

\documentclass[conference]{IEEEtran}


\usepackage{cite}
% \usepackage[pdftex]{graphicx}
% \graphicspath{{../pdf/}{../jpeg/}}
% \DeclareGraphicsExtensions{.pdf,.jpeg,.png}
\usepackage{amsmath}
%\usepackage{array}
%  \usepackage[caption=false,font=footnotesize]{subfig}
%\usepackage{url}

% correct bad hyphenation here
\hyphenation{op-tical net-works semi-conduc-tor}


\newcommand{\dop}   [1]{\ensuremath{ \mathop{\mathrm{d}#1}                 }}
\newcommand{\R}        {\ensuremath{ \mathbf{R}                            }}
\newcommand{\expp}  [1]{\ensuremath{ \exp \left( #1 \right)                }}

\newcommand{\qq}    [1]{``#1''}

\begin{document}
%
% paper title
% Titles are generally capitalized except for words such as a, an, and, as,
% at, but, by, for, in, nor, of, on, or, the, to and up, which are usually
% not capitalized unless they are the first or last word of the title.
% Linebreaks \\ can be used within to get better formatting as desired.
% Do not put math or special symbols in the title.
\title{Bare Demo of IEEEtran.cls\\ for IEEE Conferences}


% author names and affiliations
% use a multiple column layout for up to three different
% affiliations
\author{	
    
    \IEEEauthorblockN{Markus Bullmann, Toni Fetzer, Frank Ebner, and Frank Deinzer}%
    \IEEEauthorblockA{%
        Faculty of Computer Science and Business Information Systems\\ 
        University of Applied Sciences W\"urzburg-Schweinfurt\\
        W\"urzburg, Germany\\
        \{markus.bullmann, toni.fetzer, frank.ebner, frank.deinzer\}@fhws.de\\		
    }
}


\maketitle


\begin{abstract}
The abstract goes here.
\end{abstract}

% no keywords


% For peer review papers, you can put extra information on the cover
% page as needed:
% \ifCLASSOPTIONpeerreview
% \begin{center} \bfseries EDICS Category: 3-BBND \end{center}
% \fi
%
% For peerreview papers, this IEEEtran command inserts a page break and
% creates the second title. It will be ignored for other modes.
\IEEEpeerreviewmaketitle



\section{Introduction}
% KDE wellknown nonparametic estimation method
% Flexibility is paid with slow speed
% Finding optimal bandwidth
% Expensive computation

\section{Related work}
% original work rosenblatt/parzen
% binned version silverman, scott, härdle
% -> Fourier transfom
% other approaches Fast Gaussian Transform

\section{Kernel Density Estimation}
% KDE by rosenblatt and parzen
% general KDE
% Gauss Kernel
% Formula Gauss KDE
% -> complexity/operation count
% Binned KDE
% Binned Gauss KDE
% -> complexity/operation count

The histogram is a simple and for a long time the most used non-parametric estimator.
However, its inability to produce a continuous estimate dismisses it for many applications where a smooth distribution is assumed.
In contrast, the KDE is often the preferred tool because of its ability to produce a continuous estimate and its flexibility.
Given $n$ independently observed realizations of the observation set $X=(x_1,\dots,x_n)$, the kernel density estimate $\hat{f}_n$ of the density function $f$ of the underlying distribution is given with
\begin{equation}
\label{eq:kde}
\hat{f}_n = \frac{1}{nh} \sum_{i=1}^{n} K \left(  \frac{x-X_i}{h} \right)  \text{,} %= \frac{1}{n} \sum_{i=1}^{n} K_h(x-x_i)
\end{equation}
where $K$ is the kernel function and $h\in\R^+$ is an arbitrary smoothing parameter called bandwidth.
While any density function can be used as the kernel function $K$ (such that $\int K(u) \dop{u} = 1$), a variety of popular choices of the kernel function $K$ exits.
In practice the Gaussian kernel is commonly used:
\begin{equation}
K(u)=\frac{1}{\sqrt{2\pi}} \expp{- \frac{u^2}{2} }
\end{equation}

\begin{equation}
\hat{f}_n = \frac{1}{nh\sqrt{2\pi}} \sum_{i=1}^{n} \expp{-\frac{(x-X_i)^2}{2h^2}}
\end{equation}


\section{Moving Average Filter}
% Basic box filter formula
% Recursive form
% Gauss Blur Filter
% Repetitive Box filter to approx Gauss
% Simple multipass, n/m approach, extended box filter
The moving average filter is a simplistic filter which takes an input function $x$ and produces a second function $y$.
A single output value is computed by taking the average of a number of values symmetrical around a single point in the input.
The number of values in the average can also be seen as the width $w=2r+1$, where $r$ is the \qq{radius} of the filter.
The computation of an output value using a moving average filter of radius $r$ is defined as
\begin{equation}
\label{eq:symMovAvg}
y[i]=\frac{1}{2r+1} \sum_{j=-r}^{r}x[i+j] \text{.}
\end{equation}

It is well-known that a moving average filter can approximate a Gaussian filter by repetitive recursive computations.



As is known the Gaussian filter is parametrized by its standard deviation $\sigma$.
To approximate a Gaussian filter one needs to express a given $\sigma$ in terms of moving average filters.


\section{Combination}


\section{Experiments}



\section{Conclusion}
The conclusion goes here.





% use section* for acknowledgment
%\section*{Acknowledgment}
%The authors would like to thank...




% trigger a \newpage just before the given reference
% number - used to balance the columns on the last page
% adjust value as needed - may need to be readjusted if
% the document is modified later
%\IEEEtriggeratref{8}
% The "triggered" command can be changed if desired:
%\IEEEtriggercmd{\enlargethispage{-5in}}

% references section
\bibliographystyle{IEEEtran}
\bibliography{IEEEabrv,egbib}



\end{document}
