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<H2><A NAME="SECTION00512000000000000000">
Noise distributions</A>
</H2>
Besides gross errors which are discussed in Section <A HREF="node16.html#artifacts">2.2.1</A>
the two main sources of noise in a frame come from the detector system
<I>N</I> and from photon shot-noise of the image intensity <I>I</I>
(see Equation <A HREF="node12.html#eq:image">2.1</A>).
It is assumed that the digitalization is done with sufficiently high
resolution to resolve the noise.
If not, the quantization of output values gives raise to additional noise
and errors.
<P>
A large number of independent noise sources from different electronics
components normally contributes to the system noise of a detector.
Using the central limit theorem, the total noise can be approximated by
a Gaussian or normal distribution which has the frequency function :
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{equation}
P_G(x;\mu,\sigma) = \frac{1}{\sigma\sqrt{2\pi}}
\exp \left( -\frac{1}{2} \left[ \frac{x-\mu}{\sigma} \right]^2
\right)
\end{equation} -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:gaus-dist"> </A><IMG
WIDTH="383" HEIGHT="69"
SRC="img23.gif"
ALT="\begin{displaymath}P_G(x;\mu,\sigma) = \frac{1}{\sigma\sqrt{2\pi}}
\exp \left( -\frac{1}{2} \left[ \frac{x-\mu}{\sigma} \right]^2
\right)
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(2.3)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
where <IMG
WIDTH="19" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img24.gif"
ALT="$\mu$">
and <IMG
WIDTH="18" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
SRC="img25.gif"
ALT="$\sigma$">
are mean and standard deviation, respectively.
The photon noise of a source is Poisson distributed with the probability
density <I>P</I><SUB><I>P</I></SUB> for a given number of photons <I>n</I> :
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{equation}
P_P(n;\mu) = \frac{\mu^n}{n!} e^{-\mu}
\end{equation} -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:pois-dist"> </A><IMG
WIDTH="171" HEIGHT="54"
SRC="img26.gif"
ALT="\begin{displaymath}P_P(n;\mu) = \frac{\mu^n}{n!} e^{-\mu}
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(2.4)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
where <IMG
WIDTH="18" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img27.gif"
ALT="$\mu$">
is the mean intensity of the source.
It can be approximated with a Gaussian distribution when <IMG
WIDTH="19" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img28.gif"
ALT="$\mu$">
becomes
large.
For photon counting devices the number of events is normally so small
that Equation <A HREF="node13.html#eq:pois-dist">2.4</A> must be used while Gaussian approximation
often can be used for integrating systems (e.g. CCD's).
<P>
In the statistical analysis of the probability distribution of data
several estimators based on moments are used.
The <I>r</I><SUP><I>th</I></SUP> moment <I>m</I><SUB><I>r</I></SUB> about the mean <IMG
WIDTH="17" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
SRC="img29.gif"
ALT="$\bar{x}$">
and its dimensional
form <IMG
WIDTH="29" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img30.gif"
ALT="$\alpha_r$">
are defined as
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{equation}
m_r = \frac{1}{N} \sum_{i=1}^N (x_i - \bar{x}) \;\;, \;\;
\alpha_r = \frac{m_r}{\sqrt{m_2^r}}.
\end{equation} -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:moment"> </A><IMG
WIDTH="333" HEIGHT="74"
SRC="img31.gif"
ALT="\begin{displaymath}m_r = \frac{1}{N} \sum_{i=1}^N (x_i - \bar{x}) \;\;, \;\;
\alpha_r = \frac{m_r}{\sqrt{m_2^r}}.
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(2.5)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
The second moment is the variance while first is always zero.
The general shape of a distribution is characterized by the <I>skewness</I>
which denotes its asymmetry (i.e. its third moment <IMG
WIDTH="30" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img32.gif"
ALT="$\alpha_3$">)
and
the <I>kurtosis</I> showing how peaked it is
(i.e. its fourth moment <IMG
WIDTH="29" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img33.gif"
ALT="$\alpha_4$">).
For a normal distribution, these moments are
<!-- MATH: $\alpha_3 = 0$ -->
<IMG
WIDTH="70" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img34.gif"
ALT="$\alpha_3 = 0$">
and
<!-- MATH: $\alpha_4=3$ -->
<IMG
WIDTH="69" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img35.gif"
ALT="$\alpha_4=3$">while for a Poisson distribution are
<!-- MATH: $\alpha_3 = 1/\sqrt{\mu}$ -->
<IMG
WIDTH="112" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
SRC="img36.gif"
ALT="$\alpha_3 = 1/\sqrt{\mu}$">
and
<!-- MATH: $\alpha_4=3+1/\mu$ -->
<IMG
WIDTH="130" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
SRC="img37.gif"
ALT="$\alpha_4=3+1/\mu$">.
Besides these moments other estimators are used to describe a distribution
e.g. <I>median</I> and <I>mode</I>.
The <I>median</I> of a distribution is defined as the value which has
equally many values above and below it while a <I>mode</I> is the local
maximum of the probability density function.
<P>
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<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
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