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+\documentclass{sciposter}
+
+
+\usepackage{epsfig}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{multicol}
+%\usepackage{fancybullets}
+
+\newtheorem{Def}{Definition}
+
+%\definecolor{BoxCol}{rgb}{0.9,0.9,0.9}
+% uncomment for grey background to \section boxes 
+% for use with default option boxedsections
+
+%\definecolor{BoxCol}{rgb}{0.9,0.9,1}
+% uncomment for light blue background to \section boxes 
+% for use with default option boxedsections
+
+%\definecolor{SectionCol}{rgb}{0,0,0.5}
+% uncomment for dark blue \section text 
+
+
+
+
+
+
+\title{Generalized Pattern Spectra Sensitive to Spatial Information}
+
+% Note: only give author names, not institute
+\author{Michael H. F. Wilkinson}
+ 
+% insert correct institute name
+\institute{Institute for Mathematics and Computing Science,\\
+           University of Groningen\\}
+
+\email{michael@cs.rug.nl}  % shows author email address below institute
+
+%\date is unused by the current \maketitle
+
+
+% The following commands can be used to alter the default logo settings
+%\leftlogo[0.9]{logoWenI}{  % defines logo to left of title (with scale factor)
+%\rightlogo[0.52]{RuGlogo}  % same but on right
+
+% NOTE: This will require presence of files logoWenI.eps and RuGlogo.eps, 
+% or other supported format in the current directory  
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%% Begin of Document
+
+
+
+\begin{document}
+%define conference poster is presented at (appears as footer)
+
+\conference{{\bf ICPR 2002}, 16th International Conference on Pattern
+  Recognition, 11-15 August 2002, Qu\'ebec City, Canada}
+
+
+\maketitle
+
+%%% Begin of Multicols-Enviroment
+\begin{multicols}{3}
+
+%%% Abstract
+\begin{abstract}
+Morphological pattern spectra computed from granulometries are frequently used
+to classify the size classes of details in textures and images. An extension 
+of this technique, which retains information on the spatial 
+distribution of the details in each size class is developed. Algorithms for
+computation of these spatial pattern spectra for a large number of 
+granulometries on binary images are presented. 
+\end{abstract}
+
+%%% Introduction
+\section{Introduction}
+
+\PARstart{G}{ranulometries} are ordered sets of morphological openings or closings, each of
+which removes image details below a certain size. These can be used for texture
+analysis
+through the use of \emph{pattern spectra}, which show how the number of 
+foreground pixels in the image changes as a function of the size parameter 
+\cite{maragos89:_patter}.
+A drawback of the classical definition of pattern spectra is that spatial 
+information is not included in a pattern spectrum as shown below.
+ In this paper, \emph{spatial pattern spectra} are developed which retain information on the distribution of these details at different scales.
+ 
+
+\newcommand{\imsize}{0.45\columnwidth}
+\begin{figure}
+\begin{center}
+\begin{tabular}{c c}
+{\resizebox{\imsize}{!}{\includegraphics{blocks1}}} &
+{\resizebox{\imsize}{!}{\includegraphics{blocks2}}}\\
+(a) & (b) \\
+{\resizebox{\imsize}{!}{\includegraphics{blocks3}}} &
+{\resizebox{\imsize}{!}{\includegraphics{blocks1a}}}\\
+(c) & (d) \\
+\end{tabular}
+\end{center}
+\caption{ Parts (a) through (c) show three images consisting of squares of
+different sizes;
+(d) shows the pattern spectra, denoting the number of foreground pixels 
+ removed by openings by reconstruction by $\lambda \times \lambda$ squares. No 
+granulometry is capable of separating the patterns, because the only 
+differences between the images lie in the distributions of the 
+connected components. }\label{fig:blocks}
+\end{figure}
+
+
+
+
+\section{Theory}
+
+Let binary images $X$ and $Y$ be defined as a subset of the image domain 
+${\mathbf M}\subset {\mathbb Z}^n$ or ${\mathbb R}^n$ (usually $n=2$). 
+\begin{Def}
+A binary 
+granulometry is a set of operators $\{\alpha_r\}$ with $r$ from some ordered 
+set $\Lambda$ (usually $\Lambda \subset {\mathbb R}$ or ${\mathbb Z}$), with 
+the following three properties
+\begin{align}
+   \alpha_r(X) & \subset  X \label{eq:antiext} \\
+   X \subset Y & \Rightarrow \alpha_r(X) \subset \alpha_r(Y) 
+   \label{eq:increasing} \\
+   \alpha_r(\alpha_s(X)) & =  \alpha_{\max(r,s)}(X) \label{eq:idempot},
+\end{align}   
+for all $r,s \in \Lambda$.
+\end{Def}
+
+\begin{Def}
+The pattern spectrum $s_{\alpha}(X)$ obtained by applying 
+granulometry $\{\alpha_r\}$ to a binary image $X$ is defined as
+\begin{equation}
+    (s_{\alpha}(X))(u) = 
+    - \frac{\partial A(\alpha_r(X))}{\partial r}\bigg{\vert}_{r=u}
+\end{equation}
+in which $A(X)$ is a function denoting the Lebesgue measure in 
+${\mathbb R}^n$.
+\end{Def} 
+In the case of discrete images, and with $r \in \Lambda \subset {\mathbb Z}$, 
+this differentiation reduces to
+\begin{align}
+    (s_{\alpha}(X))(r)  & = \#(\alpha_{r}(X) \setminus \alpha_{r^+}(X)) \\ 
+                        & = \#(\alpha_{r}(X)) - \#(\alpha_{r^+}(X)), 
+\end{align}
+with $r^+ = \min\{ r' \in \Lambda \vert r' > r \}$, and $\#(X)$ the 
+numnber of elements of $X$.
+
+The opening transform \cite{Nacken:thesis} $\Omega_X$ of a binary image $X$ 
+for a granulometry ${\alpha_r}$ is
+\begin{equation}
+  \Omega_X(x) = \max\{ r \in \Lambda \vert x \in \alpha_r(X) \}
+\end{equation}
+
+The pattern spectrum of a binary image $X$ using granulometry 
+$\{\alpha_r\}$ is the histogram of $\Omega_X$ obtained with the same 
+size distribution \cite{Nacken:thesis}, disregarding the bin for grey level 0.
+
+
+\begin{figure}
+\begin{center}
+{\resizebox{\imsize}{!}{\includegraphics{blocks3}}}
+{\resizebox{\imsize}{!}{\includegraphics{blocks3rec}}}
+\end{center}
+\caption{ \label{fig:opentransf} Opening transform with $\{\alpha_r\}$ as in 
+ Fig. \ref{fig:blocks}: (left) original image; (right) opening transform
+(contrast stretched for clarity). 
+}
+\end{figure}
+
+
+\section{Spatial pattern spectra}
+Pattern spectra only retain the amount of detail present at  scale $r$.
+This can be amended by computing some parameterization of the spatial 
+distribution in an image $\alpha_r(X) \setminus \alpha_{r+}(X)$ as a function of $r$. 
+
+\begin{Def}
+Let ${M}(X)$ be some parameterization of the spatial distribution of detail
+in the image $X$. The spatial pattern spectrum ${S}_{{M},\alpha}$ is
+then defined as
+\begin{equation}
+  ({S}_{{M},\alpha}(X))(r) = {M}(\alpha_r(X) \setminus \alpha_{r+}(X)).
+\end{equation}  
+\end{Def}
+
+An obvious parameterization of the spatial distribution is through 
+the use of moments. Focusing on the case of 2-D binary images, the 
+moment $m_{ij}$ of order $ij$ of an image $X$ is given by
+\begin{equation}
+  m_{ij}(X) = \sum_{(x,y) \in \mathbf X} x^i y^j.
+\end{equation}  
+The spatial moment spectrum $S_{m_{ij},\alpha}$ of order $ij$ is
+\begin{equation}
+  (S_{m_{ij},\alpha}(X))(r) = m_{i,j}(\alpha_r(X) \setminus \alpha_{r^+}(X)).
+\end{equation}  
+For $i=0$ and $j=0$ we obtain the standard pattern spectrum. 
+For each $r$, $(S_{m_{ij},\alpha}(X))(r)$ is just the moment of an image, 
+therefore, derived parameters such as coordinates of the centre of mass, 
+(co-)variances, skewness and kurtosis of the distribution of details at each
+scale can be computed easily. We can then define pattern mean
+spectra, pattern (co-)variance spectra, pattern kurtosis spectra, etc. The 
+pattern mean-$x$ and variance-$x$ spectra 
+($S_{\bar x,\alpha}$ and $S_{\sigma(x),\alpha}$) are defined as: 
+\begin{align}
+  S_{\bar x,\alpha} & = \frac{S_{m_{10},\alpha}} {S_{m_{00},\alpha}} \\
+\intertext{and}
+   S_{\sigma(x),\alpha} & = \sqrt{\frac{S_{m_{20},\alpha}}
+                                 {S_{m_{00},\alpha}} 
+                                 - S_{\bar x, \alpha}}.  
+ \end{align}
+These two are shown in Figures \ref{fig:tauspect} and \ref{fig:binspect}. Note that 
+these definitions hold only where $(S_{m_{00},\alpha}(f))(r) \neq 0$. For all 
+other values of $r$ they will be defined as zero. Further post-processing can
+be done to compute central moments and moment invariant from pattern moment 
+spectra \cite{Flusser:Suk:93,Hu:62}. 
+
+\section{An Algorithm}
+
+Nacken \cite{Nacken:thesis} derived an algorithm for computation
+of pattern spectra for granulometries based on openings by discs of increasing
+radius for various metrics, using the opening transform. After the
+opening transform has been computed, it is straightforward to compute the 
+pattern spectrum:
+\begin{itemize}
+\item Set all elements of array {\tt S} to zero
+\item For all $x \in X$ increment {\tt S}[$\Omega_X(x)$] by one. 
+\end{itemize}
+
+To compute the pattern \emph{moment} spectrum, the only thing that needs to be
+changed is the way {\tt S}[$\Omega_X(x)$] is incremented. As shown in Algorithm
+\ref{alg:spect}.
+
+\begin{algorithm}
+\begin{itemize}
+\item Set all elements of array {\tt S} to zero
+\item For all $(x,y) \in X$ increment {\tt S}[$\Omega_X(x,y)$] by 
+$x^iy^j$. 
+\end{itemize}
+\caption{ Algorithm for computation of pattern moment
+spectrum of order $ij$. \label{alg:spect}}
+\end{algorithm}
+
+This algorithm can 
+readily be adapted to other granulometries, simply by computing the 
+appropriate opening transform.
+
+\begin{figure}
+\begin{center}
+\begin{tabular}{c c}
+{\resizebox{\imsize}{!}{\includegraphics{blocks3op}}} &
+{\resizebox{\imsize}{!}{\includegraphics{blocksopen3a}}}\\
+(a) & (b)\\
+{\resizebox{\imsize}{!}{\includegraphics{blocksopen3vx}}}&
+{\resizebox{\imsize}{!}{\includegraphics{blocksopen3vy}}}\\
+(c) & (d) \\
+\end{tabular}
+\end{center}
+\caption{ \label{fig:tauspect} 
+The opening transform using city-block metric: (a) opening transform of
+Fig. 1(c); (b) pattern spectrum; (c) pattern variance-$x$; 
+(d) variance-$y$ spectra.}
+\end{figure}
+
+
+
+ 
+
+
+\renewcommand{\imsize}{0.3\columnwidth}
+\begin{figure}
+\begin{center}
+{\resizebox{\imsize}{!}{\includegraphics{blocks1mx}}}
+{\resizebox{\imsize}{!}{\includegraphics{blocks2mx}}}
+{\resizebox{\imsize}{!}{\includegraphics{blocks3mx}}}\\
+{\resizebox{\imsize}{!}{\includegraphics{blocks1vx}}}
+{\resizebox{\imsize}{!}{\includegraphics{blocks1vx}}}
+{\resizebox{\imsize}{!}{\includegraphics{blocks3vx}}}
+\end{center}
+\caption{ \label{fig:binspect} Pattern mean-$x$ (top) and variance-$x$ 
+(bottom) spectra: the three collumns show spectra for Fig. 1(a), (b) and (c) 
+from left to right respectively.  Unlike the standard pattern spectra, 
+these spatial pattern spectra can distinguish the three images.}
+\end{figure}
+
+\section{Discussion}
+Spatial pattern spectra form a useful supplement to ordinary pattern 
+spectra, because of their ability to retain spatial information.  
+Pattern moment spectra, in particular, are easily computed concurrently with 
+computation of the standard pattern spectrum. Post-processing of these pattern
+moment spectra can be done to yield a number of easily interpreted spectra, 
+such as pattern mean, variance, skew, and kurtosis spectra, which have reduced 
+covariance compared to the ``raw'' pattern moment spectra. Invariance to 
+rotation, translation or scale change can also be achieved by post-processing
+\cite{Flusser:Suk:93,Hu:62}.
+
+In the future grey scale versions of these spatial pattern spectra will be 
+developed. I expect that the efficient grey level algorithms for area and
+attribute pattern spectra  
+\cite{Meijster:Wilkinson:PAMI}
+can be adapted to spatial pattern spectra as well.
+ 
+%%% References
+
+%% Note: use of BibTeX als works!!
+
+\bibliographystyle{plain}
+\begin{thebibliography}{1}
+
+\bibitem{Flusser:Suk:93}
+J.~Flusser and T.~Suk.
+\newblock Pattern recognition by affine moment invariants.
+\newblock {\em Pattern Recognition}, 26:167--174, 1993.
+
+\bibitem{Hu:62}
+M.~K. Hu.
+\newblock Visual pattern recognition by moment invariants.
+\newblock {\em IRE Transactions on Information Theory}, IT-8:179--187, 1962.
+
+\bibitem{maragos89:_patter}
+P.~Maragos.
+\newblock Pattern spectrum and multiscale shape representation.
+\newblock {\em IEEE Trans. Patt. Anal. Mach. Intell.}, 11:701--715, 1989.
+
+\bibitem{Meijster:Wilkinson:PAMI}
+A.~Meijster and M.~H.~F. Wilkinson.
+\newblock A comparison of algorithms for connected set openings and closings.
+\newblock {\em IEEE Trans. Patt. Anal. Mach. Intell.}, 24(4):484--494, 2002.
+
+\bibitem{Nacken:thesis}
+P.~F.~M. Nacken.
+\newblock {\em Image Analysis Methods Based on Hierarchies of Graphs and
+  Multi-Scale Mathematical Morphology}.
+\newblock PhD thesis, University of Amsterdam, Amsterdam, The Netherlands,
+  1994.
+
+\end{thebibliography}
+
+\end{multicols}
+
+\end{document}
+