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Apostolos Fanakis 6 years ago
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      Assignment_2/report/1.0.introduction.tex
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Assignment_2/report/1.0.introduction.tex

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\section{Εισαγωγή}
Η παρούσα εργασία εκπονήθηκε στα πλαίσια του μαθήματος ``Ψηφιακή Επεξεργασία Εικόνας'' του όγδοου (8\textsuperscript{ου}) εξαμήνου του τμήματος Ηλεκτρολόγων Μηχανικών και Μηχανικών Υπολογιστών του Αριστοτέλειου Πανεπιστημίου Θεσσαλονίκης. Ζητούμενο του δεύτερου παραδοτέου της εργασίας είναι η υλοποίηση, με χρήση του λογισμικού Matlab, τεσσάρων συνολικά συναρτήσεων που:
\begin{itemize}
\item δημιουργούν τον γράφο συγγένειας (affinity matrix) μίας εικόνας
\item κατατμίζουν την εικόνα με χρήση μεθόδου spectral clustering
\item κατατμίζουν την εικόνα με χρήση του αλγόριθμου normalized cuts
\item υπολογίζουν τη μετρική NCut για μία κατάτμηση της εικόνας
\end{itemize}
Επιπλέον έγινε συγγραφή πέντε scripts που επιδεικνύουν τη λειτουργία των παραπάνω συναρτήσεων.

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Assignment_2/report/2.0.affinity_matrix.tex

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\section{Γράφος συγγένειας}
Αρχικά υλοποιήθηκε η συνάρτηση \texttt{Image2Graph}, η οποία δέχεται σαν είσοδο μία εικόνα και παράγει έναν ``affinity matrix'' ο οποίος αναπαριστά τον μη κατευθυντικό γράφο $G = (V,E)$. Η συνάρτηση αυτή λειτουργεί για οποιοδήποτε αριθμό καναλιών εικόνας, ωστόσο υπάρχει σοβαρός περιορισμός στο μέγεθος της εικόνας καθώς ο παραγώμενος πίνακας έχει διαστάσεις $M \cdot N \times M \cdot N$, όπου $M$ και $N$ είναι οι διαστάσεις της εικόνας. Ενδεικτικά, για εικόνα διαστάσεων $100 \times 100$ ο πίνακας έχει μέγεθος 400MB. Το αξιόλογο μέγεθος του πίνακα αποτελεί πρόβλημα σε επόμενα βήματα της εργασίας όπου αυτός αντιγράφεται σε νέες μεταβλητές προκαλώντας προβλήματα σε συστήματα με περιορισμένη μνήμη.

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Assignment_2/report/3.0.spectral_clustering.tex

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\section{Spectral clustering}
Υλοποιήθηκε η συνάρτηση \texttt{mySpectralClustering} στην οποία μία εικόνα κατατμίζεται σε $k$ τμήματα με βάση τη χρωματική συγγένεια των εικονοστοιχείων, χρησιμοποιώντας τον affinity matrix που έχει δημιουργηθεί προηγουμένος. Σε αυτή τη συνάρτηση, για τη κατάτμηση της εικόνας ακολουθείται η μέθοδος spectral clustering και χρησιμοποιείται ο αλγόριθμος K-Means.

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Assignment_2/report/4.0.normalized_cuts.tex

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\section{Normalized cuts}
Στη συνάρτηση \texttt{myNCuts} γίνεται και πάλι κατάτμηση της εικόνας σε $k$ τμήματα με βάση τη χρωματική συγγένεια των εικονοστοιχείων, χρησιμοποιώντας τον affinity matrix που έχει δημιουργηθεί προηγουμένος. Αυτή τη φορά όμως ακολουθείται η μέθοδος normalized cuts.

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Assignment_2/report/5.0.demo1.tex

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\section{Πρώτη σειρά πειραμάτων}
Στη πρώτη σειρά πειραμάτων γίνεται παρουσίαση της λειτουργίας και των αποτελεσμάτων της συνάρτησης \texttt{mySpectralClustering}. Τα πειράματα εκτελούνται με κλήση του script \texttt{demo1} χωρίς ορίσματα. Το script εκτελεί μία σειρά από τρία πειράματα κατά τα οποία γίνεται κατάτμηση ενός δοσμένου γράφου συγγένειας (``\texttt{d1a}'') σε δύο, τρεις και τέσσερεις ομάδες (clusters) διαδοχικά. Τα αποτελέσματα φαίνονται παρακάτω.
Η έξοδος της συνάρτηση είναι ένα διάνυσμα (vector) με την ομάδα στην οποία ανήκει κάθε pixel της αρχικής εικόνας. Στα επόμενα πειράματα στα οποία οι διαστάσεις της αρχικής εικόνας είναι διαθέσιμες θα γίνεται πρώτα κατάλληλη μετατροπή του διανύσματος σε πίνακα με διαστάσεις ίδιες με αυτές της αρχικής εικόνας. Εδώ οι διαστάσεις δεν είναι γνωστές οπότε το βήμα αυτό παραλείπεται. Έπειτα πρέπει να αντιμετωπιστεί το πρόβλημα των τιμών του πίνακα, οι οποίες είναι οι φυσικοί $1, 2, 3,... ,k$ και αντιστοιχούν στην ομάδα που έχει ανατεθεί σε κάθε pixel. Για να γίνει απεικόνιση του πίνακα πρέπει οι τιμές αυτές να μεταφερθούν στο διάστημα $[0, 1]$, για το σκοπό αυτό εκτελείται διαίρεση των στοιχείων του πίνακα με τον αριθμό των ομάδων $k$. Έτσι προκύπτουν οι grayscale κατατμημένες εικόνες.
\begin{figure}[H]
\centering
\resizebox{\textwidth}{!}{
\colorbox{shadecolor}{\input{res/demo1_f1.pdf_tex}}
}
\caption{Κατάτμηση γράφου σε 2 ομάδες}
\end{figure}
\begin{figure}[H]
\centering
\resizebox{\textwidth}{!}{
\colorbox{shadecolor}{\input{res/demo1_f2.pdf_tex}}
}
\caption{Κατάτμηση γράφου σε 3 ομάδες}
\end{figure}
\begin{figure}[H]
\centering
\resizebox{\textwidth}{!}{
\colorbox{shadecolor}{\input{res/demo1_f3.pdf_tex}}
}
\caption{Κατάτμηση γράφου σε 4 ομάδες}
\end{figure}

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Assignment_2/report/6.0.demo2.tex

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\section{Δεύτερη σειρά πειραμάτων}
Στη δεύτερη σειρά πειραμάτων γίνεται παρουσίαση της λειτουργίας και των αποτελεσμάτων της συνάρτησης \texttt{mySpectralClustering} σε συνδυασμό με τη συνάρτηση \texttt{Image2Graph}. Τα πειράματα εκτελούνται με κλήση του script \texttt{demo2} χωρίς ορίσματα. Το script εκτελεί μία σειρά από έξι πειράματα κατά τα οποία γίνεται κατάτμηση δύο δοσμένων εικόνων (``\texttt{d2a}'' και ``\texttt{d2b}'') σε δύο, τρεις και τέσσερεις ομάδες (clusters) διαδοχικά. Τα αποτελέσματα φαίνονται και σχολιάζονται στη συνέχεια.
Στο πρώτο μέρος του script εκτελούνται τα πειράματα για την πρώτη εικόνα που δίνεται (``\texttt{d2a}''). Ο affinity matrix που προκύπτει απεικονίζεται γραφικά ώστε να γίνει εμφανής η δομή και η συμμετρία του. Τα αποτελέσματα της κατάτμησης φαίνονται παρακάτω.
\begin{figure}[H]
\centering
\resizebox{.7\textwidth}{!}{
\colorbox{shadecolor}{\input{res/demo2_e1_g.pdf_tex}}
}
\caption{Απεικόνιση affinity matrix πρώτης εικόνας (``\texttt{d2a}'')}
\end{figure}
\begin{multicols}{2}
\begin{figure}[H]
\centering
\resizebox{.5\textwidth}{!}{
\colorbox{shadecolor}{\input{res/demo2_e1_f1.pdf_tex}}
}
\caption{Αρχική εικόνα (``\texttt{d2a}'')}
\end{figure}
\begin{figure}[H]
\centering
\resizebox{.5\textwidth}{!}{
\colorbox{shadecolor}{\input{res/demo2_e1_f2.pdf_tex}}
}
\caption{Κατάτμηση πρώτης εικόνας (``\texttt{d2a}'') σε 2 ομάδες}
\end{figure}
\end{multicols}
\begin{multicols}{2}
\begin{figure}[H]
\centering
\resizebox{.5\textwidth}{!}{
\colorbox{shadecolor}{\input{res/demo2_e1_f3.pdf_tex}}
}
\caption{Κατάτμηση πρώτης εικόνας (``\texttt{d2a}'') σε 3 ομάδες}
\end{figure}
\begin{figure}[H]
\centering
\resizebox{.5\textwidth}{!}{
\colorbox{shadecolor}{\input{res/demo2_e1_f4.pdf_tex}}
}
\caption{Κατάτμηση πρώτης εικόνας (``\texttt{d2a}'') σε 4 ομάδες}
\end{figure}
\end{multicols}
Όπως φαίνεται στις εικόνες ο αλγόριθμος κάνει τέλεια κατάτμηση για τη περίπτωση των τριών ομάδων. Στη περίπτωση των δύο ομάδων η πράσινη και κόκκινη περιοχή κατατάσσονται στην ίδια ομάδα, ωστόσο αυτό μπορεί να αλλάξει ανάλογα με το σπόρο που θα δοθεί στην αρχή του script λόγω της τυχαιότητας που εισάγεται από τον αλγόριθμο K-Means. Τέλος στη περίπτωση των τεσσάρων ομάδων παρουσιάζεται ``θόρυβος'' στην ομαδοποίηση.
Στο δεύτερο μέρος του script εκτελούνται τα πειράματα για την δεύτερη εικόνα που δίνεται (``\texttt{d2b}''). Ο affinity matrix που προκύπτει απεικονίζεται και πάλι γραφικά. Τα αποτελέσματα της κατάτμησης φαίνονται παρακάτω.
\begin{figure}[H]
\centering
\resizebox{.7\textwidth}{!}{
\colorbox{shadecolor}{\input{res/demo2_e2_g.pdf_tex}}
}
\caption{Απεικόνιση affinity matrix πρώτης εικόνας (``\texttt{d2b}'')}
\end{figure}
\begin{multicols}{2}
\begin{figure}[H]
\centering
\resizebox{.5\textwidth}{!}{
\colorbox{shadecolor}{\input{res/demo2_e2_f1.pdf_tex}}
}
\caption{Αρχική εικόνα (``\texttt{d2b}'')}
\end{figure}
\begin{figure}[H]
\centering
\resizebox{.5\textwidth}{!}{
\colorbox{shadecolor}{\input{res/demo2_e2_f3.pdf_tex}}
}
\caption{Κατάτμηση πρώτης εικόνας (``\texttt{d2b}'') σε 2 ομάδες, grayscale χρώματα}
\end{figure}
\end{multicols}
\begin{multicols}{2}
\begin{figure}[H]
\centering
\resizebox{.5\textwidth}{!}{
\colorbox{shadecolor}{\input{res/demo2_e2_f5.pdf_tex}}
}
\caption{Κατάτμηση πρώτης εικόνας (``\texttt{d2b}'') σε 3 ομάδες, grayscale χρώματα}
\end{figure}
\begin{figure}[H]
\centering
\resizebox{.5\textwidth}{!}{
\colorbox{shadecolor}{\input{res/demo2_e2_f7.pdf_tex}}
}
\caption{Κατάτμηση πρώτης εικόνας (``\texttt{d2b}'') σε 4 ομάδες, grayscale χρώματα}
\end{figure}
\end{multicols}
Σε αυτό το πείραμα (όπως και σε επόμενα) θεωρήθηκε χρήσιμο να υλοποιηθεί μία διαφορετική ανάθεση χρωμάτων στις ομάδες της κατατμημένης εικόνας. Δημιουργήθηκε η συνάρτηση \texttt{meanClustersColorRGB} η οποία δέχεται σαν είσοδο την αρχική και την κατατμημένη εικόνα και παράγει στην έξοδο την κατατμημένη εικόνα με χρώματα για κάθε ομάδα το χρωματικό διάμεσο της ομάδας.
\begin{multicols}{2}
\begin{figure}[H]
\centering
\resizebox{.5\textwidth}{!}{
\colorbox{shadecolor}{\input{res/demo2_e2_f1.pdf_tex}}
}
\caption{Αρχική εικόνα (``\texttt{d2b}'')}
\end{figure}
\begin{figure}[H]
\centering
\resizebox{.5\textwidth}{!}{
\colorbox{shadecolor}{\input{res/demo2_e2_f2.pdf_tex}}
}
\caption{Κατάτμηση πρώτης εικόνας (``\texttt{d2b}'') σε 2 ομάδες, χρωματικός διάμεσος}
\end{figure}
\end{multicols}
\begin{multicols}{2}
\begin{figure}[H]
\centering
\resizebox{.5\textwidth}{!}{
\colorbox{shadecolor}{\input{res/demo2_e2_f4.pdf_tex}}
}
\caption{Κατάτμηση πρώτης εικόνας (``\texttt{d2b}'') σε 3 ομάδες, χρωματικός διάμεσος}
\end{figure}
\begin{figure}[H]
\centering
\resizebox{.5\textwidth}{!}{
\colorbox{shadecolor}{\input{res/demo2_e2_f6.pdf_tex}}
}
\caption{Κατάτμηση πρώτης εικόνας (``\texttt{d2b}'') σε 4 ομάδες, χρωματικός διάμεσος}
\end{figure}
\end{multicols}
Όπως φαίνεται στις εικόνες ο αλγόριθμος έχει ικανοποιητικά αποτελέσματα σε όλες τις περιπτώσεις αριθμού ομάδων.

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Assignment_2/report/7.0.demo3.tex

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Assignment_2/report/8.0.results.tex

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\section{Εκτέλεση και αποτελέσματα}
% Παρακάτω παρατίθενται ενδεικτικά οι χρόνοι εκτέλεσης των βασικών συναρτήσεων για επιλεγμένες διαστάσεις τελικής εικόνας 960Χ1280 καθώς:
% \begin{table}[H]
% \centering
% \begin{tabular}{l l} \toprule
% Συνάρτηση & Χρόνος (seconds) \\ \midrule
% \texttt{bayer2rgb} (``nearest'') & 0.164 \\
% \texttt{bayer2rgb} (``linear'') & 0.484 \\
% \texttt{imagequant} & 0.0315 \\
% \texttt{imagedequant} & 0.024 \\
% \texttt{saveasppm} & 0.208 \\ \bottomrule
% \end{tabular}
% \caption{Χρόνοι βασικών συναρτήσεων}
% \end{table}

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Assignment_2/report/front_page.tex

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\maketitle

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Assignment_2/report/report.tex

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\documentclass[11pt]{article}
\usepackage{geometry}
\geometry{a4paper, top=2.5cm, bottom=2.5cm, left=2.2cm, right=2.2cm}
\usepackage{fontspec}
\usepackage[nonumeralsign]{xgreek}
\usepackage{fancyhdr}
\usepackage{hyperref}
\usepackage{enumitem}
\usepackage{cite}
\usepackage{authblk}
\usepackage{textcomp}
\usepackage{listings}
\usepackage{float}
\usepackage{color}
\usepackage{graphicx}
\usepackage{booktabs}
\usepackage{multicol}
\setlength{\columnsep}{1cm}
\definecolor{shadecolor}{RGB}{180,180,180}
\graphicspath{{res/}}
%\usepackage{multirow}
%\usepackage[all]{hypcap}
\definecolor{codegreen}{rgb}{0,0.6,0}
\definecolor{codegray}{rgb}{0.5,0.5,0.5}
\definecolor{codepurple}{rgb}{0.58,0,0.82}
\definecolor{backcolour}{rgb}{0.95,0.95,0.92}
\lstdefinestyle{mystyle}{
backgroundcolor=\color{backcolour},
commentstyle=\color{codegreen},
keywordstyle=\color{magenta},
numberstyle=\tiny\color{codegray},
stringstyle=\color{codepurple},
basicstyle=\footnotesize,
breakatwhitespace=false,
breaklines=true,
captionpos=b,
keepspaces=true,
numbers=left,
numbersep=5pt,
showspaces=false,
showstringspaces=false,
showtabs=false,
tabsize=2
}
\lstset{style=mystyle}
\setmainfont{Lato}
\setmonofont{Courier}
\setlength{\parskip}{\baselineskip}
%\setlength{\parindent}{0pt}
\title{Ψηφιακή Επεξεργασία Εικόνας \\
\large Τμήμα Ηλεκτρολόγων Μηχανικών και Μηχανικών Υπολογιστών ΑΠΘ \\
8\textsuperscript{ο} εξάμηνο}
\author{Φανάκης Απόστολος, 8261}
\affil{\textit {apostolof@ece.auth.gr}}
\date{\today}
\pagestyle{fancy}
% \lhead{Kefalida}
% \rhead{Yposelido}
% \renewcommand{\headrulewidth}{0.4pt}
% \renewcommand{\footrulewidth}{0.4pt}
\setlength{\headheight}{14pt}
\hypersetup{colorlinks=true, linkcolor=black, urlcolor=blue, citecolor=blue}
\urlstyle{same}
\begin{document}
\input{front_page.tex}
\input{1.0.introduction.tex}
\input{2.0.affinity_matrix.tex}
\input{3.0.spectral_clustering.tex}
\input{4.0.normalized_cuts.tex}
\input{5.0.demo1.tex}
\input{6.0.demo2.tex}
\input{7.0.demo3.tex}
\input{8.0.results}
\end{document}

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Assignment_2/report/res/demo1_f1.pdf_tex

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%% \input{<filename>.pdf_tex}
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%% \includegraphics[width=<desired width>]{<filename>.pdf}
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%% in the preamble, and then including the image with
%% \import{<path to file>}{<filename>.pdf_tex}
%% Alternatively, one can specify
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%% Accompanies image file 'demo2_e2_g.pdf' (pdf, eps, ps)
%%
%% To include the image in your LaTeX document, write
%% \input{<filename>.pdf_tex}
%% instead of
%% \includegraphics{<filename>.pdf}
%% To scale the image, write
%% \def\svgwidth{<desired width>}
%% \input{<filename>.pdf_tex}
%% instead of
%% \includegraphics[width=<desired width>]{<filename>.pdf}
%%
%% Images with a different path to the parent latex file can
%% be accessed with the `import' package (which may need to be
%% installed) using
%% \usepackage{import}
%% in the preamble, and then including the image with
%% \import{<path to file>}{<filename>.pdf_tex}
%% Alternatively, one can specify
%% \graphicspath{{<path to file>/}}
%%
%% For more information, please see info/svg-inkscape on CTAN:
%% http://tug.ctan.org/tex-archive/info/svg-inkscape
%%
\begingroup%
\makeatletter%
\providecommand\color[2][]{%
\errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}%
\renewcommand\color[2][]{}%
}%
\providecommand\transparent[1]{%
\errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded}%
\renewcommand\transparent[1]{}%
}%
\providecommand\rotatebox[2]{#2}%
\newcommand*\fsize{\dimexpr\f@size pt\relax}%
\newcommand*\lineheight[1]{\fontsize{\fsize}{#1\fsize}\selectfont}%
\ifx\svgwidth\undefined%
\setlength{\unitlength}{1440bp}%
\ifx\svgscale\undefined%
\relax%
\else%
\setlength{\unitlength}{\unitlength * \real{\svgscale}}%
\fi%
\else%
\setlength{\unitlength}{\svgwidth}%
\fi%
\global\let\svgwidth\undefined%
\global\let\svgscale\undefined%
\makeatother%
\begin{picture}(1,0.50520833)%
\lineheight{1}%
\setlength\tabcolsep{0pt}%
\put(0,0){\includegraphics[width=\unitlength,page=1]{demo2_e2_g.pdf}}%
\end{picture}%
\endgroup%

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