1 | \secA{How Gaussian noise changes with wavelet scale} |
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2 | \label{app-scaling} |
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3 | |
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4 | The key element in the wavelet reconstruction of an array is the |
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5 | thresholding of the individual wavelet coefficient arrays. This is |
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6 | usually done by choosing a level to be some number of standard |
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7 | deviations above the mean value. |
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8 | |
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9 | However, since the wavelet arrays are produced by convolving the input |
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10 | array by an increasingly large filter, the pixels in the coefficient |
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11 | arrays become increasingly correlated as the scale of the filter |
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12 | increases. This results in the measured standard deviation from a |
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13 | given coefficient array decreasing with increasing scale. To calculate |
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14 | this, we need to take into account how many other pixels each pixel in |
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15 | the convolved array depends on. |
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16 | |
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17 | To demonstrate, suppose we have a 1-D array with $N$ pixel values |
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18 | given by $F_i,\ i=1,...,N$, and we convolve it with the B$_3$-spline |
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19 | filter, defined by the set of coefficients |
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20 | $\{1/16,1/4,3/8,1/4,1/16\}$. The flux of the $i$th pixel in the |
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21 | convolved array will be |
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22 | \[ |
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23 | F'_i = \frac{1}{16}F_{i-2} + \frac{1}{4}F_{i-1} + \frac{3}{8}F_{i} |
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24 | + \frac{1}{4}F_{i+1} + \frac{1}{16}F_{i+2} |
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25 | \] |
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26 | and the flux of the corresponding pixel in the wavelet array will be |
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27 | \[ |
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28 | W'_i = F_i - F'_i = \frac{-1}{16}F_{i-2} - \frac{1}{4}F_{i-1} |
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29 | + \frac{5}{8}F_{i} - \frac{1}{4}F_{i+1} - \frac{1}{16}F_{i+2} |
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30 | \] |
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31 | Now, assuming each pixel has the same standard deviation |
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32 | $\sigma_i=\sigma$, we can work out the standard deviation for the |
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33 | wavelet array: |
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34 | \[ |
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35 | \sigma'_i = \sigma \sqrt{\left(\frac{1}{16}\right)^2 |
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36 | + \left(\frac{1}{4}\right)^2 + \left(\frac{5}{8}\right)^2 |
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37 | + \left(\frac{1}{4}\right)^2 + \left(\frac{1}{16}\right)^2} |
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38 | = 0.72349\ \sigma |
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39 | \] |
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40 | Thus, the first scale wavelet coefficient array will have a standard |
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41 | deviation of 72.3\% of the input array. This procedure can be followed |
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42 | to calculate the necessary values for all scales, dimensions and |
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43 | filters used by \duchamp. |
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44 | |
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45 | Calculating these values is clearly a critical step in performing the |
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46 | reconstruction. \citet{starck02:book} did so by simulating data sets |
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47 | with Gaussian noise, taking the wavelet transform, and measuring the |
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48 | value of $\sigma$ for each scale. We take a different approach, by |
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49 | calculating the scaling factors directly from the filter coefficients |
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50 | by taking the wavelet transform of an array made up of a 1 in the |
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51 | central pixel and 0s everywhere else. The scaling value is then |
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52 | derived by taking the square root of the sum (in quadrature) of all |
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53 | the wavelet coefficient values at each scale. We give the scaling |
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54 | factors for the three filters available to \duchamp\ on the following |
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55 | page. These values are hard-coded into \duchamp, so no on-the-fly |
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56 | calculation of them is necessary. |
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57 | |
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58 | Memory limitations prevent us from calculating factors for large |
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59 | scales, particularly for the three-dimensional case (hence the -- |
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60 | symbols in the tables). To calculate factors for higher scales than |
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61 | those available, we note the following relationships apply for large |
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62 | scales to a sufficient level of precision: |
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63 | \begin{itemize} |
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64 | \item 1-D: factor(scale $i$) = factor(scale $i-1$)$/\sqrt{2}$. |
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65 | \item 2-D: factor(scale $i$) = factor(scale $i-1$)$/2$. |
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66 | \item 1-D: factor(scale $i$) = factor(scale $i-1$)$/\sqrt{8}$. |
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67 | \end{itemize} |
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68 | |
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69 | \newpage |
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70 | \begin{itemize} |
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71 | \item \textbf{B$_3$-Spline Function:} $\{1/16,1/4,3/8,1/4,1/16\}$ |
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72 | |
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73 | \begin{tabular}{llll} |
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74 | Scale & 1 dimension & 2 dimension & 3 dimension\\ \hline |
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75 | 1 & 0.723489806 & 0.890796310 & 0.956543592\\ |
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76 | 2 & 0.285450405 & 0.200663851 & 0.120336499\\ |
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77 | 3 & 0.177947535 & 0.0855075048 & 0.0349500154\\ |
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78 | 4 & 0.122223156 & 0.0412474444 & 0.0118164242\\ |
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79 | 5 & 0.0858113122 & 0.0204249666 & 0.00413233507\\ |
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80 | 6 & 0.0605703043 & 0.0101897592 & 0.00145703714\\ |
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81 | 7 & 0.0428107206 & 0.00509204670 & 0.000514791120\\ |
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82 | 8 & 0.0302684024 & 0.00254566946 & --\\ |
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83 | 9 & 0.0214024008 & 0.00127279050 & --\\ |
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84 | 10 & 0.0151336781 & 0.000636389722 & --\\ |
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85 | 11 & 0.0107011079 & 0.000318194170 & --\\ |
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86 | 12 & 0.00756682272 & -- & --\\ |
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87 | 13 & 0.00535055108 & -- & --\\ |
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88 | %14 & 0.00378341085 & -- & --\\ |
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89 | %15 & 0.00267527545 & -- & --\\ |
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90 | %16 & 0.00189170541 & -- & --\\ |
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91 | %17 & 0.00133763772 & -- & --\\ |
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92 | %18 & 0.000945852704 & -- & -- |
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93 | \end{tabular} |
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94 | |
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95 | \item \textbf{Triangle Function:} $\{1/4,1/2,1/4\}$ |
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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 & 0.612372436 & 0.800390530 & 0.895954449 \\ |
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100 | 2 & 0.330718914 & 0.272878894 & 0.192033014\\ |
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101 | 3 & 0.211947812 & 0.119779282 & 0.0576484078\\ |
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102 | 4 & 0.145740298 & 0.0577664785 & 0.0194912393\\ |
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103 | 5 & 0.102310944 & 0.0286163283 & 0.00681278387\\ |
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104 | 6 & 0.0722128185 & 0.0142747506 & 0.00240175885\\ |
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105 | 7 & 0.0510388224 & 0.00713319703 & 0.000848538128 \\ |
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106 | 8 & 0.0360857673 & 0.00356607618 & 0.000299949455 \\ |
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107 | 9 & 0.0255157615 & 0.00178297280 & -- \\ |
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108 | 10 & 0.0180422389 & 0.000891478237 & -- \\ |
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109 | 11 & 0.0127577667 & 0.000445738098 & -- \\ |
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110 | 12 & 0.00902109930 & 0.000222868922 & -- \\ |
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111 | 13 & 0.00637887978 & -- & -- \\ |
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112 | %14 & 0.00451054902 & -- & -- \\ |
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113 | %15 & 0.00318942978 & -- & -- \\ |
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114 | %16 & 0.00225527449 & -- & -- \\ |
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115 | %17 & 0.00159471988 & -- & -- \\ |
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116 | %18 & 0.000112763724 & -- & -- |
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117 | |
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118 | \end{tabular} |
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119 | |
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120 | \item \textbf{Haar Wavelet:} $\{0,1/2,1/2\}$ |
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121 | |
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122 | \begin{tabular}{llll} |
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123 | Scale & 1 dimension & 2 dimension & 3 dimension\\ \hline |
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124 | 1 & 0.707167810 & 0.433012702 & 0.935414347 \\ |
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125 | 2 & 0.500000000 & 0.216506351 & 0.330718914\\ |
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126 | 3 & 0.353553391 & 0.108253175 & 0.116926793\\ |
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127 | 4 & 0.250000000 & 0.0541265877 & 0.0413398642\\ |
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128 | 5 & 0.176776695 & 0.0270632939 & 0.0146158492\\ |
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129 | 6 & 0.125000000 & 0.0135316469 & 0.00516748303 |
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130 | |
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131 | \end{tabular} |
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132 | |
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133 | |
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134 | \end{itemize} |
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