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<H1><A NAME="SECTION001120000000000000000"> </A>
<A NAME="order-definition"> </A>
<BR>
Order Definition
</H1>
The dispersion relation is defined by the following
equations:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{equation}
\begin{array}{rcl}
y &=& f_1(x,m) \\
\lambda &=& f_2(x,m)
\end{array}
\end{equation} -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="echelle-1"> </A><IMG
WIDTH="145" HEIGHT="50"
SRC="img275.gif"
ALT="\begin{displaymath}\begin{array}{rcl}
y &=& f_1(x,m) \\
\lambda &=& f_2(x,m)
\end{array}\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(7.1)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
The first of the equations <A HREF="node150.html#echelle-1">7.1</A> defines the position of the
spectral orders, <I>m</I>, in the raw image, while the second equation
gives, for each order, the dispersion relation in one dimension. The
mapping between the spaces
<!-- MATH: $(\lambda,m)$ -->
<IMG
WIDTH="18" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
SRC="img276.gif"
ALT="$(\lambda,m)$">
and (<I>x</I>,<I>y</I>) is separated into
two different equations; the first one will be discussed in this
Section, while the description of the second equation will be
postponed to Section <A HREF="node158.html#wavelength-calibration">7.6</A>.
<P>
The function <I>f</I><SUB>1</SUB> is approximated by a polynomial of the form
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{equation}
y = f_1(x,m)\approx\sum_{j=0}^J\sum_{i=0}^I a_{ij} x^i m^j
\end{equation} -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="echelle-2"> </A><IMG
WIDTH="283" HEIGHT="64"
SRC="img277.gif"
ALT="\begin{displaymath}y = f_1(x,m)\approx\sum_{j=0}^J\sum_{i=0}^I a_{ij} x^i m^j
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(7.2)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
where the coefficients <I>a</I><SUB><I>ij</I></SUB> are computed using least squares
techniques on a grid
<!-- MATH: $(x_k, y_k)$ -->
(<I>x</I><SUB><I>k</I></SUB>, <I>y</I><SUB><I>k</I></SUB>), i.e. sample number and line number
of points located within the spectral orders of the image. These
points in the grid are found automatically by an order-following
algorithm, normally using the <TT>FLAT</TT> or <TT>STD</TT> image.
<P>
<UL>
<LI>A first guess of the position
of the orders is found on a trace perpendicular to the dispersion
direction done in the middle of the flat field image, in this way we define
the set of points
<!-- MATH: $(x_0,y_{0m})$ -->
(<I>x</I><SUB>0</SUB>,<I>y</I><SUB>0<I>m</I></SUB>), <I>m</I> being the relative order number.
<LI>For each order, the order-following algorithm finds the series of
points located on the order at
<!-- MATH: $x_n=x_0+n\times\Delta x$ -->
<IMG
WIDTH="132" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img278.gif"
ALT="$x_n=x_0+n\times\Delta x$">
for points on the
right half of the order, and at
<!-- MATH: $x_{-n}= x_0 - n\times\Delta x$ -->
<IMG
WIDTH="146" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
SRC="img279.gif"
ALT="$x_{-n}= x_0 - n\times\Delta x$">
for points
on the left half of the order,
<!-- MATH: $n=1,2,\ldots$ -->
<IMG
WIDTH="68" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img280.gif"
ALT="$n=1,2,\ldots$">
integer and <IMG
WIDTH="60" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
SRC="img281.gif"
ALT="$\Delta x$">
is
the step of the grid.
</UL>
<P>
This set of points forms the basic grid with the geometric positions of the
orders. Typical values of the standard deviation of the residuals of this
approximation are about 0.3 to 0.1 pixel.
<P>
It is worth mentioning here that the order<A NAME="7164"> </A> following algorithm finds
the center of the orders by taking the middle point with respect to the edges
of the orders. The edges of the orders are detected automatically by
thresholding the order profiles, perpendicular to the dispersion direction;
the level of the threshold is a function of the signal in the order.
The command <TT>DEFINE/ECHELLE</TT> performs the automatic order detection.
<P>
An alternative method is available, based on the Hough transform to perform
the order detection and involving a tracing algorithm able to estimate an
optimal threshold for each order independently.
The order definition is performed as
follows:
<UL>
<LI>A preprocessing of the frame is performed, including a median filtering
(radx,y=2,1) to remove hot pixels and bad rows from the image. Then the
background value is measured in the central area of the image
and subtracted. This preprocessing assumes that the
defaults are small enough to be corrected by a simple median filtering
and that the interorder background is basically constant all over the image.
If the above conditions are not respected, the frame must be processed by
the user. The echelle command <TT>BACKGROUND/SMOOTH</TT> enables
performance of a background correction at this early stage of the
calibration.
<LI>A first guess<A NAME="7168"> </A> of the position and the slope of the orders is found
by processing the Hough transform of a subset of columns of the input
image. The order detection by Hough transform is described in
(Ballester, 1994).
<LI>For each order, an initial threshold is estimated by measuring the
pixel values in the middle of the order. The order following algorithm
finds the series of points located on the order at regular steps on
the grid, as describd above. The threshold is optimised in order to
follow the order on the longest possible distance. If the trace of the
order is lost, the algorithm extrapolates linearly the positions and
attempts to skip the gap.
<LI>For each position, the center of the order is
defined as the first moment of the pixel values above the threshold:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{displaymath}
y_{center} = \frac{\sum_{y=y_{min}}^{y_{max}} y*(I_y - threshold)}
{\sum_{y=y_{min}}^{y_{max}} y}
\end{displaymath} -->
<IMG
WIDTH="363" HEIGHT="57"
SRC="img283.gif"
ALT="\begin{displaymath}y_{center} = \frac{\sum_{y=y_{min}}^{y_{max}} y*(I_y - threshold)}
{\sum_{y=y_{min}}^{y_{max}} y} \end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
This algorithm is implemented in the command <TT>DEFINE/HOUGH</TT>. The
algorithm can run in a fully automatic mode (no parameters
are required apart from the name of the input frame).
It is also possible to set the following parameters
to enforce a given solution:
<UL>
<LI>numbers of orders to be detected.
<LI>half-width of orders
<LI>threshold<A NAME="7176"> </A>
</UL></UL>A practical decription of the way to use this algorithm and to optimise the
parameters is described in the Appendix <A HREF="node455.html#app:echelle">D</A>
<P>
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<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
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