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%-----------------------------------------------------------------------
%			    Paper ID Code
%-----------------------------------------------------------------------
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\paperID{O3-3}
%%%% ID=O3-3

%-----------------------------------------------------------------------
%		            Paper Title 
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\title{Source Detection with Bayesian Inference on ROSAT All-Sky Survey Data Sample}
\titlemark{Source Detection with Bayesian Inference}

%-----------------------------------------------------------------------
%		          Authors of Paper
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%                 University of Illinois Urbana-Champaign, Urbana, IL
%                 61801}
%          \author{Tom Troland}
%          \affil{University of Kentucky}
%
%          \altaffiltext{1}{Astronomy Department, UIUC}
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% In this example, the first three authors, "Plante", "Roberts", and
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\author{Fabrizia Guglielmetti, Rainer Fischer, Volker Dose}
\affil{Centre for Interdisciplinary Plasma Science, Garching, Germany D-85748}
%\affil{Centre for Interdisciplinary Plasma Science, Garching, Germany D-85748\altaffilmark{1}}
%\altaffiltext{1}{The Centre for Interdisciplinary Plasma Science was founded with effect of January 1st, 2000 by the Max-Planck-Institut f$\ddot{u}$r Extraterrestrische Physik and the Max-Planck-Institut f$\ddot{u}$r Plasmaphysik, with the aim to merge and strengthen complementary research activities in complex plasmas, theoretical and numerical plasma science and scientific data analysis}

\author{Wolfgang Voges, Guenter Boese}
\affil{Max-Planck-Institut f$\ddot{u}$r Extraterrestrische Physik, Garching, Germany D-85748}

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\contact{Fabrizia Guglielmetti}
\email{fabrizia@mpe.mpg.de}

%-----------------------------------------------------------------------
%		      Author Index Specification
%-----------------------------------------------------------------------
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% \paindex{ } should be used to indicate the primary author, and the
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\paindex{Guglielmetti, F.}
\aindex{Voges, W.}
\aindex{Fischer, R.}     % Remove this line if there is only one author
\aindex{Boese, G.}
\aindex{Dose, V.}

%-----------------------------------------------------------------------
%		      Author list for page header	
%-----------------------------------------------------------------------
% Please supply a list of author last names for the page header. in
% one of these formats:
%
% EXAMPLES:
% \authormark{Lastname}
% \authormark{Lastname1 \& Lastname2}
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%
% Use the "et al." form in the case of seven or more authors, or if
% the preferred form is too long to fit in the header.

\authormark{Guglielmetti, Voges, Fischer, Boese \& Dose}

%-----------------------------------------------------------------------
%			Subject Index keywords
%-----------------------------------------------------------------------
% Enter a comma separated list of up to 6 keywords describing your
% paper.  These will NOT be printed as part of your paper; however,
% they will be used to generate the subject index for the proceedings.
% There is no standard list; however, you can consult the indices
% for past proceedings (http://adass.org/adass/proceedings/).
%
% EXAMPLE:  \keywords{visualization, astronomy: radio, parallel
%                     computing, AIPS++, Galactic Center}
%
% In this example, the author noticed that "radio astronomy" appeared
% in the ADASS VII Index as "astronomy" being the major keyword and
% "radio" as the minor keyword.  The colon is used to introduce another
% level into the index.

\keywords{data analysis, astronomy: algorithms, Bayesian inference, background estimation, source detection}

%-----------------------------------------------------------------------
%			       Abstract
%-----------------------------------------------------------------------
% Type abstract in the space below.  Consult the User Guide and Latex
% Information file for a list of supported macros (e.g. for typesetting 
% special symbols). Do not leave a blank line between \begin{abstract} 
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\begin{abstract}          % Leave intact
% Place the text of your abstract here - NO BLANK LINES
We employ Bayesian inference for the joint estimation of sources and
background on ROSAT All-Sky Survey (RASS) data. 
The probabilistic method allows for detection improvement of faint
extended celestial sources compared to the Standard Analysis Software
System (SASS).
Background maps were estimated in a single step together with the
detection of sources without pixel censoring.
Consistent uncertainties of background and sources are provided.
The source probability is evaluated for single pixels as well as for
pixel domains to enhance source detection of weak and extended
sources.
\end{abstract}

%-----------------------------------------------------------------------
%			      Main Body
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% It is recommended that you look at the sample papers, sample1.tex
% and sample2.tex, for examples for formatting references, footnotes,
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\section{Introduction}

The RASS x-ray data were largely analyzed with the Standard
Analysis Software System (SASS).
The results can be found in the Bright and Faint Source catalogues 
(Voges et al.\ 1999, 2000).
Nevertheless, SASS is known for lack of sensitivity for faint or extended
sources.
This is due to the sliding window technique which locally
searches for count enhancements relative to the intensity in a
surrounding area defining the background intensity.
In multiple steps the window width is changed to allow for the detection
of extended sources.
But faint extended sources and blended faint sources in crowded fields
may get lost.
One reason is due to the local estimation of the background in a small
region around the
sliding window which may provide only poor signal-to-noise ratios (S/N).
The sources are characterized by fitting the candidate sources in a
further step using a Maximum-Likelihood (ML) method (Boese $\&$ Doebereiner
2001). 
The ML method works properly on prominent point-like sources which account for
94$\%$ of the sources published in the RASS Bright Source catalogue.
The characteristics of the faint sources may not be properly estimated
in case of faint extended sources (Voges et al.\ 1999).

The present method using Bayesian Probability Theory (BPT) estimates the
background and sources in a single step neither employing pixel
censoring nor using a sliding window technique. 
The aim is to infer simultaneously a background map for the complete field size 
($6.4^\circ \times 6.4^\circ$ in the sky) and a
probability for having source intensity in addition to the background
intensity in a pixel cell or pixel domain. 
Bayesian inference allows reasoning on the basis of sparse data
employing additional information independent of the data.
The results are given by probability distributions quantifying our state
of knowledge.
For background estimation and source detection the additional
information is the assumption that the background is smooth, e.g.\ spatially slowly varying compared to source dimensions.
To allow for smoothness the background is modelled with a bivariate Thin-Plate spline.
The coexistence of background and sources is described with a probabilistic
two-component mixture model where one component describes
background contribution only and the other component describes
background plus signal contributions. 
Each pixel cell (or pixel domain) is characterized by the
probability of belonging to one of the two mixture components. 
For the background spline estimation the photons contained in all pixel cells
are considered including pixels containing additional source
contributions.

This technique is applied on a data sample coming from the ROSAT PSPC in 
Survey Mode (0.1-2.4 keV). The ROSAT exposure map and the observatory's point 
spread function have been properly accounted for.

\section{Method}

Given the observed data set $D=\{d_{ij}\} \in \mathbf{N}_0$, where
$d_{ij}$ expresses photon counts in a pixel cell $ij$, 
two complementary hypotheses arise:
%
\begin{displaymath}
\left\{ \begin{array}{ll}
B_{ij} \to d_{ij}=b_{ij}+\epsilon_{ij} \\
\overline{B}_{ij} \to d_{ij}=b_{ij}+s_{ij}+\epsilon_{ij} 
\end{array} \right.
\end{displaymath}
%
Hypothesis $B_{ij}$ specifies that $d_{ij}$ consists only of background 
$b_{ij}$ spoiled with noise $\epsilon_{ij}$. 
Hypothesis $\overline{B}_{ij}$ specifies the case where additional
source intensity $s_{ij}$ contributes to the background.

Additional assumptions are that no negative values for signal
and background amplitudes are allowed and that
the background is smoother than the signal. 
This is achieved by modelling the background count rate with a bivariate
Thin-Plate spline where the supporting points are chosen sparsely to
ensure that sources can not be fitted.
The spline fits the background component whereas count enhancements
classify pixel (domains) with source contributions.

The likelihood distributions for the two hypotheses are
%
\begin{eqnarray}
p(d_{ij} \mid B_{ij},b_{ij}) & = & \frac{b_{ij}^{d_{ij}}}{d_{ij}!} e^{-b_{ij}}, \\
\label{O3-3:like_bg}
%
p(d_{ij} \mid \overline{B}_{ij},b_{ij},s_{ij}) & = & \frac{(b_{ij}+s_{ij})^{d_{ij}}}{d_{ij}!} e^{-(b_{ij}+s_{ij})}.  
\label{O3-3:like_source}
\end{eqnarray}
%
For background estimation we have to marginalize over the
signal in eqn.\ \ref{O3-3:like_source} according to the sum rule of BPT. 
The prior distribution over the
signal is chosen to be exponential, $p(s_{ij} \mid \lambda) =
\exp\{-\frac{s_{ij}}{\lambda}\} / \lambda$,
assuming we know only the average value of the signal
intensity $\lambda$.
The prior probability for the two hypotheses is chosen uninformative to
be $p(B_{ij}) = p(\overline{B}_{ij}) = \beta = 0.5$.

Because we do not know if a certain pixel contains purely background or
additional signal, the likelihood for the mixture model is
%
\begin{equation}
p(D \mid b,\lambda) = \prod_{ij} [\beta \cdot p(d_{ij} \mid
B_{ij},b_{ij}) + (1-\beta) \cdot p(d_{ij} \mid
\overline{B}_{ij},b_{ij},\lambda)]
\end{equation} 
%
where %$
\begin{displaymath}
p(d_{ij} \mid \overline{B}_{ij},b_{ij},\lambda) =
\frac{e^{\frac{b_{ij}}{\lambda}}}{\lambda
(1+\frac{1}{\lambda})^{d_{ij}+1}} \cdot
\frac{\Gamma[(d_{ij}+1),b_{ij}(1+\frac{1}{\lambda})]}{\Gamma(d_{ij}+1)} %$
\end{displaymath}
is the Poisson distribution marginalized over $s_{ij}$. 
The prior probability density function for $b_{ij}$ is chosen flat for positive values and 0 elsewhere.
The posterior distribution is according to Bayes theorem proportional to
the product of the mixture likelihood and the prior.
Its maximum with respect to $b$ gives an estimate
of the background map which includes the observatory's exposure map. 

The probability of having source contribution in pixel cells or
domains is
\begin{equation}
p(\overline{B}_{ij} \mid d_{ij},b_{ij},\lambda) = \frac{
p(\overline{B}_{ij}) \cdot p(d_{ij} \mid
\overline{B}_{ij},b_{ij},\lambda)}{p(\overline{B}_{ij}) \cdot p(d_{ij}
\mid \overline{B}_{ij},b_{ij},\lambda) + p({B}_{ij}) \cdot p(d_{ij} \mid
{B}_{ij},b_{ij})}.
\end{equation}
%
Details of mixture modelling in the framework of BPT can be found in 
von der Linden et al.\ (1999) and Fischer et al.\ (2000).
In order to include the width of the instrumental PSF, a minimum of 3 by 3 pixels
domain has to be considered.
In addition, we allow for larger correlation lengths
to enhance weak and extended source detection.

\section{Results}

%
\begin{figure}[h]
\epsscale{.60}
\plottwo{O3-3_1.eps}{O3-3_2.eps}  %{O3-3_2_color.eps}
\caption{On the left, ROSAT field rs930625 at $\alpha=17^{h} 49^{\prime}
5.47''$ $\delta=+61^{\circ}52^{m}30.0^{s}$. On the right, source
probability evaluated accounting for the width of the ROSAT PSPC PSF.} 
\label{O3-3:rosatfield}
\end{figure}
%
Fig.~\ref{O3-3:rosatfield} left shows the ROSAT PSPC field RS930625
in the broad energy band ($E=0.1-2.4$~keV)
located in the ecliptic polar region. 
The observatory's exposure time varies between 1693.1-13475.5 sec. 
The average photon counts per pixel is 2.9.
The right figure shows the probability for having source contribution in
3 by 3 pixel domains.
%
\begin{figure}[h]
\plotfiddle{O3-3_3.eps}{2.7in}{-90.}{50.}{50.}{-200}{250} 
\caption{Background map extracted from rs930625 with a Thin-Plate spline
combined with the observatory's exposure map.}
\label{O3-3:tps_rep}
\end{figure}
%
The corresponding background map is shown in Fig.~\ref{O3-3:tps_rep}.
The background intensity varies between $9.37 \pm 0.02$ and $1.153 \pm
0.006$ expected counts which shows the prominent variation due to the
heterogeneous satellite exposure time.

\section{Conclusions and Further Prospects}

BPT allows to estimate background maps and to detect sources in a single
step providing consistent uncertainties of background and sources.
The source probability is evaluated for single pixels as well as for
pixel domains to enhance source detection for weak and extended
sources.
The detection sensitivity is enhanced compared to SASS results because
the full field of view is exploited for background estimation.
An extensive comparison with SASS results is beyond the scope of the present
paper and will be addressed in a forthcoming paper.


%-----------------------------------------------------------------------
%			      References
%-----------------------------------------------------------------------
% List your references below within the reference environment
% (i.e. between the \begin{references} and \end{references} tags).
% Each new reference should begin with a \reference command which sets
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% articles.  Note that many journal names are available as macros; see
% the User Guide listing "macro-ized" journals.   
%
% EXAMPLE:  \reference Hagiwara, K., \& Zeppenfeld, D.\  1986, 
%                Nucl.Phys., 274, 1
%           \reference H\'enon, M.\  1961, Ann.d'Ap., 24, 369
%           \reference King, I.\ R.\  1966, \aj, 71, 276
%           \reference King, I.\ R.\  1975, in Dynamics of Stellar 
%                Systems, ed.\ A.\ Hayli (Dordrecht: Reidel), 99
%           \reference Tody, D.\  1998, \adassvii, 146
%           \reference Zacharias, N.\ \& Zacharias, M.\ 2003,
%                \adassxii, \paperref{P7.6}
% 
% Note the following tricks used in the example above:
%
%   o  \& is used to format an ampersand symbol (&).
%   o  \'e puts an accent agu over the letter e.  See the User Guide
%      and the sample files for details on formatting special
%      characters.  
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%      as an end of a sentence.
%   o  \aj is a macro that expands to "Astron. J."  See the User Guide
%      for a full list of journal macros
%   o  \adassvii is a macro that expands to the full title, editor,
%      and publishing information for the ADASS VII conference
%      proceedings.  Such macros are defined for ADASS conferences I
%      through XI.
%   o  When referencing a paper in the current volume, use the
%      \adassxii and \paperref macros.  The argument to \paperref is
%      the paper ID code for the paper you are referencing.  See the 
%      note in the "Paper ID Code" section above for details on how to 
%      determine the paper ID code for the paper you reference.  
%
\begin{references}
\reference Boese, F.\ G.\ \& Doebereiner, S.\ 2001, \aap, 370, 649
\reference Boese, F.\ G.\ 2000,  A\&AS, 141, 507
\reference Fischer, R.\ \& Dose, V.\ 2002, AIPC, 617, 143
\reference Fischer, R.\, Hanson, K.\ M.\, Dose, V.\ \& von der Linden, W.\ 2000, PRE, 61.1152 
\reference Voges, W.\ et al.\ 2000, IAUC, 7432
\reference Voges, W.\ et al.\ 1999, \aap, 349, 389
\reference von der Linden, W.\, Dose, V.\, Padayachee, J.\ \& Prozesky, V.\ 1999, PRE, 59.6527
\end{references}

% Do not place any material after the references section

\end{document}  % Leave intact







