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\begin{center}
  The Pennsylvania State University \linebreak \\
  The Graduate School \linebreak \\
  College of Engineering \linebreak \linebreak \linebreak \linebreak\\
  \uppercase{An Evaluation of Three Endmember Extraction Algorithms: \linebreak \linebreak ATGP, ICA-EEA, and VCA}  \linebreak \linebreak \linebreak \linebreak  \\
  A Thesis in \linebreak \\
  Electrical Engineering \linebreak \\
  by \linebreak \\
  Isaac D. Gerg \linebreak \linebreak \\
  \copyright 2008 Isaac D. Gerg \linebreak \linebreak \\
  Submitted in Partial Fulfillment  \\
  of the Requirements \\
  for the Degree of \linebreak \linebreak\\
  Master of Science \linebreak \linebreak\\
  May 2008 \\
\end{center}
\title{An Evaluation of Three Endmember Extraction Algorithms: ATGP, ICA-EEA, and VCA}
\author{Isaac Gerg \\ Department of Electrical Engineering \\
        The Pennsylvania State  University \\ State College, PA 16801}

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The thesis of Isaac D. Gerg. was reviewed and approved* by the following:\linebreak \linebreak \linebreak \linebreak

Richard L. Tutwiler\linebreak
Associate Professor of Acoustics\linebreak
Thesis Advisor \linebreak \linebreak \linebreak \linebreak


Timothy J. Kane\linebreak
Professor of Electrical Engineering \linebreak \linebreak \linebreak \linebreak


John F. Doherty\linebreak
Professor of Electrical Engineering \linebreak \linebreak \linebreak \linebreak

W. Kenneth Jenkins \linebreak
Professor of Electrical Engineering\linebreak
Head of the Electrical Engineering Department\linebreak \linebreak \linebreak \linebreak

* Signatures are on file in the Graduate School

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\begin{abstract}
Hyperspectral unmixing is necessary for material abundance map (MAM) creation.  Most unmixing algorithms search for a set of pure pixels, called endmembers,
of which all other pixels in the image are linear combinations of this set. Hence, endmember extraction is an important process in the creation of useful material abundance maps.
Material abundance maps are created using a three step process: estimation of the number of materials in an image, unmixing of the image to determine the spectral signatures (endmembers)
of these fundamental materials, and finally, some type of constrained least squares algorithm using the recovered endmember spectra to generate the abundance maps.
This thesis formally evaluates the first two steps of this process using simulated and real data.  First, a common material count estimation algorithm, known as virtual dimensionality (VD),
is examined. Second, three endmember extraction algorithms are evaluated: Automatic Target Generation Process (ATGP), ICA-Based Endmember Extraction Algorithm (ICA-EEA)
and Vertex Component Analysis (VCA). Finally, the derivation of a constrained least squares technique is given from which the results of the first two steps
are used as input to create abundance maps on real data from the AVIRIS sensor.  This process serves as a means of qualitatively  evaluating the efficacy of these algorithms
on live data from which ground truth information could not be realized.
The three unmixing algorithms arise from different schools of thought.  The ICA-EEA uses independent component analysis
(ICA) to isolate endmembers present in a scene. Conversely, ATGP and VCA
work on the principle of orthogonality in that endmembers are extracted by iteratively projecting the data orthogonally
to the current span of detected endmembers.  The two methods differ in how they select the next orthogonal direction.  Additionally,
the authors of VCA state ICA based methods cannot perform well due to the \emph{sum-to-one} constraint in linear endmember mixing. The
results of this thesis prove otherwise. All methods require that \emph{pure} pixels are present in the scene.
\end{abstract}
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\section*{Acknowledgments}
\addcontentsline{toc}{section}{Acknowledgments}
Thank you again Rick for supervising another thesis of mine!  You introduced me to image processing during the fall semester of 2004 and to the problem
of hyperspectral unmixing during the spring semester of 2006.  I am very thankful for this!  Thank you Dr. Kane for serving on my committee and
enlightening me to the many other aspects of remote sensing (i.e. the atmosphere!).  I've had a great experience in your Remote Sensing of Earth
Systems class and hope it continues to grow and turns into a yearly event.  Thank you Dr. Doherty for serving on my committee and providing inputs to improve
the quality of my writing and linear algebra skills (whoops!).  You all have been selfless, motivating, but most importantly, inspirational.  With the full work schedule and perils of
classes, you have all supported me so much and I humbly thank you for it.

I would like to thank Michael Eastwood and Robert Green of the NASA Jet Propulsion Laboratory
for providing me with the AVIRIS SNR data captured in Appendix \ref{avirisSnrTable}.

I would like to thank my family; you have loved me, guided me, believed in me, and picked me up not a moment too late. Thank you!
I am eternally grateful for all of your love: Mom, Dad, and Ian.

Finally, I thank my Lord and Savior, Jesus Christ.  Without You, I would have never been able to hold it together working full time while going to school.
 You worked in me to complete the task You set out for me to do.  Thank you for revealing Yourself to me during the spring of 2006. I am Yours.

\emph{I know that You can do all things, and that no purpose of Yours can be hindered.} Job 42:2
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\section*{A Note on Notation}
\addcontentsline{toc}{section}{A Note on Notation}
Hyperspectral data is often expressed many ways to better describe the mathematical
handling of the data; mainly as a vector of pixels when referring to the data in a space or a matrix of pixels
when referring to data as an image.

For consistency, a common notation is defined to differentiate these concepts clearly.
Hyperspectral data examined like an image will be defined as a matrix $\mathbf{M}_{m \times n \times p}$
of dimension $m \times n \times p$ where
$m$ is defined as the number of rows in the image, $n$ is defined as the number of columns in the image, and
$p$ is defined as the number of bands in the image.  Therefore, a single element of such an image will be accessed using $\mathbf{M}_{i, j, k}$ and a single
pixel of an image will be accessed using $\mathbf{M}_{i, j, :}$
Hyperspectral data formed as a vector of vectors (i.e. 2D matrix) is defined as $\mathbf{M} _{(m\cdot n) \times p}$
of dimension $(m\cdot n) \times p$.  A single element is accessed using $\mathbf{M} _{i, j}$ and a single pixel is
accessed using $\mathbf{M} _{:, j}$. Notice the multi-element notation is consistent with Matlab$^{TM}$; this is intentional.

The list below provides a summary of the notation convention used throughout this thesis.
\begin{table}[h]
  \centering
  \begin{tabular}{p{1cm} p{12cm}}
    $\mathbf{M}$ & Data matrix. Defined as an image of spectral signatures or vectors:  $\mathbf{M}_{m \times n \times p}$.
    Or, defined as a long vector of spectral signatures: $\mathbf{M}_{(m\cdot n) \times p}$. \\
    $N$ & The total number of pixels. For example $N=m\cdot n$. \\
    $m$ & Number of rows in the image. \\
    $n$ & Number of columns in the image. \\
    $p$ & Number of bands. \\
    $q$ & Number of classes / endmembers. \\
    $\mathbf{U}$ & Matrix of endmembers. Each column of the matrix represents an endmember vector. \\
    $\mathbf{b}$ & Observation vector; a single pixel. \\
    $\mathbf{x}$ & Weight vector. A matrix of weight vectors forms an abundance map.
  \end{tabular}
\end{table}
\newpage
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\section*{Orthorectification}
\addcontentsline{toc}{section}{Orthorectification}
Often, the results of data analysis on hyperspectral imagery are more meaningful when fused with other data such as street maps from Google Earth.
Images obtained from an airborne platform are often skewed due to the imaging geometry.  This skewing process results in imagery
which cannot resolve pixel distances linearly (i.e. a fixed pixel distance between two points does not correspond to a fixed ground distance.)
This results in the inability to simply lay processed imagery over map data with accurate alignment. The process of \emph{orthorectification}
corrects for alignment unevenness due to the imaging geometry.  Once corrected, a fixed distance $d$ measured anywhere in the processed imagery
will correspond to a fixed distance $d'$ on the map.

An orthorectification algorithm is provided below to help obtain georegistered abundance map overlays such as that of
figure \ref{moffetFieldFusedWithGoogleMaps_Rooftops}.  It is a simple model assuming the curvature of the Earth and relative terrain altitude are negligible.
It also assumes the imagery is obtained at nadir with no sensor movement distortion.  The overall ground sampling distance (GSD) is approximated using the GSD at nadir.  The algorithm
creates a mapping from $x_g$, the coordinates on the ground in meters, to $x_p$, the coordinates in the image in pixels.

\noindent
Given:
\begin{itemize}
  \item $\mathbf{I}$ is the input image with cross-track coordinates defined as $[-\frac{x_p}{2} \frac{x_p}{2}]$ in pixels.
  \item $a$ is the distance from imager to target in meters.
  \item $\beta_{\frac{1}{2}}$ is the image field of view in radians.
  \item $w$ is the width  of the image in the cross-track direction in pixels.
  \item $x_g$ is the distance from nadir in the cross track direction for which $x_p$ is sought.
\end{itemize}
\noindent
Compute:
\begin{enumerate}
  \item $\alpha = \frac{\beta_{\frac{1}{2}}}{w}$; Radians per pixel.
  \item $GSD = a \cdot \alpha$; Ground sampling distance (GSD) at nadir.
  \item $\theta = atan(\frac{GSD \cdot x_g}{a})$; Angle of incidence.
  \item $x_p = \frac{\theta}{\alpha}$
\end{enumerate}

\newpage

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\input{virtual_dimensionality.tex}
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\input{ica.tex}
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\input{vca.tex}
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\input{fcls.tex}
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\input{results.tex}
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% Appendices
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\appendix
\section{AVIRIS SNR Versus Wavelength}
\label{avirisSnrTable}
The original SNR data for AVIRIS from \cite{green07} is listed here for reference.  The table represents the SNR of the AVIRIS sensor as
a function of wavelength.  Measured 2005.

\singlespace
\begin{longtable}{ccc}
\caption[]{AVIRIS SNR Vs. Wavelength . Measured 2005.} \label{aviris snr table} \\
%This is the header for the first page of the table...
\hline \hline \\[-2ex]
   \multicolumn{1}{c}{\textbf{Wavelength [nm]}} &
   \multicolumn{1}{c}{\textbf{SNR [linear]}} &
   \multicolumn{1}{c}{\textbf{SNR [dB]}} \\[0.5ex] \hline
   \\[-1.8ex]
\endfirsthead
%This is the header for the remaining page(s) of the table...
\multicolumn{3}{c}{{\tablename} \thetable{} -- Continued} \\[0.5ex]
  \hline \hline \\[-2ex]
   \multicolumn{1}{c}{\textbf{Wavelength [nm]}} &
   \multicolumn{1}{c}{\textbf{SNR [linear]}} &
   \multicolumn{1}{c}{\textbf{SNR [dB]}} \\[0.5ex] \hline
  \\[-1.8ex]
\endhead

3.66094183E+02  & 1.71409960E+02  & 4.46807211E+01  \\ \hline
3.75715176E+02  & 1.93949230E+02  & 4.57537612E+01  \\ \hline
3.85337344E+02  & 2.12057117E+02  & 4.65290570E+01  \\ \hline
3.94960687E+02  & 4.16684604E+02  & 5.23961491E+01  \\ \hline
4.04585206E+02  & 5.66017792E+02  & 5.50566017E+01  \\ \hline
4.14210899E+02  & 6.19763116E+02  & 5.58445145E+01  \\ \hline
4.23837768E+02  & 7.19810972E+02  & 5.71443692E+01  \\ \hline
4.33465812E+02  & 9.58843053E+02  & 5.96349505E+01  \\ \hline
4.43095032E+02  & 1.19086765E+03  & 6.15172700E+01  \\ \hline
4.52725426E+02  & 1.30349645E+03  & 6.23021971E+01  \\ \hline
4.62356996E+02  & 1.45963133E+03  & 6.32848635E+01  \\ \hline
4.71989741E+02  & 1.45199211E+03  & 6.32392852E+01  \\ \hline
4.81623661E+02  & 1.48146713E+03  & 6.34138404E+01  \\ \hline
4.91258756E+02  & 1.46070945E+03  & 6.32912768E+01  \\ \hline
5.00895027E+02  & 1.46118996E+03  & 6.32941336E+01  \\ \hline
5.10532472E+02  & 1.40101378E+03  & 6.29288482E+01  \\ \hline
5.20171093E+02  & 1.43730962E+03  & 6.31510066E+01  \\ \hline
5.29810889E+02  & 1.31324048E+03  & 6.23668852E+01  \\ \hline
5.39451861E+02  & 1.34902195E+03  & 6.26003803E+01  \\ \hline
5.49094007E+02  & 1.35383331E+03  & 6.26313039E+01  \\ \hline
5.58737329E+02  & 1.36418522E+03  & 6.26974668E+01  \\ \hline
5.68381826E+02  & 1.34417252E+03  & 6.25691002E+01  \\ \hline
5.78027498E+02  & 1.26252967E+03  & 6.20248318E+01  \\ \hline
5.87674345E+02  & 1.24279472E+03  & 6.18879880E+01  \\ \hline
5.97322368E+02  & 1.37842601E+03  & 6.27876692E+01  \\ \hline
6.06971565E+02  & 1.29503684E+03  & 6.22456425E+01  \\ \hline
6.16621938E+02  & 1.24216655E+03  & 6.18835966E+01  \\ \hline
6.26273486E+02  & 1.24935040E+03  & 6.19336852E+01  \\ \hline
6.35926210E+02  & 1.22038660E+03  & 6.17299486E+01  \\ \hline
6.45580108E+02  & 1.27207050E+03  & 6.20902236E+01  \\ \hline
6.55235182E+02  & 1.23553001E+03  & 6.18370660E+01  \\ \hline
6.64891431E+02  & 1.43582439E+03  & 6.31420265E+01  \\ \hline
6.53248750E+02  & 1.22215763E+03  & 6.17425445E+01  \\ \hline
6.62809990E+02  & 1.27969126E+03  & 6.21421041E+01  \\ \hline
6.72372750E+02  & 1.32155529E+03  & 6.24217067E+01  \\ \hline
6.81937030E+02  & 1.32796932E+03  & 6.24637608E+01  \\ \hline
6.91502830E+02  & 1.25978875E+03  & 6.20059545E+01  \\ \hline
7.01070150E+02  & 1.30818096E+03  & 6.23333565E+01  \\ \hline
7.10638990E+02  & 1.33844558E+03  & 6.25320143E+01  \\ \hline
7.20209350E+02  & 1.08637476E+03  & 6.07195933E+01  \\ \hline
7.29781230E+02  & 1.19926204E+03  & 6.15782817E+01  \\ \hline
7.39354630E+02  & 1.48068741E+03  & 6.34092677E+01  \\ \hline
7.48929550E+02  & 1.56084279E+03  & 6.38671832E+01  \\ \hline
7.58505990E+02  & 1.05138568E+03  & 6.04352412E+01  \\ \hline
7.68083950E+02  & 1.22483590E+03  & 6.17615582E+01  \\ \hline
7.77663430E+02  & 1.51091522E+03  & 6.35848019E+01  \\ \hline
7.87244430E+02  & 1.47408767E+03  & 6.33704663E+01  \\ \hline
7.96826950E+02  & 1.41536054E+03  & 6.30173417E+01  \\ \hline
8.06410990E+02  & 1.32831876E+03  & 6.24660461E+01  \\ \hline
8.15996550E+02  & 1.04476901E+03  & 6.03804057E+01  \\ \hline
8.25583630E+02  & 1.09621080E+03  & 6.07978815E+01  \\ \hline
8.35172230E+02  & 1.22292397E+03  & 6.17479891E+01  \\ \hline
8.44762350E+02  & 1.28138830E+03  & 6.21536151E+01  \\ \hline
8.54353990E+02  & 1.22292082E+03  & 6.17479668E+01  \\ \hline
8.63947150E+02  & 1.28772393E+03  & 6.21964554E+01  \\ \hline
8.73541830E+02  & 1.31472007E+03  & 6.23766659E+01  \\ \hline
8.83138030E+02  & 1.30081782E+03  & 6.22843295E+01  \\ \hline
8.92735750E+02  & 1.11053809E+03  & 6.09106691E+01  \\ \hline
9.02334990E+02  & 9.42283817E+02  & 5.94836347E+01  \\ \hline
9.11935750E+02  & 8.82309221E+02  & 5.89124164E+01  \\ \hline
9.21538030E+02  & 8.80827652E+02  & 5.88978188E+01  \\ \hline
9.29038030E+02  & 3.10011534E+02  & 4.98275570E+01  \\ \hline
9.45065635E+02  & 3.15350873E+02  & 4.99758807E+01  \\ \hline
9.54537046E+02  & 4.56180276E+02  & 5.31827301E+01  \\ \hline
9.64007861E+02  & 7.66714085E+02  & 5.76926688E+01  \\ \hline
9.73478080E+02  & 9.48553148E+02  & 5.95412334E+01  \\ \hline
9.82947703E+02  & 1.13390814E+03  & 6.10915575E+01  \\ \hline
9.92416730E+02  & 1.25089814E+03  & 6.19444390E+01  \\ \hline
1.00188516E+03  & 1.23472682E+03  & 6.18314176E+01  \\ \hline
1.01135300E+03  & 1.22496850E+03  & 6.17624984E+01  \\ \hline
1.02082024E+03  & 1.23123859E+03  & 6.18068444E+01  \\ \hline
1.03028688E+03  & 1.22164297E+03  & 6.17388860E+01  \\ \hline
1.03975293E+03  & 1.22020818E+03  & 6.17286787E+01  \\ \hline
1.04921838E+03  & 1.29469933E+03  & 6.22433784E+01  \\ \hline
1.05868323E+03  & 1.29084409E+03  & 6.22174758E+01  \\ \hline
1.06814749E+03  & 1.33355511E+03  & 6.25002194E+01  \\ \hline
1.07761115E+03  & 1.33000000E+03  & 6.24770328E+01  \\ \hline
1.08707422E+03  & 1.33081224E+03  & 6.24823357E+01  \\ \hline
1.09653669E+03  & 1.14767620E+03  & 6.11963875E+01  \\ \hline
1.10599857E+03  & 9.05993118E+02  & 5.91424980E+01  \\ \hline
1.11545985E+03  & 2.83397226E+02  & 4.90479119E+01  \\ \hline
1.12492053E+03  & 1.02975589E+02  & 4.02546857E+01  \\ \hline
1.13438062E+03  & 1.94348373E+02  & 4.57716182E+01  \\ \hline
1.14384011E+03  & 2.21749402E+02  & 4.69172492E+01  \\ \hline
1.15329900E+03  & 3.59757422E+02  & 5.11201952E+01  \\ \hline
1.16275730E+03  & 7.55323123E+02  & 5.75626556E+01  \\ \hline
1.17221500E+03  & 9.45753994E+02  & 5.95155637E+01  \\ \hline
1.18167211E+03  & 9.11768637E+02  & 5.91976930E+01  \\ \hline
1.19112862E+03  & 9.21179973E+02  & 5.92868897E+01  \\ \hline
1.20058454E+03  & 8.81441217E+02  & 5.89038671E+01  \\ \hline
1.21003986E+03  & 9.11280787E+02  & 5.91930443E+01  \\ \hline
1.21949458E+03  & 9.73333027E+02  & 5.97652292E+01  \\ \hline
1.22894871E+03  & 1.01686855E+03  & 6.01452963E+01  \\ \hline
1.23840224E+03  & 1.04719751E+03  & 6.04005720E+01  \\ \hline
1.24785517E+03  & 1.02824028E+03  & 6.02418923E+01  \\ \hline
1.25730751E+03  & 1.03596543E+03  & 6.03069052E+01  \\ \hline
1.25400065E+03  & 3.52438633E+02  & 5.09416702E+01  \\ \hline
1.26398069E+03  & 3.33741712E+02  & 5.04682098E+01  \\ \hline
1.27396047E+03  & 3.56814050E+02  & 5.10488389E+01  \\ \hline
1.28394000E+03  & 4.06046979E+02  & 5.21715257E+01  \\ \hline
1.29391927E+03  & 4.21582957E+02  & 5.24976609E+01  \\ \hline
1.30389829E+03  & 3.91839015E+02  & 5.18621535E+01  \\ \hline
1.31387705E+03  & 3.53165872E+02  & 5.09595746E+01  \\ \hline
1.32385555E+03  & 3.03650846E+02  & 4.96474899E+01  \\ \hline
1.33383380E+03  & 1.86772960E+02  & 4.54262800E+01  \\ \hline
1.34381179E+03  & 9.01318980E+01  & 3.90975703E+01  \\ \hline
1.35378953E+03  & 6.92154988E+00  & 1.68040671E+01  \\ \hline
1.36376701E+03  & 2.74474953E+00  & 8.77005440E+00  \\ \hline
1.37374423E+03  & 2.55073937E+00  & 8.13332171E+00  \\ \hline
1.38372120E+03  & 2.47787777E+00  & 7.88159758E+00  \\ \hline
1.39369791E+03  & 3.07318486E+00  & 9.75177370E+00  \\ \hline
1.40367437E+03  & 2.57881306E+00  & 8.22839723E+00  \\ \hline
1.41365057E+03  & 3.25905839E+00  & 1.02618428E+01  \\ \hline
1.42362651E+03  & 8.45606491E+00  & 1.85433662E+01  \\ \hline
1.43360220E+03  & 1.46499761E+01  & 2.33167383E+01  \\ \hline
1.44357763E+03  & 2.66836443E+01  & 2.85249029E+01  \\ \hline
1.45355281E+03  & 8.53536569E+01  & 3.86244427E+01  \\ \hline
1.46352773E+03  & 1.07354592E+02  & 4.06164125E+01  \\ \hline
1.47350239E+03  & 8.18323486E+01  & 3.82585003E+01  \\ \hline
1.48347680E+03  & 1.58216119E+02  & 4.39850146E+01  \\ \hline
1.49345095E+03  & 3.32147466E+02  & 5.04266189E+01  \\ \hline
1.50342485E+03  & 4.91103167E+02  & 5.38234547E+01  \\ \hline
1.51339849E+03  & 5.78755480E+02  & 5.52499023E+01  \\ \hline
1.52337187E+03  & 6.39958448E+02  & 5.61230355E+01  \\ \hline
1.53334500E+03  & 6.81427543E+02  & 5.66683937E+01  \\ \hline
1.54331787E+03  & 7.09647291E+02  & 5.70208510E+01  \\ \hline
1.55329049E+03  & 7.39560450E+02  & 5.73794736E+01  \\ \hline
1.56326285E+03  & 7.57627052E+02  & 5.75891095E+01  \\ \hline
1.57323495E+03  & 6.99617369E+02  & 5.68972117E+01  \\ \hline
1.58320680E+03  & 7.15196690E+02  & 5.70885099E+01  \\ \hline
1.59317839E+03  & 7.31231594E+02  & 5.72810990E+01  \\ \hline
1.60314973E+03  & 6.76314977E+02  & 5.66029801E+01  \\ \hline
1.61312081E+03  & 6.58043478E+02  & 5.63650918E+01  \\ \hline
1.62309163E+03  & 7.65111030E+02  & 5.76744893E+01  \\ \hline
1.63306220E+03  & 8.01579583E+02  & 5.80789329E+01  \\ \hline
1.64303251E+03  & 7.58368564E+02  & 5.75976064E+01  \\ \hline
1.65300257E+03  & 7.56483671E+02  & 5.75759912E+01  \\ \hline
1.66297237E+03  & 8.05498162E+02  & 5.81212911E+01  \\ \hline
1.67294191E+03  & 8.73059480E+02  & 5.88208766E+01  \\ \hline
1.68291120E+03  & 8.72858649E+02  & 5.88188784E+01  \\ \hline
1.69288023E+03  & 8.19318308E+02  & 5.82690532E+01  \\ \hline
1.70284901E+03  & 7.25666603E+02  & 5.72147427E+01  \\ \hline
1.71281753E+03  & 6.90370300E+02  & 5.67816420E+01  \\ \hline
1.72278579E+03  & 7.52504513E+02  & 5.75301822E+01  \\ \hline
1.73275380E+03  & 6.29641087E+02  & 5.59818612E+01  \\ \hline
1.74272155E+03  & 5.29767245E+02  & 5.44817021E+01  \\ \hline
1.75268905E+03  & 5.55629415E+02  & 5.48957046E+01  \\ \hline
1.76265629E+03  & 4.81421838E+02  & 5.36505157E+01  \\ \hline
1.77262327E+03  & 3.70282207E+02  & 5.13706569E+01  \\ \hline
1.78259000E+03  & 2.29641644E+02  & 4.72210129E+01  \\ \hline
1.79255647E+03  & 1.12367859E+02  & 4.10128421E+01  \\ \hline
1.80252269E+03  & 2.36009960E+01  & 2.74586066E+01  \\ \hline
1.81248865E+03  & 2.51989647E+00  & 8.02765396E+00  \\ \hline
1.82245435E+03  & 1.00447693E+00  & 3.87993394E-02  \\ \hline
1.83241980E+03  & 1.03064798E+00  & 2.62207106E-01  \\ \hline
1.84238499E+03  & 9.01369974E-01  & -9.01938265E-01 \\ \hline
1.85234993E+03  & 9.02022314E-01  & -8.95654374E-01 \\ \hline
1.86231461E+03  & 7.64712093E-01  & -2.33004084E+00 \\ \hline
1.87227903E+03  & 9.00263226E-01  & -9.12609788E-01 \\ \hline
1.88224320E+03  & 6.11002968E-01  & -4.27913361E+00 \\ \hline
1.87923220E+03  & 1.67421124E+00  & 4.47620506E+00  \\ \hline
1.88929280E+03  & 1.73369509E+00  & 4.77945439E+00  \\ \hline
1.89935180E+03  & 1.47839901E+00  & 3.39583329E+00  \\ \hline
1.90940920E+03  & 1.58019236E+00  & 3.97419915E+00  \\ \hline
1.91946500E+03  & 1.54452530E+00  & 3.77590051E+00  \\ \hline
1.92951920E+03  & 1.57944251E+00  & 3.97007647E+00  \\ \hline
1.93957180E+03  & 4.89274539E+00  & 1.37910523E+01  \\ \hline
1.94962280E+03  & 2.02349680E+01  & 2.61220504E+01  \\ \hline
1.95967220E+03  & 6.21483228E+01  & 3.58685883E+01  \\ \hline
1.96972000E+03  & 2.67708766E+02  & 4.85532518E+01  \\ \hline
1.97976620E+03  & 4.78191465E+02  & 5.35920364E+01  \\ \hline
1.98981080E+03  & 4.62792920E+02  & 5.33077341E+01  \\ \hline
1.99985380E+03  & 1.89655899E+02  & 4.55593271E+01  \\ \hline
2.00989520E+03  & 1.22509697E+02  & 4.17634093E+01  \\ \hline
2.01993500E+03  & 2.28847211E+02  & 4.71909125E+01  \\ \hline
2.02997320E+03  & 5.07340590E+02  & 5.41059922E+01  \\ \hline
2.04000980E+03  & 5.61452610E+02  & 5.49862621E+01  \\ \hline
2.05004480E+03  & 3.85349876E+02  & 5.17171045E+01  \\ \hline
2.06007820E+03  & 3.36999712E+02  & 5.05525906E+01  \\ \hline
2.07011000E+03  & 3.74738557E+02  & 5.14745676E+01  \\ \hline
2.08014020E+03  & 5.20552405E+02  & 5.43292891E+01  \\ \hline
2.09016880E+03  & 5.50233199E+02  & 5.48109358E+01  \\ \hline
2.10019580E+03  & 5.78134306E+02  & 5.52405748E+01  \\ \hline
2.11022120E+03  & 6.21615563E+02  & 5.58704376E+01  \\ \hline
2.12024500E+03  & 6.01407743E+02  & 5.55833803E+01  \\ \hline
2.13026720E+03  & 6.22449219E+02  & 5.58820785E+01  \\ \hline
2.14028780E+03  & 6.27187956E+02  & 5.59479542E+01  \\ \hline
2.15030680E+03  & 5.81111426E+02  & 5.52851883E+01  \\ \hline
2.16032420E+03  & 5.54856647E+02  & 5.48836159E+01  \\ \hline
2.17034000E+03  & 5.54611972E+02  & 5.48797848E+01  \\ \hline
2.18035420E+03  & 5.40123334E+02  & 5.46498588E+01  \\ \hline
2.19036680E+03  & 5.42665956E+02  & 5.46906515E+01  \\ \hline
2.20037780E+03  & 5.08875795E+02  & 5.41322359E+01  \\ \hline
2.21038720E+03  & 5.81866238E+02  & 5.52964632E+01  \\ \hline
2.22039500E+03  & 5.76006532E+02  & 5.52085482E+01  \\ \hline
2.23040120E+03  & 5.73608193E+02  & 5.51723069E+01  \\ \hline
2.24040580E+03  & 5.54500136E+02  & 5.48780331E+01  \\ \hline
2.25040880E+03  & 5.56042790E+02  & 5.49021643E+01  \\ \hline
2.26041020E+03  & 5.41539289E+02  & 5.46725994E+01  \\ \hline
2.27041000E+03  & 5.39240021E+02  & 5.46356423E+01  \\ \hline
2.28040820E+03  & 5.22409031E+02  & 5.43602135E+01  \\ \hline
2.29040480E+03  & 5.17100070E+02  & 5.42714919E+01  \\ \hline
2.30039980E+03  & 4.75581185E+02  & 5.35444933E+01  \\ \hline
2.31039320E+03  & 4.78941938E+02  & 5.36056574E+01  \\ \hline
2.32038500E+03  & 3.80809649E+02  & 5.16141589E+01  \\ \hline
2.33037520E+03  & 4.12905071E+02  & 5.23170043E+01  \\ \hline
2.34036380E+03  & 3.36707706E+02  & 5.05450611E+01  \\ \hline
2.35035080E+03  & 2.83041459E+02  & 4.90370011E+01  \\ \hline
2.36033620E+03  & 3.14285152E+02  & 4.99464773E+01  \\ \hline
2.37032000E+03  & 2.35979966E+02  & 4.74575027E+01  \\ \hline
2.38030220E+03  & 1.99795868E+02  & 4.60117300E+01  \\ \hline
2.39028280E+03  & 1.93373994E+02  & 4.57279613E+01  \\ \hline
2.40026180E+03  & 2.09168287E+02  & 4.64099168E+01  \\ \hline
2.41023920E+03  & 1.36226831E+02  & 4.26852531E+01  \\ \hline
2.42021500E+03  & 1.01466157E+02  & 4.01264242E+01  \\ \hline
2.43018920E+03  & 1.50774481E+02  & 4.35665568E+01  \\ \hline
2.44016180E+03  & 1.17187629E+02  & 4.13776353E+01  \\ \hline
2.45013280E+03  & 4.64186155E+01  & 3.33338437E+01  \\ \hline
2.46010220E+03  & 8.47493741E+01  & 3.85627300E+01  \\ \hline
2.47007000E+03  & 4.07071833E+01  & 3.21934211E+01  \\ \hline
2.48003620E+03  & 1.35927714E+01  & 2.26661602E+01  \\ \hline
2.49000080E+03  & 1.44823893E+00  & 3.21680434E+00  \\ \hline
2.49996380E+03  & 1.75295314E+00  & 4.87540613E+00  \\ \hline
2.50992520E+03  & 2.71175883E+00  & 8.66502128E+00  \\
\hline
\end{longtable}
\doublespace
\newpage

%------------------------------------------------------------------------------------------------------------------------------------------
% Affinity matrices of datasets.
%------------------------------------------------------------------------------------------------------------------------------------------
\section{Affinity Matrices of Endmembers for SMDE Dataset}
\label{affinityMatriciesNYC}
\begin{tabular}{ c c c}
  \includegraphics[width=50mm]{../../thesis/results/thesis/nyc5/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/nyc10/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/nyc15/affinityMatrix/affinityMatrix.png} \\
    5 endmembers & 10 endmembers & 15 endmembers \\ \hline
  \includegraphics[width=50mm]{../../thesis/results/thesis/nyc20/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/nyc25/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/nyc30/affinityMatrix/affinityMatrix.png} \\
    20 endmembers & 25 endmembers & 30 endmembers \\ \hline
  \includegraphics[width=50mm]{../../thesis/results/thesis/nyc35/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/nyc40/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/nyc45/affinityMatrix/affinityMatrix.png} \\
    35 endmembers & 40 endmembers & 45 endmembers \\ \hline
  \includegraphics[width=50mm]{../../thesis/results/thesis/nyc50/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/nyc55/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/nyc60/affinityMatrix/affinityMatrix.png} \\
    50 endmembers & 55 endmembers & 60 endmembers \\
\end{tabular}
\newpage
\section{Histograms of Endmember Affinity for SMDE Dataset}
\label{histogramsNYC}
\begin{tabular}{ c c c}
  \includegraphics[width=50mm]{../../thesis/results/thesis/nyc5/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/nyc10/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/nyc15/affinityMatrix/histogramOfAffinityMatrix.png} \\
    5 endmembers & 10 endmembers & 15 endmembers \\ \hline
  \includegraphics[width=50mm]{../../thesis/results/thesis/nyc20/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/nyc25/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/nyc30/affinityMatrix/histogramOfAffinityMatrix.png} \\
    20 endmembers & 25 endmembers & 30 endmembers \\ \hline
  \includegraphics[width=50mm]{../../thesis/results/thesis/nyc35/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/nyc40/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/nyc45/affinityMatrix/histogramOfAffinityMatrix.png} \\
    35 endmembers & 40 endmembers & 45 endmembers \\ \hline
  \includegraphics[width=50mm]{../../thesis/results/thesis/nyc50/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/nyc55/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/nyc60/affinityMatrix/histogramOfAffinityMatrix.png} \\
    50 endmembers & 55 endmembers & 60 endmembers \\
\end{tabular}
\newpage

\section{Affinity Matrices of Endmembers for RMDE Dataset}
\label{affinityMatriciesRMDE}
\begin{tabular}{ c c c}
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image5/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image10/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image15/affinityMatrix/affinityMatrix.png} \\
    5 endmembers & 10 endmembers & 15 endmembers \\ \hline
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image20/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image25/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image30/affinityMatrix/affinityMatrix.png} \\
    20 endmembers & 25 endmembers & 30 endmembers \\ \hline
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image35/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image40/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image45/affinityMatrix/affinityMatrix.png} \\
    35 endmembers & 40 endmembers & 45 endmembers \\ \hline
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image50/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image55/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image60/affinityMatrix/affinityMatrix.png} \\
    50 endmembers & 55 endmembers & 60 endmembers \\
\end{tabular}
\newpage
\section{Histograms of Endmember Affinity for RMDE Dataset}
\label{histogramsRMDE}
\begin{tabular}{ c c c}
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image5/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image10/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image15/affinityMatrix/histogramOfAffinityMatrix.png} \\
    5 endmembers & 10 endmembers & 15 endmembers \\ \hline
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image20/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image25/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image30/affinityMatrix/histogramOfAffinityMatrix.png} \\
    20 endmembers & 25 endmembers & 30 endmembers \\ \hline
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image35/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image40/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image45/affinityMatrix/histogramOfAffinityMatrix.png} \\
    35 endmembers & 40 endmembers & 45 endmembers \\ \hline
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image50/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image55/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image60/affinityMatrix/histogramOfAffinityMatrix.png} \\
    50 endmembers & 55 endmembers & 60 endmembers \\
\end{tabular}
\newpage

\section{Affinity Matrices of Endmembers for RMRE Dataset}
\label{affinityMatriciesRMRE}
\begin{tabular}{ c c c}
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_mix_random_endmembers5/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_mix_random_endmembers10/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_mix_random_endmembers15/affinityMatrix/affinityMatrix.png} \\
    5 endmembers & 10 endmembers & 15 endmembers \\ \hline
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_mix_random_endmembers20/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_mix_random_endmembers25/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_mix_random_endmembers30/affinityMatrix/affinityMatrix.png} \\
    20 endmembers & 25 endmembers & 30 endmembers \\ \hline
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_mix_random_endmembers35/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_mix_random_endmembers40/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_mix_random_endmembers45/affinityMatrix/affinityMatrix.png} \\
    35 endmembers & 40 endmembers & 45 endmembers \\ \hline
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_mix_random_endmembers50/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_mix_random_endmembers55/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_mix_random_endmembers60/affinityMatrix/affinityMatrix.png} \\
    50 endmembers & 55 endmembers & 60 endmembers \\
\end{tabular}
\newpage
\section{Histograms of Endmember Affinity for RMRE Dataset}
\label{histogramsRMRE}
\begin{tabular}{ c c c}
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_mix_random_endmembers5/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_mix_random_endmembers10/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_mix_random_endmembers15/affinityMatrix/histogramOfAffinityMatrix.png} \\
    5 endmembers & 10 endmembers & 15 endmembers \\ \hline
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_mix_random_endmembers20/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_mix_random_endmembers25/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_mix_random_endmembers30/affinityMatrix/histogramOfAffinityMatrix.png} \\
    20 endmembers & 25 endmembers & 30 endmembers \\ \hline
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_mix_random_endmembers35/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_mix_random_endmembers40/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_mix_random_endmembers45/affinityMatrix/histogramOfAffinityMatrix.png} \\
    35 endmembers & 40 endmembers & 45 endmembers \\ \hline
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_mix_random_endmembers50/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_mix_random_endmembers55/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_mix_random_endmembers60/affinityMatrix/histogramOfAffinityMatrix.png} \\
    50 endmembers & 55 endmembers & 60 endmembers \\
\end{tabular}
\newpage

\section{Affinity Matrices of Endmembers for RMDEI Dataset}
\label{affinityMatriciesRMDEI}
\begin{tabular}{ c c c}
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image_illumination5/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image_illumination10/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image_illumination15/affinityMatrix/affinityMatrix.png} \\
    5 endmembers & 10 endmembers & 15 endmembers \\ \hline
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image_illumination20/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image_illumination25/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image_illumination30/affinityMatrix/affinityMatrix.png} \\
    20 endmembers & 25 endmembers & 30 endmembers \\ \hline
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image_illumination35/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image_illumination40/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image_illumination45/affinityMatrix/affinityMatrix.png} \\
    35 endmembers & 40 endmembers & 45 endmembers \\ \hline
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image_illumination50/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image_illumination55/affinityMatrix/affinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image_illumination60/affinityMatrix/affinityMatrix.png} \\
    50 endmembers & 55 endmembers & 60 endmembers \\
\end{tabular}
\newpage
\section{Histograms of Endmember Affinity for RMDEI Dataset}
\label{histogramsRMDEI}
\begin{tabular}{ c c c}
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image_illumination5/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image_illumination10/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image_illumination15/affinityMatrix/histogramOfAffinityMatrix.png} \\
    5 endmembers & 10 endmembers & 15 endmembers \\ \hline
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image_illumination20/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image_illumination25/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image_illumination30/affinityMatrix/histogramOfAffinityMatrix.png} \\
    20 endmembers & 25 endmembers & 30 endmembers \\ \hline
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image_illumination35/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image_illumination40/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image_illumination45/affinityMatrix/histogramOfAffinityMatrix.png} \\
    35 endmembers & 40 endmembers & 45 endmembers \\ \hline
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image_illumination50/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image_illumination55/affinityMatrix/histogramOfAffinityMatrix.png} &
  \includegraphics[width=50mm]{../../thesis/results/thesis/random_image_illumination60/affinityMatrix/histogramOfAffinityMatrix.png} \\
    50 endmembers & 55 endmembers & 60 endmembers \\
\end{tabular}
\newpage

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