一元函数插值

nshukwrd

1. function yi = interp1(varargin)
   
2. % yi=interp1(x,y,xi)根据数据(x,y)给出在xi的线性插值结果yi.
   
3. % yi=interp1(x,y,xi,'spline')使用三次样条插值.
   
4. % yi=interp1(x,y,xi,'cubic')使用三次插值.
   
5. % 例如
   
6. %   clear;close;fplot('sin',[0,2*pi]);hold on;
   
7. %   x=0:2*pi;y=sin(x);
   
8. %   h1=plot(x,y,'ko');
   
9. %   xi=[1:5]+0.5;
   
10. %   yi=interp1(x,y,xi,'linear');
    
11. %   h2=plot(xi,yi,'kx');
    
12. %   yi=interp1(x,y,xi,'spline');
    
13. %   h3=plot(xi,yi,'k*');
    
14. %   legend([h1;h2;h3],'data','linear','spline');
    
15. %   hold off;
    
16. %
    
17. %INTERP1 1-D interpolation (table lookup).
    
18. %   YI = INTERP1(X,Y,XI) interpolates to find YI, the values of
    
19. %   the underlying function Y at the points in the vector XI.
    
20. %   The vector X specifies the points at which the data Y is
    
21. %   given. If Y is a matrix, then the interpolation is performed
    
22. %   for each column of Y and YI will be length(XI)-by-size(Y,2).
    
23. %   Out of range values are returned as NaN.
    
24. %
    
25. %   YI = INTERP1(Y,XI) assumes X = 1:N, where N is the length(Y)
    
26. %   for vector Y or SIZE(Y,1) for matrix Y.
    
27. %
    
28. %   Interpolation is the same operation as "table lookup".  Described in
    
29. %   "table lookup" terms, the "table" is [X,Y] and INTERP1 "looks-up"
    
30. %   the elements of XI in X, and, based upon their location, returns
    
31. %   values YI interpolated within the elements of Y.
    
32. %
    
33. %   YI = INTERP1(X,Y,XI,'method') specifies alternate methods.
    
34. %   The default is linear interpolation.  Available methods are:
    
35. %
    
36. %     'nearest' - nearest neighbor interpolation
    
37. %     'linear'  - linear interpolation
    
38. %     'spline'  - cubic spline interpolation
    
39. %     'cubic'   - cubic interpolation
    
40. %
    
41. %   All the interpolation methods require that X be monotonic. X can be
    
42. %   non-uniformly spaced.  For faster interpolation when X is equally
    
43. %   spaced and monotonic, use the methods '*linear', '*cubic', '*nearest'.
    
44. %   For faster linear interpolation when X is non-uniformly spaced
    
45. %   see INTERP1Q.
    
46. %
    
47. %   For example, generate a coarse sine curve and interpolate over a
    
48. %   finer abscissa:
    
49. %       x = 0:10; y = sin(x); xi = 0:.25:10;
    
50. %       yi = interp1(x,y,xi); plot(x,y,'o',xi,yi)
    
51. %
    
52. %   See also INTERP1Q, INTERPFT, SPLINE, INTERP2, INTERP3, INTERPN.
    
53. 
    
54. %   Copyright (c) 1984-98 by The MathWorks, Inc.
    
55. %   $Revision: 5.25 $  $Date: 1997/11/21 23:40:40 $
    
56. 
    
57. error(nargchk(2,4,nargin));
    
58. 
    
59. bypass = 0;
    
60. uniform = 1;
    
61. if isstr(varargin{end}),
    
62.   narg = nargin-1;
    
63.   method = [varargin{end} '    ']; % Protect against short string.
    
64.   if method(1)=='*', % Direct call bypass.
    
65.     if method(2)=='l', % linear interpolation.
    
66.       yi = linear(varargin{1:end-1});
    
67.       return
    
68. 
    
69.     elseif method(2)=='c', % cubic interpolation
    
70.       yi = cubic(varargin{1:end-1});
    
71.       return
    
72. 
    
73.     elseif method(2)=='n', % Nearest neighbor interpolation
    
74.       yi = nearest(varargin{1:end-1});
    
75.       return
    
76. 
    
77.     elseif method(2)=='s', % Spline interpolation
    
78.       method = 'spline'; bypass = 1;
    
79. 
    
80.     else
    
81.       error([deblank(method),' is an invalid method.']);
    
82. 
    
83.     end
    
84.   elseif method(1)=='s', % Spline interpolation
    
85.     method = 'spline'; bypass = 1;
    
86.   end
    
87.   
    
88. else
    
89.   narg = nargin;
    
90.   method = 'linear';
    
91. end
    
92. 
    
93. if narg==2, % interp1(y,xi)
    
94.   y = varargin{1};
    
95.   if min(size(y))==1, x = 1:length(y); else x = 1:size(y,1); end
    
96.   [msg,x,y,xi] = xychk(x,varargin{1:2});
    
97. 
    
98. elseif narg==3, % interp1(x,y,xi)
    
99.   [msg,x,y,xi] = xychk(varargin{1:3});
    
100. 
     
101. end
     
102. 
     
103. if ~isempty(msg), error(msg); end
     
104. 
     
105. if isempty(xi), yi = []; return, end
     
106. if min(size(xi))~=1, error('XI must be a vector.'); end
     
107. 
     
108. x = x(:); % Make sure x is a column vector.
     
109. if min(size(y))==1, y = y(:); end % Make sure y is a column vector.
     
110. siz = size(xi); xi = xi(:); % Make sure xi is a column vector.
     
111. 
     
112. %
     
113. % Check for non-equally spaced data.  If so, map x and
     
114. % xi to matrix (row,col) coordinate system.
     
115. %
     
116. if length(x)>2 & ~bypass,
     
117.   dx = diff(x);
     
118.   if max(abs(diff(dx))) > eps*max(x),
     
119.     if any(dx < 0), % Flip orientation of data so x is increasing.
     
120.       if size(x,1)==1,
     
121.         x = fliplr(x); y = fliplr(y);
     
122.         dx = -fliplr(dx);
     
123.       else
     
124.         x(:) = flipud(x); y(:) = flipud(y);
     
125.         dx(:) = -flipud(dx);
     
126.       end
     
127.     end
     
128. 
     
129.     if any(dx<=0),
     
130.       error('X must be monotonic.');
     
131.       return
     
132.     end
     
133. 
     
134.     % Only nearest needs this mapping code now
     
135.    if method(1)=='n',
     
136. 
     
137.       % Determine the nearest location of xi in x
     
138.       [xxi,j] = sort(xi(:));
     
139.       [dum,i] = sort([x;xxi]);
     
140.       ui(i) = 1:length(i);
     
141.       ui = (ui(length(x)+1:end) - (1:length(xxi)))';
     
142.       ui(j) = ui;
     
143.    
     
144.       % Map values in xi to index offset (ui) via linear interpolation
     
145.       ui(ui<1) = 1;
     
146.       ui(ui>length(x)-1) = length(x)-1;
     
147.       ui = ui + (xi(:)-x(ui))./(x(ui+1)-x(ui));
     
148.    
     
149.       x = (1:length(x)).';
     
150.       xi = ui;
     
151.     else
     
152.       uniform = 0;
     
153.     end
     
154.   end
     
155. end
     
156. 
     
157. %
     
158. % Now do the interpolation based on the method flag
     
159. %
     
160. if method(1)=='n', % Nearest neighbor interpolation
     
161.   yi = nearest(x,y,xi);
     
162. 
     
163. elseif method(1)=='l', % Linear interpolation
     
164.   if uniform
     
165.     yi = linear(x,y,xi);
     
166.   else
     
167.     yi = interp1q(x,y,xi);
     
168.   end
     
169. 
     
170. elseif method(1)=='s', % Spline interpolation
     
171.   yi = spline(x,y.',xi(:).').';
     
172. 
     
173. elseif method(1)=='c', % Cubic interpolation
     
174.   if uniform
     
175.     yi = cubic(x,y,xi);
     
176.   else
     
177.     d = find(xi < min(x) | xi > max(x));
     
178.     yi = spline(x,y.',xi(:).').';
     
179.     if min(size(yi))==1, yi(d) = NaN; else yi(d,:) = NaN; end
     
180.   end
     
181. 
     
182. else
     
183.   error([deblank(method),' is an invalid method.']);
     
184. 
     
185. end
     
186. 
     
187. if (min(size(yi))==1) & (prod(siz)>1), yi = reshape(yi,siz); end
     
188. 
     
189. 
     
190. %------------------------------------------------------
     
191. function F=linear(x,y,u)
     
192. %LINEAR Linear Interpolation of a 1-D function.
     
193. %   F=LINEAR(Y,XI) returns the value of the 1-D function Y at the
     
194. %   points XI using linear interpolation. length(F)=length(XI). XI is
     
195. %   an index into the vector Y. Y is the value of the function
     
196. %   evaluated uniformly on a interval. If Y is a matrix, then
     
197. %   the interpolation is performed for each column of Y in which
     
198. %   case F is length(XI)-by-size(Y,2).
     
199. %
     
200. %   If Y is of length N then XI must contain values between 1 and N.
     
201. %   The value NaN is returned if this is not the case.
     
202. %
     
203. %   F = LINEAR(X,Y,XI) uses the vector X to specify the coordinates
     
204. %   of the underlying interval. X must be equally spaced and
     
205. %   monotonic. NaN's are returned for values of XI outside the
     
206. %   coordinates in X.
     
207. %
     
208. %   See also CUBIC, INTERP1.
     
209. 
     
210. %   Clay M. Thompson 7-4-91
     
211. 
     
212. if nargin==2,   % No X specified.
     
213.   u = y; y = x;
     
214.   % Check for vector problem.  If so, make everything a column vector.
     
215.   if min(size(y))==1, y = y(:); end
     
216.   [nrows,ncols] = size(y);
     
217. 
     
218. elseif nargin==3, % X specified.
     
219.   % Check for vector problem.  If so, make everything a column vector.
     
220.   if min(size(y))==1, y = y(:); end
     
221.   if min(size(x))==1, x = x(:); end
     
222.   [nrows,ncols] = size(y);
     
223.   % Scale and shift u to be indices into Y.
     
224.   if (min(size(x))~=1), error('X must be a vector.'); end
     
225.   [m,n] = size(x);
     
226.   if m ~= nrows,
     
227.     error('The length of X must match the number of rows of Y.');
     
228.   end
     
229.   u = 1 + (u-x(1))/(x(m)-x(1))*(nrows-1);
     
230.   
     
231. else
     
232.   error('Wrong number of input arguments.');
     
233. end
     
234. 
     
235. if isempty(u), F = []; return, end
     
236. if nrows<2, error('Y must have at least 2 rows.'); end
     
237. 
     
238. siz = size(u);
     
239. u = u(:); % Make sure u is a vector
     
240. u = u(:,ones(1,ncols)); % Expand u
     
241. [m,n] = size(u);
     
242. 
     
243. % Check for out of range values of u and set to 1
     
244. uout = find(u<1 | u>nrows);
     
245. if ~isempty(uout), u(uout) = 1; end
     
246. 
     
247. % Interpolation parameters, check for boundary value.
     
248. s = (u - floor(u));
     
249. u = floor(u);
     
250. if isempty(u), d = u; else d = find(u==nrows); end
     
251. if length(d)>0, u(d) = u(d)-1; s(d) = s(d)+1; end
     
252. 
     
253. % Now interpolate.
     
254. v = (0:n-1)*nrows;
     
255. ndx = u+v(ones(m,1),:);
     
256. F =  ( y(ndx).*(1-s) + y(ndx+1).*s );
     
257. 
     
258. % Now set out of range values to NaN.
     
259. if ~isempty(uout), F(uout) = NaN; end
     
260. 
     
261. if (min(size(F))==1) & (prod(siz)>1), F = reshape(F,siz); end
     
262. 
     
263. 
     
264. %------------------------------------------------------
     
265. function F=cubic(x,y,u)
     
266. %CUBIC Cubic Interpolation of a 1-D function.
     
267. %   YI=CUBIC(Y,XI) returns the value of the 1-D function Y at the
     
268. %   points XI using cubic interpolation. length(YI)=length(XI). XI is
     
269. %   an index into the vector Y. Y is the value of the function
     
270. %   evaluated uniformly on a interval. If Y is a matrix, then
     
271. %   the interpolation is performed for each column of Y in which
     
272. %   case F is length(XI)-by-size(Y,2).
     
273. %
     
274. %   If Y is of length N then XI must contain values between 1 and N.
     
275. %   The value NaN is returned if this is not the case.
     
276. %
     
277. %   YI = CUBIC(X,Y,XI) uses the vector X to specify the coordinates
     
278. %   of the underlying interval. X must be equally spaced and
     
279. %   monotonic. NaN's are returned for values of XI outside the
     
280. %   coordinates in X.
     
281. %
     
282. %   See also LINEAR, INTERP1.
     
283. 
     
284. %   Clay M. Thompson 7-4-91
     
285. 
     
286. %   Based on "Cubic Convolution Interpolation for Digital Image
     
287. %   Processing", Robert G. Keys, IEEE Trans. on Acoustics, Speech, and
     
288. %   Signal Processing, Vol. 29, No. 6, Dec. 1981, pp. 1153-1160.
     
289. 
     
290. if nargin==2,   % No X specified.
     
291.   u = y; y = x;
     
292.   % Check for vector problem.  If so, make everything a column vector.
     
293.   if min(size(y))==1, y = y(:); end
     
294.   [nrows,ncols] = size(y);
     
295. 
     
296. elseif nargin==3, % X specified.
     
297.   % Check for vector problem.  If so, make everything a column vector.
     
298.   if min(size(y))==1, y = y(:); end
     
299.   if min(size(x))==1, x = x(:); end
     
300.   [nrows,ncols] = size(y);
     
301.   % Scale and shift u to be indices into Y.
     
302.   if (min(size(x))~=1), error('X must be a vector.'); end
     
303.   [m,n] = size(x);
     
304.   if m ~= nrows,
     
305.     error('The length of X must match the number of rows of Y.');
     
306.   end
     
307.   u = 1 + (u-x(1))/(x(m)-x(1))*(nrows-1);
     
308.   
     
309. else
     
310.   error('Wrong number of input arguments.');
     
311. end
     
312. 
     
313. if isempty(u), F = []; return, end
     
314. if nrows<3, error('Y must have at least 3 rows.'); end
     
315. 
     
316. siz = size(u); u = u(:); % Make sure u is a vector.
     
317. u = u(:,ones(1,ncols)); % Expand u
     
318. [m,n] = size(u);
     
319. 
     
320. % Check for out of range values of u and set to 1
     
321. if isempty(u), uout = u; else uout = find(u<1 | u>nrows); end
     
322. if ~isempty(uout), u(uout) = 1; end
     
323. 
     
324. % Interpolation parameters, check for boundary value.
     
325. s = (u - floor(u));
     
326. u = floor(u);
     
327. if isempty(u), d = u; else d = find(u==nrows); end
     
328. if length(d)>0, u(d) = u(d)-1; s(d) = s(d)+1; end
     
329. 
     
330. % Expand y so interpolation is valid at the boundary.
     
331. y = [3*y(1,:)-3*y(2,:)+y(3,:);y;3*y(nrows,:)-3*y(nrows-1,:)+y(nrows-2,:)];
     
332. nrows = nrows + 2;
     
333. 
     
334. % Now interpolate using computationally efficient algorithm.
     
335. s2 = s.*s; s3 = s.*s2;
     
336. v = (0:n-1)*nrows;
     
337. ndx = u+v(ones(m,1),:);
     
338. F = y(ndx).*(-s3+2*s2-s) + y(ndx+1).*(3*s3-5*s2+2) + ...
     
339.     y(ndx+2).*(-3*s3+4*s2+s) + y(ndx+3).*(s3-s2);
     
340. F = F/2;
     
341. 
     
342. % Now set out of range values to NaN.
     
343. if ~isempty(uout), F(uout) = NaN; end
     
344. 
     
345. if (min(size(F))==1) & (prod(siz)>1), F = reshape(F,siz); end
     
346. 
     
347. 
     
348. %------------------------------------------------------
     
349. function F = nearest(x,y,u)
     
350. %NEAREST Nearest Neighbor Interpolation of a 1-D function.
     
351. %   YI=NEAREST(Y,XI) returns the value of the 1-D function Y at
     
352. %   the points XI using nearest neighbor interpolation,
     
353. %   length(YI)=length(XI). XI is an index into the vector Y. Y is
     
354. %   the value of the function evaluated uniformly on a interval. If
     
355. %   Y is a matrix, then the interpolation is performed for each
     
356. %   column of Y in which case F is length(XI)-by-size(Y,2).
     
357. %
     
358. %   If Y is of length N then XI must contain values between 1 and N.
     
359. %   The value NaN is returned if this is not the case.
     
360. %
     
361. %   YI = NEAREST(X,Y,XI) uses the vector X to specify the
     
362. %   coordinates of the underlying interval. X must be equally spaced
     
363. %   and monotonic. NaN's are returned for values of XI outside the
     
364. %   coordinates in X.
     
365. %
     
366. %   See also LINEAR, INTERP1.
     
367. 
     
368. %   Clay M. Thompson 7-4-91
     
369. 
     
370. if nargin==2,   % No X specified.
     
371.   u = y; y = x;
     
372.   % Check for vector problem.  If so, make everything a column vector.
     
373.   if min(size(y))==1, y = y(:); end
     
374.   [nrows,ncols] = size(y);
     
375. 
     
376. elseif nargin==3, % X specified.
     
377.   % Check for vector problem.  If so, make everything a column vector.
     
378.   if min(size(y))==1, y = y(:); end
     
379.   if min(size(x))==1, x = x(:); end
     
380.   [nrows,ncols] = size(y);
     
381.   % Scale and shift u to be indices into Y.
     
382.   if (min(size(x))~=1), error('X must be a vector.'); end
     
383.   [m,n] = size(x);
     
384.   if m ~= nrows,
     
385.     error('The length of X must match the number of rows of Y.');
     
386.   end
     
387.   u = 1 + (u-x(1))/(x(m)-x(1))*(nrows-1);
     
388.   
     
389. else
     
390.   error('Wrong number of input arguments.');
     
391. end
     
392. 
     
393. if isempty(u), F = []; return, end
     
394. if nrows<2, error('Y must have at least 2 rows.'); end
     
395. 
     
396. siz = size(u); u = u(:); % Make sure u is a vector.
     
397. u = u(:,ones(1,ncols)); % Expand u
     
398. [m,n] = size(u);
     
399. 
     
400. % Check for out of range values of u and set to 1
     
401. uout = find(u<.5 | u>=nrows+.5);
     
402. if ~isempty(uout), u(uout) = 1; end
     
403. 
     
404. % Interpolation parameters
     
405. s = (u - round(u));
     
406. u = round(u);
     
407. 
     
408. % Now interpolate
     
409. v = (0:n-1)*nrows;
     
410. ndx = u+v(ones(m,1),:);
     
411. F = y(ndx);
     
412. 
     
413. % Now set out of range values to NaN.
     
414. if ~isempty(uout), F(uout) = NaN; end
     
415. 
     
416. if (min(size(F))==1) & (prod(siz)>1), F = reshape(F,siz); end
     

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