<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
<!--Converted with LaTeX2HTML 98.1p1 release (March 2nd, 1998)
originally by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds
* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
* with significant contributions from:
Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
<HTML>
<HEAD>
<TITLE>Image sampling</TITLE>
<META NAME="description" CONTENT="Image sampling">
<META NAME="keywords" CONTENT="vol2">
<META NAME="resource-type" CONTENT="document">
<META NAME="distribution" CONTENT="global">
<META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1">
<LINK REL="STYLESHEET" HREF="vol2.css">
<LINK REL="next" HREF="node13.html">
<LINK REL="previous" HREF="node11.html">
<LINK REL="up" HREF="node11.html">
<LINK REL="next" HREF="node13.html">
</HEAD>
<BODY >
<!--Navigation Panel-->
<A NAME="tex2html1607"
HREF="node13.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
SRC="icons.gif/next_motif.gif"></A>
<A NAME="tex2html1604"
HREF="node11.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
SRC="icons.gif/up_motif.gif"></A>
<A NAME="tex2html1598"
HREF="node11.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
SRC="icons.gif/previous_motif.gif"></A>
<A NAME="tex2html1606"
HREF="node1.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
SRC="icons.gif/contents_motif.gif"></A>
<BR>
<B> Next:</B> <A NAME="tex2html1608"
HREF="node13.html">Noise distributions</A>
<B> Up:</B> <A NAME="tex2html1605"
HREF="node11.html">Basic Concepts</A>
<B> Previous:</B> <A NAME="tex2html1599"
HREF="node11.html">Basic Concepts</A>
<BR>
<BR>
<!--End of Navigation Panel-->
<H2><A NAME="SECTION00511000000000000000">
Image sampling</A>
</H2>
The acquisition of a data frame involves a spatial sampling and digitalization
of the continuous image formed in the focus plane of a telescope.
The image may be recorded analog (e.g. on photographic plates) for later
measurements or acquired directly when digital detectors such as diode
arrays and CCD's are used.
The individual pixel values are obtained by convolving the continuous
image <I>I</I>(<I>x</I>,<I>y</I>) with the pixel response function <I>R</I>(<I>x</I>,<I>y</I>).
With a sampling step of <IMG
WIDTH="36" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
SRC="img15.gif"
ALT="$\Delta x$">
and <IMG
WIDTH="36" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img16.gif"
ALT="$\Delta y$">
the digital frame
is given by
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{equation}
F_{i,j} = \int I(x,y) R(x-i\Delta x, y-j\Delta y) \; dx dy
+ N_{i,j}
\end{equation} -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:image"> </A><IMG
WIDTH="460" HEIGHT="54"
SRC="img17.gif"
ALT="\begin{displaymath}F_{i,j} = \int I(x,y) R(x-i\Delta x, y-j\Delta y) \; dx dy
+ N_{i,j}
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(2.1)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
where <I>N</I> is the acquisition noise.
This convolution is done analog in most detectors except for imaging
photon counting systems where it partly is performed digitally.
The sampling step and response function are determined normally by the
physical properties of the detector and the acquisition setup.
The variation of the response function may be very sharp as for most
semi-conductor detectors or more smooth as in image dissector tubes.
If the original image <I>I</I> is band width limited (i.e. only contains
features with spatial frequencies less than a cutoff value <IMG
WIDTH="28" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img18.gif"
ALT="$\omega_c$">)
all information is retained in the digitized frame when the sampling
frequency
<!-- MATH: $\omega_s = 2\pi/\Delta x$ -->
<IMG
WIDTH="123" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
SRC="img19.gif"
ALT="$\omega_s = 2\pi/\Delta x$">
satisfies the Nyquist criterion:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{equation}
\omega_s = 2 \; \omega_c.
\end{equation} -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:nyquist"> </A><IMG
WIDTH="95" HEIGHT="38"
SRC="img20.gif"
ALT="\begin{displaymath}\omega_s = 2 \; \omega_c.
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(2.2)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
In Equation <A HREF="node12.html#eq:nyquist">2.2</A> it is assumed that <I>R</I> is a Dirac delta
function.
This means that only features which are larger than <IMG
WIDTH="47" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
SRC="img21.gif"
ALT="$2\Delta x$">
can
be resolved.
A frame is oversampled when
<!-- MATH: $\omega_s > 2\omega_c$ -->
<IMG
WIDTH="90" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img22.gif"
ALT="$\omega_s > 2\omega_c$">
while for
smaller sample rates it is undersampled.
<P>
In astronomy the band width of an image is determined by the point spread
function (PSF) and has often no sharp cutoff frequency.
Many modern detector systems are designed to have a sampling step only
a few times smaller than the typical full width half maximum (FWHM) of
seeing disk or PSF.
Therefore they will not fully satisfy Equation <A HREF="node12.html#eq:nyquist">2.2</A> and
tend to be undersampled especially in good seeing conditions.
<P>
A typical assumption in image processing algorithms is that the pixel
response function <I>R</I> can be approximated by a Dirac delta function.
This is reasonable when the image intensity does not vary significantly
over <I>R</I> as for well oversampled frames where the effective size of <I>R</I>is roughly equal to the sample step.
If it is not the case, the effects on the algorithm used should be checked.
Interpolation of values between existing pixels is often necessary e.g.
for rebinning.
Depending on the shape of <I>R</I> and band width of the image different schemes
may be chosen to give the best reproduction of the original intensity
distribution.
In many cases low order polynomial functions are used (e.g. zero or first
order) while sinc, spline or gaussian weighted interpolation may be more
appropriate for some applications.
<P>
<HR>
<!--Navigation Panel-->
<A NAME="tex2html1607"
HREF="node13.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
SRC="icons.gif/next_motif.gif"></A>
<A NAME="tex2html1604"
HREF="node11.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
SRC="icons.gif/up_motif.gif"></A>
<A NAME="tex2html1598"
HREF="node11.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
SRC="icons.gif/previous_motif.gif"></A>
<A NAME="tex2html1606"
HREF="node1.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
SRC="icons.gif/contents_motif.gif"></A>
<BR>
<B> Next:</B> <A NAME="tex2html1608"
HREF="node13.html">Noise distributions</A>
<B> Up:</B> <A NAME="tex2html1605"
HREF="node11.html">Basic Concepts</A>
<B> Previous:</B> <A NAME="tex2html1599"
HREF="node11.html">Basic Concepts</A>
<!--End of Navigation Panel-->
<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
</ADDRESS>
</BODY>
</HTML>
