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\section{Deblending}
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\section{Deblending}
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\label{chap:deblending}
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\label{chap:deblending}
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Each time an object extraction is completed, the connected set of
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Each time an object extraction is completed, the connected set of
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pixels passes through a \hide{sort of} filter that tries to split it into
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pixels passes through a \hide{sort of} filter that tries to split it into
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eventual overlapping components. This case appears more frequently
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eventual overlapping components. This case appears more frequently
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when the field is crowded or when the detection threshold is set very
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when the field is crowded or when the detection \index{threshold} threshold is set very
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low. The deblending method adopted in {\sc SExtractor}, is based on
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low. The \index{deblending} deblending method adopted in {\sc SExtractor}, is based on
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{\em multi-thresholding}, and works on any kind of object; but it is
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{\em \index{multi-thresholding} multi-thresholding}, and works on any kind of object; but it is
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unable to deblend components that are so close that no saddle is
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unable to deblend components that are so close that no saddle is
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present in their profile. However, as no assumption has to be made on
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present in their profile. However, as no assumption has to be made on
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the shape of the objects, it is perfectly suited for galaxies as well
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the shape of the objects, it is perfectly suited for galaxies as well
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as for high galactic latitude stellar fields.
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as for high galactic latitude stellar fields.
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Typical problematic cases for deblending include patchy, extended {\bf
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Typical problematic cases for \index{deblending} deblending include patchy, extended {\bf
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Sc} galaxies (which must be considered as single entities), and
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Sc} galaxies (which must be considered as single entities), and
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close or interacting pairs of optically faint galaxies (which should
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close or interacting pairs of optically faint galaxies (which should
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be considered as separate objects). Basically, the multi-thresholding
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be considered as separate objects). Basically, the \index{multi-thresholding} multi-thresholding
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algorithm employs a multiple isophotal analysis technique similar to
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algorithm employs a multiple isophotal analysis technique similar to
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those in use at the APM \gam{Reference?} and the COSMOS machines
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those in use at the \index{APM} APM \gam{Reference?} and the \index{COSMOS} COSMOS machines
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\cite{beard:al:1991}; in a first pass, each extracted set of connected
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\cite{beard:al:1991}; in a first pass, each extracted set of connected
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pixels is re-thresholded at $N$ levels linearly or exponentially
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pixels is re-thresholded at $N$ levels linearly or exponentially
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spaced between its primary extraction threshold and its peak value.
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spaced between its primary extraction \index{threshold} threshold and its peak value.
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This gives us a 2-dimensional ``model'' of the light
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This gives us a 2-dimensional ``model'' of the light
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distribution within the object(s), which is stored in the form of a
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distribution within the object(s), which is stored in the form of a
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tree structure (fig. \ref{figsegmentmeth}). Then the algorithm goes
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tree structure (fig. \ref{figsegmentmeth}). Then the algorithm goes
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downwards, from the tips of branches to the trunk, and decides at each
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downwards, from the tips of branches to the trunk, and decides at each
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junction whether it shall extract two (or more) objects or continue
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junction whether it shall extract two (or more) objects or continue
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its way down. To meet the conditions described earlier, the following
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its way down. To meet the conditions described earlier, the following
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simple decision criteria are adopted: at any junction threshold
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simple decision criteria are adopted: at any junction \index{threshold} threshold
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$t_{i}$, any branch will be considered as a separate component if
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$t_{i}$, any branch will be considered as a separate component if
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\begin{enumerate}
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\begin{enumerate}
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\item[(1)] the integrated pixel intensity (above $t_{i}$) of the
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\item[(1)] the integrated pixel intensity (above $t_{i}$) of the
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branch is greater than a certain fraction $\delta_{c}$ of the total
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branch is greater than a certain fraction $\delta_{c}$ of the total
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intensity of the composite object;
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intensity of the composite object;
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\item[(2)] condition (1) is verified for at least one more branch at the same level $i$.
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\item[(2)] condition (1) is verified for at least one more branch at the same level $i$.
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\end{enumerate}
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\end{enumerate}
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Note that ideally, condition (1) is both flux- and scale-invariant.
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Note that ideally, condition (1) is both flux- and scale-invariant.
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However for faint, poorly resolved objects, the efficiency of the
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However for faint, poorly resolved objects, the efficiency of the
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deblending is limited mostly by seeing and sampling. From the analysis
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\index{deblending} deblending is limited mostly by seeing and sampling. From the analysis
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of both small and extended galaxy images, a compromise value for the
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of both small and extended galaxy \index{image} images, a compromise value for the
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contrast parameter $\delta_{c}$ $\sim$ 0.005 proved to be optimum.
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contrast parameter $\delta_{c}$ $\sim$ 0.005 proved to be optimum.
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This should normally separate objects with a difference in
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This should normally separate objects with a difference in
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magnitude greater than $\approx 6$.
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magnitude greater than $\approx 6$.
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%---------------------------------- Fig. segmentmeth --------------------------------
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%---------------------------------- Fig. segmentmeth --------------------------------
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\begin{figure}[htbp]
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\begin{figure}[htbp]
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\centerline{\includegraphics[width=10cm]{ps/segment.ps}}
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\centerline{\includegraphics[width=10cm]{ps/segment.ps}}
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\caption{ A schematic diagram of the method used to deblend a
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\caption{ A schematic diagram of the method used to deblend a
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composite object. The area profile of the object (\emph{smooth curve}) can be
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composite object. The \index{area} area profile of the object (\emph{smooth curve}) can be
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described in a tree-structured way (\emph{thick lines}). The decision to
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described in a tree-structured way (\emph{thick lines}). The decision to
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regard or not a branch as a distinct object is determined according to
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regard or not a branch as a distinct object is determined according to
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its relative integrated intensity (\emph{tinted area}). In that case above,
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its relative integrated intensity (\emph{tinted \index{area} area}). In that case above,
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the original object shall split into two components A and B. Remaining
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the original object shall split into two components A and B. Remaining
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pixels are assigned to their most credible ``progenitors'' afterwards.
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pixels are assigned to their most credible ``progenitors'' afterwards.
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}
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}
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\label{figsegmentmeth}
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\label{figsegmentmeth}
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\end{figure}
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\end{figure}
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The outlying pixels with flux lower than the separation thresholds
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The outlying pixels with flux lower than the separation \index{threshold} thresholds
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have to be reallocated to the proper components of the merger. To do
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have to be reallocated to the proper components of the merger. To do
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so, we have opted for a {\em statistical} approach: at each faint
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so, we have opted for a {\em statistical} approach: at each faint
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pixel, we compute the contribution expected from each
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pixel, we compute the contribution expected from each
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sub-object, using a bivariate Gaussian fit to its profile, and then derive
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sub-object, using a bivariate Gaussian fit to its profile, and then derive
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the probability for that pixel to belong to the sub-object. For
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the probability for that pixel to belong to the sub-object. For
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instance, a faint pixel lying halfway between two close bright stars
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instance, a faint pixel lying halfway between two close bright \index{stars} stars
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having the same magnitude will be appended to one of these with equal
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having the same magnitude will be appended to one of these with equal
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probabilities. One important advantage of this technique is that the
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probabilities. One important advantage of this technique is that the
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morphology of any object is completely defined simply through its list
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morphology of any object is completely defined simply through its list
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of pixels.
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of pixels.
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To test the effects of deblending on photometry and astrometry
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To test the effects of \index{deblending} deblending on photometry and astrometry
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measurements, we made several simulations of photographic images of
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measurements, we made several simulations of photographic \index{image} images of
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double stars with different separations and magnitudes under typical
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double \index{stars} stars with different separations and magnitudes under typical
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observational conditions (fig. \ref{figsegmentsim}). It is obvious
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observational conditions (fig. \ref{figsegmentsim}). It is obvious
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that multiple isophotal techniques fail when there is no saddle point
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that multiple isophotal techniques fail when there is no saddle point
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present in profiles (i.e. for distance between stars $< 2 \sigma $ in
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present in profiles (i.e. for distance between \index{stars} stars $< 2 \sigma $ in
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the case of Gaussian images). We measured a magnitude error $ \leq
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the case of Gaussian \index{image} images). We measured a \index{magnitude error} magnitude error $ \leq
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0.2$ mag and a shift of the centroid ($\leq 0.4$ pixels) for the
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0.2$ mag and a shift of the \index{centroid} centroid ($\leq 0.4$ pixels) for the
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fainter star in the very worst cases, but no other systematic effects
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fainter star in the very worst cases, but no other systematic effects
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were noticeable.
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were noticeable.
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%---------------------------------- Fig. segmentsim ---------------------------------
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%---------------------------------- Fig. segmentsim ---------------------------------
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\begin{figure}[htbp]
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\begin{figure}[htbp]
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\centerline{\includegraphics[width=10cm]{ps/sepsim_pos.ps}}
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\centerline{\includegraphics[width=10cm]{ps/sepsim_pos.ps}}
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\centerline{\includegraphics[width=10cm]{ps/sepsim_mag.ps}}
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\centerline{\includegraphics[width=10cm]{ps/sepsim_mag.ps}}
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\caption{
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\caption{
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Centroid and corrected isophotal magnitude errors for a
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Centroid and corrected isophotal \index{magnitude error} \index{magnitude errors} magnitude errors for a
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simulated $19^{th}$ magnitude star blended with a $11, 15, 19$ and
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simulated $19^{th}$ magnitude star blended with a $11, 15, 19$ and
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$21^{th}$ mag. companion as a function of distance (expressed in
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$21^{th}$ mag. companion as a function of distance (expressed in
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pixels). Lines stop at the left when the objects are too close to be
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pixels). Lines stop at the left when the objects are too close to be
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deblended. The \emph{dashed vertical line} is the theoretical limit for
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deblended. The \emph{dashed vertical line} is the theoretical limit for
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unsaturated stars with equal magnitudes. In the centroid plot, the
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unsaturated \index{stars} stars with equal magnitudes. In the \index{centroid} centroid plot, the
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\emph{arrow} indicates the direction of the neighbour. The simulation assumes
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\emph{arrow} indicates the direction of the \index{neighbour} neighbour. The simulation assumes
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a 1 hour exposure with the CERGA telescope on a IIIaJ plate and Moffat
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a 1 hour exposure with the CERGA telescope on a IIIaJ plate and Moffat
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profiles with a seeing FWHM of 3 pixels ($2''$). }
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profiles with a seeing \index{FWHM} FWHM of 3 pixels ($2''$). }
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\label{figsegmentsim}
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\label{figsegmentsim}
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\end{figure}
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\end{figure}
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The user can control the multi-thresholding operation through 3
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The user can control the \index{multi-thresholding} multi-thresholding operation through 3
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parameters. The first one is the number of deblending thresholds ({\tt
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parameters. The first one is the number of \index{deblending} deblending \index{threshold} thresholds ({\tt
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DEBLEND\_NTHRESH}). A good value is 32. Higher values are generally
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DEBLEND\_NTHRESH}). A good value is 32. Higher values are generally
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useless, except perhaps for images having an unusually high dynamic
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useless, except perhaps for \index{image} images having an unusually high dynamic
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range. In case of memory problems, decreasing the number of thresholds
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range. In case of \index{memory} memory problems, decreasing the number of \index{threshold} thresholds
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to say, 8 or even less may be a solution. But then, of course, a
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to say, 8 or even less may be a solution. But then, of course, a
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degradation of the deblending performances may occur. The second
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degradation of the \index{deblending} deblending performances may occur. The second
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parameter is the contrast parameter ({\tt DEBLEND\_MINCONT}). As
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parameter is the contrast parameter ({\tt DEBLEND\_MINCONT}). As
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described above, values from 0.001 to 0.01 give the best results. Putting
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described above, values from 0.001 to 0.01 give the best results. Putting
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{\tt DEBLEND\_MINCONT} to 0 means that even the faintest local peaks
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{\tt DEBLEND\_MINCONT} to 0 \index{mean} means that even the faintest local peaks
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in the profile will be considered as separate objects. Putting it to 1
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in the profile will be considered as separate objects. Putting it to 1
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means that no deblending will be authorized. The last parameter
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\index{mean} means that no \index{deblending} deblending will be authorized. The last parameter
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concerns the kind of scale used for the thresholds. If the image comes
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concerns the kind of scale used for the \index{threshold} thresholds. If the \index{image} image comes
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from photographic material, then a linear scale has to be used ({\tt
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from photographic material, then a linear scale has to be used ({\tt
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DETECTION\_TYPE PHOTO}). Otherwise, for an image obtained with a
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DETECTION\_TYPE PHOTO}). Otherwise, for an \index{image} image obtained with a
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linear device like a CCD, an exponential scale is more appropriate
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linear device like a \index{CCD} CCD, an exponential scale is more appropriate
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({\tt DETECTION\_TYPE CCD}).
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({\tt DETECTION\_TYPE \index{CCD} CCD}).
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