不规则数据的曲面插值
function = 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 | 8484 * * 86
% 使用
% x=;
% y=;
% z=;
% subplot(2,1,1);stem3(x,y,z);title('RAW DATA');
% xi=0:0.1:4;yi=2:0.2:4;
% =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.
%
% = GRIDDATA(X,Y,Z,XI,YI) also returns the XI and YI
% formed this way (the results of = 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))
= 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 = sxyz(:,1);
y = sxyz(:,2);
z = sxyz(:,3);
ind = ;
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 = ;
t2 = ;
t3 = ;
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 = ; i2 = ; i3 = ;
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);
Wna = 1/4*(-2*yy(:,i2).*yy(:,i3)+yy(:,i2).^2+yy(:,i3).^2+xx(:,i2).^2 - ...
2*xx(:,i2).*xx(:,i3)+xx(:,i3).^2).*zz;
Wna(:) = Wna-1/16.*(yy(:,i2).^2-2*yy(:,i2).*yy(:,i3)+yy(:,i3).^2+xx(:,i2).^2- ...
2.*xx(:,i2).*xx(:,i3)+xx(:,i3).^2).*(-xx(:,i2)+2.*xx(:,i1)-xx(:,i3)).*gx;
Wna(:) = Wna-1/16.*(yy(:,i2).^2-2.*yy(:,i2).*yy(:,i3)+yy(:,i3).^2+xx(:,i2).^2- ...
2.*xx(:,i2).*xx(:,i3)+xx(:,i3).^2).*(-yy(:,i2)+2.*yy(:,i1)-yy(:,i3)).*gy;
Wna(:) = Wna./Area./len(:,i1);
Wnb = 1/4*(yy(:,i1).^2+yy(:,i1).*yy(:,i3)-3.*yy(:,i2).*yy(:,i1)+ ...
3.*yy(:,i2).*yy(:,i3)-2.*yy(:,i3).^2+xx(:,i1).^2+xx(:,i1).*xx(:,i3)- ...
3.*xx(:,i2).*xx(:,i1)+3.*xx(:,i2).*xx(:,i3)-2.*xx(:,i3).^2).*zz;
Wnb(:) = Wnb-1/16*(6*yy(:,i1).*xx(:,i2).*yy(:,i3)-3*yy(:,i1).^2.*xx(:,i2)- ...
2*yy(:,i1).*xx(:,i1).*yy(:,i3)+2*yy(:,i1).^2.*xx(:,i1)- ...
4*yy(:,i1).*xx(:,i3).*yy(:,i3)+yy(:,i1).^2.*xx(:,i3)- ...
2*yy(:,i2).*xx(:,i3).*yy(:,i3)+2*yy(:,i2).*xx(:,i3).*yy(:,i1)+ ...
2*yy(:,i2).*xx(:,i1).*yy(:,i3)-2*yy(:,i2).*xx(:,i1).*yy(:,i1)+ ...
3*yy(:,i3).^2.*xx(:,i3)-3*yy(:,i3).^2.*xx(:,i2)-xx(:,i1).^2.*xx(:,i3)+ ...
2*xx(:,i1).^3+10*xx(:,i2).*xx(:,i1).*xx(:,i3)-5*xx(:,i2).*xx(:,i1).^2- ...
4*xx(:,i1).*xx(:,i3).^2-5*xx(:,i3).^2.*xx(:,i2)+3*xx(:,i3).^3).*gx;
Wnb(:) = Wnb-1/16*(-yy(:,i1).^2.*yy(:,i3)+2*yy(:,i1).^3+ ...
10*yy(:,i2).*yy(:,i1).*yy(:,i3)-5*yy(:,i2).*yy(:,i1).^2- ...
4*yy(:,i1).*yy(:,i3).^2-5*yy(:,i3).^2.*yy(:,i2)+3*yy(:,i3).^3+ ...
6*yy(:,i2).*xx(:,i1).*xx(:,i3)-3*yy(:,i2).*xx(:,i1).^2- ...
2*yy(:,i1).*xx(:,i1).*xx(:,i3)+2*yy(:,i1).*xx(:,i1).^2- ...
4*yy(:,i3).*xx(:,i1).*xx(:,i3)+yy(:,i3).*xx(:,i1).^2- ...
2*yy(:,i3).*xx(:,i2).*xx(:,i3)+2*yy(:,i3).*xx(:,i2).*xx(:,i1)+ ...
2*yy(:,i1).*xx(:,i2).*xx(:,i3)-2*yy(:,i1).*xx(:,i2).*xx(:,i1)+ ...
3*xx(:,i3).^2.*yy(:,i3)-3*yy(:,i2).*xx(:,i3).^2).*gy;
Wnb(:) = Wnb./Area./len(:,i2);
Wnc = 1/4*(yy(:,i2).*yy(:,i1)+yy(:,i1).^2-2*yy(:,i2).^2+3*yy(:,i2).*yy(:,i3)- ...
3.*yy(:,i1).*yy(:,i3)+xx(:,i2).*xx(:,i1)+xx(:,i1).^2-2.*xx(:,i2).^2+ ...
3.*xx(:,i2).*xx(:,i3)-3.*xx(:,i1).*xx(:,i3)).*zz;
Wnc(:) = Wnc-1/16*(yy(:,i1).^2.*xx(:,i2)-4*yy(:,i1).*xx(:,i2).*yy(:,i2)+ ...
2*yy(:,i1).^2.*xx(:,i1)-2*yy(:,i2).*xx(:,i1).*yy(:,i1)- ...
3*yy(:,i1).^2.*xx(:,i3)+6*yy(:,i2).*xx(:,i3).*yy(:,i1)- ...
3*yy(:,i2).^2.*xx(:,i3)+3*yy(:,i2).^2.*xx(:,i2)+ ...
2*yy(:,i1).*xx(:,i2).*yy(:,i3)-2*yy(:,i2).*xx(:,i2).*yy(:,i3)- ...
2*yy(:,i1).*xx(:,i1).*yy(:,i3)+2*yy(:,i2).*xx(:,i1).*yy(:,i3)+ ...
2*xx(:,i1).^3-xx(:,i2).*xx(:,i1).^2-4*xx(:,i2).^2.*xx(:,i1)- ...
5*xx(:,i1).^2.*xx(:,i3)+10*xx(:,i2).*xx(:,i1).*xx(:,i3)+ ...
3*xx(:,i2).^3-5*xx(:,i2).^2.*xx(:,i3)).*gx;
Wnc(:) = Wnc-1/16*(2*yy(:,i1).^3-yy(:,i2).*yy(:,i1).^2- ...
4*yy(:,i2).^2.*yy(:,i1)-5*yy(:,i1).^2.*yy(:,i3)+ ...
10*yy(:,i2).*yy(:,i1).*yy(:,i3)+3*yy(:,i2).^3- ...
5*yy(:,i2).^2.*yy(:,i3)+yy(:,i2).*xx(:,i1).^2- ...
4*yy(:,i2).*xx(:,i1).*xx(:,i2)+2*yy(:,i1).*xx(:,i1).^2- ...
2*yy(:,i1).*xx(:,i2).*xx(:,i1)-3*yy(:,i3).*xx(:,i1).^2+ ...
6*yy(:,i3).*xx(:,i2).*xx(:,i1)-3*yy(:,i3).*xx(:,i2).^2+ ...
3*yy(:,i2).*xx(:,i2).^2+2*yy(:,i2).*xx(:,i1).*xx(:,i3)- ...
2*yy(:,i2).*xx(:,i2).*xx(:,i3)-2*yy(:,i1).*xx(:,i1).*xx(:,i3)+ ...
2*yy(:,i1).*xx(:,i2).*xx(:,i3)).*gy;
Wnc(:) = Wnc./Area./len(:,i3);
Wn = Wna(:,) + Wnb(:,) + Wnc(:,);
% 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,:);
Area = Area(t,:);
len = len(t,:);
xx = xx(t,:);
yy = yy(t,:);
zz = zz(t,:);
gx = gx(t,:);
gy = gy(t,:);
gn = gn(t,:);
Wn = Wn(t,:);
% Compute Barycentric coordinates (w).P. 78 in Watson.
w = 1/2.*((xx(:,i2)-repmat(xi,1,3)).*(yy(:,i3)-repmat(yi,1,3)) - ...
(xx(:,i3)-repmat(xi,1,3)).*(yy(:,i2)-repmat(yi,1,3)))./Area;
w(out,:) = ones(length(out),3);
N1 = w(:,i1) + w(:,i1).^2.*w(:,i2) + w(:,i1).^2.*w(:,i3) - ...
w(:,i1).*w(:,i2).^2 - w(:,i1).*w(:,i3).^2;
N2 = (xx(:,i2)-xx(:,i1)).*(w(:,i1).^2.*w(:,i2)+1/2.*w(:,i1).*w(:,i2).*w(:,i3))+ ...
(xx(:,i3)-xx(:,i1)).*(w(:,i1).^2.*w(:,i3)+1/2.*w(:,i1).*w(:,i2).*w(:,i3));
N3 = (yy(:,i2)-yy(:,i1)).*(w(:,i1).^2.*w(:,i2)+1/2.*w(:,i1).*w(:,i2).*w(:,i3))+ ...
(yy(:,i3)-yy(:,i1)).*(w(:,i1).^2.*w(:,i3)+1/2.*w(:,i1).*w(:,i2).*w(:,i3));
N1(out) = zeros(size(out));
N2(out) = zeros(size(out));
N3(out) = zeros(size(out));
M = 8*Area./len.*w(:,i1).*w(:,i2).^2.*w(:,i3).^2 ./ ...
(w(:,i1)+w(:,i2)+(w(:,i1)+w(:,i2)==0)) ./ ...
(w(:,i1)+w(:,i3)+(w(:,i1)+w(:,i3)==0));
M(out,:) = zeros(length(out),3);
zi = sum((N1.*zz + N2.*gx + N3.*gy + M.*(gn - Wn)).').';
zi = reshape(zi,siz);
if ~isempty(out), zi(out) = NaN; end
%------------------------------------------------------------
%------------------------------------------------------------
function zi = nearest(x,y,z,xi,yi)
%NEAREST Triangle-based nearest neightbor 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 a 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
% Find the nearest vertex
k = dsearch(x,y,tri,xi,yi);
zi = k;
d = find(isfinite(k));
zi(d) = z(k(d));
zi = reshape(zi,siz);
%----------------------------------------------------------
%----------------------------------------------------------
function = gdatav4(x,y,z,xi,yi)
%GDATAV4 MATLAB 4 GRIDDATA interpolation
% Reference:David T. Sandwell, Biharmonic spline
% interpolation of GEOS-3 and SEASAT altimeter
% data, Geophysical Research Letters, 2, 139-142,
% 1987.Describes interpolation using value or
% gradient of value in any dimension.
xy = x(:) + y(:)*sqrt(-1);
% Determine distances between points
d = xy(:,ones(1,length(xy)));
d = abs(d - d.');
n = size(d,1);
% Replace zeros along diagonal with ones (so these don't show up in the
% find below or in the Green's function calculation).
d(1:n+1:prod(size(d))) = ones(1,n);
non = find(d == 0);
if ~isempty(non),
% If we've made it to here, then some points aren't distinct.Remove
% the non-distinct points by averaging.
= find(d == 0);
k = find(r < c);
r = r(k); c = c(k); % Extract unique (row,col) pairs
v = (z(r) + z(c))/2; % Average non-distinct pairs
rep = find(diff(c)==0);
if ~isempty(rep), % More than two points need to be averaged.
runs = find(diff(diff(c)==0)==1)+1;
for i=1:length(runs),
k = find(c==c(runs(i))); % All the points in a run
v(runs(i)) = mean(z()); % Average (again)
end
end
z(r) = v;
if ~isempty(rep),
z(r(runs)) = v(runs); % Make sure average is in the dataset
end
% Now remove the extra points.
x(c) = [];
y(c) = [];
z(c) = [];
xy(c,:) = [];
xy(:,c) = [];
d(c,:) = [];
d(:,c) = [];
% Determine the non distinct points
ndp = sort();
ndp(find(ndp(1:length(ndp)-1)==ndp(2:length(ndp)))) = [];
warning(sprintf(['Averaged %d non-distinct points.\n' ...
' Indices are: %s.'],length(ndp),num2str(ndp')))
end
% Determine weights for interpolation
g = (d.^2) .* (log(d)-1); % Green's function.
% Fixup value of Green's function along diagonal
g(1:size(d,1)+1:prod(size(d))) = zeros(size(d,1),1);
weights = g \ z(:);
= size(xi);
zi = zeros(size(xi));
jay = sqrt(-1);
xy = xy.';
% Evaluate at requested points (xi,yi).Loop to save memory.
for i=1:m
for j=1:n
d = abs(xi(i,j)+jay*yi(i,j) - xy);
mask = find(d == 0);
if length(mask)>0, d(mask) = ones(length(mask),1); end
g = (d.^2) .* (log(d)-1); % Green's function.
% Value of Green's function at zero
if length(mask)>0, g(mask) = zeros(length(mask),1); end
zi(i,j) = g * weights;
end
end
if nargout<=1,
xi = zi;
end
%----------------------------------------------------------
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