1 | #!/usr/bin/env python |
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2 | r""" |
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3 | Multiple scattering calculator |
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4 | |
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5 | Calculate multiple scattering using 2D FFT convolution. |
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6 | |
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7 | Usage: |
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8 | |
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9 | -f, --fraction: the scattering probability |
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10 | -q, --qmax: that max q that you care about |
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11 | -w, --window: the extension window (q is calculated for qmax*window) |
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12 | -n, --nq: the number of mesh points (dq = qmax*window/nq) |
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13 | -r, --random: generate a random parameter set |
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14 | -2, --2d: perform the calculation for an oriented pattern |
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15 | model_name |
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16 | model_par=value ... |
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17 | |
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18 | Assume the probability of scattering is $p$. After each scattering event, |
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19 | $1-p$ neutrons will leave the system and go to the detector, and the remaining |
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20 | $p$ will scatter again. |
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21 | |
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22 | Let the scattering probability for a single scattering event at $q$ be $f(q)$, |
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23 | where |
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24 | .. math:: f(q) = p \frac{I_1(q)}{\int I_1(q) dq} |
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25 | for $I_1(q)$, the single scattering from the system. After two scattering |
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26 | events, the scattering will be the convolution of the first scattering |
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27 | and itself, or $(f*f)(q)$. After $n$ events it will be |
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28 | $(f* \cdots * f)(q)$. |
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29 | |
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30 | The total scattering after multiple events will then be the number that didn't |
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31 | scatter the first time $(=1-p)$ plus the number that scattered only once |
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32 | $(=(1-p)f)$ plus the number that scattered only twice $(=(1-p)(f*f))$, etc., |
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33 | so |
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34 | .. math:: |
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35 | I(q) = (1-p)\sum_{k=0}^{\infty} f^{*k}(q) |
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36 | Since convolution is linear, the contributions from the individual terms |
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37 | will fall off as $p^k$, so we can cut off the series |
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38 | at $n = \lceil \ln C / \ln p \rceil$ for some small fraction $C$ (default is |
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39 | $C = 0.001$). |
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40 | |
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41 | Using the convolution theorem, where |
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42 | $F = \mathcal{F}(f)$ is the Fourier transform, |
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43 | .. math:: |
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44 | |
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45 | f * g = \mathcal{F}^{-1}\{\mathcal{F}\{f\} \cdot \mathcal{F}\{g\}\} |
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46 | |
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47 | so |
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48 | .. math:: |
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49 | |
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50 | f * \ldots * f = \mathcal{F}^{-1}\{ F^n \} |
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51 | |
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52 | Since the Fourier transform is a linear operator, we can move the polynomial |
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53 | expression for the convolution into the transform, giving |
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54 | .. math:: |
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55 | |
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56 | I(q) = \mathcal{F}^{-1}\left\{ (1-p) \sum_{k=0}^{n} F^k \right\} |
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57 | |
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58 | We drop the transmission term, $k=0$, and rescale the result by the |
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59 | total scattering $\int I_1(q) dq$. |
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60 | |
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61 | For speed we use the fast fourier transform for a power of two. The resulting |
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62 | $I(q)$ will be linearly spaced and likely heavily oversampled. The usual |
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63 | pinhole or slit resolution calculation can performed from these calculated |
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64 | values. |
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65 | """ |
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66 | |
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67 | from __future__ import print_function |
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68 | |
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69 | import argparse |
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70 | import time |
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71 | |
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72 | import numpy as np |
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73 | |
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74 | from sasmodels import core |
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75 | from sasmodels import compare |
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76 | from sasmodels import resolution2d |
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77 | from sasmodels.resolution import Resolution, bin_edges |
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78 | from sasmodels.data import empty_data1D, empty_data2D, plot_data |
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79 | from sasmodels.direct_model import call_kernel |
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80 | |
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81 | |
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82 | class MultipleScattering(Resolution): |
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83 | def __init__(self, qmax, nq, fraction, is2d, window=2, power=0): |
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84 | self.qmax = qmax |
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85 | self.nq = nq |
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86 | self.power = power |
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87 | self.fraction = fraction |
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88 | self.is2d = is2d |
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89 | self.window = window |
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90 | |
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91 | q_range = qmax * window |
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92 | q = np.linspace(-q_range, q_range, nq) |
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93 | qx, qy = np.meshgrid(q, q) |
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94 | |
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95 | if is2d: |
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96 | q_calc = [qx.flatten(), qy.flatten()] |
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97 | else: |
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98 | q_range_corners = np.sqrt(2.) * q_range |
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99 | nq_corners = int(np.sqrt(2.) * nq/2) |
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100 | q_corners = np.linspace(0, q_range_corners, nq_corners+1)[1:] |
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101 | q_calc = [q_corners] |
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102 | self._qxy = np.sqrt(qx**2 + qy**2) |
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103 | self._edges = bin_edges(q_corners) |
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104 | self._norm = np.histogram(self._qxy, bins=self._edges)[0] |
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105 | |
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106 | self.q_calc = q_calc |
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107 | self.q_range = q_range |
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108 | |
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109 | def apply(self, theory): |
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110 | t0 = time.time() |
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111 | if self.is2d: |
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112 | Iq_calc = theory |
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113 | else: |
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114 | q_corners = self.q_calc[0] |
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115 | Iq_calc = np.interp(self._qxy, q_corners, theory) |
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116 | Iq_calc = Iq_calc.reshape(self.nq, self.nq) |
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117 | #plotxy(Iq_calc); import pylab; pylab.figure() |
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118 | if self.power > 0: |
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119 | Iqxy = autoconv_n(Iq_calc, self.power) |
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120 | else: |
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121 | Iqxy = multiple_scattering(Iq_calc, self.fraction, cutoff=0.001) |
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122 | print("multiple scattering calc time", time.time()-t0) |
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123 | #plotxy(Iqxy); import pylab; pylab.figure() |
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124 | if self.is2d: |
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125 | if 1: |
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126 | import pylab |
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127 | plotxy(Iq_calc) |
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128 | pylab.title("single scattering") |
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129 | pylab.figure() |
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130 | |
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131 | return Iqxy |
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132 | else: |
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133 | # circular average, no anti-aliasing |
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134 | Iq = np.histogram(self._qxy, bins=self._edges, weights=Iqxy)[0]/self._norm |
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135 | |
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136 | if 1: |
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137 | import pylab |
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138 | pylab.loglog(q_corners, theory, label="single scattering") |
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139 | if self.power > 0: |
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140 | label = "scattering power %d"%self.power |
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141 | else: |
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142 | label = "scattering fraction %d"%self.fraction |
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143 | pylab.loglog(q_corners, Iq, label=label) |
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144 | pylab.legend() |
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145 | pylab.figure() |
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146 | return Iqxy |
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147 | return q_corners, Iq |
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148 | |
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149 | def multiple_scattering(Iq_calc, frac, cutoff=0.001): |
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150 | #plotxy(Iq_calc) |
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151 | num_scatter = int(np.ceil(np.log(cutoff)/np.log(frac))) |
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152 | |
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153 | # Prepare padded array for transform |
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154 | nq = Iq_calc.shape[0] |
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155 | half_nq = nq//2 |
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156 | frame = np.zeros((2*nq, 2*nq)) |
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157 | frame[:half_nq, :half_nq] = Iq_calc[half_nq:, half_nq:] |
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158 | frame[-half_nq:, :half_nq] = Iq_calc[:half_nq, half_nq:] |
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159 | frame[:half_nq, -half_nq:] = Iq_calc[half_nq:, :half_nq] |
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160 | frame[-half_nq:, -half_nq:] = Iq_calc[:half_nq, :half_nq] |
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161 | #plotxy(frame) |
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162 | |
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163 | # Compute multiple scattering via convolution. |
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164 | scale = np.sum(Iq_calc) |
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165 | fourier_frame = np.fft.fft2(frame/scale) |
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166 | #plotxy(abs(frame)) |
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167 | # total = (1-a)f + (1-a)af^2 + (1-a)a^2f^3 + ... |
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168 | # = (1-a)f[1 + af + (af)^2 + (af)^3 + ...] |
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169 | multiple_scattering = ( |
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170 | (1-frac)*fourier_frame |
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171 | *np.polyval(np.ones(num_scatter), frac*fourier_frame)) |
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172 | conv_frame = scale*np.fft.ifft2(multiple_scattering).real |
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173 | |
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174 | # Recover the transformed data |
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175 | #plotxy(conv_frame) |
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176 | Iq_conv = np.empty((nq, nq)) |
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177 | Iq_conv[half_nq:, half_nq:] = conv_frame[:half_nq, :half_nq] |
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178 | Iq_conv[:half_nq, half_nq:] = conv_frame[-half_nq:, :half_nq] |
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179 | Iq_conv[half_nq:, :half_nq] = conv_frame[:half_nq, -half_nq:] |
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180 | Iq_conv[:half_nq, :half_nq] = conv_frame[-half_nq:, -half_nq:] |
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181 | #plotxy(Iq_conv) |
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182 | return Iq_conv |
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183 | |
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184 | def multiple_scattering_cl(Iq_calc, frac, cutoff=0.001): |
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185 | raise NotImplementedError("no support for opencl calculations at this time") |
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186 | |
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187 | import pyopencl as cl |
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188 | import pyopencl.array as cla |
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189 | from gpyfft.fft import FFT |
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190 | context = cl.create_some_context() |
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191 | queue = cl.CommandQueue(context) |
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192 | |
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193 | #plotxy(Iq_calc) |
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194 | num_scatter = int(np.ceil(np.log(cutoff)/np.log(frac))) |
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195 | |
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196 | # Prepare padded array for transform |
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197 | nq = Iq_calc.shape[0] |
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198 | half_nq = nq//2 |
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199 | frame = np.zeros((2*nq, 2*nq), dtype='float32') |
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200 | frame[:half_nq, :half_nq] = Iq_calc[half_nq:, half_nq:] |
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201 | frame[-half_nq:, :half_nq] = Iq_calc[:half_nq, half_nq:] |
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202 | frame[:half_nq, -half_nq:] = Iq_calc[half_nq:, :half_nq] |
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203 | frame[-half_nq:, -half_nq:] = Iq_calc[:half_nq, :half_nq] |
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204 | #plotxy(frame) |
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205 | |
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206 | # Compute multiple scattering via convolution (OpenCL operations) |
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207 | frame_gpu = cla.to_device(queue, frame) |
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208 | fourier_frame_gpu = cla.zeros(frame.shape, dtype='complex64') |
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209 | scale = frame_gpu.sum() |
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210 | frame_gpu /= scale |
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211 | transform = FFT(context, queue, frame_gpu, fourier_frame_gpu, axes=(0,1)) |
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212 | event, = transform.enqueue() |
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213 | event.wait() |
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214 | fourier_frame_gpu *= frac |
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215 | multiple_scattering_gpu = fourier_frame_gpu.copy() |
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216 | for _ in range(num_scatter-1): |
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217 | multiple_scattering_gpu += 1 |
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218 | multiple_scattering_gpu *= fourier_frame_gpu |
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219 | multiple_scattering_gpu *= (1 - frac)/frac |
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220 | transform = FFT(context, queue, multiple_scattering_gpu, frame_gpu, axes=(0,1)) |
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221 | event, = transform.enqueue(forward=False) |
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222 | event.wait() |
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223 | conv_frame = cla.from_device(queue, frame_gpu) |
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224 | |
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225 | # Recover the transformed data |
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226 | #plotxy(conv_frame) |
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227 | Iq_conv = np.empty((nq, nq)) |
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228 | Iq_conv[half_nq:, half_nq:] = conv_frame[:half_nq, :half_nq] |
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229 | Iq_conv[:half_nq, half_nq:] = conv_frame[-half_nq:, :half_nq] |
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230 | Iq_conv[half_nq:, :half_nq] = conv_frame[:half_nq, -half_nq:] |
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231 | Iq_conv[:half_nq, :half_nq] = conv_frame[-half_nq:, -half_nq:] |
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232 | #plotxy(Iq_conv) |
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233 | return Iq_conv |
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234 | |
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235 | def autoconv_n(Iq_calc, power): |
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236 | # Compute multiple scattering via convolution. |
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237 | #plotxy(Iq_calc) |
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238 | scale = np.sum(Iq_calc) |
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239 | nq = Iq_calc.shape[0] |
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240 | frame = np.zeros((2*nq, 2*nq)) |
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241 | half_nq = nq//2 |
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242 | frame[:half_nq, :half_nq] = Iq_calc[half_nq:, half_nq:] |
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243 | frame[-half_nq:, :half_nq] = Iq_calc[:half_nq, half_nq:] |
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244 | frame[:half_nq, -half_nq:] = Iq_calc[half_nq:, :half_nq] |
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245 | frame[-half_nq:, -half_nq:] = Iq_calc[:half_nq, :half_nq] |
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246 | #plotxy(frame) |
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247 | fourier_frame = np.fft.fft2(frame/scale) |
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248 | #plotxy(abs(frame)) |
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249 | fourier_frame = fourier_frame**power |
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250 | conv_frame = scale*np.fft.ifft2(fourier_frame).real |
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251 | #plotxy(conv_frame) |
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252 | Iq_conv = np.empty((nq, nq)) |
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253 | Iq_conv[half_nq:, half_nq:] = conv_frame[:half_nq, :half_nq] |
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254 | Iq_conv[:half_nq, half_nq:] = conv_frame[-half_nq:, :half_nq] |
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255 | Iq_conv[half_nq:, :half_nq] = conv_frame[:half_nq, -half_nq:] |
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256 | Iq_conv[:half_nq, :half_nq] = conv_frame[-half_nq:, -half_nq:] |
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257 | #plotxy(Iq_conv) |
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258 | return Iq_conv |
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259 | |
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260 | def parse_pars(model, opts): |
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261 | # type: (ModelInfo, argparse.Namespace) -> Dict[str, float] |
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262 | |
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263 | seed = np.random.randint(1000000) if opts.random and opts.seed < 0 else opts.seed |
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264 | compare_opts = { |
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265 | 'info': (model.info, model.info), |
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266 | 'use_demo': False, |
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267 | 'seed': seed, |
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268 | 'mono': True, |
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269 | 'magnetic': False, |
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270 | 'values': opts.pars, |
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271 | 'show_pars': True, |
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272 | 'is2d': opts.is2d, |
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273 | } |
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274 | pars, pars2 = compare.parse_pars(compare_opts) |
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275 | return pars |
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276 | |
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277 | |
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278 | def main(): |
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279 | parser = argparse.ArgumentParser( |
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280 | description="Compute multiple scattering", |
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281 | formatter_class=argparse.ArgumentDefaultsHelpFormatter, |
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282 | ) |
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283 | parser.add_argument('-p', '--power', type=int, default=0, help="show pattern for nth scattering") |
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284 | parser.add_argument('-f', '--fraction', type=float, default=0.1, help="scattering fraction") |
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285 | parser.add_argument('-n', '--nq', type=int, default=1024, help='number of mesh points') |
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286 | parser.add_argument('-q', '--qmax', type=float, default=0.5, help='max q') |
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287 | parser.add_argument('-w', '--window', type=float, default=2.0, help='q calc = q max * window') |
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288 | parser.add_argument('-2', '--2d', dest='is2d', action='store_true', help='oriented sample') |
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289 | parser.add_argument('-s', '--seed', default=-1, help='random pars with given seed') |
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290 | parser.add_argument('-r', '--random', action='store_true', help='random pars with random seed') |
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291 | parser.add_argument('model', type=str, help='sas model') |
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292 | parser.add_argument('pars', type=str, nargs='*', help='sas model') |
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293 | opts = parser.parse_args() |
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294 | assert opts.nq%2 == 0, "require even # points" |
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295 | |
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296 | model = core.load_model(opts.model) |
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297 | pars = parse_pars(model, opts) |
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298 | res = MultipleScattering(opts.qmax, opts.nq, opts.fraction, opts.is2d, |
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299 | window=opts.window, power=opts.power) |
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300 | kernel = model.make_kernel(res.q_calc) |
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301 | #print(pars) |
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302 | bg = pars.get('background', 0.0) |
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303 | pars['background'] = 0.0 |
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304 | Iq_calc = call_kernel(kernel, pars) |
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305 | t0 = time.time() |
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306 | for i in range(10): |
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307 | Iq_calc = call_kernel(kernel, pars) |
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308 | print("single scattering calc time", (time.time()-t0)/10) |
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309 | Iq = res.apply(Iq_calc) + bg |
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310 | plotxy(Iq) |
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311 | import pylab |
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312 | if opts.power > 0: |
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313 | pylab.title('scattering power %d'%opts.power) |
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314 | else: |
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315 | pylab.title('multiple scattering with fraction %g'%opts.fraction) |
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316 | pylab.show() |
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317 | |
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318 | def plotxy(Iq): |
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319 | import pylab |
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320 | if isinstance(Iq, tuple): |
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321 | q, Iq = Iq |
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322 | pylab.loglog(q, Iq) |
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323 | else: |
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324 | data = Iq+0. |
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325 | data[Iq <= 0] = np.min(Iq[Iq>0])/2 |
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326 | pylab.imshow(np.log10(data)) |
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327 | #import pylab; pylab.show() |
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328 | |
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329 | if __name__ == "__main__": |
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330 | main() |
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