Mercurial > hg > octave-nkf
annotate doc/interpreter/interp.txi @ 14291:c39c7f2b1e46 gui
Replaced local-native with local in octave-gui.pro for building on MacOS.
* octave-gui.pro: Modified INCFLAGS and LFLAGS.
| author | Jacob Dawid <jacob.dawid@googlemail.com> |
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| date | Tue, 31 Jan 2012 02:12:56 +0100 |
| parents | 72c96de7a403 |
| children | c3fd61c59e9c |
| rev | line source |
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1 @c Copyright (C) 2007-2012 John W. Eaton |
| 7018 | 2 @c |
| 3 @c This file is part of Octave. | |
| 4 @c | |
| 5 @c Octave is free software; you can redistribute it and/or modify it | |
| 6 @c under the terms of the GNU General Public License as published by the | |
| 7 @c Free Software Foundation; either version 3 of the License, or (at | |
| 8 @c your option) any later version. | |
| 9 @c | |
| 10 @c Octave is distributed in the hope that it will be useful, but WITHOUT | |
| 11 @c ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or | |
| 12 @c FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License | |
| 13 @c for more details. | |
| 14 @c | |
| 15 @c You should have received a copy of the GNU General Public License | |
| 16 @c along with Octave; see the file COPYING. If not, see | |
| 17 @c <http://www.gnu.org/licenses/>. | |
| 6549 | 18 |
| 19 @node Interpolation | |
| 20 @chapter Interpolation | |
| 21 | |
| 6702 | 22 @menu |
| 23 * One-dimensional Interpolation:: | |
| 24 * Multi-dimensional Interpolation:: | |
| 25 @end menu | |
| 26 | |
| 27 @node One-dimensional Interpolation | |
| 28 @section One-dimensional Interpolation | |
| 6549 | 29 |
| 6850 | 30 Octave supports several methods for one-dimensional interpolation, most |
| 31 of which are described in this section. @ref{Polynomial Interpolation} | |
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32 and @ref{Interpolation on Scattered Data} describe further methods. |
| 6850 | 33 |
| 6549 | 34 @DOCSTRING(interp1) |
| 35 | |
| 6721 | 36 There are some important differences between the various interpolation |
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37 methods. The 'spline' method enforces that both the first and second |
| 6721 | 38 derivatives of the interpolated values have a continuous derivative, |
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39 whereas the other methods do not. This means that the results of the |
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40 'spline' method are generally smoother. If the function to be |
| 6743 | 41 interpolated is in fact smooth, then 'spline' will give excellent |
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42 results. However, if the function to be evaluated is in some manner |
| 6743 | 43 discontinuous, then 'pchip' interpolation might give better results. |
| 44 | |
| 45 This can be demonstrated by the code | |
| 6721 | 46 |
| 47 @example | |
| 48 @group | |
| 6743 | 49 t = -2:2; |
| 50 dt = 1; | |
| 51 ti =-2:0.025:2; | |
| 52 dti = 0.025; | |
| 53 y = sign(t); | |
| 54 ys = interp1(t,y,ti,'spline'); | |
| 55 yp = interp1(t,y,ti,'pchip'); | |
| 56 ddys = diff(diff(ys)./dti)./dti; | |
| 57 ddyp = diff(diff(yp)./dti)./dti; | |
| 58 figure(1); | |
| 59 plot (ti, ys,'r-', ti, yp,'g-'); | |
| 60 legend('spline','pchip',4); | |
| 61 figure(2); | |
| 62 plot (ti, ddys,'r+', ti, ddyp,'g*'); | |
| 63 legend('spline','pchip'); | |
| 6721 | 64 @end group |
| 65 @end example | |
| 66 | |
| 67 @ifnotinfo | |
| 68 @noindent | |
| 6743 | 69 The result of which can be seen in @ref{fig:interpderiv1} and |
| 70 @ref{fig:interpderiv2}. | |
| 6721 | 71 |
| 6743 | 72 @float Figure,fig:interpderiv1 |
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73 @center @image{interpderiv1,4in} |
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74 @caption{Comparison of 'pchip' and 'spline' interpolation methods for a |
| 6743 | 75 step function} |
| 76 @end float | |
| 77 | |
| 78 @float Figure,fig:interpderiv2 | |
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79 @center @image{interpderiv2,4in} |
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80 @caption{Comparison of the second derivative of the 'pchip' and 'spline' |
| 6743 | 81 interpolation methods for a step function} |
| 6721 | 82 @end float |
| 83 @end ifnotinfo | |
| 84 | |
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85 A simplified version of @code{interp1} that performs only linear |
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86 interpolation is available in @code{interp1q}. This argument is slightly |
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87 faster than @code{interp1} as to performs little error checking. |
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88 |
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89 @DOCSTRING(interp1q) |
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90 |
| 6702 | 91 Fourier interpolation, is a resampling technique where a signal is |
| 92 converted to the frequency domain, padded with zeros and then | |
| 93 reconverted to the time domain. | |
| 6549 | 94 |
| 95 @DOCSTRING(interpft) | |
| 96 | |
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97 There are two significant limitations on Fourier interpolation. Firstly, |
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98 the function signal is assumed to be periodic, and so non-periodic |
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99 signals will be poorly represented at the edges. Secondly, both the |
| 6702 | 100 signal and its interpolation are required to be sampled at equispaced |
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101 points. An example of the use of @code{interpft} is |
| 6549 | 102 |
| 6702 | 103 @example |
| 104 @group | |
| 105 t = 0 : 0.3 : pi; dt = t(2)-t(1); | |
| 106 n = length (t); k = 100; | |
| 107 ti = t(1) + [0 : k-1]*dt*n/k; | |
| 108 y = sin (4*t + 0.3) .* cos (3*t - 0.1); | |
| 109 yp = sin (4*ti + 0.3) .* cos (3*ti - 0.1); | |
| 110 plot (ti, yp, 'g', ti, interp1(t, y, ti, 'spline'), 'b', ... | |
| 111 ti, interpft (y, k), 'c', t, y, 'r+'); | |
| 112 legend ('sin(4t+0.3)cos(3t-0.1','spline','interpft','data'); | |
| 113 @end group | |
| 114 @end example | |
| 115 | |
| 8828 | 116 @noindent |
| 6721 | 117 @ifinfo |
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118 which demonstrates the poor behavior of Fourier interpolation for non-periodic |
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119 functions. |
| 6721 | 120 @end ifinfo |
| 121 @ifnotinfo | |
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122 which demonstrates the poor behavior of Fourier interpolation for non-periodic |
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123 functions, as can be seen in @ref{fig:interpft}. |
| 6721 | 124 |
| 125 @float Figure,fig:interpft | |
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126 @center @image{interpft,4in} |
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127 @caption{Comparison of @code{interp1} and @code{interpft} for non-periodic data} |
| 6721 | 128 @end float |
| 129 @end ifnotinfo | |
| 6702 | 130 |
| 131 In additional the support function @code{spline} and @code{lookup} that | |
| 132 underlie the @code{interp1} function can be called directly. | |
| 9367 | 133 @ref{Finding Elements and Checking Conditions} |
| 6549 | 134 |
| 6558 | 135 @DOCSTRING(spline) |
| 136 | |
| 6702 | 137 @node Multi-dimensional Interpolation |
| 138 @section Multi-dimensional Interpolation | |
| 139 | |
| 6939 | 140 There are three multi-dimensional interpolation functions in Octave, with |
| 6850 | 141 similar capabilities. Methods using Delaunay tessellation are described |
| 142 in @ref{Interpolation on Scattered Data}. | |
| 6702 | 143 |
| 144 @DOCSTRING(interp2) | |
| 145 | |
| 146 @DOCSTRING(interp3) | |
| 147 | |
| 148 @DOCSTRING(interpn) | |
| 149 | |
| 150 A significant difference between @code{interpn} and the other two | |
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151 multi-dimensional interpolation functions is the fashion in which the |
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152 dimensions are treated. For @code{interp2} and @code{interp3}, the 'y' |
| 6702 | 153 axis is considered to be the columns of the matrix, whereas the 'x' |
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154 axis corresponds to the rows of the array. As Octave indexes arrays in |
| 6702 | 155 column major order, the first dimension of any array is the columns, and |
| 156 so @code{interpn} effectively reverses the 'x' and 'y' dimensions. | |
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157 Consider the example, |
| 6702 | 158 |
| 159 @example | |
| 160 @group | |
| 161 x = y = z = -1:1; | |
| 162 f = @@(x,y,z) x.^2 - y - z.^2; | |
| 163 [xx, yy, zz] = meshgrid (x, y, z); | |
| 164 v = f (xx,yy,zz); | |
| 6721 | 165 xi = yi = zi = -1:0.1:1; |
| 6702 | 166 [xxi, yyi, zzi] = meshgrid (xi, yi, zi); |
| 6721 | 167 vi = interp3(x, y, z, v, xxi, yyi, zzi, 'spline'); |
| 6702 | 168 [xxi, yyi, zzi] = ndgrid (xi, yi, zi); |
| 6721 | 169 vi2 = interpn(x, y, z, v, xxi, yyi, zzi, 'spline'); |
| 6723 | 170 mesh (zi, yi, squeeze (vi2(1,:,:))); |
| 6702 | 171 @end group |
| 172 @end example | |
| 173 | |
| 174 @noindent | |
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175 where @code{vi} and @code{vi2} are identical. The reversal of the |
| 6702 | 176 dimensions is treated in the @code{meshgrid} and @code{ndgrid} functions |
| 177 respectively. | |
| 6721 | 178 @ifnotinfo |
| 179 The result of this code can be seen in @ref{fig:interpn}. | |
| 180 | |
| 181 @float Figure,fig:interpn | |
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182 @center @image{interpn,4in} |
| 6721 | 183 @caption{Demonstration of the use of @code{interpn}} |
| 184 @end float | |
| 185 @end ifnotinfo | |
| 6702 | 186 |
| 187 In additional the support function @code{bicubic} that underlies the | |
| 188 cubic interpolation of @code{interp2} function can be called directly. | |
| 189 | |
| 190 @DOCSTRING(bicubic) |