1 | """Least squares interpolation. |
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2 | |
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3 | Implements a least-squares interpolation. |
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4 | |
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5 | Ole Nielsen, Stephen Roberts, Duncan Gray, Christopher Zoppou |
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6 | Geoscience Australia, 2004. |
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7 | |
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8 | DESIGN ISSUES |
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9 | * what variables should be global? |
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10 | - if there are no global vars functions can be moved around alot easier |
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11 | |
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12 | * What will be the public interface to this class? |
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13 | |
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14 | TO DO |
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15 | * remove points outside the mesh ?(in interpolate_block)? |
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16 | * geo-ref (in interpolate_block) |
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17 | * add functional interpolate interface - in mesh and points, out interp data |
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18 | """ |
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19 | |
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20 | import time |
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21 | |
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22 | from Numeric import zeros, array, Float, Int, dot, transpose, concatenate, \ |
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23 | ArrayType, allclose, take |
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24 | |
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25 | from pyvolution.mesh import Mesh |
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26 | from pyvolution.sparse import Sparse, Sparse_CSR |
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27 | from pyvolution.cg_solve import conjugate_gradient, VectorShapeError |
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28 | from coordinate_transforms.geo_reference import Geo_reference |
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29 | from pyvolution.quad import build_quadtree |
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30 | from utilities.numerical_tools import ensure_numeric |
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31 | from utilities.polygon import inside_polygon |
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32 | |
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33 | from search_functions import search_tree_of_vertices |
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34 | |
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35 | class Interpolate: |
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36 | |
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37 | def __init__(self, |
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38 | vertex_coordinates, |
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39 | triangles, |
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40 | mesh_origin = None, |
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41 | verbose=False, |
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42 | max_vertices_per_cell=30): |
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43 | |
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44 | |
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45 | """ Build interpolation matrix mapping from |
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46 | function values at vertices to function values at data points |
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47 | |
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48 | Inputs: |
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49 | |
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50 | vertex_coordinates: List of coordinate pairs [xi, eta] of |
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51 | points constituting a mesh (or an m x 2 Numeric array) |
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52 | Points may appear multiple times |
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53 | (e.g. if vertices have discontinuities) |
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54 | |
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55 | triangles: List of 3-tuples (or a Numeric array) of |
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56 | integers representing indices of all vertices in the mesh. |
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57 | |
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58 | mesh_origin: 3-tuples consisting of |
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59 | UTM zone, easting and northing. |
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60 | If specified vertex coordinates are assumed to be |
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61 | relative to their respective origins. |
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62 | |
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63 | max_vertices_per_cell: Number of vertices in a quad tree cell |
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64 | at which the cell is split into 4. |
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65 | |
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66 | """ |
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67 | |
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68 | # Initialise variabels |
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69 | self._A_can_be_reused = False |
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70 | self._point_coordinates = None |
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71 | |
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72 | #Convert input to Numeric arrays |
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73 | triangles = ensure_numeric(triangles, Int) |
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74 | vertex_coordinates = ensure_numeric(vertex_coordinates, Float) |
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75 | |
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76 | #Build underlying mesh |
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77 | if verbose: print 'Building mesh' |
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78 | #self.mesh = General_mesh(vertex_coordinates, triangles, |
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79 | #FIXME: Trying the normal mesh while testing precrop, |
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80 | # The functionality of boundary_polygon is needed for that |
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81 | |
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82 | #FIXME - geo ref does not have to go into mesh. |
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83 | # Change the point co-ords to conform to the |
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84 | # mesh co-ords early in the code |
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85 | |
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86 | #FIXME: geo_ref can also be a geo_ref object |
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87 | #FIXME: move this to interpolate_block |
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88 | if mesh_origin is None: |
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89 | geo = None |
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90 | else: |
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91 | geo = Geo_reference(mesh_origin[0],mesh_origin[1],mesh_origin[2]) |
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92 | self.mesh = Mesh(vertex_coordinates, triangles, |
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93 | geo_reference = geo) |
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94 | |
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95 | self.mesh.check_integrity() |
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96 | |
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97 | self.root = build_quadtree(self.mesh, |
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98 | max_points_per_cell = max_vertices_per_cell) |
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99 | |
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100 | |
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101 | def _build_interpolation_matrix_A(self, |
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102 | point_coordinates, |
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103 | verbose = False): |
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104 | """Build n x m interpolation matrix, where |
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105 | n is the number of data points and |
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106 | m is the number of basis functions phi_k (one per vertex) |
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107 | |
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108 | This algorithm uses a quad tree data structure for fast binning |
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109 | of data points |
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110 | origin is a 3-tuple consisting of UTM zone, easting and northing. |
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111 | If specified coordinates are assumed to be relative to this origin. |
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112 | |
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113 | This one will override any data_origin that may be specified in |
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114 | instance interpolation |
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115 | |
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116 | Preconditions |
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117 | Point_coordindates and mesh vertices have the same origin. |
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118 | """ |
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119 | |
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120 | |
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121 | |
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122 | #Convert point_coordinates to Numeric arrays, in case it was a list. |
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123 | point_coordinates = ensure_numeric(point_coordinates, Float) |
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124 | |
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125 | #Remove points falling outside mesh boundary |
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126 | # do this bit later - that sorta means this becomes an object |
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127 | # get a list of what indices are outside the boundary |
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128 | # maybe fill these rows with n/a? |
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129 | |
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130 | |
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131 | #Build n x m interpolation matrix |
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132 | m = self.mesh.coordinates.shape[0] #Nbr of basis functions (1/vertex) |
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133 | n = point_coordinates.shape[0] #Nbr of data points |
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134 | |
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135 | if verbose: print 'Number of datapoints: %d' %n |
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136 | if verbose: print 'Number of basis functions: %d' %m |
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137 | |
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138 | A = Sparse(n,m) |
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139 | |
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140 | #Compute matrix elements |
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141 | for i in range(n): |
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142 | #For each data_coordinate point |
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143 | if verbose and i%((n+10)/10)==0: print 'Doing %d of %d' %(i, n) |
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144 | x = point_coordinates[i] |
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145 | element_found, sigma0, sigma1, sigma2, k = \ |
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146 | search_tree_of_vertices(self.root, self.mesh, x) |
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147 | #Update interpolation matrix A if necessary |
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148 | if element_found is True: |
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149 | #Assign values to matrix A |
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150 | |
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151 | j0 = self.mesh.triangles[k,0] #Global vertex id for sigma0 |
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152 | j1 = self.mesh.triangles[k,1] #Global vertex id for sigma1 |
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153 | j2 = self.mesh.triangles[k,2] #Global vertex id for sigma2 |
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154 | |
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155 | sigmas = {j0:sigma0, j1:sigma1, j2:sigma2} |
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156 | js = [j0,j1,j2] |
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157 | |
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158 | for j in js: |
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159 | A[i,j] = sigmas[j] |
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160 | else: |
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161 | print 'Could not find triangle for point', x |
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162 | return A |
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163 | |
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164 | def _search_tree_of_vertices_OBSOLETE(self, root, mesh, x): |
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165 | """ |
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166 | Find the triangle (element) that the point x is in. |
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167 | |
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168 | root: A quad tree of the vertices |
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169 | Return the associated sigma and k values |
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170 | (and if the element was found) . |
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171 | """ |
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172 | #Find triangle containing x: |
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173 | element_found = False |
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174 | |
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175 | # This will be returned if element_found = False |
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176 | sigma2 = -10.0 |
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177 | sigma0 = -10.0 |
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178 | sigma1 = -10.0 |
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179 | k = -10.0 |
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180 | |
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181 | #Find vertices near x |
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182 | candidate_vertices = root.search(x[0], x[1]) |
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183 | is_more_elements = True |
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184 | |
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185 | element_found, sigma0, sigma1, sigma2, k = \ |
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186 | self._search_triangles_of_vertices(mesh, |
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187 | candidate_vertices, x) |
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188 | while not element_found and is_more_elements: |
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189 | candidate_vertices, branch = root.expand_search() |
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190 | if branch == []: |
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191 | # Searching all the verts from the root cell that haven't |
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192 | # been searched. This is the last try |
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193 | element_found, sigma0, sigma1, sigma2, k = \ |
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194 | self._search_triangles_of_vertices(mesh, |
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195 | candidate_vertices, x) |
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196 | is_more_elements = False |
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197 | else: |
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198 | element_found, sigma0, sigma1, sigma2, k = \ |
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199 | self._search_triangles_of_vertices(mesh, |
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200 | candidate_vertices, x) |
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201 | |
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202 | return element_found, sigma0, sigma1, sigma2, k |
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203 | |
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204 | def _search_triangles_of_vertices_OBSOLETE(self, mesh, candidate_vertices, x): |
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205 | #Find triangle containing x: |
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206 | element_found = False |
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207 | |
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208 | # This will be returned if element_found = False |
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209 | sigma2 = -10.0 |
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210 | sigma0 = -10.0 |
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211 | sigma1 = -10.0 |
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212 | k = -10.0 |
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213 | #print "*$* candidate_vertices", candidate_vertices |
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214 | #For all vertices in same cell as point x |
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215 | for v in candidate_vertices: |
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216 | #FIXME (DSG-DSG): this catches verts with no triangle. |
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217 | #Currently pmesh is producing these. |
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218 | #this should be stopped, |
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219 | if mesh.vertexlist[v] is None: |
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220 | continue |
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221 | #for each triangle id (k) which has v as a vertex |
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222 | for k, _ in mesh.vertexlist[v]: |
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223 | #Get the three vertex_points of candidate triangle |
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224 | xi0 = mesh.get_vertex_coordinate(k, 0) |
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225 | xi1 = mesh.get_vertex_coordinate(k, 1) |
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226 | xi2 = mesh.get_vertex_coordinate(k, 2) |
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227 | |
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228 | #Get the three normals |
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229 | n0 = mesh.get_normal(k, 0) |
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230 | n1 = mesh.get_normal(k, 1) |
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231 | n2 = mesh.get_normal(k, 2) |
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232 | |
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233 | #Compute interpolation |
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234 | sigma2 = dot((x-xi0), n2)/dot((xi2-xi0), n2) |
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235 | sigma0 = dot((x-xi1), n0)/dot((xi0-xi1), n0) |
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236 | sigma1 = dot((x-xi2), n1)/dot((xi1-xi2), n1) |
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237 | |
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238 | #FIXME: Maybe move out to test or something |
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239 | epsilon = 1.0e-6 |
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240 | assert abs(sigma0 + sigma1 + sigma2 - 1.0) < epsilon |
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241 | |
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242 | #Check that this triangle contains the data point |
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243 | |
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244 | #Sigmas can get negative within |
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245 | #machine precision on some machines (e.g nautilus) |
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246 | #Hence the small eps |
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247 | eps = 1.0e-15 |
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248 | if sigma0 >= -eps and sigma1 >= -eps and sigma2 >= -eps: |
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249 | element_found = True |
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250 | break |
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251 | |
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252 | if element_found is True: |
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253 | #Don't look for any other triangle |
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254 | break |
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255 | return element_found, sigma0, sigma1, sigma2, k |
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256 | |
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257 | |
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258 | |
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259 | # FIXME: What is a good start_blocking_count value? |
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260 | def interpolate(self, f, point_coordinates = None, |
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261 | start_blocking_len = 500000, verbose=False): |
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262 | """Interpolate mesh data f to determine values, z, at points. |
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263 | |
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264 | f is the data on the mesh vertices. |
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265 | |
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266 | The mesh values representing a smooth surface are |
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267 | assumed to be specified in f. |
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268 | |
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269 | Inputs: |
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270 | f: Vector or array of data at the mesh vertices. |
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271 | If f is an array, interpolation will be done for each column as |
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272 | per underlying matrix-matrix multiplication |
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273 | point_coordinates: Interpolate mesh data to these positions. |
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274 | List of coordinate pairs [x, y] of |
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275 | data points (or an nx2 Numeric array) |
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276 | If point_coordinates is absent, the points inputted last time |
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277 | this method was called are used, if possible. |
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278 | start_blocking_len: If the # of points is more or greater than this, |
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279 | start blocking |
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280 | |
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281 | Output: |
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282 | Interpolated values at inputted points (z). |
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283 | """ |
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284 | |
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285 | # Can I interpolate, based on previous point_coordinates? |
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286 | if point_coordinates is None: |
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287 | if self._A_can_be_reused is True and \ |
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288 | len(self._point_coordinates) < start_blocking_len: |
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289 | z = self._get_point_data_z(f, |
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290 | verbose=verbose) |
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291 | elif self._point_coordinates is not None: |
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292 | # if verbose, give warning |
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293 | if verbose: |
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294 | print 'WARNING: Recalculating A matrix, due to blocking.' |
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295 | point_coordinates = self._point_coordinates |
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296 | else: |
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297 | #There are no good point_coordinates. import sys; sys.exit() |
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298 | msg = 'ERROR (interpolate.py): No point_coordinates inputted' |
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299 | raise msg |
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300 | |
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301 | if point_coordinates is not None: |
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302 | self._point_coordinates = point_coordinates |
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303 | if len(point_coordinates) < start_blocking_len or \ |
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304 | start_blocking_len == 0: |
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305 | self._A_can_be_reused = True |
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306 | z = self.interpolate_block(f, point_coordinates, |
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307 | verbose=verbose) |
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308 | else: |
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309 | #Handle blocking |
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310 | self._A_can_be_reused = False |
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311 | start=0 |
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312 | z = self.interpolate_block(f, point_coordinates[0:0]) |
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313 | for end in range(start_blocking_len |
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314 | ,len(point_coordinates) |
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315 | ,start_blocking_len): |
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316 | t = self.interpolate_block(f, point_coordinates[start:end], |
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317 | verbose=verbose) |
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318 | z = concatenate((z,t)) |
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319 | start = end |
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320 | end = len(point_coordinates) |
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321 | t = self.interpolate_block(f, point_coordinates[start:end], |
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322 | verbose=verbose) |
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323 | z = concatenate((z,t)) |
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324 | return z |
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325 | |
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326 | def interpolate_block(self, f, point_coordinates = None, verbose=False): |
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327 | """ |
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328 | Call this if you want to control the blocking or make sure blocking |
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329 | doesn't occur. |
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330 | |
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331 | See interpolate for doc info. |
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332 | """ |
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333 | if point_coordinates is not None: |
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334 | self._A =self._build_interpolation_matrix_A(point_coordinates, |
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335 | verbose=verbose) |
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336 | return self._get_point_data_z(f) |
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337 | |
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338 | def _get_point_data_z(self, f, verbose=False): |
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339 | return self._A * f |
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340 | |
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341 | #------------------------------------------------------------- |
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342 | if __name__ == "__main__": |
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343 | a = [0.0, 0.0] |
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344 | b = [0.0, 2.0] |
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345 | c = [2.0,0.0] |
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346 | points = [a, b, c] |
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347 | vertices = [ [1,0,2] ] #bac |
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348 | |
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349 | data = [ [2.0/3, 2.0/3] ] #Use centroid as one data point |
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350 | |
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351 | interp = Interpolate(points, vertices) #, data) |
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352 | A = interp._build_interpolation_matrix_A(data, verbose=True) |
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353 | A = A.todense() |
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354 | print "A",A |
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355 | assert allclose(A, [[1./3, 1./3, 1./3]]) |
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356 | print "finished" |
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