1 | \secA{What \duchamp is doing} |
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2 | \label{sec-flow} |
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3 | |
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4 | Each of the steps that \duchamp goes through in the course of its |
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5 | execution are discussed here in more detail. This should provide |
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6 | enough background information to fully understand what \duchamp is |
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7 | doing and what all the output information is. For those interested in |
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8 | the programming side of things, \duchamp is written in C/C++ and makes |
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9 | use of the \textsc{cfitsio}, \textsc{wcslib} and \textsc{pgplot} |
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10 | libraries. |
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11 | |
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12 | \secB{Image input} |
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13 | \label{sec-input} |
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14 | |
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15 | The cube is read in using basic \textsc{cfitsio} commands, and stored |
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16 | as an array in a special C++ class. This class keeps track of the list |
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17 | of detected objects, as well as any reconstructed arrays that are made |
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18 | (see \S\ref{sec-recon}). The World Coordinate System |
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19 | (WCS)\footnote{This is the information necessary for translating the |
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20 | pixel locations to quantities such as position on the sky, frequency, |
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21 | velocity, and so on.} information for the cube is also obtained from |
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22 | the FITS header by \textsc{wcslib} functions \citep{greisen02, |
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23 | calabretta02}, and this information, in the form of a \texttt{wcsprm} |
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24 | structure, is also stored in the same class. |
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25 | |
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26 | A sub-section of a cube can be requested by defining the subsection |
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27 | with the \texttt{subsection} parameter and setting |
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28 | \texttt{flagSubsection=true} -- this can be a good idea if the cube |
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29 | has very noisy edges, which may produce many spurious detections. |
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30 | |
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31 | There are two ways of specifying the \texttt{subsection} string. The |
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32 | first is the generalised form |
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33 | \texttt{[x1:x2:dx,y1:y2:dy,z1:z2:dz,...]}, as used by the |
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34 | \textsc{cfitsio} library. This has one set of colon-separated numbers |
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35 | for each axis in the FITS file. In this manner, the x-coordinates run |
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36 | from \texttt{x1} to \texttt{x2} (inclusive), with steps of |
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37 | \texttt{dx}. The step value can be omitted, so a subsection of the |
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38 | form \texttt{[2:50,2:50,10:1000]} is still valid. In fact, \duchamp |
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39 | does not make use of any step value present in the subsection string, |
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40 | and any that are present are removed before the file is opened. |
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41 | |
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42 | If the entire range of a coordinate is required, one can replace the |
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43 | range with a single asterisk, \eg \texttt{[2:50,2:50,*]}. Thus, the |
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44 | subsection string \texttt{[*,*,*]} is simply the entire cube. A |
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45 | complete description of this section syntax can be found at the |
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46 | \textsc{fitsio} web site% |
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47 | \footnote{% |
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48 | \href% |
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49 | {http://heasarc.gsfc.nasa.gov/docs/software/fitsio/c/c\_user/node91.html}% |
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50 | {http://heasarc.gsfc.nasa.gov/docs/software/fitsio/c/c\_user/node91.html}}. |
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51 | |
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52 | |
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53 | Making full use of the subsection requires knowledge of the size of |
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54 | each of the dimensions. If one wants to, for instance, trim a certain |
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55 | number of pixels off the edges of the cube, without examining the cube |
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56 | to obtain the actual size, one can use the second form of the |
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57 | subsection string. This just gives a number for each axis, \eg |
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58 | \texttt{[5,5,5]} (which would trim 5 pixels from the start \emph{and} |
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59 | end of each axis). |
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60 | |
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61 | All types of subsections can be combined \eg \texttt{[5,2:98,*]}. |
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62 | |
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63 | |
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64 | %A sub-section of an image can be requested via the \texttt{subsection} |
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65 | %parameter -- this can be a good idea if the cube has very noisy edges, |
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66 | %which may produce many spurious detections. The generalised form of |
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67 | %the subsection that is used by \textsc{cfitsio} is |
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68 | %\texttt{[x1:x2:dx,y1:y2:dy,z1:z2:dz,...]}, such that the x-coordinates run |
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69 | %from \texttt{x1} to \texttt{x2} (inclusive), with steps of |
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70 | %\texttt{dx}. The step value can be omitted (so a subsection of the |
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71 | %form \texttt{[2:50,2:50,10:1000]} is still valid). \duchamp does not |
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72 | %make use of any step value present in the subsection string, and any |
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73 | %that are present are removed before the file is opened. |
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74 | % |
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75 | %If one wants the full range of a coordinate then replace the range |
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76 | %with an asterisk, \eg \texttt{[2:50,2:50,*]}. If one wants to use a |
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77 | %subsection, one must set \texttt{flagSubsection = 1}. A complete |
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78 | %description of the section syntax can be found at the \textsc{fitsio} |
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79 | %web site% |
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80 | %\footnote{% |
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81 | %\href% |
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82 | %{http://heasarc.gsfc.nasa.gov/docs/software/fitsio/c/c\_user/node91.html}% |
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83 | %{http://heasarc.gsfc.nasa.gov/docs/software/fitsio/c/c\_user/node91.html}}. |
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84 | % |
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85 | |
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86 | \secB{Image modification} |
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87 | \label{sec-modify} |
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88 | |
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89 | Several modifications to the cube can be made that improve the |
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90 | execution and efficiency of \duchamp (their use is optional, governed |
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91 | by the relevant flags in the parameter file). |
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92 | |
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93 | \secC{BLANK pixel removal} |
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94 | |
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95 | If the imaged area of a cube is non-rectangular (see the example in |
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96 | Fig.~\ref{fig-moment}, a cube from the HIPASS survey), BLANK pixels are |
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97 | used to pad it out to a rectangular shape. The value of these pixels |
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98 | is given by the FITS header keywords BLANK, BSCALE and BZERO. While |
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99 | these pixels make the image a nice shape, they will unnecessarily |
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100 | interfere with the processing (as well as taking up needless |
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101 | memory). The first step, then, is to trim them from the edge. This is |
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102 | done when the parameter \texttt{flagBlankPix=true}. If the above |
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103 | keywords are not present, the user can specify the BLANK value by the |
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104 | parameter \texttt{blankPixValue}. |
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105 | |
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106 | Removing BLANK pixels is particularly important for the reconstruction |
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107 | step, as lots of BLANK pixels on the edges will smooth out features in |
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108 | the wavelet calculation stage. The trimming will also reduce the size |
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109 | of the cube's array, speeding up the execution. The amount of trimming |
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110 | is recorded, and these pixels are added back in once the |
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111 | source-detection is completed (so that quoted pixel positions are |
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112 | applicable to the original cube). |
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113 | |
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114 | Rows and columns are trimmed one at a time until the first non-BLANK |
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115 | pixel is reached, so that the image remains rectangular. In practice, |
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116 | this means that there will be some BLANK pixels left in the trimmed |
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117 | image (if the non-BLANK region is non-rectangular). However, these are |
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118 | ignored in all further calculations done on the cube. |
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119 | |
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120 | \secC{Baseline removal} |
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121 | |
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122 | Second, the user may request the removal of baselines from the |
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123 | spectra, via the parameter \texttt{flagBaseline}. This may be |
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124 | necessary if there is a strong baseline ripple present, which can |
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125 | result in spurious detections at the high points of the ripple. The |
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126 | baseline is calculated from a wavelet reconstruction procedure (see |
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127 | \S\ref{sec-recon}) that keeps only the two largest scales. This is |
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128 | done separately for each spatial pixel (\ie for each spectrum in the |
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129 | cube), and the baselines are stored and added back in before any |
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130 | output is done. In this way the quoted fluxes and displayed spectra |
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131 | are as one would see from the input cube itself -- even though the |
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132 | detection (and reconstruction if applicable) is done on the |
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133 | baseline-removed cube. |
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134 | |
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135 | The presence of very strong signals (for instance, masers at several |
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136 | hundred Jy) could affect the determination of the baseline, and would |
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137 | lead to a large dip centred on the signal in the baseline-subtracted |
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138 | spectrum. To prevent this, the signal is trimmed prior to the |
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139 | reconstruction process at some standard threshold (at $8\sigma$ above |
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140 | the mean). The baseline determined should thus be representative of |
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141 | the true, signal-free baseline. Note that this trimming is only a |
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142 | temporary measure which does not affect the source-detection. |
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143 | |
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144 | \secC{Ignoring bright Milky Way emission} |
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145 | |
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146 | Finally, a single set of contiguous channels can be ignored -- these |
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147 | may exhibit very strong emission, such as that from the Milky Way as |
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148 | seen in extragalactic \hi cubes (hence the references to ``Milky |
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149 | Way'' in relation to this task -- apologies to Galactic |
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150 | astronomers!). Such dominant channels will produce many detections |
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151 | that are unnecessary, uninteresting (if one is interested in |
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152 | extragalactic \hi) and large (in size and hence in memory usage), and |
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153 | so will slow the program down and detract from the interesting |
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154 | detections. |
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155 | |
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156 | The use of this feature is controlled by the \texttt{flagMW} |
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157 | parameter, and the exact channels concerned are able to be set by the |
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158 | user (using \texttt{maxMW} and \texttt{minMW} -- these give an |
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159 | inclusive range of channels). When employed, these channels are |
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160 | ignored for the searching, and the scaling of the spectral output (see |
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161 | Fig.~\ref{fig-spect}) will not take them into account. They will be |
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162 | present in the reconstructed array, however, and so will be included |
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163 | in the saved FITS file (see \S\ref{sec-reconIO}). When the final |
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164 | spectra are plotted, the range of channels covered by these parameters |
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165 | is indicated by a green hashed box. |
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166 | |
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167 | \secB{Image reconstruction} |
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168 | \label{sec-recon} |
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169 | |
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170 | The user can direct \duchamp to reconstruct the data cube using the |
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171 | \atrous wavelet procedure. A good description of the procedure can be |
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172 | found in \citet{starck02:book}. The reconstruction is an effective way |
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173 | of removing a lot of the noise in the image, allowing one to search |
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174 | reliably to fainter levels, and reducing the number of spurious |
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175 | detections. This is an optional step, but one that greatly enhances |
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176 | the source-detection process, with the payoff that it can be |
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177 | relatively time- and memory-intensive. |
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178 | |
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179 | \secC{Algorithm} |
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180 | |
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181 | The steps in the \atrous reconstruction are as follows: |
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182 | \begin{enumerate} |
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183 | \item The reconstructed array is set to 0 everywhere. |
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184 | \item The input array is discretely convolved with a given filter |
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185 | function. This is determined from the parameter file via the |
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186 | \texttt{filterCode} parameter -- see Appendix~\ref{app-param} for |
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187 | details on the filters available. |
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188 | \item The wavelet coefficients are calculated by taking the difference |
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189 | between the convolved array and the input array. |
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190 | \item If the wavelet coefficients at a given point are above the |
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191 | requested threshold (given by \texttt{snrRecon} as the number of |
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192 | $\sigma$ above the mean and adjusted to the current scale -- see |
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193 | Appendix~\ref{app-scaling}), add these to the reconstructed array. |
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194 | \item The separation of the filter coefficients is doubled. (Note that |
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195 | this step provides the name of the procedure\footnote{\atrous means |
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196 | ``with holes'' in French.}, as gaps or holes are created in the |
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197 | filter coverage.) |
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198 | \item The procedure is repeated from step 2, using the convolved array |
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199 | as the input array. |
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200 | \item Continue until the required maximum number of scales is reached. |
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201 | \item Add the final smoothed (\ie convolved) array to the |
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202 | reconstructed array. This provides the ``DC offset'', as each of the |
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203 | wavelet coefficient arrays will have zero mean. |
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204 | \end{enumerate} |
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205 | |
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206 | The reconstruction has at least two iterations. The first iteration |
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207 | makes a first pass at the wavelet reconstruction (the process outlined |
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208 | in the 8 stages above), but the residual array will likely have some |
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209 | structure still in it, so the wavelet filtering is done on the |
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210 | residual, and any significant wavelet terms are added to the final |
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211 | reconstruction. This step is repeated until the change in the measured |
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212 | standard deviation of the background (see note below on the evaluation |
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213 | of this quantity) is less than some fiducial amount. |
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214 | |
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215 | It is important to note that the \atrous decomposition is an example |
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216 | of a ``redundant'' transformation. If no thresholding is performed, |
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217 | the sum of all the wavelet coefficient arrays and the final smoothed |
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218 | array is identical to the input array. The thresholding thus removes |
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219 | only the unwanted structure in the array. |
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220 | |
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221 | Note that any BLANK pixels that are still in the cube will not be |
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222 | altered by the reconstruction -- they will be left as BLANK so that |
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223 | the shape of the valid part of the cube is preserved. |
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224 | |
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225 | \secC{Note on Statistics} |
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226 | |
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227 | The correct calculation of the reconstructed array needs good |
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228 | estimators of the underlying mean and standard deviation of the |
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229 | background noise distribution. These statistics are estimated using |
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230 | robust methods, to avoid corruption by strong outlying points. The |
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231 | mean of the distribution is actually estimated by the median, while |
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232 | the median absolute deviation from the median (MADFM) is calculated |
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233 | and corrected assuming Gaussianity to estimate the underlying standard |
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234 | deviation $\sigma$. The Gaussianity (or Normality) assumption is |
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235 | critical, as the MADFM does not give the same value as the usual rms |
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236 | or standard deviation value -- for a Normal distribution |
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237 | $N(\mu,\sigma)$ we find MADFM$=0.6744888\sigma$, but this will change |
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238 | for different distributions. Since this ratio is corrected for, the |
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239 | user need only think in the usual multiples of the rms when setting |
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240 | \texttt{snrRecon}. See Appendix~\ref{app-madfm} for a derivation of |
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241 | this value. |
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242 | |
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243 | When thresholding the different wavelet scales, the value of the rms |
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244 | as measured from the wavelet array needs to be scaled to account for |
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245 | the increased amount of correlation between neighbouring pixels (due |
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246 | to the convolution). See Appendix~\ref{app-scaling} for details on |
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247 | this scaling. |
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248 | |
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249 | \secC{User control of reconstruction parameters} |
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250 | |
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251 | The most important parameter for the user to select in relation to the |
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252 | reconstruction is the threshold for each wavelet array. This is set |
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253 | using the \texttt{snrRecon} parameter, and is given as a multiple of |
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254 | the rms (estimated by the MADFM) above the mean (which for the wavelet |
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255 | arrays should be approximately zero). There are several other |
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256 | parameters that can be altered as well that affect the outcome of the |
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257 | reconstruction. |
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258 | |
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259 | By default, the cube is reconstructed in three dimensions, using a |
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260 | 3-dimensional filter and 3-dimensional convolution. This can be |
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261 | altered, however, using the parameter \texttt{reconDim}. If set to 1, |
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262 | this means the cube is reconstructed by considering each spectrum |
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263 | separately, whereas \texttt{reconDim=2} will mean the cube is |
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264 | reconstructed by doing each channel map separately. The merits of |
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265 | these choices are discussed in \S\ref{sec-notes}, but it should be |
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266 | noted that a 2-dimensional reconstruction can be susceptible to edge |
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267 | effects if the spatial shape of the pixel array is not rectangular. |
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268 | |
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269 | The user can also select the minimum scale to be used in the |
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270 | reconstruction. The first scale exhibits the highest frequency |
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271 | variations, and so ignoring this one can sometimes be beneficial in |
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272 | removing excess noise. The default is to use all scales |
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273 | (\texttt{minscale = 1}). |
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274 | |
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275 | Finally, the filter that is used for the convolution can be selected |
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276 | by using \texttt{filterCode} and the relevant code number -- the |
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277 | choices are listed in Appendix~\ref{app-param}. A larger filter will |
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278 | give a better reconstruction, but take longer and use more memory when |
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279 | executing. When multi-dimensional reconstruction is selected, this |
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280 | filter is used to construct a 2- or 3-dimensional equivalent. |
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281 | |
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282 | \secB{Smoothing the cube} |
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283 | \label{sec-smoothing} |
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284 | |
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285 | An alternative to doing the wavelet reconstruction is to smooth the |
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286 | cube. This technique can be useful in reducing the noise level |
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287 | slightly (at the cost of making neighbouring pixels correlated and |
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288 | blurring any signal present), and is particularly well suited to the |
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289 | case where a particular signal size (\ie a certain channel width or |
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290 | spatial size) is believed to be present in the data. |
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291 | |
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292 | There are two alternative methods that can be used: spectral |
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293 | smoothing, using the Hanning filter; or spatial smoothing, using a 2D |
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294 | Gaussian kernel. These alternatives are outlined below. To utilise the |
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295 | smoothing option, set the parameter \texttt{flagSmooth=true} and set |
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296 | \texttt{smoothType} to either \texttt{spectral} or \texttt{spatial}. |
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297 | |
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298 | \secC{Spectral smoothing} |
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299 | |
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300 | When \texttt{smoothType=spectral} is selected, the cube is smoothed |
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301 | only in the spectral domain. Each spectrum is independently smoothed |
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302 | by a Hanning filter, and then put back together to form the smoothed |
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303 | cube, which is then used by the searching algorithm (see below). Note |
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304 | that in the case of both the reconstruction and the smoothing options |
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305 | being requested, the reconstruction will take precedence and the |
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306 | smoothing will \emph{not} be done. |
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307 | |
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308 | There is only one parameter necessary to define the degree of |
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309 | smoothing -- the Hanning width $a$ (given by the user parameter |
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310 | \texttt{hanningWidth}). The coefficients $c(x)$ of the Hanning filter |
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311 | are defined by |
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312 | \[ |
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313 | c(x) = |
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314 | \begin{cases} |
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315 | \frac{1}{2}\left(1+\cos(\frac{\pi x}{a})\right) &|x| \leq (a+1)/2\\ |
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316 | 0 &|x| > (a+1)/2. |
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317 | \end{cases} |
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318 | ,\ a,x \in \mathbb{Z} |
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319 | \] |
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320 | Note that the width specified must be an |
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321 | odd integer (if the parameter provided is even, it is incremented by |
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322 | one). |
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323 | |
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324 | \secC{Spatial smoothing} |
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325 | |
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326 | When \texttt{smoothType=spatial} is selected, the cube is smoothed |
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327 | only in the spatial domain. Each channel map is independently smoothed |
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328 | by a two-dimensional Gaussian kernel, and then put back together to |
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329 | form the smoothed cube, and used in the searching algorithm (see |
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330 | below). Again, reconstruction is always done by preference if both |
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331 | techniques are requested. |
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332 | |
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333 | The two-dimensional Gaussian has three parameters to define it, |
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334 | governed by the elliptical cross-sectional shape of the gaussian |
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335 | function: the FWHM (full-width at half-maximum) of the major and minor |
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336 | axes, and the position angle of the major axis. These are given by the |
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337 | user parameters \texttt{kernMaj, kernMin \& kernPA}. If we define |
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338 | these parameters as $a,b,\theta$ respectively, we can define the |
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339 | kernel by the function |
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340 | \[ |
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341 | k(x,y) = \exp\left[-0.5 \left(\frac{X^2}{\sigma_X^2} + |
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342 | \frac{Y^2}{\sigma_Y^2} \right) \right] |
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343 | \] |
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344 | where $(x,y)$ are the offsets from the central pixel of the gaussian |
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345 | function, and |
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346 | \begin{align*} |
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347 | X& = x\sin\theta - y\cos\theta& |
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348 | Y&= x\cos\theta + y\sin\theta\\ |
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349 | \sigma_X^2& = \frac{(a/2)^2}{2\ln2}& |
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350 | \sigma_Y^2& = \frac{(b/2)^2}{2\ln2}\\ |
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351 | \end{align*} |
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352 | |
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353 | |
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354 | \secB{Input/Output of reconstructed arrays} |
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355 | \label{sec-reconIO} |
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356 | |
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357 | The smoothing and reconstruction stages can be relatively |
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358 | time-consuming, particularly for large cubes and reconstructions in |
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359 | 3-D (or even spatial smoothing). To get around this, \duchamp provides |
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360 | a shortcut to allow users to perform multiple searches (\eg with |
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361 | different thresholds) on the same reconstruction/smoothing setup |
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362 | without re-doing the calculations each time. |
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363 | |
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364 | To save the reconstructed array as a FITS file, set |
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365 | \texttt{flagOutputRecon = true}. The file will be saved in the same |
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366 | directory as the input image, so the user needs to have write |
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367 | permissions for that directory. |
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368 | |
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369 | The filename will be derived from the input filename, with extra |
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370 | information detailing the reconstruction that has been done. For |
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371 | example, suppose \texttt{image.fits} has been reconstructed using a |
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372 | 3-dimensional reconstruction with filter \#2, thresholded at $4\sigma$ |
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373 | using all scales. The output filename will then be |
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374 | \texttt{image.RECON-3-2-4-1.fits} (\ie it uses the four parameters |
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375 | relevant for the \atrous reconstruction as listed in |
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376 | Appendix~\ref{app-param}). The new FITS file will also have these |
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377 | parameters as header keywords. If a subsection of the input image has |
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378 | been used (see \S\ref{sec-input}), the format of the output filename |
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379 | will be \texttt{image.sub.RECON-3-2-4-1.fits}, and the subsection that |
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380 | has been used is also stored in the FITS header. |
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381 | |
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382 | Likewise, the residual image, defined as the difference between the |
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383 | input and reconstructed arrays, can also be saved in the same manner |
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384 | by setting \texttt{flagOutputResid = true}. Its filename will be the |
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385 | same as above, with \texttt{RESID} replacing \texttt{RECON}. |
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386 | |
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387 | If a reconstructed image has been saved, it can be read in and used |
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388 | instead of redoing the reconstruction. To do so, the user should set |
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389 | the parameter \texttt{flagReconExists = true}. The user can indicate |
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390 | the name of the reconstructed FITS file using the \texttt{reconFile} |
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391 | parameter, or, if this is not specified, \duchamp searches for the |
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392 | file with the name as defined above. If the file is not found, the |
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393 | reconstruction is performed as normal. Note that to do this, the user |
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394 | needs to set \texttt{flagAtrous = true} (obviously, if this is |
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395 | \texttt{false}, the reconstruction is not needed). |
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396 | |
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397 | To save the smoothed array, set \texttt{flagOutputSmooth = true}. The |
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398 | name of the saved file will depend on the method of smoothing used. It |
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399 | will be either \texttt{image.SMOOTH-1D-a.fits}, where a is replaced by |
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400 | the Hanning width used, or \texttt{image.SMOOTH-2D-a-b-c.fits}, where |
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401 | the Gaussian kernel parameters are a,b,c. Similarly to the |
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402 | reconstruction case, a saved file can be read in by setting |
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403 | \texttt{flagSmoothExists = true} and either specifying a file to be |
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404 | read with the \texttt{smoothFile} parameter or relying on \duchamp to |
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405 | find the file with the name as given above. |
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406 | |
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407 | |
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408 | \secB{Searching the image} |
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409 | \label{sec-detection} |
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410 | |
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411 | \secC{Technique} |
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412 | |
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413 | The basic idea behind detection is to locate sets of contiguous voxels |
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414 | that lie above some threshold. One threshold is calculated for the |
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415 | entire cube, enabling calculation of signal-to-noise ratios for each |
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416 | source (see Section~\ref{sec-output} for details). The user can |
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417 | manually specify a value (using the parameter |
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418 | \texttt{threshold}) for the threshold, which will override the |
---|
419 | calculated value. Note that this only applies for the first of the two |
---|
420 | cases discussed below -- the FDR case ignores any manually-set |
---|
421 | threshold value. |
---|
422 | |
---|
423 | The cube is searched one channel map at a time, using the |
---|
424 | 2-dimensional raster-scanning algorithm of \citet{lutz80} that |
---|
425 | connects groups of neighbouring pixels. Such an algorithm cannot be |
---|
426 | applied directly to a 3-dimensional case, as it requires that objects |
---|
427 | are completely nested in a row (when scanning along a row, if an |
---|
428 | object finishes and other starts, you won't get back to the first |
---|
429 | until the second is completely finished for the |
---|
430 | row). Three-dimensional data does not have this property, hence the |
---|
431 | need to treat the data on a 2-dimensional basis. |
---|
432 | |
---|
433 | Although there are parameters that govern the minimum number of pixels |
---|
434 | in a spatial and spectral sense that an object must have |
---|
435 | (\texttt{minPix} and \texttt{minChannels} respectively), these |
---|
436 | criteria are not applied at this point. It is only after the merging |
---|
437 | and growing (see \S\ref{sec-merger}) is done that objects are rejected |
---|
438 | for not meeting these criteria. |
---|
439 | |
---|
440 | Finally, the search only looks for positive features. If one is |
---|
441 | interested instead in negative features (such as absorption lines), |
---|
442 | set the parameter \texttt{flagNegative = true}. This will invert the |
---|
443 | cube (\ie multiply all pixels by $-1$) prior to the search, and then |
---|
444 | re-invert the cube (and the fluxes of any detections) after searching |
---|
445 | is complete. All outputs are done in the same manner as normal, so |
---|
446 | that fluxes of detections will be negative. |
---|
447 | |
---|
448 | \secC{Calculating statistics} |
---|
449 | |
---|
450 | A crucial part of the detection process is estimating the statistics |
---|
451 | that define the detection threshold. To determine a threshold, we need |
---|
452 | to estimate from the data two parameters: the middle of the noise |
---|
453 | distribution (the ``noise level''), and the width of the distribution |
---|
454 | (the ``noise spread''). For both cases, we again use robust methods, |
---|
455 | using the median and MADFM. |
---|
456 | |
---|
457 | The choice of pixels to be used depend on the analysis method. If the |
---|
458 | wavelet reconstruction has been done, the residuals (defined |
---|
459 | in the sense of original $-$ reconstruction) are used to estimate the |
---|
460 | noise spread of the cube, since the reconstruction should pick out |
---|
461 | all significant structure. The noise level (the middle of the |
---|
462 | distribution) is taken from the original array. |
---|
463 | |
---|
464 | If smoothing of the cube has been done instead, all noise parameters |
---|
465 | are measured from the smoothed array, and detections are made with |
---|
466 | these parameters. When the signal-to-noise level is quoted for each |
---|
467 | detection (see \S\ref{sec-output}), the noise parameters of the |
---|
468 | original array are used, since the smoothing process correlates |
---|
469 | neighbouring pixels, reducing the noise level. |
---|
470 | |
---|
471 | If neither reconstruction nor smoothing has been done, then the |
---|
472 | statistics are calculated from the original, input array. |
---|
473 | |
---|
474 | The parameters that are estimated should be representative of the |
---|
475 | noise in the cube. For the case of small objects embedded in many |
---|
476 | noise pixels (\eg the case of \hi surveys), using the full cube will |
---|
477 | provide good estimators. It is possible, however, to use only a |
---|
478 | subsection of the cube by setting the parameter \texttt{flagStatSec = |
---|
479 | true} and providing the desired subsection to the \texttt{StatSec} |
---|
480 | parameter. This subsection works in exactly the same way as the pixel |
---|
481 | subsection discussed in \S\ref{sec-input}. Note that this subsection |
---|
482 | applies only to the statistics used to determine the threshold. It |
---|
483 | does not affect the calculation of statistics in the case of the |
---|
484 | wavelet reconstruction. |
---|
485 | |
---|
486 | \secC{Determining the threshold} |
---|
487 | |
---|
488 | Once the statistics have been calculated, the threshold is determined |
---|
489 | in one of two ways. The first way is a simple sigma-clipping, where a |
---|
490 | threshold is set at a fixed number $n$ of standard deviations above |
---|
491 | the mean, and pixels above this threshold are flagged as detected. The |
---|
492 | value of $n$ is set with the parameter \texttt{snrCut}. As before, the |
---|
493 | value of the standard deviation is estimated by the MADFM, and |
---|
494 | corrected by the ratio derived in Appendix~\ref{app-madfm}. |
---|
495 | |
---|
496 | The second method uses the False Discovery Rate (FDR) technique |
---|
497 | \citep{miller01,hopkins02}, whose basis we briefly detail here. The |
---|
498 | false discovery rate (given by the number of false detections divided |
---|
499 | by the total number of detections) is fixed at a certain value |
---|
500 | $\alpha$ (\eg $\alpha=0.05$ implies 5\% of detections are false |
---|
501 | positives). In practice, an $\alpha$ value is chosen, and the ensemble |
---|
502 | average FDR (\ie $\langle FDR \rangle$) when the method is used will |
---|
503 | be less than $\alpha$. One calculates $p$ -- the probability, |
---|
504 | assuming the null hypothesis is true, of obtaining a test statistic as |
---|
505 | extreme as the pixel value (the observed test statistic) -- for each |
---|
506 | pixel, and sorts them in increasing order. One then calculates $d$ |
---|
507 | where |
---|
508 | \[ |
---|
509 | d = \max_j \left\{ j : P_j < \frac{j\alpha}{c_N N} \right\}, |
---|
510 | \] |
---|
511 | and then rejects all hypotheses whose $p$-values are less than or |
---|
512 | equal to $P_d$. (So a $P_i<P_d$ will be rejected even if $P_i \geq |
---|
513 | j\alpha/c_N N$.) Note that ``reject hypothesis'' here means ``accept |
---|
514 | the pixel as an object pixel'' (\ie we are rejecting the null |
---|
515 | hypothesis that the pixel belongs to the background). |
---|
516 | |
---|
517 | The $c_N$ value here is a normalisation constant that depends on the |
---|
518 | correlated nature of the pixel values. If all the pixels are |
---|
519 | uncorrelated, then $c_N=1$. If $N$ pixels are correlated, then their |
---|
520 | tests will be dependent on each other, and so $c_N = \sum_{i=1}^N |
---|
521 | i^{-1}$. \citet{hopkins02} consider real radio data, where the pixels |
---|
522 | are correlated over the beam. For the calculations done in \duchamp, |
---|
523 | $N=2B$, where $B$ is the beam size in pixels, calculated from the FITS |
---|
524 | header (if the correct keywords -- BMAJ, BMIN -- are not present, the |
---|
525 | size of the beam is taken from the parameter \texttt{beamSize}). The |
---|
526 | factor of 2 comes about because we treat neighbouring channels as |
---|
527 | correlated. |
---|
528 | |
---|
529 | The theory behind the FDR method implies a direct connection between |
---|
530 | the choice of $\alpha$ and the fraction of detections that will be |
---|
531 | false positives. These detections, however, are individual pixels, |
---|
532 | which undergo a process of merging and rejection (\S\ref{sec-merger}), |
---|
533 | and so the fraction of the final list of detected objects that are |
---|
534 | false positives will be much smaller than $\alpha$. See the discussion |
---|
535 | in \S\ref{sec-notes}. |
---|
536 | |
---|
537 | %\secC{Storage of detected objects in memory} |
---|
538 | % |
---|
539 | %It is useful to understand how \duchamp stores the detected objects in |
---|
540 | %memory while it is running. This makes use of nested C++ classes, so |
---|
541 | %that an object is stored as a class that includes the set of detected |
---|
542 | %pixels, plus all the various calculated parameters (fluxes, WCS |
---|
543 | %coordinates, pixel centres and extrema, flags,...). The set of pixels |
---|
544 | %are stored using another class, that stores 3-dimensional objects as a |
---|
545 | %set of channel maps, each consisting of a $z$-value and a |
---|
546 | %2-dimensional object (a spatial map if you like). This 2-dimensional |
---|
547 | %object is recorded using ``run-length'' encoding, where each row (a |
---|
548 | %fixed $y$ value) is stored by the starting $x$-value and the length |
---|
549 | |
---|
550 | \secB{Merging detected objects} |
---|
551 | \label{sec-merger} |
---|
552 | |
---|
553 | The searching step produces a list of detected objects that will have |
---|
554 | many repeated detections of a given object -- for instance, spectral |
---|
555 | detections in adjacent pixels of the same object and/or spatial |
---|
556 | detections in neighbouring channels. These are then combined in an |
---|
557 | algorithm that matches all objects judged to be ``close'', according |
---|
558 | to one of two criteria. |
---|
559 | |
---|
560 | One criterion is to define two thresholds -- one spatial and one in |
---|
561 | velocity -- and say that two objects should be merged if there is at |
---|
562 | least one pair of pixels that lie within these threshold distances of |
---|
563 | each other. These thresholds are specified by the parameters |
---|
564 | \texttt{threshSpatial} and \texttt{threshVelocity} (in units of pixels |
---|
565 | and channels respectively). |
---|
566 | |
---|
567 | Alternatively, the spatial requirement can be changed to say that |
---|
568 | there must be a pair of pixels that are \emph{adjacent} -- a stricter, |
---|
569 | but perhaps more realistic requirement, particularly when the spatial |
---|
570 | pixels have a large angular size (as is the case for |
---|
571 | \hi surveys). This |
---|
572 | method can be selected by setting the parameter |
---|
573 | \texttt{flagAdjacent} to 1 (\ie \texttt{true}) in the parameter |
---|
574 | file. The velocity thresholding is done in the same way as the first |
---|
575 | option. |
---|
576 | |
---|
577 | Once the detections have been merged, they may be ``grown''. This is a |
---|
578 | process of increasing the size of the detection by adding nearby |
---|
579 | pixels (according to the \texttt{threshSpatial} and |
---|
580 | \texttt{threshVelocity} parameters) that are above some secondary |
---|
581 | threshold. This threshold is lower than the one used for the initial |
---|
582 | detection, but above the noise level, so that faint pixels are only |
---|
583 | detected when they are close to a bright pixel. The value of this |
---|
584 | threshold is a possible input parameter (\texttt{growthCut}), with a |
---|
585 | default value of $1.5\sigma$. |
---|
586 | |
---|
587 | The use of the growth algorithm is controlled by the |
---|
588 | \texttt{flagGrowth} parameter -- the default value of which is |
---|
589 | \texttt{false}. If the detections are grown, they are sent through the |
---|
590 | merging algorithm a second time, to pick up any detections that now |
---|
591 | overlap or have grown over each other. |
---|
592 | |
---|
593 | Finally, to be accepted, the detections must span \emph{both} a |
---|
594 | minimum number of channels (to remove any spurious single-channel |
---|
595 | spikes that may be present), and a minimum number of spatial |
---|
596 | pixels. These numbers, as for the original detection step, are set |
---|
597 | with the \texttt{minChannels} and \texttt{minPix} parameters. The |
---|
598 | channel requirement means there must be at least one set of |
---|
599 | \texttt{minChannels} consecutive channels in the source for it to be |
---|
600 | accepted. |
---|