Mercurial > octave-nkf
annotate doc/interpreter/interp.txi @ 8920:eb63fbe60fab
update copyright notices
| author | John W. Eaton <jwe@octave.org> |
|---|---|
| date | Sat, 07 Mar 2009 10:41:27 -0500 |
| parents | 8463d1a2e544 |
| children | e9dc2ed2ec0f |
| rev | line source |
|---|---|
| 8920 | 1 @c Copyright (C) 2007, 2008, 2009 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} | |
|
8347
fa78cb8d8a5c
corrections for typos
Brian Gough<bjg@network-theory.co.uk>
parents:
8325
diff
changeset
|
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 |
| 37 methods. The 'spline' method enforces that both the first and second | |
| 38 derivatives of the interpolated values have a continuous derivative, | |
| 6743 | 39 whereas the other methods do not. This means that the results of the |
| 40 'spline' method are generally smoother. If the function to be | |
| 41 interpolated is in fact smooth, then 'spline' will give excellent | |
| 42 results. However, if the function to be evaluated is in some manner | |
| 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 |
| 73 @image{interpderiv1,8cm} | |
| 74 @caption{Comparison of 'phcip' and 'spline' interpolation methods for a | |
| 75 step function} | |
| 76 @end float | |
| 77 | |
| 78 @float Figure,fig:interpderiv2 | |
| 79 @image{interpderiv2,8cm} | |
|
8325
b93ac0586e4b
spelling corrections
Brian Gough<bjg@network-theory.co.uk>
parents:
7984
diff
changeset
|
80 @caption{Comparison of the second derivative of the 'phcip' and 'spline' |
| 6743 | 81 interpolation methods for a step function} |
| 6721 | 82 @end float |
| 83 @end ifnotinfo | |
| 84 | |
|
7984
bbaa5d7d0143
Some documentation updates
David Bateman <dbateman@free.fr>
parents:
7018
diff
changeset
|
85 A simplified version of @code{interp1} that performs only linear |
|
bbaa5d7d0143
Some documentation updates
David Bateman <dbateman@free.fr>
parents:
7018
diff
changeset
|
86 interpolation is available in @code{interp1q}. This argument is slightly |
|
bbaa5d7d0143
Some documentation updates
David Bateman <dbateman@free.fr>
parents:
7018
diff
changeset
|
87 faster than @code{interp1} as to performs little error checking. |
|
bbaa5d7d0143
Some documentation updates
David Bateman <dbateman@free.fr>
parents:
7018
diff
changeset
|
88 |
|
bbaa5d7d0143
Some documentation updates
David Bateman <dbateman@free.fr>
parents:
7018
diff
changeset
|
89 @DOCSTRING(interp1q) |
|
bbaa5d7d0143
Some documentation updates
David Bateman <dbateman@free.fr>
parents:
7018
diff
changeset
|
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 | |
| 6702 | 97 There are two significant limitations on Fourier interpolation. Firstly, |
|
8495
e7b4de25ecb6
[docs] non periodic => non-periodic
Brian Gough <bjg@gnu.org>
parents:
8347
diff
changeset
|
98 the function signal is assumed to be periodic, and so non-periodic |
| 6702 | 99 signals will be poorly represented at the edges. Secondly, both the |
| 100 signal and its interpolation are required to be sampled at equispaced | |
| 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 |
|
8495
e7b4de25ecb6
[docs] non periodic => non-periodic
Brian Gough <bjg@gnu.org>
parents:
8347
diff
changeset
|
118 which demonstrates the poor behavior of Fourier interpolation for non-periodic functions. |
| 6721 | 119 @end ifinfo |
| 120 @ifnotinfo | |
|
8495
e7b4de25ecb6
[docs] non periodic => non-periodic
Brian Gough <bjg@gnu.org>
parents:
8347
diff
changeset
|
121 which demonstrates the poor behavior of Fourier interpolation for non-periodic functions, as can be seen in @ref{fig:interpft}. |
| 6721 | 122 |
| 123 @float Figure,fig:interpft | |
| 124 @image{interpft,8cm} | |
|
8495
e7b4de25ecb6
[docs] non periodic => non-periodic
Brian Gough <bjg@gnu.org>
parents:
8347
diff
changeset
|
125 @caption{Comparison of @code{interp1} and @code{interpft} for non-periodic data} |
| 6721 | 126 @end float |
| 127 @end ifnotinfo | |
| 6702 | 128 |
| 129 In additional the support function @code{spline} and @code{lookup} that | |
| 130 underlie the @code{interp1} function can be called directly. | |
| 6549 | 131 |
| 6558 | 132 @DOCSTRING(spline) |
| 133 | |
|
8347
fa78cb8d8a5c
corrections for typos
Brian Gough<bjg@network-theory.co.uk>
parents:
8325
diff
changeset
|
134 The @code{lookup} function is used by other interpolation functions to identify |
| 6702 | 135 the points of the original data that are closest to the current point |
| 136 of interest. | |
| 137 | |
| 6549 | 138 @DOCSTRING(lookup) |
| 6702 | 139 |
| 140 @node Multi-dimensional Interpolation | |
| 141 @section Multi-dimensional Interpolation | |
| 142 | |
| 6939 | 143 There are three multi-dimensional interpolation functions in Octave, with |
| 6850 | 144 similar capabilities. Methods using Delaunay tessellation are described |
| 145 in @ref{Interpolation on Scattered Data}. | |
| 6702 | 146 |
| 147 @DOCSTRING(interp2) | |
| 148 | |
| 149 @DOCSTRING(interp3) | |
| 150 | |
| 151 @DOCSTRING(interpn) | |
| 152 | |
| 153 A significant difference between @code{interpn} and the other two | |
| 6939 | 154 multidimensional interpolation functions is the fashion in which the |
| 6702 | 155 dimensions are treated. For @code{interp2} and @code{interp3}, the 'y' |
| 156 axis is considered to be the columns of the matrix, whereas the 'x' | |
| 6939 | 157 axis corresponds to the rows of the array. As Octave indexes arrays in |
| 6702 | 158 column major order, the first dimension of any array is the columns, and |
| 159 so @code{interpn} effectively reverses the 'x' and 'y' dimensions. | |
| 160 Consider the example | |
| 161 | |
| 162 @example | |
| 163 @group | |
| 164 x = y = z = -1:1; | |
| 165 f = @@(x,y,z) x.^2 - y - z.^2; | |
| 166 [xx, yy, zz] = meshgrid (x, y, z); | |
| 167 v = f (xx,yy,zz); | |
| 6721 | 168 xi = yi = zi = -1:0.1:1; |
| 6702 | 169 [xxi, yyi, zzi] = meshgrid (xi, yi, zi); |
| 6721 | 170 vi = interp3(x, y, z, v, xxi, yyi, zzi, 'spline'); |
| 6702 | 171 [xxi, yyi, zzi] = ndgrid (xi, yi, zi); |
| 6721 | 172 vi2 = interpn(x, y, z, v, xxi, yyi, zzi, 'spline'); |
| 6723 | 173 mesh (zi, yi, squeeze (vi2(1,:,:))); |
| 6702 | 174 @end group |
| 175 @end example | |
| 176 | |
| 177 @noindent | |
| 178 where @code{vi} and @code{vi2} are identical. The reversal of the | |
| 179 dimensions is treated in the @code{meshgrid} and @code{ndgrid} functions | |
| 180 respectively. | |
| 6721 | 181 @ifnotinfo |
| 182 The result of this code can be seen in @ref{fig:interpn}. | |
| 183 | |
| 184 @float Figure,fig:interpn | |
| 185 @image{interpn,8cm} | |
| 186 @caption{Demonstration of the use of @code{interpn}} | |
| 187 @end float | |
| 188 @end ifnotinfo | |
| 6702 | 189 |
| 190 In additional the support function @code{bicubic} that underlies the | |
| 191 cubic interpolation of @code{interp2} function can be called directly. | |
| 192 | |
| 193 @DOCSTRING(bicubic) |