1 | """Least squares fitting. |
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2 | |
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3 | Implements a penalised least-squares fit. |
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4 | |
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5 | The penalty term (or smoothing term) is controlled by the smoothing |
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6 | parameter alpha. |
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7 | With a value of alpha=0, the fit function will attempt |
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8 | to interpolate as closely as possible in the least-squares sense. |
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9 | With values alpha > 0, a certain amount of smoothing will be applied. |
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10 | A positive alpha is essential in cases where there are too few |
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11 | data points. |
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12 | A negative alpha is not allowed. |
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13 | A typical value of alpha is 1.0e-6 |
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14 | |
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15 | |
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16 | Ole Nielsen, Stephen Roberts, Duncan Gray, Christopher Zoppou |
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17 | Geoscience Australia, 2004. |
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18 | |
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19 | TO DO |
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20 | * test geo_ref, geo_spatial |
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21 | """ |
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22 | |
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23 | from Numeric import zeros, Float, ArrayType,take |
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24 | |
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25 | from geospatial_data.geospatial_data import Geospatial_data, ensure_absolute |
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26 | from fit_interpolate.general_fit_interpolate import FitInterpolate |
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27 | from utilities.sparse import Sparse, Sparse_CSR |
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28 | from utilities.polygon import in_and_outside_polygon |
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29 | from fit_interpolate.search_functions import search_tree_of_vertices |
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30 | from utilities.cg_solve import conjugate_gradient |
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31 | from utilities.numerical_tools import ensure_numeric, gradient |
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32 | |
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33 | import exceptions |
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34 | class ToFewPointsError(exceptions.Exception): pass |
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35 | class VertsWithNoTrianglesError(exceptions.Exception): pass |
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36 | |
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37 | DEFAULT_ALPHA = 0.001 |
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38 | |
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39 | |
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40 | class Fit(FitInterpolate): |
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41 | |
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42 | def __init__(self, |
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43 | vertex_coordinates, |
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44 | triangles, |
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45 | mesh_origin=None, |
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46 | alpha = None, |
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47 | verbose=False, |
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48 | max_vertices_per_cell=30): |
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49 | |
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50 | |
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51 | """ |
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52 | Fit data at points to the vertices of a mesh. |
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53 | |
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54 | Inputs: |
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55 | |
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56 | vertex_coordinates: List of coordinate pairs [xi, eta] of |
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57 | points constituting a mesh (or an m x 2 Numeric array or |
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58 | a geospatial object) |
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59 | Points may appear multiple times |
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60 | (e.g. if vertices have discontinuities) |
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61 | |
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62 | triangles: List of 3-tuples (or a Numeric array) of |
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63 | integers representing indices of all vertices in the mesh. |
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64 | |
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65 | mesh_origin: A geo_reference object or 3-tuples consisting of |
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66 | UTM zone, easting and northing. |
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67 | If specified vertex coordinates are assumed to be |
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68 | relative to their respective origins. |
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69 | |
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70 | max_vertices_per_cell: Number of vertices in a quad tree cell |
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71 | at which the cell is split into 4. |
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72 | |
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73 | Note: Don't supply a vertex coords as a geospatial object and |
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74 | a mesh origin, since geospatial has its own mesh origin. |
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75 | """ |
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76 | # Initialise variabels |
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77 | |
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78 | if alpha is None: |
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79 | |
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80 | self.alpha = DEFAULT_ALPHA |
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81 | else: |
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82 | self.alpha = alpha |
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83 | FitInterpolate.__init__(self, |
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84 | vertex_coordinates, |
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85 | triangles, |
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86 | mesh_origin, |
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87 | verbose, |
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88 | max_vertices_per_cell) |
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89 | |
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90 | m = self.mesh.coordinates.shape[0] #Nbr of basis functions (vertices) |
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91 | |
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92 | self.AtA = None |
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93 | self.Atz = None |
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94 | |
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95 | self.point_count = 0 |
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96 | if self.alpha <> 0: |
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97 | if verbose: print 'Building smoothing matrix' |
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98 | self._build_smoothing_matrix_D() |
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99 | |
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100 | def _build_coefficient_matrix_B(self, |
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101 | verbose = False): |
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102 | """ |
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103 | Build final coefficient matrix |
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104 | |
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105 | Precon |
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106 | If alpha is not zero, matrix D has been built |
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107 | Matrix Ata has been built |
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108 | """ |
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109 | |
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110 | if self.alpha <> 0: |
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111 | #if verbose: print 'Building smoothing matrix' |
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112 | #self._build_smoothing_matrix_D() |
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113 | self.B = self.AtA + self.alpha*self.D |
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114 | else: |
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115 | self.B = self.AtA |
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116 | |
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117 | #Convert self.B matrix to CSR format for faster matrix vector |
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118 | self.B = Sparse_CSR(self.B) |
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119 | |
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120 | def _build_smoothing_matrix_D(self): |
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121 | """Build m x m smoothing matrix, where |
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122 | m is the number of basis functions phi_k (one per vertex) |
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123 | |
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124 | The smoothing matrix is defined as |
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125 | |
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126 | D = D1 + D2 |
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127 | |
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128 | where |
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129 | |
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130 | [D1]_{k,l} = \int_\Omega |
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131 | \frac{\partial \phi_k}{\partial x} |
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132 | \frac{\partial \phi_l}{\partial x}\, |
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133 | dx dy |
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134 | |
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135 | [D2]_{k,l} = \int_\Omega |
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136 | \frac{\partial \phi_k}{\partial y} |
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137 | \frac{\partial \phi_l}{\partial y}\, |
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138 | dx dy |
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139 | |
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140 | |
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141 | The derivatives \frac{\partial \phi_k}{\partial x}, |
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142 | \frac{\partial \phi_k}{\partial x} for a particular triangle |
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143 | are obtained by computing the gradient a_k, b_k for basis function k |
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144 | """ |
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145 | |
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146 | #FIXME: algorithm might be optimised by computing local 9x9 |
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147 | #"element stiffness matrices: |
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148 | |
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149 | m = self.mesh.coordinates.shape[0] #Nbr of basis functions (1/vertex) |
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150 | |
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151 | self.D = Sparse(m,m) |
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152 | |
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153 | #For each triangle compute contributions to D = D1+D2 |
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154 | for i in range(len(self.mesh)): |
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155 | |
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156 | #Get area |
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157 | area = self.mesh.areas[i] |
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158 | |
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159 | #Get global vertex indices |
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160 | v0 = self.mesh.triangles[i,0] |
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161 | v1 = self.mesh.triangles[i,1] |
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162 | v2 = self.mesh.triangles[i,2] |
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163 | |
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164 | #Get the three vertex_points |
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165 | xi0 = self.mesh.get_vertex_coordinate(i, 0) |
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166 | xi1 = self.mesh.get_vertex_coordinate(i, 1) |
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167 | xi2 = self.mesh.get_vertex_coordinate(i, 2) |
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168 | |
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169 | #Compute gradients for each vertex |
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170 | a0, b0 = gradient(xi0[0], xi0[1], xi1[0], xi1[1], xi2[0], xi2[1], |
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171 | 1, 0, 0) |
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172 | |
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173 | a1, b1 = gradient(xi0[0], xi0[1], xi1[0], xi1[1], xi2[0], xi2[1], |
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174 | 0, 1, 0) |
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175 | |
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176 | a2, b2 = gradient(xi0[0], xi0[1], xi1[0], xi1[1], xi2[0], xi2[1], |
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177 | 0, 0, 1) |
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178 | |
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179 | #Compute diagonal contributions |
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180 | self.D[v0,v0] += (a0*a0 + b0*b0)*area |
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181 | self.D[v1,v1] += (a1*a1 + b1*b1)*area |
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182 | self.D[v2,v2] += (a2*a2 + b2*b2)*area |
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183 | |
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184 | #Compute contributions for basis functions sharing edges |
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185 | e01 = (a0*a1 + b0*b1)*area |
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186 | self.D[v0,v1] += e01 |
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187 | self.D[v1,v0] += e01 |
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188 | |
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189 | e12 = (a1*a2 + b1*b2)*area |
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190 | self.D[v1,v2] += e12 |
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191 | self.D[v2,v1] += e12 |
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192 | |
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193 | e20 = (a2*a0 + b2*b0)*area |
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194 | self.D[v2,v0] += e20 |
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195 | self.D[v0,v2] += e20 |
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196 | |
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197 | |
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198 | def get_D(self): |
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199 | return self.D.todense() |
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200 | |
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201 | |
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202 | def _build_matrix_AtA_Atz(self, |
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203 | point_coordinates, |
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204 | z, |
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205 | verbose = False): |
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206 | """Build: |
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207 | AtA m x m interpolation matrix, and, |
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208 | Atz m x a interpolation matrix where, |
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209 | m is the number of basis functions phi_k (one per vertex) |
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210 | a is the number of data attributes |
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211 | |
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212 | This algorithm uses a quad tree data structure for fast binning of |
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213 | data points. |
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214 | |
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215 | If Ata is None, the matrices AtA and Atz are created. |
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216 | |
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217 | This function can be called again and again, with sub-sets of |
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218 | the point coordinates. Call fit to get the results. |
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219 | |
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220 | Preconditions |
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221 | z and points are numeric |
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222 | Point_coordindates and mesh vertices have the same origin. |
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223 | |
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224 | The number of attributes of the data points does not change |
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225 | """ |
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226 | #Build n x m interpolation matrix |
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227 | |
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228 | if self.AtA == None: |
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229 | # AtA and Atz need ot be initialised. |
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230 | m = self.mesh.coordinates.shape[0] #Nbr of vertices |
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231 | if len(z.shape) > 1: |
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232 | att_num = z.shape[1] |
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233 | self.Atz = zeros((m,att_num), Float) |
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234 | else: |
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235 | att_num = 1 |
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236 | self.Atz = zeros((m,), Float) |
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237 | assert z.shape[0] == point_coordinates.shape[0] |
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238 | |
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239 | self.AtA = Sparse(m,m) |
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240 | self.point_count += point_coordinates.shape[0] |
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241 | #print "_build_matrix_AtA_Atz - self.point_count", self.point_count |
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242 | if verbose: print 'Getting indices inside mesh boundary' |
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243 | #print 'point_coordinates.shape', point_coordinates.shape |
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244 | #print 'self.mesh.get_boundary_polygon()',\ |
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245 | # self.mesh.get_boundary_polygon() |
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246 | |
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247 | self.inside_poly_indices, self.outside_poly_indices = \ |
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248 | in_and_outside_polygon(point_coordinates, |
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249 | self.mesh.get_boundary_polygon(), |
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250 | closed = True, verbose = verbose) |
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251 | #print "self.inside_poly_indices",self.inside_poly_indices |
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252 | #print "self.outside_poly_indices",self.outside_poly_indices |
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253 | |
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254 | |
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255 | n = len(self.inside_poly_indices) |
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256 | #Compute matrix elements for points inside the mesh |
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257 | for i in self.inside_poly_indices: |
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258 | #For each data_coordinate point |
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259 | if verbose and i%((n+10)/10)==0: print 'Doing %d of %d' %(i, n) |
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260 | x = point_coordinates[i] |
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261 | element_found, sigma0, sigma1, sigma2, k = \ |
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262 | search_tree_of_vertices(self.root, self.mesh, x) |
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263 | |
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264 | if element_found is True: |
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265 | j0 = self.mesh.triangles[k,0] #Global vertex id for sigma0 |
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266 | j1 = self.mesh.triangles[k,1] #Global vertex id for sigma1 |
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267 | j2 = self.mesh.triangles[k,2] #Global vertex id for sigma2 |
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268 | |
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269 | sigmas = {j0:sigma0, j1:sigma1, j2:sigma2} |
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270 | js = [j0,j1,j2] |
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271 | |
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272 | for j in js: |
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273 | self.Atz[j] += sigmas[j]*z[i] |
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274 | #print "self.Atz building", self.Atz |
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275 | #print "self.Atz[j]", self.Atz[j] |
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276 | #print " sigmas[j]", sigmas[j] |
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277 | #print "z[i]",z[i] |
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278 | #print "result", sigmas[j]*z[i] |
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279 | |
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280 | for k in js: |
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281 | self.AtA[j,k] += sigmas[j]*sigmas[k] |
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282 | else: |
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283 | msg = 'Could not find triangle for point', x |
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284 | raise Exception(msg) |
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285 | |
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286 | |
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287 | def fit(self, point_coordinates=None, z=None, |
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288 | verbose = False, |
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289 | point_origin = None): |
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290 | """Fit a smooth surface to given 1d array of data points z. |
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291 | |
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292 | The smooth surface is computed at each vertex in the underlying |
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293 | mesh using the formula given in the module doc string. |
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294 | |
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295 | Inputs: |
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296 | point_coordinates: The co-ordinates of the data points. |
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297 | List of coordinate pairs [x, y] of |
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298 | data points or an nx2 Numeric array or a Geospatial_data object |
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299 | z: Single 1d vector or array of data at the point_coordinates. |
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300 | |
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301 | """ |
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302 | if point_coordinates is None: |
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303 | assert self.AtA <> None |
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304 | assert self.Atz <> None |
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305 | #FIXME (DSG) - do a message |
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306 | else: |
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307 | point_coordinates = ensure_absolute(point_coordinates, |
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308 | geo_reference=point_origin) |
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309 | self.build_fit_subset(point_coordinates, z, verbose) |
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310 | |
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311 | #Check sanity |
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312 | m = self.mesh.coordinates.shape[0] #Nbr of basis functions (1/vertex) |
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313 | n = self.point_count |
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314 | if n<m and self.alpha == 0.0: |
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315 | msg = 'ERROR (least_squares): Too few data points\n' |
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316 | msg += 'There are only %d data points and alpha == 0. ' %n |
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317 | msg += 'Need at least %d\n' %m |
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318 | msg += 'Alternatively, set smoothing parameter alpha to a small ' |
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319 | msg += 'positive value,\ne.g. 1.0e-3.' |
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320 | raise ToFewPointsError(msg) |
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321 | |
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322 | self._build_coefficient_matrix_B(verbose) |
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323 | loners = self.mesh.get_lone_vertices() |
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324 | # FIXME - make this as error message. |
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325 | # test with |
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326 | # Not_yet_test_smooth_att_to_mesh_with_excess_verts. |
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327 | if len(loners)>0: |
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328 | msg = 'WARNING: (least_squares): \nVertices with no triangles\n' |
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329 | msg += 'All vertices should be part of a triangle.\n' |
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330 | msg += 'In the future this will be inforced.\n' |
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331 | msg += 'The following vertices are not part of a triangle;\n' |
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332 | msg += str(loners) |
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333 | print msg |
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334 | #raise VertsWithNoTrianglesError(msg) |
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335 | |
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336 | |
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337 | return conjugate_gradient(self.B, self.Atz, self.Atz, |
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338 | imax=2*len(self.Atz) ) |
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339 | |
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340 | |
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341 | def build_fit_subset(self, point_coordinates, z, |
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342 | verbose = False): |
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343 | """Fit a smooth surface to given 1d array of data points z. |
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344 | |
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345 | The smooth surface is computed at each vertex in the underlying |
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346 | mesh using the formula given in the module doc string. |
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347 | |
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348 | Inputs: |
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349 | point_coordinates: The co-ordinates of the data points. |
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350 | List of coordinate pairs [x, y] of |
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351 | data points or an nx2 Numeric array or a Geospatial_data object |
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352 | z: Single 1d vector or array of data at the point_coordinates. |
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353 | |
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354 | """ |
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355 | #Note: Don't get the z info from Geospatial_data.attributes yet. |
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356 | # If we did fit would have to handle attribute title info. |
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357 | |
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358 | #FIXME(DSG-DSG): Check that the vert and point coords |
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359 | #have the same zone. |
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360 | if isinstance(point_coordinates,Geospatial_data): |
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361 | point_coordinates = point_coordinates.get_data_points( \ |
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362 | absolute = True) |
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363 | |
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364 | #Convert input to Numeric arrays |
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365 | z = ensure_numeric(z, Float) |
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366 | point_coordinates = ensure_numeric(point_coordinates, Float) |
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367 | |
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368 | self._build_matrix_AtA_Atz(point_coordinates, z, verbose) |
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369 | |
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370 | |
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371 | ############################################################################ |
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372 | |
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373 | def fit_to_mesh(vertex_coordinates, |
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374 | triangles, |
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375 | point_coordinates, |
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376 | point_attributes, |
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377 | alpha = DEFAULT_ALPHA, |
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378 | verbose = False, |
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379 | acceptable_overshoot = 1.01, |
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380 | mesh_origin = None, |
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381 | data_origin = None, |
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382 | use_cache = False): |
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383 | """ |
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384 | Fit a smooth surface to a triangulation, |
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385 | given data points with attributes. |
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386 | |
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387 | |
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388 | Inputs: |
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389 | vertex_coordinates: List of coordinate pairs [xi, eta] of |
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390 | points constituting a mesh (or an m x 2 Numeric array or |
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391 | a geospatial object) |
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392 | Points may appear multiple times |
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393 | (e.g. if vertices have discontinuities) |
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394 | |
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395 | triangles: List of 3-tuples (or a Numeric array) of |
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396 | integers representing indices of all vertices in the mesh. |
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397 | |
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398 | point_coordinates: List of coordinate pairs [x, y] of data points |
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399 | (or an nx2 Numeric array) |
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400 | |
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401 | alpha: Smoothing parameter. |
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402 | |
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403 | acceptable overshoot: controls the allowed factor by which fitted values |
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404 | may exceed the value of input data. The lower limit is defined |
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405 | as min(z) - acceptable_overshoot*delta z and upper limit |
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406 | as max(z) + acceptable_overshoot*delta z |
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407 | |
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408 | mesh_origin: A geo_reference object or 3-tuples consisting of |
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409 | UTM zone, easting and northing. |
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410 | If specified vertex coordinates are assumed to be |
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411 | relative to their respective origins. |
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412 | |
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413 | |
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414 | point_attributes: Vector or array of data at the |
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415 | point_coordinates. |
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416 | |
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417 | """ |
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418 | #Since this is a wrapper for fit, lets handle the geo_spatial att's |
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419 | if use_cache is True: |
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420 | interp = cache(_fit, |
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421 | (vertex_coordinates, |
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422 | triangles), |
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423 | {'verbose': verbose, |
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424 | 'mesh_origin': mesh_origin, |
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425 | 'alpha':alpha}, |
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426 | verbose = verbose) |
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427 | |
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428 | else: |
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429 | interp = Fit(vertex_coordinates, |
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430 | triangles, |
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431 | verbose = verbose, |
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432 | mesh_origin = mesh_origin, |
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433 | alpha=alpha) |
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434 | |
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435 | vertex_attributes = interp.fit(point_coordinates, |
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436 | point_attributes, |
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437 | point_origin = data_origin, |
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438 | verbose = verbose) |
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439 | |
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440 | |
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441 | # Add the value checking stuff that's in least squares. |
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442 | # Maybe this stuff should get pushed down into Fit. |
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443 | # at least be a method of Fit. |
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444 | # Or intigrate it into the fit method, saving teh max and min's |
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445 | # as att's. |
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446 | |
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447 | return vertex_attributes |
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448 | |
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449 | |
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450 | def fit_to_mesh_file(mesh_file, point_file, mesh_output_file, |
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451 | alpha=DEFAULT_ALPHA, verbose= False, |
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452 | display_errors = True): |
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453 | """ |
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454 | Given a mesh file (tsh) and a point attribute file (xya), fit |
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455 | point attributes to the mesh and write a mesh file with the |
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456 | results. |
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457 | |
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458 | |
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459 | If data_origin is not None it is assumed to be |
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460 | a 3-tuple with geo referenced |
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461 | UTM coordinates (zone, easting, northing) |
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462 | |
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463 | NOTE: Throws IOErrors, for a variety of file problems. |
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464 | |
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465 | """ |
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466 | |
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467 | # Question |
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468 | # should data_origin and mesh_origin be passed in? |
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469 | # No they should be in the data structure |
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470 | # |
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471 | #Should the origin of the mesh be changed using this function? |
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472 | # That is overloading this function. Have it as a seperate |
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473 | # method, at least initially. |
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474 | |
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475 | from load_mesh.loadASCII import import_mesh_file, \ |
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476 | import_points_file, export_mesh_file, \ |
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477 | concatinate_attributelist |
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478 | |
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479 | # FIXME: Use geospatial instead of import_points_file |
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480 | try: |
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481 | mesh_dict = import_mesh_file(mesh_file) |
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482 | except IOError,e: |
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483 | if display_errors: |
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484 | print "Could not load bad file. ", e |
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485 | raise IOError #Re-raise exception |
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486 | |
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487 | vertex_coordinates = mesh_dict['vertices'] |
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488 | triangles = mesh_dict['triangles'] |
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489 | if type(mesh_dict['vertex_attributes']) == ArrayType: |
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490 | old_point_attributes = mesh_dict['vertex_attributes'].tolist() |
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491 | else: |
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492 | old_point_attributes = mesh_dict['vertex_attributes'] |
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493 | |
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494 | if type(mesh_dict['vertex_attribute_titles']) == ArrayType: |
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495 | old_title_list = mesh_dict['vertex_attribute_titles'].tolist() |
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496 | else: |
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497 | old_title_list = mesh_dict['vertex_attribute_titles'] |
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498 | |
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499 | if verbose: print 'tsh file %s loaded' %mesh_file |
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500 | |
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501 | # load in the .pts file |
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502 | try: |
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503 | point_dict = import_points_file(point_file, verbose=verbose) |
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504 | except IOError,e: |
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505 | if display_errors: |
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506 | print "Could not load bad file. ", e |
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507 | raise IOError #Re-raise exception |
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508 | |
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509 | point_coordinates = point_dict['pointlist'] |
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510 | title_list,point_attributes = concatinate_attributelist(point_dict['attributelist']) |
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511 | |
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512 | if point_dict.has_key('geo_reference') and not point_dict['geo_reference'] is None: |
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513 | data_origin = point_dict['geo_reference'].get_origin() |
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514 | else: |
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515 | data_origin = (56, 0, 0) #FIXME(DSG-DSG) |
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516 | |
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517 | if mesh_dict.has_key('geo_reference') and not mesh_dict['geo_reference'] is None: |
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518 | mesh_origin = mesh_dict['geo_reference'].get_origin() |
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519 | else: |
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520 | mesh_origin = (56, 0, 0) #FIXME(DSG-DSG) |
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521 | |
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522 | if verbose: print "points file loaded" |
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523 | if verbose: print "fitting to mesh" |
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524 | f = fit_to_mesh(vertex_coordinates, |
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525 | triangles, |
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526 | point_coordinates, |
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527 | point_attributes, |
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528 | alpha = alpha, |
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529 | verbose = verbose, |
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530 | data_origin = data_origin, |
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531 | mesh_origin = mesh_origin) |
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532 | if verbose: print "finished fitting to mesh" |
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533 | |
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534 | # convert array to list of lists |
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535 | new_point_attributes = f.tolist() |
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536 | #FIXME have this overwrite attributes with the same title - DSG |
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537 | #Put the newer attributes last |
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538 | if old_title_list <> []: |
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539 | old_title_list.extend(title_list) |
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540 | #FIXME can this be done a faster way? - DSG |
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541 | for i in range(len(old_point_attributes)): |
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542 | old_point_attributes[i].extend(new_point_attributes[i]) |
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543 | mesh_dict['vertex_attributes'] = old_point_attributes |
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544 | mesh_dict['vertex_attribute_titles'] = old_title_list |
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545 | else: |
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546 | mesh_dict['vertex_attributes'] = new_point_attributes |
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547 | mesh_dict['vertex_attribute_titles'] = title_list |
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548 | |
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549 | if verbose: print "exporting to file ", mesh_output_file |
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550 | |
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551 | try: |
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552 | export_mesh_file(mesh_output_file, mesh_dict) |
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553 | except IOError,e: |
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554 | if display_errors: |
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555 | print "Could not write file. ", e |
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556 | raise IOError |
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557 | |
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558 | |
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559 | def _fit(*args, **kwargs): |
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560 | """Private function for use with caching. Reason is that classes |
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561 | may change their byte code between runs which is annoying. |
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562 | """ |
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563 | |
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564 | return Fit(*args, **kwargs) |
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565 | |
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