Setting up the structure

diff --git a/.gitignore b/.gitignore
new file mode 100644 (file)
index 0000000..dba8159
--- /dev/null
+++ b/.gitignore
@@ -0,0 +1,7 @@
+*.aux
+*.bbl
+*.blg
+*.log
+*.out
+*.pdf
+*.toc

diff --git a/Makefile b/Makefile
new file mode 100644 (file)
index 0000000..c2023ad
--- /dev/null
+++ b/Makefile
@@ -0,0 +1,6 @@
+all:
+       xelatex main
+       bibtex main
+       xelatex main
+       xelatex main
+       cp main.pdf ~/Dropbox/master/

diff --git a/appendix.tex b/appendix.tex
new file mode 100644 (file)
index 0000000..87c013b
--- /dev/null
+++ b/appendix.tex
@@ -0,0 +1,4 @@
+\chapter{\cpp\ implementation}
+
+Some code.
+

diff --git a/bib.bib b/bib.bib
new file mode 100644 (file)
index 0000000..6d7589b
--- /dev/null
+++ b/bib.bib
@@ -0,0 +1,294 @@
+<bibtex>\r
+\r
+@article{goldberg1988new,\r
+       AUTHOR = {Goldberg, Andrew V. and Tarjan, Robert E.},\r
+       TITLE = {A new approach to the maximum-flow problem},\r
+       JOURNAL = {J. Assoc. Comput. Mach.},\r
+       FJOURNAL = {Journal of the Association for Computing Machinery},\r
+       VOLUME = {35},\r
+       YEAR = {1988},\r
+       NUMBER = {4},\r
+       PAGES = {921--940},\r
+       ISSN = {0004-5411},\r
+       CODEN = {JACOAH},\r
+       MRCLASS = {90B10 (68Q25)},\r
+       MRNUMBER = {1072405 (92c:90050)},\r
+       MRREVIEWER = {Robert E. Beck},\r
+       DOI = {10.1145/48014.61051},\r
+       URL = {http://dx.doi.org/10.1145/48014.61051},\r
+}\r
+\r
+@article{cherkassky1997implementing,\r
+       AUTHOR = {Cherkassky, B. V. and Goldberg, A. V.},\r
+       TITLE = {On implementing the push-relabel method for the maximum flow problem},\r
+       JOURNAL = {Algorithmica},\r
+       FJOURNAL = {Algorithmica. An International Journal in Computer Science},\r
+       VOLUME = {19},\r
+       YEAR = {1997},\r
+       NUMBER = {4},\r
+       PAGES = {390--410},\r
+       ISSN = {0178-4617},\r
+       CODEN = {ALGOEJ},\r
+       MRCLASS = {68Q25 (68R05)},\r
+       MRNUMBER = {1470042},\r
+       DOI = {10.1007/PL00009180},\r
+       URL = {http://dx.doi.org/10.1007/PL00009180},\r
+}\r
+\r
+@incollection{boykov2006graph,\r
+       AUTHOR = {Boykov, Y. and Veksler, O.},\r
+       TITLE = {Graph cuts in vision and graphics: theories and applications},\r
+       BOOKTITLE = {Handbook of mathematical models in computer vision},\r
+       PAGES = {79--96},\r
+       PUBLISHER = {Springer, New York},\r
+       YEAR = {2006},\r
+       MRCLASS = {68U05 (94A08)},\r
+       MRNUMBER = {2232525},\r
+       DOI = {10.1007/0-387-28831-7_5},\r
+       URL = {http://dx.doi.org/10.1007/0-387-28831-7_5},\r
+}\r
+\r
+@article{derigs1989implementing,\r
+       AUTHOR = {Derigs, U. and Meier, W.},\r
+       TITLE = {Implementing {G}oldberg's max-flow-algorithm---a computational\r
+               investigation},\r
+       JOURNAL = {Z. Oper. Res.},\r
+       FJOURNAL = {Zeitschrift f\"ur Operations Research. Mathematical Methods of\r
+               Operations Research},\r
+       VOLUME = {33},\r
+       YEAR = {1989},\r
+       NUMBER = {6},\r
+       PAGES = {383--403},\r
+       ISSN = {0340-9422},\r
+       MRCLASS = {90C35 (90B10)},\r
+       MRNUMBER = {1030791 (90k:90163)},\r
+       DOI = {10.1007/BF01415937},\r
+       URL = {http://dx.doi.org/10.1007/BF01415937},\r
+}\r
+\r
+@incollection{caselles2011total,\r
+       year={2011},\r
+       isbn={978-0-387-92919-4},\r
+       booktitle={Handbook of Mathematical Methods in Imaging},\r
+       editor={Scherzer, Otmar},\r
+       doi={10.1007/978-0-387-92920-0_23},\r
+       title={Total Variation in Imaging},\r
+       url={http://dx.doi.org/10.1007/978-0-387-92920-0_23},\r
+       publisher={Springer New York},\r
+       author={Caselles, V. and Chambolle, A. and Novaga, M.},\r
+       pages={1016-1057}\r
+}\r
+\r
+@incollection{chan2011numerical,\r
+       title={Numerical methods and applications in total variation image restoration},\r
+       author={Chan, Raymond and Chan, Tony and Yip, Andy},\r
+       booktitle={Handbook of Mathematical Methods in Imaging},\r
+       pages={1059--1094},\r
+       year={2011},\r
+       publisher={Springer}\r
+}\r
+\r
+@article{rudin1992nonlinear,\r
+       title={Nonlinear total variation based noise removal algorithms},\r
+       author={Rudin, Leonid I and Osher, Stanley and Fatemi, Emad},\r
+       journal={Physica D: Nonlinear Phenomena},\r
+       volume={60},\r
+       number={1--4},\r
+       pages={259--268},\r
+       year={1992},\r
+       publisher={Elsevier}\r
+}\r
+\r
+@article{darbon2006image,\r
+       AUTHOR = {Darbon, J{\'e}r{\^o}me and Sigelle, Marc},\r
+       TITLE = {Image restoration with discrete constrained total variation.\r
+               {P}art {I}: {F}ast and exact optimization},\r
+       JOURNAL = {J. Math. Imaging Vision},\r
+       FJOURNAL = {Journal of Mathematical Imaging and Vision},\r
+       VOLUME = {26},\r
+       YEAR = {2006},\r
+       NUMBER = {3},\r
+       PAGES = {261--276},\r
+       ISSN = {0924-9907},\r
+       CODEN = {JMIVEK},\r
+       MRCLASS = {68U10 (94A08)},\r
+       MRNUMBER = {2286448 (2008b:68117)},\r
+       MRREVIEWER = {Vassileios Drakopoulos},\r
+       DOI = {10.1007/s10851-006-8803-0},\r
+       URL = {http://dx.doi.org/10.1007/s10851-006-8803-0},\r
+}\r
+\r
+@inproceedings{boykov2003computing,\r
+       title={Computing geodesics and minimal surfaces via graph cuts},\r
+       author={Boykov, Yuri and Kolmogorov, Vladimir},\r
+       booktitle={Computer Vision, 2003. Proceedings. Ninth IEEE International Conference on},\r
+       pages={26--33},\r
+       year={2003},\r
+       organization={IEEE}\r
+}\r
+\r
+@book{weickert1998anisotropic,\r
+       AUTHOR = {Weickert, Joachim},\r
+       TITLE = {Anisotropic diffusion in image processing},\r
+       SERIES = {European Consortium for Mathematics in Industry},\r
+       PUBLISHER = {B. G. Teubner, Stuttgart},\r
+       YEAR = {1998},\r
+       ISBN = {3-519-02606-6},\r
+       MRCLASS = {94A08 (65M30 68U10)},\r
+       MRNUMBER = {1666943 (2000a:94003)},\r
+       MRREVIEWER = {Gilbert Crombez},\r
+}\r
+\r
+@article{chambolle2004algorithm,\r
+       AUTHOR = {Chambolle, Antonin},\r
+       TITLE = {An algorithm for total variation minimization and\r
+               applications},\r
+       NOTE = {Special issue on mathematics and image analysis},\r
+       JOURNAL = {J. Math. Imaging Vision},\r
+       FJOURNAL = {Journal of Mathematical Imaging and Vision},\r
+       VOLUME = {20},\r
+       YEAR = {2004},\r
+       NUMBER = {1-2},\r
+       PAGES = {89--97},\r
+       ISSN = {0924-9907},\r
+       CODEN = {JMIVEK},\r
+       MRCLASS = {49M30 (65R32 68T45 68U10 90C30)},\r
+       MRNUMBER = {2049783 (2005m:49058)},\r
+       DOI = {10.1023/B:JMIV.0000011320.81911.38},\r
+       URL = {http://dx.doi.org/10.1023/B:JMIV.0000011320.81911.38},\r
+}\r
+\r
+@incollection{kolmogorov2002energy,\r
+       title={What energy functions can be minimized via graph cuts?},\r
+       author={Kolmogorov, Vladimir and Zabih, Ramin},\r
+       booktitle={Computer Vision—ECCV 2002},\r
+       pages={65--81},\r
+       year={2002},\r
+       publisher={Springer}\r
+}\r
+\r
+@book{cormen2009introduction,\r
+       title={Introduction to algorithms},\r
+       author={Cormen, Thomas H. and Leiserson, Charles E. and Rivest, Ronald L. and Stein, Clifford},\r
+       edition={Third},\r
+       year={2009},\r
+       publisher={MIT press}\r
+}\r
+\r
+@book{scherzer2008variational,\r
+       AUTHOR = {Scherzer, Otmar and Grasmair, Markus and Grossauer, Harald and\r
+               Haltmeier, Markus and Lenzen, Frank},\r
+       TITLE = {Variational methods in imaging},\r
+       SERIES = {Applied Mathematical Sciences},\r
+       VOLUME = {167},\r
+       PUBLISHER = {Springer, New York},\r
+       YEAR = {2009},\r
+       ISBN = {978-0-387-30931-6},\r
+       MRCLASS = {49-02 (49J10 68U10 94A08)},\r
+       MRNUMBER = {2455620 (2009j:49001)},\r
+       MRREVIEWER = {Bogdan G. Nita},\r
+}\r
+\r
+@article{opencv_library,\r
+       author = {Bradski, G.},\r
+       citeulike-article-id = {2236121},\r
+       journal = {Dr. Dobb's Journal of Software Tools},\r
+       keywords = {bibtex-import},\r
+       posted-at = {2008-01-15 19:21:54},\r
+       priority = {4},\r
+       title = {OpenCV Library},\r
+       year = {2000}\r
+}\r
+\r
+@incollection{dinitz2006dinitz,\r
+       AUTHOR = {Dinitz, Yefim},\r
+       TITLE = {Dinitz' algorithm: the original version and {E}ven's version},\r
+       BOOKTITLE = {Theoretical computer science},\r
+       SERIES = {Lecture Notes in Comput. Sci.},\r
+       VOLUME = {3895},\r
+       PAGES = {218--240},\r
+       PUBLISHER = {Springer, Berlin},\r
+       YEAR = {2006},\r
+       MRCLASS = {90C35 (01A60 05C85 90-03)},\r
+       MRNUMBER = {2248665 (2007f:90123)},\r
+       MRREVIEWER = {Mechthild Opperud},\r
+       DOI = {10.1007/11685654_10},\r
+       URL = {http://dx.doi.org/10.1007/11685654_10},\r
+}\r
+\r
+@article{boykov2004experimental,\r
+       title={An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision},\r
+       author={Boykov, Yuri and Kolmogorov, Vladimir},\r
+       journal={Pattern Analysis and Machine Intelligence, IEEE Transactions on},\r
+       volume={26},\r
+       number={9},\r
+       pages={1124--1137},\r
+       year={2004},\r
+       publisher={IEEE}\r
+}\r
+\r
+@article{cheriyan1989analysis,\r
+       title={Analysis of preflow push algorithms for maximum network flow},\r
+       author={Cheriyan, Joseph and Maheshwari, S. N.},\r
+       journal={SIAM Journal on Computing},\r
+       volume={18},\r
+       number={6},\r
+       pages={1057--1086},\r
+       year={1989},\r
+       publisher={SIAM}\r
+}\r
+\r
+@article{goldfarb2009parametric,\r
+       title={Parametric maximum flow algorithms for fast total variation minimization},\r
+       author={Goldfarb, Donald and Yin, Wotao},\r
+       journal={SIAM Journal on Scientific Computing},\r
+       volume={31},\r
+       number={5},\r
+       pages={3712--3743},\r
+       year={2009},\r
+       publisher={SIAM}\r
+}\r
+\r
+@misc{darbonsigellebinary,\r
+       title={{J}{\'e}r{\^o}me {D}arbon, {T}otal {V}ariation {M}inimization},\r
+       howpublished = {\url{http://jerome.berbiqui.org/total-variation-code/}},\r
+       note={Accessed: 2014-02-10}\r
+}\r
+\r
+@article {hochbaum2001efficient,\r
+       AUTHOR = {Hochbaum, Dorit S.},\r
+       TITLE = {An efficient algorithm for image segmentation, {M}arkov random\r
+               fields and related problems},\r
+       JOURNAL = {J. ACM},\r
+       FJOURNAL = {Journal of the ACM},\r
+       VOLUME = {48},\r
+       YEAR = {2001},\r
+       NUMBER = {4},\r
+       PAGES = {686--701 (electronic)},\r
+       ISSN = {0004-5411},\r
+       MRCLASS = {68U10 (62M40 68Q25)},\r
+       MRNUMBER = {2144926 (2005m:68226)},\r
+       DOI = {10.1145/502090.502093},\r
+       URL = {http://dx.doi.org/10.1145/502090.502093},\r
+}\r
+\r
+@article {gallo1989fast,\r
+       AUTHOR = {Gallo, Giorgio and Grigoriadis, Michael D. and Tarjan, Robert\r
+               E.},\r
+       TITLE = {A fast parametric maximum flow algorithm and applications},\r
+       JOURNAL = {SIAM J. Comput.},\r
+       FJOURNAL = {SIAM Journal on Computing},\r
+       VOLUME = {18},\r
+       YEAR = {1989},\r
+       NUMBER = {1},\r
+       PAGES = {30--55},\r
+       ISSN = {0097-5397},\r
+       CODEN = {SMJCAT},\r
+       MRCLASS = {68Q25 (05C35 68R10 90B10)},\r
+       MRNUMBER = {978165 (90b:68038)},\r
+       MRREVIEWER = {Andreas Brandst{\"a}dt},\r
+       DOI = {10.1137/0218003},\r
+       URL = {http://dx.doi.org/10.1137/0218003},\r
+}\r
+\r
+</bibtex>\r

diff --git a/commands.tex b/commands.tex
new file mode 100644 (file)
index 0000000..4b6a8a8
--- /dev/null
+++ b/commands.tex
@@ -0,0 +1,8 @@
+\newcommand{\abs}[1]{\lvert #1 \rvert}
+\newcommand{\norm}[1]{\lVert #1 \rVert}
+\newcommand{\TV}{\mathit{TV}}
+\newcommand{\BV}{\mathit{BV}}
+\newcommand{\idfun}{\mathbbm{1}}
+\definecolor{fixme-color}{rgb}{0.7,0.2,0}
+\newcommand{\fixme}[1]{\textcolor{fixme-color}{FIXME: #1}}
+\def\cpp{C{}\texttt{++}}

diff --git a/introduction.tex b/introduction.tex
new file mode 100644 (file)
index 0000000..56bfd85
--- /dev/null
+++ b/introduction.tex
@@ -0,0 +1,50 @@
+\chapter{Introduction}
+
+So here maybe we introduce the field of image restoration. Ideas can be
+found in the introduction of the project. Maybe we also say something
+about the different noise models, and common assumptions concerning the
+noise.
+
+\section{Diffusion filtering}
+
+First we say something about normal Gaussian filtering, how it is
+related to solving the heat equation etc. Weickert
+\cite{weickert1998anisotropic} is a good reference here, and in many of
+the other sections. End with the motivation of controlling the diffusion
+coefficient.
+
+\subsection{Non-linear diffusion}
+
+This section is for methods that control the diffusion coefficient
+$\alpha$. Here I guess Perona-Malik is the most noteworthy. We write
+something about the motivation, its well-posedness (or lack thereof) and
+other alternatives. But we can control the diffusion even more!
+
+\subsection{Anisotropic diffusion}
+
+Section for methods introducing a diffusion \emph{tensor} $A$ that
+varies around the image. This means we can control diffusion strength in
+different directions. More specifically we can make the diffusion strong
+along edges, and weak across edges.
+
+We must also say something about what this method does well, and where
+it fails. Anisotropic filtering might introduce structures that do not
+exist in the original image. Some example images are possible here.
+
+\section{Total variation filtering}
+
+Describe briefly the functional we minimize in this method, and
+intuitively how this leads to a filtered image. But how much do we write
+about how we find our minimizer? We have to at least mention that there
+are different ways to do it, and that one of them involves finding a
+minimum cut for each level.
+
+We also write something about the limitations of the method. It is good
+at preserving sharp edges, but suffers from the so-called stair-casing
+effect. Thin objects and corners might gradually lose contrast depending
+on the strength of the regularization. Example images here as well.
+
+Then finish the chapter in some way saying that we will try to combine
+these two methods, retaining their strenghts but eliminating their
+weaknesses.
+

diff --git a/main.tex b/main.tex
new file mode 100644 (file)
index 0000000..7b75885
--- /dev/null
+++ b/main.tex
@@ -0,0 +1,112 @@
+\documentclass[a4paper, titlepage, twoside, 12pt]{report}
+
+\usepackage{geometry}
+
+\usepackage{amsmath}
+\usepackage{unicode-math}
+\usepackage{fontspec}
+\usepackage{amsthm}
+\usepackage{graphicx}
+\usepackage{mathtools}
+
+\usepackage{polyglossia}
+\setmainlanguage[variant=american]{english}
+
+\usepackage[hidelinks]{hyperref}
+
+\usepackage[binary-units=true]{siunitx}
+
+\usepackage{tikz}
+\usetikzlibrary{matrix}
+\usetikzlibrary{arrows}
+\usetikzlibrary{positioning}
+\usetikzlibrary{calc}
+
+\usepackage{color}
+\usepackage{bbm}    % Indicator function \mathbbm{1}
+%\usepackage[margin=0.8cm]{caption}
+\usepackage{caption}
+%\usepackage[subrefformat=parens,labelformat=parens]{subcaption}
+\usepackage{subcaption}
+\usepackage{xfrac}
+\usepackage{listings}
+
+\lstset{
+    % FIXME: Courier 9pt anbefales
+    basicstyle=\footnotesize\ttfamily,
+    keepspaces=true,
+    tabsize=4,
+    %inputencoding=utf8/latin1,
+    extendedchars=true,
+    breakatwhitespace=false,
+    breaklines=true, 
+}
+
+\usepackage{algorithm}
+\usepackage{algpseudocode}
+
+\newtheorem{theorem}{Theorem}
+\newtheorem{lemma}{Lemma}
+\newtheorem{definition}{Definition}
+                                 
+% Indeksgenerering er kommentert ut her. Ta bort prosenttegnene
+% hvis du vil ha en indeks:
+%\usepackage{makeidx}     
+%\makeindex              
+
+\input{commands}
+
+% Selve dokumentet begynner:
+
+\begin{document}
+
+% På forsida skal vi ikke ha noen sidenummerering:
+
+\pagestyle{empty}
+\pagenumbering{roman}
+
+% Inkluder forsida:
+\input{titlepage}
+
+\cleardoublepage
+
+% Romerske tall på alt før selve rapporten starter er pent.
+\pagenumbering{roman}
+
+% For å ikke begynne innholdslista på baksida av forsida:
+\cleardoublepage
+% (kun aktuelt når man har twoside som global opsjon)
+
+% Nå vi vil ha noe i topp- og bunnteksten
+\pagestyle{headings}
+
+% Si til LaTeX at vi vil ha ei innholdsliste generert akkurat her:
+\tableofcontents
+
+% Pass på at neste side ikke begynner på baksida av en annen side.
+\cleardoublepage
+
+% Arabisk (vanlige tall) sidenummerering. Starter på side 1 igjen.
+\pagenumbering{arabic}
+
+% Inkluder alle de andre kildefilene
+\input{introduction}
+\input{theory}
+\input{results}
+
+% Bibliografi/referanseliste skal komme før appendiks
+\bibliography{bib}
+\bibliographystyle{unsrt}
+
+\newpage
+% En latex-kommando for å si fra at kapitlene/seksjonene fra nå 
+% av skal nummereres med store bokstaver:
+\appendix
+\input{appendix}
+
+% Indeks for rapporten. Ta bort prosenttegn hvis du vil ha det med.
+%\printindex
+
+% Avslutter dokumentet vårt:
+\end{document}
+

diff --git a/results.tex b/results.tex
new file mode 100644 (file)
index 0000000..3a9bc06
--- /dev/null
+++ b/results.tex
@@ -0,0 +1,4 @@
+\chapter{Results}
+
+\section{Comparison}
+

diff --git a/theory.tex b/theory.tex
new file mode 100644 (file)
index 0000000..1457eea
--- /dev/null
+++ b/theory.tex
@@ -0,0 +1,53 @@
+\chapter{Theory}
+
+This is the chapter containing all the theory of the presented
+restoration method. The order in which everything is presented is not
+fixed. For example how much should we discuss the details of the
+anisotropy matrix $M$ before introducing the coarea and perimiter
+theory? Because the coarea formula \emph{can} be presented in a very
+general form. We will figure that out as we go.
+
+\section{Anisotropic total variation}
+
+Here we start out with the normal total variation (?), give some
+motivation and introduce the anisotropy. We probably need the definition
+of total variation, and the space of functions of bounded variation. The
+order is not clear to me, yet!
+
+\section{Anisotropy tensor}
+
+We discuss what kind of properties we want from the anisotropy tensor,
+and also introduce the structure tensor (?) described by Weickert. This
+tensor can be visualized, and can be used for edge detection. There is
+probably a bit to be said about the smoothing we do. We also discuss
+whether to use the noisy image (yes, probably), or the smoothed image
+(implicitly, complicated, possible iteratively) in the structure tensor.
+We then describe how the structure tensor is decomposed and transformed
+into our anisotropy tensor. Then there are some problems related to
+stability which we can discuss here, or maybe we should leave it for a
+later, more implementation-focused chapter.
+
+\section{Continuous formulation}
+
+In spirit of my project I have included a section with this title. It
+will be a theory heavy chapter, probably including the anisotropic
+coarea formula and the perimeters and all that. Should probably include
+some proofs and derivations as it is not easy to find this in the
+literature. This is one of the most important parts of the theory
+chapter, I guess.
+
+\section{Discretization}
+
+This is where we discretize! It will also be an important chapter, as
+the choices and possibilities are endless. The easiest way out is to
+choose the same stencil in every point, and only let the weights depend
+on the anisotropy. This can work, but size has to be discussed (and
+tested later). Then the stencil shape can be adjusted as well.
+
+\section{Graph theory}
+
+\subsection{Intro}
+\subsection{Graphification}
+
+\section{Flow}
+

diff --git a/titlepage.tex b/titlepage.tex
new file mode 100644 (file)
index 0000000..678469b
--- /dev/null
+++ b/titlepage.tex
@@ -0,0 +1,8 @@
+\title{Anisotropic total variation based image restoration using graph
+cuts}
+
+\date{\today}
+
+\author{Bjørn Rustad}
+
+\maketitle