\secA{What \duchamp is doing}
\label{sec-flow}

The execution flow of \duchamp is detailed here, indicating the main
algorithmic steps that are used. The program is written in C/C++ and
makes use of the \textsc{cfitsio}, \textsc{wcslib} and \textsc{pgplot}
libraries.

\secB{Image input}
\label{sec-input}

The cube is read in using basic \textsc{cfitsio} commands, and stored
as an array in a special C++ class. This class keeps track of the list
of detected objects, as well as any reconstructed arrays that are made
(see \S\ref{sec-recon}). The World Coordinate System
(WCS)\footnote{This is the information necessary for translating the
pixel locations to quantities such as position on the sky, frequency,
velocity, and so on.} information for the cube is also obtained from
the FITS header by \textsc{wcslib} functions \citep{greisen02,
calabretta02}, and this information, in the form of a \texttt{wcsprm}
structure, is also stored in the same class.

A sub-section of a cube can be requested by defining the subsection
with the \texttt{subsection} parameter and setting
\texttt{flagSubsection=true} -- this can be a good idea if the cube
has very noisy edges, which may produce many spurious detections.

There are two ways of specifying the \texttt{subsection} string. The
first is the generalised form
\texttt{[x1:x2:dx,y1:y2:dy,z1:z2:dz,...]}, as used by the
\textsc{cfitsio} library. This has one set of colon-separated numbers
for each axis in the FITS file. In this manner, the x-coordinates run
from \texttt{x1} to \texttt{x2} (inclusive), with steps of
\texttt{dx}. The step value can be omitted, so a subsection of the
form \texttt{[2:50,2:50,10:1000]} is still valid. In fact, \duchamp
does not make use of any step value present in the subsection string,
and any that are present are removed before the file is opened.

If the entire range of a coordinate is required, one can replace the
range with a single asterisk, \eg \texttt{[2:50,2:50,*]}. Thus, the
subsection string \texttt{[*,*,*]} is simply the entire cube. A
complete description of this section syntax can be found at the
\textsc{fitsio} web site%
\footnote{%
\href%
{http://heasarc.gsfc.nasa.gov/docs/software/fitsio/c/c\_user/node91.html}%
{http://heasarc.gsfc.nasa.gov/docs/software/fitsio/c/c\_user/node91.html}}.


Making full use of the subsection requires knowledge of the size of
each of the dimensions. If one wants to, for instance, trim a certain
number of pixels off the edges of the cube, without examining the cube
to obtain the actual size, one can use the second form of the
subsection string. This just gives a number for each axis, \eg
\texttt{[5,5,5]} (which would trim 5 pixels from the start \emph{and}
end of each axis).

All types of subsections can be combined \eg \texttt{[5,2:98,*]}.


%A sub-section of an image can be requested via the \texttt{subsection}
%parameter -- this can be a good idea if the cube has very noisy edges,
%which may produce many spurious detections. The generalised form of
%the subsection that is used by \textsc{cfitsio} is
%\texttt{[x1:x2:dx,y1:y2:dy,z1:z2:dz,...]}, such that the x-coordinates run
%from \texttt{x1} to \texttt{x2} (inclusive), with steps of
%\texttt{dx}. The step value can be omitted (so a subsection of the
%form \texttt{[2:50,2:50,10:1000]} is still valid). \duchamp does not
%make use of any step value present in the subsection string, and any
%that are present are removed before the file is opened.
%
%If one wants the full range of a coordinate then replace the range
%with an asterisk, \eg \texttt{[2:50,2:50,*]}. If one wants to use a
%subsection, one must set \texttt{flagSubsection = 1}. A complete
%description of the section syntax can be found at the \textsc{fitsio}
%web site%
%\footnote{%
%\href%
%{http://heasarc.gsfc.nasa.gov/docs/software/fitsio/c/c\_user/node91.html}%
%{http://heasarc.gsfc.nasa.gov/docs/software/fitsio/c/c\_user/node91.html}}.
%

\secB{Image modification}
\label{sec-modify}

Several modifications to the cube can be made that improve the
execution and efficiency of \duchamp (their use is optional, governed
by the relevant flags in the parameter file).

\secC{BLANK pixel removal}

If the imaged area of a cube is non-rectangular (see the example in
Fig.~\ref{fig-moment}, a cube from the HIPASS survey), BLANK pixels are
used to pad it out to a rectangular shape. The value of these pixels
is given by the FITS header keywords BLANK, BSCALE and BZERO. While
these pixels make the image a nice shape, they will unnecessarily
interfere with the processing (as well as taking up needless
memory). The first step, then, is to trim them from the edge. This is
done when the parameter \texttt{flagBlankPix=true}. If the above
keywords are not present, the user can specify the BLANK value by the
parameter \texttt{blankPixValue}.

Removing BLANK pixels is particularly important for the reconstruction
step, as lots of BLANK pixels on the edges will smooth out features in
the wavelet calculation stage. The trimming will also reduce the size
of the cube's array, speeding up the execution. The amount of trimming
is recorded, and these pixels are added back in once the
source-detection is completed (so that quoted pixel positions are
applicable to the original cube).

Rows and columns are trimmed one at a time until the first non-BLANK
pixel is reached, so that the image remains rectangular. In practice,
this means that there will be some BLANK pixels left in the trimmed
image (if the non-BLANK region is non-rectangular). However, these are
ignored in all further calculations done on the cube.

\secC{Baseline removal}

Second, the user may request the removal of baselines from the
spectra, via the parameter \texttt{flagBaseline}. This may be
necessary if there is a strong baseline ripple present, which can
result in spurious detections at the high points of the ripple. The
baseline is calculated from a wavelet reconstruction procedure (see
\S\ref{sec-recon}) that keeps only the two largest scales. This is
done separately for each spatial pixel (\ie for each spectrum in the
cube), and the baselines are stored and added back in before any
output is done. In this way the quoted fluxes and displayed spectra
are as one would see from the input cube itself -- even though the
detection (and reconstruction if applicable) is done on the
baseline-removed cube.

The presence of very strong signals (for instance, masers at several
hundred Jy) could affect the determination of the baseline, and would
lead to a large dip centred on the signal in the baseline-subtracted
spectrum. To prevent this, the signal is trimmed prior to the
reconstruction process at some standard threshold (at $8\sigma$ above
the mean). The baseline determined should thus be representative of
the true, signal-free baseline. Note that this trimming is only a
temporary measure which does not affect the source-detection.

\secC{Ignoring bright Milky Way emission}

Finally, a single set of contiguous channels can be ignored -- these
may exhibit very strong emission, such as that from the Milky Way as
seen in extragalactic \hi cubes (hence the references to ``Milky
Way'' in relation to this task -- apologies to Galactic
astronomers!). Such dominant channels will produce many detections
that are unnecessary, uninteresting (if one is interested in
extragalactic \hi) and large (in size and hence in memory usage), and
so will slow the program down and detract from the interesting
detections. 

The use of this feature is controlled by the \texttt{flagMW}
parameter, and the exact channels concerned are able to be set by the
user (using \texttt{maxMW} and \texttt{minMW} -- these give an
inclusive range of channels). When employed, these channels are
ignored for the searching, and the scaling of the spectral output (see
Fig.~\ref{fig-spect}) will not take them into account. They will be
present in the reconstructed array, however, and so will be included
in the saved FITS file (see \S\ref{sec-reconIO}). When the final
spectra are plotted, the range of channels covered by these parameters
is indicated by a green hashed box.

\secB{Image reconstruction}
\label{sec-recon}

The user can direct \duchamp to reconstruct the data cube using the
\atrous wavelet procedure. A good description of the procedure can be
found in \citet{starck02:book}. The reconstruction is an effective way
of removing a lot of the noise in the image, allowing one to search
reliably to fainter levels, and reducing the number of spurious
detections. This is an optional step, but one that greatly enhances
the source-detection process, with the payoff that it can be
relatively time- and memory-intensive.

\secC{Algorithm}

The steps in the \atrous reconstruction are as follows:
\begin{enumerate}
\item The reconstructed array is set to 0 everywhere.
\item The input array is discretely convolved with a given filter
  function. This is determined from the parameter file via the
  \texttt{filterCode} parameter -- see Appendix~\ref{app-param} for
  details on the filters available.
\item The wavelet coefficients are calculated by taking the difference
  between the convolved array and the input array.
\item If the wavelet coefficients at a given point are above the
  requested threshold (given by \texttt{snrRecon} as the number of
  $\sigma$ above the mean and adjusted to the current scale -- see
  Appendix~\ref{app-scaling}), add these to the reconstructed array.
\item The separation of the filter coefficients is doubled. (Note that
  this step provides the name of the procedure\footnote{\atrous means
  ``with holes'' in French.}, as gaps or holes are created in the
  filter coverage.)
\item The procedure is repeated from step 2, using the convolved array
  as the input array.
\item Continue until the required maximum number of scales is reached.
\item Add the final smoothed (\ie convolved) array to the
  reconstructed array. This provides the ``DC offset'', as each of the
  wavelet coefficient arrays will have zero mean.
\end{enumerate}

The reconstruction has at least two iterations. The first iteration
makes a first pass at the wavelet reconstruction (the process outlined
in the 8 stages above), but the residual array will likely have some
structure still in it, so the wavelet filtering is done on the
residual, and any significant wavelet terms are added to the final
reconstruction. This step is repeated until the change in the measured
standard deviation of the background (see note below on the evaluation
of this quantity) is less than some fiducial amount.

It is important to note that the \atrous decomposition is an example
of a ``redundant'' transformation. If no thresholding is performed,
the sum of all the wavelet coefficient arrays and the final smoothed
array is identical to the input array. The thresholding thus removes
only the unwanted structure in the array.

Note that any BLANK pixels that are still in the cube will not be
altered by the reconstruction -- they will be left as BLANK so that
the shape of the valid part of the cube is preserved.

\secC{Note on Statistics}

The correct calculation of the reconstructed array needs good
estimators of the underlying mean and standard deviation of the
background noise distribution. These statistics are estimated using
robust methods, to avoid corruption by strong outlying points. The
mean of the distribution is actually estimated by the median, while
the median absolute deviation from the median (MADFM) is calculated
and corrected assuming Gaussianity to estimate the underlying standard
deviation $\sigma$. The Gaussianity (or Normality) assumption is
critical, as the MADFM does not give the same value as the usual rms
or standard deviation value -- for a Normal distribution
$N(\mu,\sigma)$ we find MADFM$=0.6744888\sigma$. Since this ratio is
corrected for, the user need only think in the usual multiples of
$\sigma$ when setting \texttt{snrRecon}. See Appendix~\ref{app-madfm}
for a derivation of this value.

When thresholding the different wavelet scales, the value of $\sigma$
as measured from the wavelet array needs to be scaled to account for
the increased amount of correlation between neighbouring pixels (due
to the convolution). See Appendix~\ref{app-scaling} for details on
this scaling.

\secC{User control of reconstruction parameters}

The most important parameter for the user to select in relation to the
reconstruction is the threshold for each wavelet array. This is set
using the \texttt{snrRecon} parameter, and is given as a multiple of
the rms (estimated by the MADFM) above the mean (which for the wavelet
arrays should be approximately zero). There are several other
parameters that can be altered as well that affect the outcome of the
reconstruction.

By default, the cube is reconstructed in three dimensions, using a
3-dimensional filter and 3-dimensional convolution. This can be
altered, however, using the parameter \texttt{reconDim}. If set to 1,
this means the cube is reconstructed by considering each spectrum
separately, whereas \texttt{reconDim=2} will mean the cube is
reconstructed by doing each channel map separately. The merits of
these choices are discussed in \S\ref{sec-notes}, but it should be
noted that a 2-dimensional reconstruction can be susceptible to edge
effects if the spatial shape of the pixel array is not rectangular.

The user can also select the minimum scale to be used in the
reconstruction. The first scale exhibits the highest frequency
variations, and so ignoring this one can sometimes be beneficial in
removing excess noise. The default is to use all scales
(\texttt{minscale = 1}).

Finally, the filter that is used for the convolution can be selected
by using \texttt{filterCode} and the relevant code number -- the
choices are listed in Appendix~\ref{app-param}. A larger filter will
give a better reconstruction, but take longer and use more memory when
executing. When multi-dimensional reconstruction is selected, this
filter is used to construct a 2- or 3-dimensional equivalent.

\secB{Input/Output of reconstructed arrays}
\label{sec-reconIO}

The reconstruction stage can be relatively time-consuming,
particularly for large cubes and reconstructions in 3-D. To get around
this, \duchamp provides a shortcut to allow users to perform multiple
searches (\eg with different thresholds) on the same reconstruction
without calculating the reconstruction each time.

The first step is to choose to save the reconstructed array as a FITS
file by setting \texttt{flagOutputRecon = true}. The file will be
saved in the same directory as the input image, so the user needs to
have write permissions for that directory.

The filename will be derived from the input filename, with extra
information detailing the reconstruction that has been done. For
example, suppose \texttt{image.fits} has been reconstructed using a
3-dimensional reconstruction with filter \#2, thresholded at $4\sigma$
using all scales. The output filename will then be
\texttt{image.RECON-3-2-4-1.fits} (\ie it uses the four parameters
relevant for the \atrous reconstruction as listed in
Appendix~\ref{app-param}). The new FITS file will also have these
parameters as header keywords. If a subsection of the input image has
been used (see \S\ref{sec-input}), the format of the output filename
will be \texttt{image.sub.RECON-3-2-4-1.fits}, and the subsection that
has been used is also stored in the FITS header.

Likewise, the residual image, defined as the difference between the
input and reconstructed arrays, can also be saved in the same manner
by setting \texttt{flagOutputResid = true}. Its filename will be the
same as above, with \texttt{RESID} replacing \texttt{RECON}.

If a reconstructed image has been saved, it can be read in and used
instead of redoing the reconstruction. To do so, the user should set
the parameter \texttt{flagReconExists = true}. The user can indicate
the name of the reconstructed FITS file using the \texttt{reconFile}
parameter, or, if this is not specified, \duchamp searches for the
file with the name as defined above. If the file is not found, the
reconstruction is performed as normal. Note that to do this, the user
needs to set
\texttt{flagAtrous = true} (obviously, if this is \texttt{false}, the
reconstruction is not needed).

\secB{Smoothing the cube}
\label{sec-smoothing}

An alternative to doing the wavelet reconstruction is to Hanning
smooth the cube. This technique can be useful in reducing the noise
level slightly (at the cost of making neighbouring pixels correlated
and blurring any signal present), and is particularly well suited to
the case where a particular signal width is believed to be present in
the data. It is also substantially faster than the wavelet
reconstruction.

The cube is smoothed only in the spectral domain. That is, each
spectrum is independently smoothed, and then put back together to form
the smoothed cube. This is then treated in the same way as the
reconstructed cube, and is used for the searching algorithm (see
below). Note that in the case of both the reconstruction and the
smoothing options being requested, the reconstruction will take
precedence and the smoothing will \emph{not} be done.

There is only one parameter necessary to define the degree of
smoothing -- the Hanning width $a$ (given by the user parameter
\texttt{hanningWidth}). The coefficients $c(x)$ of the Hanning filter
are defined by
\[
c(x) = 
  \begin{cases}
   \frac{1}{2}\left(1+\cos(\frac{\pi x}{a})\right) &|x| \leq (a+1)/2\\
   0                                               &|x| > (a+1)/2.
  \end{cases}
\]
where $a,x \in \mathbb{Z}$. Note that the width specified must be an
odd integer (if the parameter provided is even, it is incremented by
one).

The user is able to save the smoothed array in exactly the same manner
as for the reconstructed array -- set \texttt{flagOutputSmooth =
true}, and then the smoothed array will be saved in
\texttt{image.SMOOTH-a.fits}, where a is replaced by the Hanning width
used. Similarly, a saved file can be read in by setting
\texttt{flagSmoothExists = true} and either specifying a file to be
read with the \texttt{smoothFile} parameter or relying on \duchamp to
find the file with the name as given above. 

\secB{Searching the image}
\label{sec-detection}

The basic idea behind detection is to locate sets of contiguous voxels
that lie above some threshold. \duchamp now calculates one threshold
for the entire cube (previous versions calculated thresholds for each
spectrum and image). This enables calculation of signal-to-noise
ratios for each source (see Section~\ref{sec-output} for details). The
user can manually specify a value (using the parameter
\texttt{threshold}) for the threshold, which will override the
calculated value. Note that this only applies for the first of the two
cases discussed below -- the FDR case ignores any manually-set
threshold value.

The image is searched for detections in two ways: spectrally (\ie a
1-dimensional search in the spectrum in each spatial pixel), and
spatially (a 2-dimensional search in the spatial image in each
channel). In both cases, the algorithm finds connected pixels that are
above the user-specified threshold. In the case of the spatial image
search, the algorithm of \citet{lutz80} is used to raster-scan through
the image and connect groups of pixels on neighbouring rows.

Note that this algorithm cannot be applied directly to a 3-dimensional
case, as it requires that objects are completely nested in a row: that
is, if you are scanning along a row, and one object finishes and
another starts, you know that you will not get back to the first one
(if at all) until the second is completely finished for that
row. Three-dimensional data does not have this property, which is why
we break up the searching into 1- and 2-dimensional cases.

Detections must have a minimum number of pixels to be counted. This
number is given by the input parameters \texttt{minPix} (for
2-dimensional searches) and \texttt{minChannels} (for 1-dimensional
searches).

Finally, the search only looks for positive features. If one is
interested instead in negative features (such as absorption lines),
set the parameter \texttt{flagNegative = true}. This will invert the
cube (\ie multiply all pixels by $-1$) prior to the search, and then
re-invert the cube (and the fluxes of any detections) after searching
is complete. All outputs are done in the same manner as normal, so
that fluxes of detections will be negative.

\secC{Calculating statistics}

A crucial part of the detection process is estimating the statistics
that define the detection threshold. To determine a threshold, we need
to estimate from the data two parameters: the middle of the pixel
distribution, and its spread. For both cases, we again use robust
methods, using the median and MADFM. If the cube has been
reconstructed or smoothed, the residuals (defined in the sense of
original $-$ reconstruction) are used to estimate the noise parameters
of the cube. Otherwise they are estimated directly from the cube
itself. In both cases, robust estimators are used.

The parameters that are estimated should be representative of the
noise in the cube. For the case of small objects embedded in many
noise pixels (\eg the case of \hi surveys), using the full cube will
provide good estimators. It is possible, however, to use only a
subsection of the cube by setting the parameter \texttt{flagStatSec =
true} and providing the desired subsection to the \texttt{StatSec}
parameter. This subsection works in exactly the same way as the pixel
subsection discussed in \S\ref{sec-input}.

\secC{Determining the threshold}

Once the statistics have been calculated, the threshold is determined
in one of two ways. The first way is a simple sigma-clipping, where a
threshold is set at a fixed number $n$ of standard deviations above
the mean, and pixels above this threshold are flagged as detected. The
value of $n$ is set with the parameter \texttt{snrCut}. As before, the
value of the standard deviation is estimated by the MADFM, and
corrected by the ratio derived in Appendix~\ref{app-madfm}.

The second method uses the False Discovery Rate (FDR) technique
\citep{miller01,hopkins02}, whose basis we briefly detail here. The
false discovery rate (given by the number of false detections divided
by the total number of detections) is fixed at a certain value
$\alpha$ (\eg $\alpha=0.05$ implies 5\% of detections are false
positives). In practice, an $\alpha$ value is chosen, and the ensemble
average FDR (\ie $\langle FDR \rangle$) when the method is used will
be less than $\alpha$.  One calculates $p$ -- the probability,
assuming the null hypothesis is true, of obtaining a test statistic as
extreme as the pixel value (the observed test statistic) -- for each
pixel, and sorts them in increasing order. One then calculates $d$
where
\[
d = \max_j \left\{ j : P_j < \frac{j\alpha}{c_N N} \right\},
\]
and then rejects all hypotheses whose $p$-values are less than or
equal to $P_d$. (So a $P_i<P_d$ will be rejected even if $P_i \geq
j\alpha/c_N N$.) Note that ``reject hypothesis'' here means ``accept
the pixel as an object pixel'' (\ie we are rejecting the null
hypothesis that the pixel belongs to the background).

The $c_N$ values here are normalisation constants that depend on the
correlated nature of the pixel values. If all the pixels are
uncorrelated, then $c_N=1$. If $N$ pixels are correlated, then their
tests will be dependent on each other, and so $c_N = \sum_{i=1}^N
i^{-1}$. \citet{hopkins02} consider real radio data, where the pixels
are correlated over the beam. In this case the sum is made over the
$N$ pixels that make up the beam. The value of $N$ is calculated from
the FITS header (if the correct keywords -- BMAJ, BMIN -- are not
present, the size of the beam is taken from the parameter
\texttt{beamSize}). 

The theory behind the FDR method implies a direct connection between
the choice of $\alpha$ and the fraction of detections that will be
false positives. However, due to the merging process, this direct
connection is lost when looking at the final number of detections --
see discussion in \S\ref{sec-notes}. The effect is that the number of
false detections will be less than indicated by the $\alpha$ value
used.


\secB{Merging detected objects}
\label{sec-merger}

The searching step produces a list of detected objects that will have
many repeated detections of a given object -- for instance, spectral
detections in adjacent pixels of the same object and/or spatial
detections in neighbouring channels. These are then combined in an
algorithm that matches all objects judged to be ``close'', according
to one of two criteria.

One criterion is to define two thresholds -- one spatial and one in
velocity -- and say that two objects should be merged if there is at
least one pair of pixels that lie within these threshold distances of
each other. These thresholds are specified by the parameters
\texttt{threshSpatial} and \texttt{threshVelocity} (in units of pixels
and channels respectively).

Alternatively, the spatial requirement can be changed to say that
there must be a pair of pixels that are \emph{adjacent} -- a stricter,
but perhaps more realistic requirement, particularly when the spatial
pixels have a large angular size (as is the case for 
\hi surveys). This 
method can be selected by setting the parameter
\texttt{flagAdjacent} to 1 (\ie \texttt{true}) in the parameter
file. The velocity thresholding is done in the same way as the first
option.

Once the detections have been merged, they may be ``grown''. This is a
process of increasing the size of the detection by adding adjacent
pixels that are above some secondary threshold. This threshold is
lower than the one used for the initial detection, but above the noise
level, so that faint pixels are only detected when they are close to a
bright pixel. The value of this threshold is a possible input
parameter (\texttt{growthCut}), with a default value of
$1.5\sigma$. The use of the growth algorithm is controlled by the
\texttt{flagGrowth} parameter -- the default value of which is
\texttt{false}. If the detections are grown, they are sent through the
merging algorithm a second time, to pick up any detections that now
overlap or have grown over each other.

Finally, to be accepted, the detections must span \emph{both} a
minimum number of channels (to remove any spurious single-channel
spikes that may be present), and a minimum number of spatial
pixels. These numbers, as for the original detection step, are set
with the \texttt{minChannels} and \texttt{minPix} parameters. The
channel requirement means there must be at least one set of
\texttt{minChannels} consecutive channels in the source for it to be
accepted.