--- /dev/null
+*.aux
+*.bbl
+*.blg
+*.dvi
+*.log
+*.out
+*.pdf
+*.toc
+
--- /dev/null
+all:
+ pdflatex main
+ bibtex main
+ pdflatex main
+ pdflatex main
--- /dev/null
+% Dette eksempelet er laget for article-dokumentklassen. Hvis
+% skriver i 'book'-dokumentklassen vil du kanskje bytte ut
+% \section med \chapter, \subsection med \section, osv...
+
+\section{C++ implementation}
+
+int main()
+
--- /dev/null
+<bibtex>\r
+\r
+@article{goldberg1988new,\r
+ title={A new approach to the maximum-flow problem},\r
+ author={Goldberg, Andrew V and Tarjan, Robert E},\r
+ journal={Journal of the ACM (JACM)},\r
+ volume={35},\r
+ number={4},\r
+ pages={921--940},\r
+ year={1988},\r
+ publisher={ACM}\r
+}\r
+\r
+@article{Mazzoni1991257,\r
+ title = "The maximum flow problem: A max-preflow approach ",\r
+ journal = "European Journal of Operational Research ",\r
+ volume = "53",\r
+ number = "3",\r
+ pages = "257 - 278",\r
+ year = "1991",\r
+ note = "",\r
+ issn = "0377-2217",\r
+ doi = "http://dx.doi.org/10.1016/0377-2217(91)90060-9",\r
+ url = "http://www.sciencedirect.com/science/article/pii/0377221791900609",\r
+ author = "Giuseppe Mazzoni and Stefano Pallottino and Maria Grazia Scutellà",\r
+ keywords = "Maximum flow",\r
+ keywords = "maximum preflow",\r
+ keywords = "distance function",\r
+ keywords = "algorithm",\r
+ keywords = "experimentation "\r
+}\r
+\r
+@article{cherkassky1997implementing,\r
+ title={On implementing the push—relabel method for the maximum flow problem},\r
+ author={Cherkassky, Boris V and Goldberg, Andrew V},\r
+ journal={Algorithmica},\r
+ volume={19},\r
+ number={4},\r
+ pages={390--410},\r
+ year={1997},\r
+ publisher={Springer}\r
+}\r
+\r
+@article{ahuja1997computational,\r
+ title={Computational investigations of maximum flow algorithms},\r
+ author={Ahuja, Ravindra K and Kodialam, Murali and Mishra, Ajay K and Orlin, James B},\r
+ journal={European Journal of operational research},\r
+ volume={97},\r
+ number={3},\r
+ pages={509--542},\r
+ year={1997},\r
+ publisher={Elsevier}\r
+}\r
+\r
+@incollection{boykov2006graph,\r
+ title={Graph cuts in vision and graphics: Theories and applications},\r
+ author={Boykov, Yuri and Veksler, Olga},\r
+ booktitle={Handbook of mathematical models in computer vision},\r
+ pages={79--96},\r
+ year={2006},\r
+ publisher={Springer}\r
+}\r
+\r
+@article{derigs1989implementing,\r
+ title={Implementing Goldberg's max-flow-algorithm—A computational investigation},\r
+ author={Derigs, Ulrich and Meier, Wolfgang},\r
+ journal={Zeitschrift f{\"u}r Operations Research},\r
+ volume={33},\r
+ number={6},\r
+ pages={383--403},\r
+ year={1989},\r
+ publisher={Springer}\r
+}\r
+\r
+@incollection{caselles2011total,\r
+ title={Total variation in imaging},\r
+ author={Caselles, Vicent and Chambolle, Antonin and Novaga, Matteo},\r
+ journal={Handbook of Mathematical Methods in Imaging},\r
+ pages={1016--1057},\r
+ year={2011},\r
+ publisher={Springer}\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
+</bibtex>\r
--- /dev/null
+% Dette eksempelet er laget for article-dokumentklassen. Hvis
+% skriver i 'book'-dokumentklassen vil du kanskje bytte ut
+% \section med \chapter, \subsection med \section, osv...
+
+\section{Introduction}
+
+Introduce the field field of early vision (?), image restoration, image
+restoration. What methods exists, and what are they good at, what are
+their applications. What kinds of noise exists.
+
--- /dev/null
+% Først spesifiserer vi hvilken dokumentklasse vi vil ha og noen
+% globale opsjoner. Bytt ut 'article' med 'book' hvis du vil ha
+% med kapitler.
+\documentclass[a4paper, twoside, titlepage, 11pt]{article}
+
+% Så sier vi fra om hvilke tilleggspakker vi trenger
+% til dokumentet vårt. De som du ikke trenger (se kommentaren)
+% kan det være en fordel å kommentere ut (sett prosenttegn foran),
+% da vil kompilering gå raskere.
+
+\usepackage[T1]{fontenc} % norsk tegnsett (æøå)
+\usepackage[utf8]{inputenc} % norsk tegnsett
+\usepackage{geometry} % anbefalt pakke for å styre marger.
+
+\usepackage{amsmath,amsfonts,amssymb} % matematikksymboler
+\usepackage{amsthm} % for å lage teoremer og lignende.
+\usepackage{graphicx} % inkludering av grafikk
+
+\usepackage{hyperref} % Lager hyperlinker i evt. pdf-dokument
+ % men har noen bugs, så den er kommentert
+ % bort her.
+\usepackage{algorithm}
+\usepackage{algpseudocode}
+
+% Indeksgenerering er kommentert ut her. Ta bort prosenttegnene
+% hvis du vil ha en indeks:
+%\usepackage{makeidx}
+%\makeindex
+
+% Selve dokumentet begynner:
+
+\begin{document}
+
+% På forsida skal vi ikke ha noen sidenummerering:
+
+\pagestyle{empty}
+\pagenumbering{roman}
+
+% Inkluder forsida:
+\input{titlepage}
+
+% 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:
+
+% NB: Vi trenger ikke ta med filendelsen .tex her. Den vet
+% LaTeX om selv!
+
+\input{introduction}
+\input{theory}
+\input{results}
+
+% Bibliografi/referanseliste skal komme før appendiks
+\bibliography{bib}
+\bibliographystyle{plain}
+
+% 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}
+
--- /dev/null
+o Relabeling
+ - Goldberg and Tarjan introduced global relabeling (every 2n discharge)
+ - Derigs and Meier introduced gap relabeling
+ - Setubal proves that global is better
+ - Goldberg recommends using both
--- /dev/null
+% Dette eksempelet er laget for article-dokumentklassen. Hvis
+% skriver i 'book'-dokumentklassen vil du kanskje bytte ut
+% \section med \chapter, \subsection med \section, osv...
+
+\section{Results}
+Method noise is the original image subtracted from the output image. It
+should not containt too many features.
+
+\subsection{Image segmentation}
+Some figures maybe. Showing the segmentation for different parameters.
+
+\subsection{Image restoration}
+Different kinds of noise, different parameters. Compare it with other
+methods maybe.
+
+\subsubsection{Impulse noise}
+\subsubsection{Gaussian noise}
+
--- /dev/null
+% Dette eksempelet er laget for article-dokumentklassen. Hvis
+% skriver i 'book'-dokumentklassen vil du kanskje bytte ut
+% \section med \chapter, \subsection med \section, osv...
+
+\section{Theory}
+
+\subsection{Methods}
+
+\subsubsection{Anistrophic diffusion}
+Some partial differential equation magic.
+
+\subsubsection{Gaussian filtering}
+Convolve the image with Gaussian function. Adds blur.
+
+\subsubsection{Non-local means}
+Somehow averaging over all the pixels in the image. Has a very good
+method noise.
+
+\subsubsection{Non-linear filters}
+Median filter for example.
+
+\subsubsection{Total variation}
+Our method. WHAT IS SPLIT BREGMAN?
+
+\subsection{Continuous formulation}
+Formulate the energy function, describe the different terms. Total
+variation. \cite{caselles2011total} has a nice introduction to total
+variation. It describes the foundations, the spaces, and some methods,
+but mostly continuous.
+
+\subsection{Discrete problem}
+How do we get to the discrete formulation. Why is it OK to do what we do
+with the gradient. How do we decompose the energy function into
+different levels and why is it OK to optimize each level separately. How
+do we choose the different neighbourhoods, and what could this mean for
+the result. Darbon and Sigelle. \cite{chan2011numerical} describes
+different methods for using total variation, and especially how one can
+use it in the discrete setting. It also goes into the min-cut method.
+Yay. Good book.
+
+\subsection{Graph cut formulation}
+Short introduction to graphs. Notation. What is a cut, what is a minimal
+s-t-cut. How do we get from the discrete energy to the graph and why
+does finding the graph cut result in minimizing the energy of the
+current label. Boykov and Kolmogorov.
+
+\subsubsection{Min-cut max-flow theorem}
+Want to find the minimal s-t-cut, so we introduce flow here maybe? Or
+earlier? Finding the max flow results in also finding the minimal cut,
+and there are «good» algorithms for finding the maximal flow.
+
+\subsection{Maximum flow algorithms}
+Two different ideas. One is to always maintain a valid flow in the
+network, Ford Foulkerson, always looking for new paths from s to t along
+which we can send additional flow. This leads to Edmonds-Karp, and
+Dinic/Dinitz. Second idea is push-relabel, allowing positive excess in
+the nodes. We lift the source, and let as much flow as possible flow
+towards the sink.
+
+\subsubsection{Dinic's algorithm}
+The «best» of the augmenting-path algorithms (citation needed). Makes a
+level graph with BFS, and finds a blocking flow through this graph which
+only goes from nodes with one label to nodes with greater labels. When
+this blocking flow is found we restart at the BFS.
+
+\subsubsection{The push-relabel algorithm}
+The nodes are allowed to have a positive excess but we still follow the
+capacity constraints. The nodes also have a labeling, which has to be
+valid. Push flow from active nodes, and relabel, preferrably in a
+specific order, until it is not possible anymore. The minimal cut is
+actually found before the flow becomes valid (has to be explained).
+
+Because of the way the graph evolves, we can reuse the graph between the
+different labels.
+
+Initially proposed in \cite{goldberg1988new} where one can also find
+proof that the algorithm maintains a valid labeling and that it
+terminates at at maximal flow. Also has the lemma that if you have a
+preflow, then there is no path from $s$ to $t$ in the residual graph.
+But what is a preflow? Also, the distance label never decreases. Also
+propose the global relabeling heuristic.
+
+Shiloach and Vishkin originally proposed the FIFO way of doing things
+(according to Golberg and Tarjan)?
+
+Cherkassky and Goldberg present some good implementation tips in
+\cite{cherkassky1997implementing}, for example the global and gap
+relabeling heuristics. Also has OK presentations of the different ways
+of choosing the order of discharge operations. In the results I should
+include some results with and without the heuristics.
+
+\cite{ahuja1997computational} has some good performance analysis.
+
+\cite{boykov2006graph} is a good source on how graph cuts are used in
+image analysis. It considers both one level graphs, and multilevel
+graphs. Which is nice. It has just a small section on the different
+algorithms that exist and that the special Boykov Kolmogorov algorithm
+is the best.
+
+\cite{derigs1989implementing} is also a good source for the different
+heuristics. And also on the neighbor list that should not be reset after
+each discharge (should try to implement this again, and reset at every
+relabel!). Also has some semi-usable notes on how the minimal cut is
+found after phase 1 of the algorithm is finished. Note that in this
+article the gap relabeling heuristic is also called RELABEL-GLOBAL.
+
+\begin{algorithm}
+\begin{algorithmic}
+ \Function{Push}{$u$}
+ \ForAll{$v$ neighbour of $u$}
+ \If{$res(u, v) > 0$}
+ \State $f \gets min(res(u, v), excess[u])$
+ \State $flow(u, v) \mathrel{+}= f$
+ \State $flow(v, u) \mathrel{-}= f$
+ \State $excess[u] \mathrel{-}= f$
+ \State $excess[v] \mathrel{+}= f$
+ \EndIf
+ \EndFor
+ \EndFunction
+\end{algorithmic}
+\caption{\sf The push procedure of the Push-Relabel algorithm}
+\end{algorithm}
+
+\begin{algorithm}
+\begin{algorithmic}
+ \Function{Relabel}{$u$}
+ \If{$u$ is only node at its height}
+ \Call{Gap}{$u$}
+ \Else
+ \State $height[u] \gets min(height[v] \, \forall v \in
+ neighbors[u] : res(u, v) > 0) + 1$
+ \EndIf
+ \EndFunction
+\end{algorithmic}
+\caption{\sf The relabel procedure of the Push-Relabel algorithm}
+\end{algorithm}
+
+\begin{algorithm}
+\begin{algorithmic}
+ \Function{Gap}{$u$}
+ \State $k \gets height[u]$
+ \ForAll{$v$ with height $\geq k$}
+ \State $height[v] \gets N$
+ \EndFor
+ \EndFunction
+\end{algorithmic}
+\caption{\sf The gap procedure of the Push-Relabel algorithm}
+\end{algorithm}
+
+\subsubsection{Other algorithms}
+Special made augmenting flow algorithm for this type of graphs. Used in
+software1, software2, etc. Is fast.
+
+\subsection{Noise theory}
+What kinds of noise occur in what kinds of settings. Which methods
+should be good for which
+
+\subsubsection{Gaussian noise}
+Common. Thermal noise? Central limit theorem. Read in Jähne.
+
+\subsubsection{Poisson noise}
+
+\subsubsection{Salt and pepper noise}
+
+\subsubsection{Shot noise}
+Same as Poisson noise? Quantum effects in the way light is emitted at
+the source. For regular photographs it will only be a determining factor
+in low light conditions.
+
+Dark current, or reverse bias leakage current, which is due to the
+random generation of electrons in the sensor, can also be modelled by a
+Poisson distribution.
+
--- /dev/null
+% Enkel forside som bruker latex sin \titlepage kommando:
+% NB: Bruken av \and mellom navn!
+
+\titlepage
+\title{Image restoration using graph cuts}
+\author{Bjørn Rustad}
+\date{\today}
+\maketitle
+
+% Local Variables:
+% TeX-master: "master"
+% End:
projects / prosjektoppgave / commitdiff