griddata2.m

function [xi,yi,zi] = griddata2(x,y,z,xi,yi,method,options)
%GRIDDATA Data gridding and surface fitting.
%
%   GRIDDATA is not recommended. Use TriScatteredInterp instead.
%
%   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(X,Y,Z,XI,YI,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.
%   If METHOD is [], then the default 'linear' method will be used.
%
%   [...] = GRIDDATA(X,Y,Z,XI,YI,METHOD,OPTIONS) specifies a cell array of 
%   strings OPTIONS that were previously used by Qhull. Qhull-specific 
%   OPTIONS are no longer required and are currently ignored. Support for 
%   these options will be removed in a future release. 
%
%   Example:
%      x = rand(100,1)*4-2; y = rand(100,1)*4-2; z = x.*exp(-x.^2-y.^2);
%      ti = -2:.25:2; 
%      [xi,yi] = meshgrid(ti,ti);
%      zi = griddata(x,y,z,xi,yi);
%      mesh(xi,yi,zi), hold on, plot3(x,y,z,'o'), hold off
%
%   See also TriScatteredInterp, DelaunayTri, GRIDDATAN, DELAUNAY, 
%   INTERP2, MESHGRID, DELAUNAYN.
%
% Note, this is the same as the Matlab function griddata.m save that
% "&&~strcmp(method,'invdist')" has been added to the line "if ( nargin < 6
% || isempty(method) ) &&~strcmp(method,'invdist'),  method = 'linear';
% end".  This is necessary because in Matlab V12 they changed griddata in
% such a way that it is no longer compatible with EEGLAB's topoplot
% function (or my topoplotMK.m version of topoplot.m).

%   Copyright 1984-2009 The MathWorks, Inc. 
%   $Revision: 5.33.4.11 $  $Date: 2009/04/21 03:25:31 $

error(nargchk(5,7,nargin,'struct'));

[msg,x,y,z,xi,yi] = xyzchk(x,y,z,xi,yi);
if ~isempty(msg), error(msg); end
if ndims(x) > 2 || ndims(y) > 2 || ndims(xi) > 2 || ndims(yi) > 2
    error('MATLAB:griddata:HigherDimArray',...
          'X,Y and XI,YI cannot be arrays of dimension greater than two.');
end

if ( issparse(x) || issparse(y) || issparse(z) || issparse(xi) || issparse(yi) )
    error('MATLAB:griddata:InvalidDataSparse',...
        'Input data cannot be sparse.');
end

if ( ~isreal(x) || ~isreal(y) || ~isreal(xi) || ~isreal(yi) )
    error('MATLAB:griddata:InvalidDataComplex',...
        'Input data cannot be complex.');
end

if ( nargin < 6 || isempty(method) ) &&~strcmp(method,'invdist'),  method = 'linear'; end
if ~ischar(method), 
  error('MATLAB:griddata:InvalidMethod',...
        'METHOD must be one of ''linear'',''cubic'',''nearest'', or ''v4''.');
end

if nargin == 7
    if ~iscellstr(options)
        error('MATLAB:OptsNotStringCell',...
              'OPTIONS should be cell array of strings.');           
    end
    opt = options;
else
    opt = [];
end

if numel(x) < 3 || numel(y) < 3
  error('MATLAB:griddata:NotEnoughSamplePts',...
        'Not enough unique sample points specified.');
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 = numel(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);
myepsx = eps(0.5 * (max(x) - min(x)))^(1/3);
myepsy = eps(0.5 * (max(y) - min(y)))^(1/3);
ind = [0; ((abs(diff(y)) < myepsy) & (abs(diff(x)) < myepsx)); 0];

if sum(ind) > 0
  warning('MATLAB:griddata:DuplicateDataPoints',['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

if numel(x) < 3
  error('MATLAB:griddata:NotEnoughSamplePts',...
        'Not enough unique sample points specified.');
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('MATLAB:griddata:UnknownMethod', '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(:); z = z(:);

dt = DelaunayTri(x,y);
scopedWarnOff = warning('off', 'MATLAB:TriRep:EmptyTri2DWarnId');
restoreWarnOff = onCleanup(@()warning(scopedWarnOff));
dtt = dt.Triangulation;
if isempty(dtt)
  error('MATLAB:griddata:EmptyTriangulation','Error computing Delaunay triangulation. The sample datapoints may be collinear.');
end


if(isreal(z))
    F = TriScatteredInterp(dt,z);
    zi = F(xi,yi);
else
    zre = real(z);
    zim = imag(z);
    F = TriScatteredInterp(dt,zre);
    zire = F(xi,yi);
    F.V = zim;
    ziim = F(xi,yi);
    zi = complex(zire,ziim);
end
zi = reshape(zi,siz);




%------------------------------------------------------------

%------------------------------------------------------------
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.

% Triangularize the data

dt = DelaunayTri([x(:) y(:)]);
scopedWarnOff = warning('off', 'MATLAB:TriRep:EmptyTri2DWarnId');
restoreWarnOff = onCleanup(@()warning(scopedWarnOff));
tri = dt.Triangulation;
if isempty(tri), 
  error('MATLAB:griddata:EmptyTriangulation','Error computing Delaunay triangulation. The sample datapoints may be collinear.');
end

% Find the enclosing triangle (t)
siz = size(xi);
t = dt.pointLocation(xi(:),yi(:));
t = reshape(t,siz);

if(isreal(z))
    zi = cubicmx(x,y,z,xi,yi,tri,t);
else
    zre = real(z);
    zim = imag(z); 
    zire = cubicmx(x,y,zre,xi,yi,tri,t);
    ziim = cubicmx(x,y,zim,xi,yi,tri,t);
    zi = complex(zire,ziim);
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
dt = DelaunayTri(x,y);
scopedWarnOff = warning('off', 'MATLAB:TriRep:EmptyTri2DWarnId');
restoreWarnOff = onCleanup(@()warning(scopedWarnOff));
dtt = dt.Triangulation;
if isempty(dtt)
  error('MATLAB:griddata:EmptyTriangulation','Error computing Delaunay triangulation. The sample datapoints may be collinear.');
end

k = dt.nearestNeighbor(xi,yi);
zi = k;
d = find(isfinite(k));
zi(d) = z(k(d));
zi = reshape(zi,siz);


%----------------------------------------------------------


%----------------------------------------------------------
function [xi,yi,zi] = 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:numel(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.
  [r,c] = 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([r(k);c(runs(i))])); % 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([r;c]);
  ndp(find(ndp(1:length(ndp)-1)==ndp(2:length(ndp)))) = [];

  warning('MATLAB:griddata:NonDistinctPoints',['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:numel(d)) = zeros(size(d,1),1);
weights = g \ z(:);

[m,n] = 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