griddata.m

数学建模的源代码

function [xi,yi,zi] = griddata(x,y,z,xi,yi,method)
%不规则数据的曲面插值
%ZI=griddata(x,y,z,XI,YI)
%  这里x,y,z均为向量(不必单调)表示数据.
%  XI,YI为网格数据矩阵.griddata采用三角形线性插值.
%ZI=griddata(x,y,z,XI,YI,'cubic') 采用三角形三次插值
%例题  如果数据残缺不全(x,y,z)
%      | 0   1    2     3    4 
%-----|------------------------
%  2  | *    *   80    82   84 
%  3  | 79   *   61    65   * 
%  4  | 84  84    *     *   86
% 使用
%   x=[2,3,4,0,2,3,0,1,4];
%   y=[2,2,2,3,3,3,4,4,4];
%   z=[80,82,84,79,61,65,84,84,86];
%   subplot(2,1,1);stem3(x,y,z);title('RAW DATA');
%   xi=0:0.1:4;yi=2:0.2:4;
%   [XI,YI]=meshgrid(xi,yi);
%   ZI=griddata(x,y,z,XI,YI,'cubic');
%   subplot(2,1,2);mesh(XI,YI,ZI);title('GRIDDATA');
%
%GRIDDATA Data gridding and surface fitting.
%   ZI = GRIDDATA(X,Y,Z,XI,YI) fits a surface of the form Z = F(X,Y)
%   to the data in the (usually) nonuniformly-spaced vectors (X,Y,Z)
%   GRIDDATA interpolates this surface at the points specified by
%   (XI,YI) to produce ZI.  The surface always goes through the data
%   points.  XI and YI are usually a uniform grid (as produced by
%   MESHGRID) and is where GRIDDATA gets its name.
%
%   XI can be a row vector, in which case it specifies a matrix with
%   constant columns. Similarly, YI can be a column vector and it 
%   specifies a matrix with constant rows. 
%
%   [XI,YI,ZI] = GRIDDATA(X,Y,Z,XI,YI) also returns the XI and YI
%   formed this way (the results of [XI,YI] = MESHGRID(XI,YI)).
%
%   [...] = GRIDDATA(...,'method') where 'method' is one of
%       'linear'    - Triangle-based linear interpolation (default).
%       'cubic'     - Triangle-based cubic interpolation.
%       'nearest'   - Nearest neighbor interpolation.
%       'v4'        - MATLAB 4 griddata method.
%   defines the type of surface fit to the data. The 'cubic' and 'v4'
%   methods produce smooth surfaces while 'linear' and 'nearest' have
%   discontinuities in the first and zero-th derivative respectively.  All
%   the methods except 'v4' are based on a Delaunay triangulation of the
%   data.
%
%   See also INTERP2, DELAUNAY, MESHGRID.

%   Clay M. Thompson 8-21-95
%   Copyright (c) 1984-98 by The MathWorks, Inc.
%   $Revision: 5.22 $  $Date: 1997/11/21 23:40:37 $

error(nargchk(5,6,nargin))

[msg,x,y,z,xi,yi] = xyzchk(x,y,z,xi,yi);
if ~isempty(msg), error(msg); end

if nargin<6, method = 'linear'; end
if ~isstr(method), 
  error('METHOD must be one of ''linear'',''cubic'',''nearest'', or ''v4''.');
end


% Sort x and y so duplicate points can be averaged before passing to delaunay

% Need x,y and z to be column vectors
sz = prod(size(x));
x = reshape(x,sz,1);
y = reshape(y,sz,1);
z = reshape(z,sz,1);
sxyz = sortrows([x y z],[2 1]);
x = sxyz(:,1);
y = sxyz(:,2);
z = sxyz(:,3);
ind = [0; y(2:end) == y(1:end-1) & x(2:end) == x(1:end-1); 0];
if sum(ind) > 0
  warning('Duplicate x-y data points detected: using average of the z values');
  fs = find(ind(1:end-1) == 0 & ind(2:end) == 1);
  fe = find(ind(1:end-1) == 1 & ind(2:end) == 0);
  for i = 1 : length(fs)
    % averaging z values
    z(fe(i)) = mean(z(fs(i):fe(i)));
  end
  x = x(~ind(2:end));
  y = y(~ind(2:end));
  z = z(~ind(2:end));
end

switch lower(method),
  case 'linear'
    zi = linear(x,y,z,xi,yi);
  case 'cubic'
    zi = cubic(x,y,z,xi,yi);
  case 'nearest'
    zi = nearest(x,y,z,xi,yi);
  case {'invdist','v4'}
    zi = gdatav4(x,y,z,xi,yi);
  otherwise
    error('Unknown method.');
end
  
if nargout<=1, xi = zi; end


%------------------------------------------------------------
function zi = linear(x,y,z,xi,yi)
%LINEAR Triangle-based linear interpolation

%   Reference: David F. Watson, "Contouring: A guide
%   to the analysis and display of spacial data", Pergamon, 1994.

siz = size(xi);
xi = xi(:); yi = yi(:); % Treat these as columns
x = x(:); y = y(:); % Treat these as columns

% Triangularize the data
tri = delaunay(x,y,'sorted');
if isempty(tri),
  warning('Data cannot be triangulated.');
  zi = repmat(NaN,size(xi));
  return
end

% Find the nearest triangle (t)
t = tsearch(x,y,tri,xi,yi);

% Only keep the relevant triangles.
out = find(isnan(t));
if ~isempty(out), t(out) = ones(size(out)); end
tri = tri(t,:);

% Compute Barycentric coordinates (w).  P. 78 in Watson.
del = (x(tri(:,2))-x(tri(:,1))) .* (y(tri(:,3))-y(tri(:,1))) - ...
      (x(tri(:,3))-x(tri(:,1))) .* (y(tri(:,2))-y(tri(:,1)));
w(:,3) = ((x(tri(:,1))-xi).*(y(tri(:,2))-yi) - ...
          (x(tri(:,2))-xi).*(y(tri(:,1))-yi)) ./ del;
w(:,2) = ((x(tri(:,3))-xi).*(y(tri(:,1))-yi) - ...
          (x(tri(:,1))-xi).*(y(tri(:,3))-yi)) ./ del;
w(:,1) = ((x(tri(:,2))-xi).*(y(tri(:,3))-yi) - ...
          (x(tri(:,3))-xi).*(y(tri(:,2))-yi)) ./ del;
w(out,:) = zeros(length(out),3);

z = z(:).'; % Treat z as a row so that code below involving
            % z(tri) works even when tri is 1-by-3.
zi = sum(z(tri) .* w,2);

zi = reshape(zi,siz);

if ~isempty(out), zi(out) = NaN; end
%------------------------------------------------------------

%------------------------------------------------------------
function zi = cubic(x,y,z,xi,yi)
%TRIANGLE Triangle-based cubic interpolation

%   Reference: T. Y. Yang, "Finite Element Structural Analysis",
%   Prentice Hall, 1986.  pp. 446-449.
%
%   Reference: David F. Watson, "Contouring: A guide
%   to the analysis and display of spacial data", Pergamon, 1994.

siz = size(xi);
xi = xi(:); yi = yi(:); % Treat these as columns
x = x(:); y = y(:); z = z(:); % Treat these as columns

% Triangularize the data
tri = delaunay(x,y,'sorted');
if isempty(tri), 
  warning('Data cannot be triangulated.');
  zi = repmat(NaN,size(xi));
  return
end

%
% Estimate the gradient as the average the triangle slopes connected
% to each vertex
%
t1 = [x(tri(:,1)) y(tri(:,1)) z(tri(:,1))];
t2 = [x(tri(:,2)) y(tri(:,2)) z(tri(:,2))];
t3 = [x(tri(:,3)) y(tri(:,3)) z(tri(:,3))];
Area = ((x(tri(:,2))-x(tri(:,1))) .* (y(tri(:,3))-y(tri(:,1))) - ...
       (x(tri(:,3))-x(tri(:,1))) .* (y(tri(:,2))-y(tri(:,1))))/2;
nv = cross((t3-t1).',(t2-t1).').';

% Normalize normals
nv = nv ./ repmat(nv(:,3),1,3);

% Sparse matrix is non-zero if the triangle specified the row
% index is connected to the point specified by the column index.
% Gradient estimate is area weighted average of triangles 
% around a datum.
m = size(tri,1);
n = length(x);
i = repmat((1:m)',1,3);
T = sparse(i,tri,repmat(-nv(1:m,1).*Area,1,3),m,n);
A = sparse(i,tri,repmat(Area,1,3),m,n);
s = full(sum(A));
gx = (full(sum(T))./(s + (s==0)))';
T = sparse(i,tri,repmat(-nv(1:m,2).*Area,1,3),m,n);
gy = (full(sum(T))./(s + (s==0)))';

% Compute triangle areas and side lengths
i1 = [1 2 3]; i2 = [2 3 1]; i3 = [3 1 2];
xx = x(tri);
yy = y(tri);
zz = z(tri);
gx = gx(tri);
gy = gy(tri);
len = sqrt((xx(:,i3)-xx(:,i2)).^2 + (yy(:,i3)-yy(:,i2)).^2);

% Compute average normal slope
gn = ((gx(:,i2)+gx(:,i3)).*(yy(:,i2)-yy(:,i3)) - ...
      (gy(:,i2)+gy(:,i3)).*(xx(:,i2)-xx(:,i3)))/2./len;

% Compute triangle normal edge gradient at the center of each side (Wn)
Area = repmat(Area,1,3);