[303] | 1 | % ----------------------------------------------------------------------- |
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| 2 | % app-waveletNoise.tex: Section detailing how the rms noise scales |
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| 3 | % with wavelet scale in the a trous method. |
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| 4 | % ----------------------------------------------------------------------- |
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| 5 | % Copyright (C) 2006, Matthew Whiting, ATNF |
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| 6 | % |
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| 7 | % This program is free software; you can redistribute it and/or modify it |
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| 8 | % under the terms of the GNU General Public License as published by the |
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| 9 | % Free Software Foundation; either version 2 of the License, or (at your |
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| 10 | % option) any later version. |
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| 11 | % |
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| 12 | % Duchamp is distributed in the hope that it will be useful, but WITHOUT |
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| 13 | % ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or |
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| 14 | % FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License |
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| 15 | % for more details. |
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| 16 | % |
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| 17 | % You should have received a copy of the GNU General Public License |
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| 18 | % along with Duchamp; if not, write to the Free Software Foundation, |
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| 19 | % Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307, USA |
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| 20 | % |
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| 21 | % Correspondence concerning Duchamp may be directed to: |
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| 22 | % Internet email: Matthew.Whiting [at] atnf.csiro.au |
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| 23 | % Postal address: Dr. Matthew Whiting |
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| 24 | % Australia Telescope National Facility, CSIRO |
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| 25 | % PO Box 76 |
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| 26 | % Epping NSW 1710 |
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| 27 | % AUSTRALIA |
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| 28 | % ----------------------------------------------------------------------- |
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[158] | 29 | \secA{How Gaussian noise changes with wavelet scale} |
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| 30 | \label{app-scaling} |
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| 31 | |
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| 32 | The key element in the wavelet reconstruction of an array is the |
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| 33 | thresholding of the individual wavelet coefficient arrays. This is |
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| 34 | usually done by choosing a level to be some number of standard |
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| 35 | deviations above the mean value. |
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| 36 | |
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| 37 | However, since the wavelet arrays are produced by convolving the input |
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| 38 | array by an increasingly large filter, the pixels in the coefficient |
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| 39 | arrays become increasingly correlated as the scale of the filter |
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| 40 | increases. This results in the measured standard deviation from a |
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| 41 | given coefficient array decreasing with increasing scale. To calculate |
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| 42 | this, we need to take into account how many other pixels each pixel in |
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| 43 | the convolved array depends on. |
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| 44 | |
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| 45 | To demonstrate, suppose we have a 1-D array with $N$ pixel values |
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| 46 | given by $F_i,\ i=1,...,N$, and we convolve it with the B$_3$-spline |
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| 47 | filter, defined by the set of coefficients |
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| 48 | $\{1/16,1/4,3/8,1/4,1/16\}$. The flux of the $i$th pixel in the |
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| 49 | convolved array will be |
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| 50 | \[ |
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| 51 | F'_i = \frac{1}{16}F_{i-2} + \frac{1}{4}F_{i-1} + \frac{3}{8}F_{i} |
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| 52 | + \frac{1}{4}F_{i+1} + \frac{1}{16}F_{i+2} |
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| 53 | \] |
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| 54 | and the flux of the corresponding pixel in the wavelet array will be |
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| 55 | \[ |
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| 56 | W'_i = F_i - F'_i = \frac{-1}{16}F_{i-2} - \frac{1}{4}F_{i-1} |
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| 57 | + \frac{5}{8}F_{i} - \frac{1}{4}F_{i+1} - \frac{1}{16}F_{i+2} |
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| 58 | \] |
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| 59 | Now, assuming each pixel has the same standard deviation |
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| 60 | $\sigma_i=\sigma$, we can work out the standard deviation for the |
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| 61 | wavelet array: |
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| 62 | \[ |
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| 63 | \sigma'_i = \sigma \sqrt{\left(\frac{1}{16}\right)^2 |
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| 64 | + \left(\frac{1}{4}\right)^2 + \left(\frac{5}{8}\right)^2 |
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| 65 | + \left(\frac{1}{4}\right)^2 + \left(\frac{1}{16}\right)^2} |
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| 66 | = 0.72349\ \sigma |
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| 67 | \] |
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| 68 | Thus, the first scale wavelet coefficient array will have a standard |
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| 69 | deviation of 72.3\% of the input array. This procedure can be followed |
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| 70 | to calculate the necessary values for all scales, dimensions and |
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| 71 | filters used by \duchamp. |
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| 72 | |
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| 73 | Calculating these values is clearly a critical step in performing the |
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[208] | 74 | reconstruction. The method used by \citet{starck02:book} was to |
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| 75 | simulate data sets with Gaussian noise, take the wavelet transform, |
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[298] | 76 | and measure the value of $\sigma$ for each scale. We take a different |
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| 77 | approach, by calculating the scaling factors directly from the filter |
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| 78 | coefficients by taking the wavelet transform of an array made up of a |
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| 79 | 1 in the central pixel and 0s everywhere else. The scaling value is |
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| 80 | then derived by taking the square root of the sum (in quadrature) of |
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| 81 | all the wavelet coefficient values at each scale. We give the scaling |
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| 82 | factors for the three filters available to \duchamp below. These |
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| 83 | values are hard-coded into \duchamp, so no on-the-fly calculation of |
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| 84 | them is necessary. |
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[158] | 85 | |
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| 86 | Memory limitations prevent us from calculating factors for large |
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| 87 | scales, particularly for the three-dimensional case (hence the -- |
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| 88 | symbols in the tables). To calculate factors for higher scales than |
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[298] | 89 | those available, we divide the previous scale's factor by either |
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| 90 | $\sqrt{2}$, $2$, or $\sqrt{8}$ for 1D, 2D and 3D respectively. |
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[158] | 91 | |
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[298] | 92 | %\newpage |
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| 93 | {\small |
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[158] | 94 | \begin{itemize} |
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| 95 | \item \textbf{B$_3$-Spline Function:} $\{1/16,1/4,3/8,1/4,1/16\}$ |
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| 96 | |
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| 97 | \begin{tabular}{llll} |
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| 98 | Scale & 1 dimension & 2 dimension & 3 dimension\\ \hline |
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| 99 | 1 & 0.723489806 & 0.890796310 & 0.956543592\\ |
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| 100 | 2 & 0.285450405 & 0.200663851 & 0.120336499\\ |
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| 101 | 3 & 0.177947535 & 0.0855075048 & 0.0349500154\\ |
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| 102 | 4 & 0.122223156 & 0.0412474444 & 0.0118164242\\ |
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| 103 | 5 & 0.0858113122 & 0.0204249666 & 0.00413233507\\ |
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| 104 | 6 & 0.0605703043 & 0.0101897592 & 0.00145703714\\ |
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| 105 | 7 & 0.0428107206 & 0.00509204670 & 0.000514791120\\ |
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| 106 | 8 & 0.0302684024 & 0.00254566946 & --\\ |
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| 107 | 9 & 0.0214024008 & 0.00127279050 & --\\ |
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| 108 | 10 & 0.0151336781 & 0.000636389722 & --\\ |
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| 109 | 11 & 0.0107011079 & 0.000318194170 & --\\ |
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| 110 | 12 & 0.00756682272 & -- & --\\ |
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| 111 | 13 & 0.00535055108 & -- & --\\ |
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| 112 | %14 & 0.00378341085 & -- & --\\ |
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| 113 | %15 & 0.00267527545 & -- & --\\ |
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| 114 | %16 & 0.00189170541 & -- & --\\ |
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| 115 | %17 & 0.00133763772 & -- & --\\ |
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| 116 | %18 & 0.000945852704 & -- & -- |
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| 117 | \end{tabular} |
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| 118 | |
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| 119 | \item \textbf{Triangle Function:} $\{1/4,1/2,1/4\}$ |
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| 120 | |
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| 121 | \begin{tabular}{llll} |
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| 122 | Scale & 1 dimension & 2 dimension & 3 dimension\\ \hline |
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| 123 | 1 & 0.612372436 & 0.800390530 & 0.895954449 \\ |
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| 124 | 2 & 0.330718914 & 0.272878894 & 0.192033014\\ |
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| 125 | 3 & 0.211947812 & 0.119779282 & 0.0576484078\\ |
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| 126 | 4 & 0.145740298 & 0.0577664785 & 0.0194912393\\ |
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| 127 | 5 & 0.102310944 & 0.0286163283 & 0.00681278387\\ |
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| 128 | 6 & 0.0722128185 & 0.0142747506 & 0.00240175885\\ |
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| 129 | 7 & 0.0510388224 & 0.00713319703 & 0.000848538128 \\ |
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| 130 | 8 & 0.0360857673 & 0.00356607618 & 0.000299949455 \\ |
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| 131 | 9 & 0.0255157615 & 0.00178297280 & -- \\ |
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| 132 | 10 & 0.0180422389 & 0.000891478237 & -- \\ |
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| 133 | 11 & 0.0127577667 & 0.000445738098 & -- \\ |
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| 134 | 12 & 0.00902109930 & 0.000222868922 & -- \\ |
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| 135 | 13 & 0.00637887978 & -- & -- \\ |
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| 136 | %14 & 0.00451054902 & -- & -- \\ |
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| 137 | %15 & 0.00318942978 & -- & -- \\ |
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| 138 | %16 & 0.00225527449 & -- & -- \\ |
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| 139 | %17 & 0.00159471988 & -- & -- \\ |
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| 140 | %18 & 0.000112763724 & -- & -- |
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| 141 | |
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| 142 | \end{tabular} |
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| 143 | |
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| 144 | \item \textbf{Haar Wavelet:} $\{0,1/2,1/2\}$ |
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| 145 | |
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| 146 | \begin{tabular}{llll} |
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| 147 | Scale & 1 dimension & 2 dimension & 3 dimension\\ \hline |
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| 148 | 1 & 0.707167810 & 0.433012702 & 0.935414347 \\ |
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| 149 | 2 & 0.500000000 & 0.216506351 & 0.330718914\\ |
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| 150 | 3 & 0.353553391 & 0.108253175 & 0.116926793\\ |
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| 151 | 4 & 0.250000000 & 0.0541265877 & 0.0413398642\\ |
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| 152 | 5 & 0.176776695 & 0.0270632939 & 0.0146158492\\ |
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| 153 | 6 & 0.125000000 & 0.0135316469 & 0.00516748303 |
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| 154 | |
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| 155 | \end{tabular} |
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| 156 | |
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| 157 | \end{itemize} |
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[298] | 158 | } |
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