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1 ## Copyright (C) 2000 Kai Habel |
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2 ## |
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3 ## This file is part of Octave. |
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4 ## |
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5 ## Octave is free software; you can redistribute it and/or modify it |
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6 ## under the terms of the GNU General Public License as published by |
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7 ## the Free Software Foundation; either version 2, or (at your option) |
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8 ## any later version. |
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9 ## |
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10 ## Octave is distributed in the hope that it will be useful, but |
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11 ## WITHOUT ANY WARRANTY; without even the implied warranty of |
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12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
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13 ## General Public License for more details. |
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14 ## |
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15 ## You should have received a copy of the GNU General Public License |
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16 ## along with Octave; see the file COPYING. If not, write to the Free |
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17 ## Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA |
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18 ## 02110-1301, USA. |
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19 |
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20 ## -*- texinfo -*- |
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21 ## @deftypefn {Function File} {@var{zi}=} interp2 (@var{x}, @var{y}, @var{z}, @var{xi}, @var{yi}) |
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22 ## @deftypefnx {Function File} {@var{zi}=} interp2 (@var{Z}, @var{xi}, @var{yi}) |
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23 ## @deftypefnx {Function File} {@var{zi}=} interp2 (@var{Z}, @var{n}) |
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24 ## @deftypefnx {Function File} {@var{zi}=} interp2 (@dots{}, @var{method}) |
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25 ## @deftypefnx {Function File} {@var{zi}=} interp2 (@dots{}, @var{method}, @var{extrapval}) |
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26 ## |
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27 ## Two-dimensional interpolation. @var{x}, @var{y} and @var{z} describe a |
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28 ## surface function. If @var{x} and @var{y} are vectors their length |
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29 ## must correspondent to the size of @var{z}. @var{x} and @var{Yy} must be |
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30 ## monotonic. If they are matrices they must have the @code{meshgrid} |
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31 ## format. |
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32 ## |
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33 ## @table @code |
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34 ## @item interp2 (@var{x}, @var{y}, @var{Z}, @var{xi}, @var{yi}, @dots{}) |
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35 ## Returns a matrix corresponding to the points described by the |
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36 ## matrices @var{XI}, @var{YI}. |
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37 ## |
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38 ## If the last argument is a string, the interpolation method can |
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39 ## be specified. The method can be 'linear', 'nearest' or 'cubic'. |
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40 ## If it is omitted 'linear' interpolation is assumed. |
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41 ## |
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42 ## @item interp2 (@var{z}, @var{xi}, @var{yi}) |
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43 ## Assumes @code{@var{x} = 1:rows (@var{z})} and @code{@var{y} = |
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44 ## 1:columns (@var{z})} |
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45 ## |
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46 ## @item interp2 (@var{z}, @var{n}) |
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47 ## Interleaves the Matrix @var{z} n-times. If @var{n} is omitted a value |
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48 ## of @code{@var{n} = 1} is assumed. |
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49 ## @end table |
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50 ## |
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51 ## The variable @var{method} defines the method to use for the |
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52 ## interpolation. It can take one of the values |
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53 ## |
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54 ## @table @asis |
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55 ## @item 'nearest' |
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56 ## Return the nearest neighbor. |
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57 ## @item 'linear' |
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58 ## Linear interpolation from nearest neighbors. |
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59 ## @item 'pchip' |
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60 ## Piece-wise cubic hermite interpolating polynomial (not implemented yet). |
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61 ## @item 'cubic' |
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62 ## Cubic interpolation from four nearest neighbors. |
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63 ## @item 'spline' |
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64 ## Cubic spline interpolation--smooth first and second derivatives |
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65 ## throughout the curve. |
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66 ## @end table |
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67 ## |
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68 ## If a scalar value @var{extrapval} is defined as the final value, then |
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69 ## values outside the mesh as set to this value. Note that in this case |
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70 ## @var{method} must be defined as well. If @var{extrapval} is not |
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71 ## defined then NA is assumed. |
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72 ## |
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73 ## @seealso{interp1} |
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74 ## @end deftypefn |
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75 |
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76 ## Author: Kai Habel <kai.habel@gmx.de> |
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77 ## 2005-03-02 Thomas Weber <weber@num.uni-sb.de> |
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78 ## * Add test cases |
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79 ## 2005-03-02 Paul Kienzle <pkienzle@users.sf.net> |
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80 ## * Simplify |
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81 ## 2005-04-23 Dmitri A. Sergatskov <dasergatskov@gmail.com> |
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82 ## * Modified demo and test for new gnuplot interface |
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83 ## 2005-09-07 Hoxide <hoxide_dirac@yahoo.com.cn> |
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84 ## * Add bicubic interpolation method |
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85 ## * Fix the eat line bug when the last element of XI or YI is |
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86 ## negative or zero. |
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87 ## 2005-11-26 Pierre Baldensperger <balden@libertysurf.fr> |
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88 ## * Rather big modification (XI,YI no longer need to be |
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89 ## "meshgridded") to be consistent with the help message |
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90 ## above and for compatibility. |
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91 |
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92 function ZI = interp2 (varargin) |
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93 Z = X = Y = XI = YI = n = []; |
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94 method = "linear"; |
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95 extrapval = NA; |
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96 |
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97 switch (nargin) |
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98 case 1 |
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99 Z = varargin{1}; |
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100 case 2 |
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101 if (ischar (varargin{2})) |
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102 [Z, method] = deal (varargin{:}); |
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103 else |
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104 [Z, n] = deal (varargin{:}); |
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105 endif |
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106 case 3 |
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107 if (ischar (varargin{3})) |
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108 [Z, n, method] = deal (varargin{:}); |
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109 else |
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110 [Z, XI, YI] = deal (varargin{:}); |
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111 endif |
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112 case 4 |
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113 if (ischar (varargin{4})) |
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114 [Z, XI, YI, method] = deal (varargin{:}); |
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115 else |
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116 [Z, n, method, extrapval] = deal (varargin{:}); |
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117 endif |
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118 case 5 |
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119 if (ischar (varargin{4})) |
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120 [Z, XI, YI, method, extrapval] = deal (varargin{:}); |
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121 else |
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122 [X, Y, Z, XI, YI] = deal (varargin{:}); |
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123 endif |
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124 case 6 |
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125 [X, Y, Z, XI, YI, method] = deal (varargin{:}); |
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126 case 7 |
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127 [X, Y, Z, XI, YI, method, extrapval] = deal (varargin{:}); |
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128 otherwise |
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129 print_usage (); |
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130 endswitch |
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131 |
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132 ## Type checking. |
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133 if (!ismatrix (Z)) |
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134 error ("interp2 expected matrix Z"); |
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135 endif |
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136 if (!isempty (n) && !isscalar (n)) |
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137 error ("interp2 expected scalar n"); |
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138 endif |
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139 if (!ischar (method)) |
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140 error ("interp2 expected string 'method'"); |
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141 endif |
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142 if (!isscalar (extrapval)) |
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143 error ("interp2 expected n extrapval"); |
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144 endif |
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145 |
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146 ## Define X, Y, XI, YI if needed |
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147 [zr, zc] = size (Z); |
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148 if (isempty (X)) |
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149 X = 1:zc; |
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150 Y = 1:zr; |
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151 endif |
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152 if (! isnumeric (X) || ! isnumeric (Y)) |
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153 error ("interp2 expected numeric X, Y"); |
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154 endif |
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155 if (! isempty (n)) |
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156 p = 2^n; |
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157 XI = (p:p*zc)/p; |
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158 YI = (p:p*zr)'/p; |
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159 endif |
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160 if (! isnumeric (XI) || ! isnumeric (YI)) |
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161 error ("interp2 expected numeric XI, YI"); |
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162 endif |
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163 |
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164 |
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165 if (strcmp (method, "linear") || strcmp (method, "nearest")) |
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166 |
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167 ## If X and Y vectors produce a grid from them |
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168 if (isvector (X) && isvector (Y)) |
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169 [X, Y] = meshgrid (X, Y); |
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170 elseif (! size_equal (X, Y)) |
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171 error ("X and Y must be matrices of same size"); |
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172 endif |
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173 if (! size_equal (X, Z)) |
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174 error ("X and Y size must match Z dimensions"); |
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175 endif |
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176 |
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177 ## If Xi and Yi are vectors of different orientation build a grid |
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178 if ((rows (XI) == 1 && columns (YI) == 1) |
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179 || (columns (XI) == 1 && rows (YI) == 1)) |
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180 [XI, YI] = meshgrid (XI, YI); |
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181 elseif (! size_equal (XI, YI)) |
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182 error ("XI and YI must be matrices of same size"); |
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183 endif |
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184 |
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185 shape = size (XI); |
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186 XI = reshape (XI, 1, prod (shape)); |
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187 YI = reshape (YI, 1, prod (shape)); |
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188 |
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189 xidx = lookup (X(1, 2:end-1), XI) + 1; |
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190 yidx = lookup (Y(2:end-1, 1), YI) + 1; |
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191 |
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192 if (strcmp (method, "linear")) |
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193 ## each quad satisfies the equation z(x,y)=a+b*x+c*y+d*xy |
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194 ## |
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195 ## a-b |
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196 ## | | |
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197 ## c-d |
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198 a = Z(1:(zr - 1), 1:(zc - 1)); |
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199 b = Z(1:(zr - 1), 2:zc) - a; |
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200 c = Z(2:zr, 1:(zc - 1)) - a; |
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201 d = Z(2:zr, 2:zc) - a - b - c; |
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202 |
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203 idx = sub2ind (size (a), yidx, xidx); |
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204 |
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205 ## scale XI, YI values to a 1-spaced grid |
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206 Xsc = (XI - X(1, xidx)) ./ (X(1, xidx + 1) - X(1, xidx)); |
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207 Ysc = (YI - Y(yidx, 1)') ./ (Y(yidx + 1, 1) - Y(yidx, 1))'; |
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208 |
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209 ## apply plane equation |
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210 ZI = a(idx) + b(idx).*Xsc + c(idx).*Ysc + d(idx).*Xsc.*Ysc; |
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211 |
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212 elseif (strcmp (method, "nearest")) |
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213 xtable = X(1, :); |
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214 ytable = Y(:, 1)'; |
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215 ii = (XI - xtable(xidx) > xtable(xidx + 1) - XI); |
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216 jj = (YI - ytable(yidx) > ytable(yidx + 1) - YI); |
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217 idx = sub2ind (size (Z), yidx+jj, xidx+ii); |
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218 ZI = Z(idx); |
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219 endif |
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220 |
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221 ## set points outside the table to 'extrapval' |
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222 if (X (1, 1) < X (1, end)) |
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223 if (Y (1, 1) < Y (end, 1)) |
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224 ZI (XI < X(1,1) | XI > X(1,end) | YI < Y(1,1) | YI > Y(end,1)) = ... |
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225 extrapval; |
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226 else |
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227 ZI (XI < X(1,1) | XI > X(1,end) | YI < Y(end,1) | YI > Y(1,1)) = ... |
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228 extrapval; |
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229 endif |
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230 else |
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231 if (Y (1, 1) < Y (end, 1)) |
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232 ZI (XI < X(1,end) | XI > X(1,1) | YI < Y(1,1) | YI > Y(end,1)) = ... |
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233 extrapval; |
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234 else |
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235 ZI (XI < X(1,end) | XI > X(1,1) | YI < Y(end,1) | YI > Y(1,1)) = ... |
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236 extrapval; |
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237 endif |
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238 endif |
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239 |
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240 ZI = reshape (ZI, shape); |
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241 else |
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242 |
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243 ## If X and Y vectors produce a grid from them |
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244 if (isvector (X) && isvector (Y)) |
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245 X = X(:).'; |
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246 Y = Y(:); |
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247 if (!isequal ([length(X), length(Y)], size(Z))) |
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248 error ("X and Y size must match Z dimensions"); |
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249 endif |
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250 elseif (!size_equal (X, Y)) |
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251 error ("X and Y must be matrices of same size"); |
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252 if (! size_equal (X, Z)) |
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253 error ("X and Y size must match Z dimensions"); |
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254 endif |
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255 endif |
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256 |
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257 ## If Xi and Yi are vectors of different orientation build a grid |
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258 if ((rows (XI) == 1 && columns (YI) == 1) |
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259 || (columns (XI) == 1 && rows (YI) == 1)) |
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260 ## Do nothing |
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261 elseif (! size_equal (XI, YI)) |
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262 error ("XI and YI must be matrices of same size"); |
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263 endif |
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264 |
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265 ## FIXME bicubic/__splinen__ don't handle arbitrary XI, YI |
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266 if (strcmp (method, "cubic")) |
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267 ZI = bicubic (X, Y, Z, XI(1,:), YI(:,1), extrapval); |
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268 |
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269 elseif (strcmp (method, "spline")) |
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270 ZI = __splinen__ ({Y(:,1).', X(1,:)}, Z, {YI(:,1), XI(1,:)}, extrapval, |
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271 "spline"); |
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272 else |
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273 error ("interpolation method not recognized"); |
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274 endif |
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275 |
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276 endif |
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277 endfunction |
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278 |
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279 %!demo |
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280 %! A=[13,-1,12;5,4,3;1,6,2]; |
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281 %! x=[0,1,4]; y=[10,11,12]; |
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282 %! xi=linspace(min(x),max(x),17); |
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283 %! yi=linspace(min(y),max(y),26)'; |
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284 %! mesh(xi,yi,interp2(x,y,A,xi,yi,'linear')); |
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285 %! [x,y] = meshgrid(x,y); |
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286 %! hold on; plot3(x(:),y(:),A(:),"b*"); hold off; |
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287 |
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288 %!demo |
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289 %! A=[13,-1,12;5,4,3;1,6,2]; |
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290 %! x=[0,1,4]; y=[10,11,12]; |
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291 %! xi=linspace(min(x),max(x),17); |
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292 %! yi=linspace(min(y),max(y),26)'; |
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293 %! mesh(xi,yi,interp2(x,y,A,xi,yi,'nearest')); |
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294 %! [x,y] = meshgrid(x,y); |
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295 %! hold on; plot3(x(:),y(:),A(:),"b*"); hold off; |
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296 |
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297 %!demo |
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298 %! A=[13,-1,12;5,4,3;1,6,2]; |
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299 %! x=[0,1,2]; y=[10,11,12]; |
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300 %! xi=linspace(min(x),max(x),17); |
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301 %! yi=linspace(min(y),max(y),26)'; |
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302 %! mesh(xi,yi,interp2(x,y,A,xi,yi,'cubic')); |
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303 %! [x,y] = meshgrid(x,y); |
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304 %! hold on; plot3(x(:),y(:),A(:),"b*"); hold off; |
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305 |
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306 %!demo |
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307 %! A=[13,-1,12;5,4,3;1,6,2]; |
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308 %! x=[0,1,2]; y=[10,11,12]; |
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309 %! xi=linspace(min(x),max(x),17); |
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310 %! yi=linspace(min(y),max(y),26)'; |
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311 %! mesh(xi,yi,interp2(x,y,A,xi,yi,'spline')); |
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312 %! [x,y] = meshgrid(x,y); |
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313 %! hold on; plot3(x(:),y(:),A(:),"b*"); hold off; |
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314 |
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315 %!test % simple test |
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316 %! x = [1,2,3]; |
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317 %! y = [4,5,6,7]; |
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318 %! [X, Y] = meshgrid(x,y); |
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319 %! Orig = X.^2 + Y.^3; |
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320 %! xi = [1.2,2, 1.5]; |
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321 %! yi = [6.2, 4.0, 5.0]'; |
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322 %! |
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323 %! Expected = ... |
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324 %! [243, 245.4, 243.9; |
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325 %! 65.6, 68, 66.5; |
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326 %! 126.6, 129, 127.5]; |
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327 %! Result = interp2(x,y,Orig, xi, yi); |
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328 %! |
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329 %! assert(Result, Expected, 1000*eps); |
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330 |
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331 %!test % 2^n form |
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332 %! x = [1,2,3]; |
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333 %! y = [4,5,6,7]; |
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334 %! [X, Y] = meshgrid(x,y); |
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335 %! Orig = X.^2 + Y.^3; |
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336 %! xi = [1:0.25:3]; yi = [4:0.25:7]'; |
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337 %! Expected = interp2(x,y,Orig, xi, yi); |
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338 %! Result = interp2(Orig,2); |
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339 %! |
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340 %! assert(Result, Expected, 10*eps); |
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341 |
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342 %!test % matrix slice |
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343 %! A = eye(4); |
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344 %! assert(interp2(A,[1:4],[1:4]),[1,1,1,1]); |
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345 |
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346 %!test % non-gridded XI,YI |
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347 %! A = eye(4); |
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348 %! assert(interp2(A,[1,2;3,4],[1,3;2,4]),[1,0;0,1]); |
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349 |
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350 %!test % for values outside of boundaries |
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351 %! x = [1,2,3]; |
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352 %! y = [4,5,6,7]; |
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353 %! [X, Y] = meshgrid(x,y); |
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354 %! Orig = X.^2 + Y.^3; |
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355 %! xi = [0,4]; |
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356 %! yi = [3,8]'; |
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357 %! assert(interp2(x,y,Orig, xi, yi),[NA,NA;NA,NA]); |
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358 %! assert(interp2(x,y,Orig, xi, yi,'linear', 0),[0,0;0,0]); |
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359 |
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360 %!test % for values at boundaries |
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361 %! A=[1,2;3,4]; |
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362 %! x=[0,1]; |
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363 %! y=[2,3]'; |
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364 %! assert(interp2(x,y,A,x,y,'linear'), A); |
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365 %! assert(interp2(x,y,A,x,y,'nearest'), A); |
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366 |
