negative @@ -284,7 +285,7 @@ The basic things we ask of our functional \begin{equation} F(u) = \int_\Omega (u - f)^2 + \beta \, \TVA(u) \end{equation} -to have a \emph{well-posed} problem are lower semicontinuity +to have a \emph{well-posed} problem are lower semi-continuity and coercivity for existence, and convexity for uniqueness. We restrict ourself to $L^2(\Omega)$ which makes sense with our fidelity term, assuming that $f \in L^2(\Omega)$ initially. @@ -351,8 +352,8 @@ weakly sequentially pre-compact. \subsection{Lower semi-continuity} -The lower semicontinuity is the most tricky part, and this is where we -will take some shortcuts. Lower semicontinuity for a functional $F$ at a +The lower semi-continuity is the most tricky part, and this is where we +will take some shortcuts. Lower semi-continuity for a functional $F$ at a point $u$ means that at points $u_\epsilon$ close to $u$, the functional takes values either close to or above $F(u)$. More specifically, for every sequence $u_k$ converging to $u$, we have $F(u) \leq \liminf_k @@ -452,7 +453,7 @@ semi-continuous functional since \end{equation} and thus our functional is sequentially weakly lower semi-continuous. -The usual ways of going from coercivity and lower semicontinuity to +The usual ways of going from coercivity and lower semi-continuity to existence do not work in infinite dimensions. But with sequential coercivity and sequential lower semi-continuity in the weak topology we can conclude that we have existence from \cite[Theorem @@ -471,7 +472,7 @@ can conclude that we have existence from \cite[Theorem % \min_{u \in L^2(\Omega)} F_n(u). % \end{equation} % Assume that each of the functionals $F_n$ is weakly sequentially -% lower-semicontinuous, has a unique minimizer $u_n$, and in addition that they are +% lower semi-continuous, has a unique minimizer $u_n$, and in addition that they are % uniformly coercive in the sense that for any sequence $v_n \in % L^2(\Omega)$ % \begin{equation} @@ -513,7 +514,7 @@ can conclude that we have existence from \cite[Theorem % Assume that the convergence in \eqref{eq:un_u_weak} does not hold, % and denote the weakly convergent subsequence by $u_{n_j} % \rightharpoonup \bar{u} \neq u^*$. By the weak sequential lower -% semicontinuity we have +% semi-continuity we have % \begin{equation} % \begin{aligned} % F(\bar{u}) &\leq \liminf_{j \to \infty} F\left(u_{n_j}\right) \\ diff --git a/discrete.tex b/discrete.tex index 257fb6d..4704c51 100644 --- a/discrete.tex +++ b/discrete.tex @@ -5,7 +5,7 @@ The whole transformation from the initial functional in \eqref{eq:first_anisotropic_functional} through the anisotropic coarea formula and the Cauchy--Crofton formula was motivated by the discrete formulation which will be described here. After discretizing -the functional, we will se how a graph cut approach can be used to find +the functional, we will see how a graph cut approach can be used to find a global minimizer in polynomial time. \section{Discretization} @@ -95,7 +95,7 @@ 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} +\subsubsection{Discrete anisotropic Cauchy--Crofton formula} \begin{figure} \input{fig/line_disc} @@ -615,8 +615,8 @@ term coming from our aim to minimize the total variation. \begin{tabular}{ccccc} \hline $u^\lambda_x$ & $F_\lambda^x(u^\lambda_x)$ & Min.\ cut $(S, T)$ - & Graph \subref{fig:norm_subgraph_pos} cut cap. & Graph - \subref{fig:norm_subgraph_neg} cut cap. \\ + & Graph (\subref{fig:norm_subgraph_pos}) cut cap. & Graph + (\subref{fig:norm_subgraph_neg}) cut cap. \\ \hline $0$ & $0$ & $(\{s\}, \{u^\lambda_x, t\})$ & $0$ & $-F_\lambda^x(1)$ \\ @@ -699,8 +699,8 @@ of $(u^\lambda_x, u^\lambda_y)$ corresponds to cuts in the graph. $(u^\lambda_x, u^\lambda_y)$ & $F^{x,y}(u^\lambda_x, u^\lambda_y)$ & Min.\ cut $(S, T)$ - & Alt.\ \subref{fig:neigh_subgraph_alt1} cut cap. & Alt.\ - \subref{fig:neigh_subgraph_alt2} cut cap. 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