二元函数网格数据的插值法
function zi = interp2(varargin)%二元函数网格数据的插值
%ZI=interp2(X,Y,Z,XI,YI,'方法')求二元函数z=f(x,y)的插值.
% 这里X,Y,Z是同维数矩阵表示网格数据,XI,YI,ZI是同维数矩阵表示插值点.
%或ZI=interp2(x,y,z,xi,yi)其中,x,xi为行向量,y,yi为列向量.
%'bilinear',使用双线性插值(默认)
%'spline' 使用二元三次样条插值.
%'cubic' 使用二元三次插值.
%例如
% clear;close;x=0:4;y=2:4;
% Z=;
% =meshgrid(x,y);
% subplot(2,1,1);
% mesh(X,Y,Z);title('RAW DATA');
% xi=0:0.1:4;yi=2:0.2:4;
% =meshgrid(xi,yi);
% zspline=interp2(x,y,z,XI,YI,'spline');
% subplot(2,1,2);
% mesh(XI,YI,zspline);
% title('SPLINE');
%
%INTERP2 2-D interpolation (table lookup).
% ZI = INTERP2(X,Y,Z,XI,YI) interpolates to find ZI, the values of the
% underlying 2-D function Z at the points in matrices XI and YI.
% Matrices X and Y specify the points at which the data Z is given.
% Out of range values are returned as NaN.
%
% 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.
%
% ZI = INTERP2(Z,XI,YI) assumes X=1:N and Y=1:M where =SIZE(Z).
% ZI = INTERP2(Z,NTIMES) expands Z by interleaving interpolates between
% every element, working recursively for NTIMES.INTERP2(Z) is the
% same as INTERP2(Z,1).
%
% ZI = INTERP2(...,'method') specifies alternate methods.The default
% is linear interpolation.Available methods are:
%
% 'nearest' - nearest neighbor interpolation
% 'linear'- bilinear interpolation
% 'cubic' - bicubic interpolation
% 'spline'- spline interpolation
%
% All the interpolation methods require that X and Y be monotonic and
% plaid (as if they were created using MESHGRID).X and Y can be
% non-uniformly spaced.For faster interpolation when X and Y are
% equally spaced and monotonic, use the methods '*linear', '*cubic', or
% '*nearest'.
%
% For example, to generate a coarse approximation of PEAKS and
% interpolate over a finer mesh:
% = peaks(10); = meshgrid(-3:.1:3,-3:.1:3);
% zi = interp2(x,y,z,xi,yi); mesh(xi,yi,zi)
%
% See also INTERP1, INTERP3, INTERPN, MESHGRID, GRIDDATA.
% Copyright (c) 1984-98 by The MathWorks, Inc.
% $Revision: 5.26 $
error(nargchk(1,6,nargin));
bypass = 0;
uniform = 1;
if isstr(varargin{end}),
narg = nargin-1;
method = ; % Protect against short string.
if method(1)=='*', % Direct call bypass.
if method(2)=='l' | all(method(2:4)=='bil'), % bilinear interpolation.
zi = linear(varargin{1:end-1});
return
elseif method(2)=='c' | all(method(2:4)=='bic'), % bicubic interpolation
zi = cubic(varargin{1:end-1});
return
elseif method(2)=='n', % Nearest neighbor interpolation
zi = nearest(varargin{1:end-1});
return
elseif method(2)=='s', % spline interpolation
method = 'spline'; bypass = 1;
else
error();
end
elseif method(1)=='s', % Spline interpolation
method = 'spline'; bypass = 1;
end
else
narg = nargin;
method = 'linear';
end
if narg==1, % interp2(z), % Expand Z
= size(varargin{1});
xi = 1:.5:ncols; yi = (1:.5:nrows)';
x = 1:ncols; y = (1:nrows);
= xyzchk(x,y,varargin{1},xi,yi);
elseif narg==2. % interp2(z,n), Expand Z n times
= size(varargin{1});
ntimes = floor(varargin{2}(1));
xi = 1:1/(2^ntimes):ncols; yi = (1:1/(2^ntimes):nrows)';
x = 1:ncols; y = (1:nrows);
= xyzchk(x,y,varargin{1},xi,yi);
elseif narg==3, % interp2(z,xi,yi)
= size(varargin{1});
x = 1:ncols; y = (1:nrows);
= xyzchk(x,y,varargin{1:3});
elseif narg==4,
error('Wrong number of input arguments.');
elseif narg==5, % linear(x,y,z,xi,yi)
= xyzchk(varargin{1:5});
end
if ~isempty(msg), error(msg); end
%
% Check for plaid data.
%
xx = x(1,:); yy = y(:,1);
if (size(x,2)>1 & ~isequal(repmat(xx,size(x,1),1),x)) | ...
(size(y,1)>1 & ~isequal(repmat(yy,1,size(y,2)),y)),
error(sprintf(['X and Y must be matrices produced by MESHGRID. Use' ...
' GRIDDATA instead \nof INTERP2 for scattered data.']));
end
%
% Check for non-equally spaced data.If so, map (x,y) and
% (xi,yi) to matrix (row,col) coordinate system.
%
if ~bypass,
xx = xx.'; % Make sure it's a column.
dx = diff(xx); dy = diff(yy);
xdiff = max(abs(diff(dx))); if isempty(xdiff), xdiff = 0; end
ydiff = max(abs(diff(dy))); if isempty(ydiff), ydiff = 0; end
if (xdiff > eps*max(abs(xx))) | (ydiff > eps*max(abs(yy))),
if any(dx < 0), % Flip orientation of data so x is increasing.
x = fliplr(x); y = fliplr(y); z = fliplr(z);
xx = flipud(xx); dx = -flipud(dx);
end
if any(dy < 0), % Flip orientation of data so y is increasing.
x = flipud(x); y = flipud(y); z = flipud(z);
yy = flipud(yy); dy = -flipud(dy);
end
if any(dx<=0) | any(dy<=0),
error('X and Y must be monotonic vectors or matrices produced by MESHGRID.');
end
% Bypass mapping code for cubic
if method(1)~='c',
% Determine the nearest location of xi in x
= sort(xi(:));
= sort();
ui(i) = (1:length(i));
ui = (ui(length(xx)+1:end)-(1:length(xxi)))';
ui(j) = ui;
% Map values in xi to index offset (ui) via linear interpolation
ui(ui<1) = 1;
ui(ui>length(xx)-1) = length(xx)-1;
ui = ui + (xi(:)-xx(ui))./(xx(ui+1)-xx(ui));
% Determine the nearest location of yi in y
= sort(yi(:));
= sort();
vi(i) = (1:length(i));
vi = (vi(length(yy)+1:end)-(1:length(yyi)))';
vi(j) = vi;
% Map values in yi to index offset (vi) via linear interpolation
vi(vi<1) = 1;
vi(vi>length(yy)-1) = length(yy)-1;
vi = vi + (yi(:)-yy(vi))./(yy(vi+1)-yy(vi));
= meshgrid(1:size(x,2),1:size(y,1));
xi(:) = ui; yi(:) = vi;
else
uniform = 0;
end
end
end
% Now do the interpolation based on method.
method = ; % Protect against short string
if method(1)=='l' | all(method(1:3)=='bil'), % bilinear interpolation.
zi = linear(x,y,z,xi,yi);
elseif method(1)=='c' | all(method(1:3)=='bic'), % bicubic interpolation
if uniform
zi = cubic(x,y,z,xi,yi);
else
d = find(xi < min(x(:)) | xi > max(x(:)) | ...
yi < min(y(:)) | yi > max(y(:)));
zi = spline2(x,y,z,xi,yi);
zi(d) = NaN;
end
elseif method(1)=='n', % Nearest neighbor interpolation
zi = nearest(x,y,z,xi,yi);
elseif method(1)=='s', % Spline interpolation
zi = spline2(x,y,z,xi,yi);
else
error();
end
%------------------------------------------------------
function F = linear(arg1,arg2,arg3,arg4,arg5)
%LINEAR 2-D bilinear data interpolation.
% ZI = LINEAR(X,Y,Z,XI,YI) uses bilinear interpolation to
% find ZI, the values of the underlying 2-D function in Z at the points
% in matrices XI and YI.Matrices X and Y specify the points at which
% the data Z is given.X and Y can also be vectors specifying the
% abscissae for the matrix Z as for MESHGRID. In both cases, X
% and Y must be equally spaced and monotonic.
%
% Values of NaN are returned in ZI for values of XI and YI that are
% outside of the range of X and Y.
%
% If XI and YI are vectors, LINEAR returns vector ZI containing
% the interpolated values at the corresponding points (XI,YI).
%
% ZI = LINEAR(Z,XI,YI) assumes X = 1:N and Y = 1:M, where
% = SIZE(Z).
%
% ZI = LINEAR(Z,NTIMES) returns the matrix Z expanded by interleaving
% bilinear interpolates between every element, working recursively
% for NTIMES.LINEAR(Z) is the same as LINEAR(Z,1).
%
% This function needs about 4 times SIZE(XI) memory to be available.
%
% See also INTERP2, CUBIC.
% Clay M. Thompson 4-26-91, revised 7-3-91, 3-22-93 by CMT.
if nargin==1, % linear(z), Expand Z
= size(arg1);
s = 1:.5:ncols; sizs = size(s);
t = (1:.5:nrows)'; sizt = size(t);
s = s(ones(sizt),:);
t = t(:,ones(sizs));
elseif nargin==2, % linear(z,n), Expand Z n times
= size(arg1);
ntimes = floor(arg2);
s = 1:1/(2^ntimes):ncols; sizs = size(s);
t = (1:1/(2^ntimes):nrows)'; sizt = size(t);
s = s(ones(sizt),:);
t = t(:,ones(sizs));
elseif nargin==3, % linear(z,s,t), No X or Y specified.
= size(arg1);
s = arg2; t = arg3;
elseif nargin==4,
error('Wrong number of input arguments.');
elseif nargin==5, % linear(x,y,z,s,t), X and Y specified.
= size(arg3);
mx = prod(size(arg1)); my = prod(size(arg2));
if any( ~= ) & ...
~isequal(size(arg1),size(arg2),size(arg3))
error('The lengths of the X and Y vectors must match Z.');
end
if any(<), error('Z must be at least 2-by-2.'); end
s = 1 + (arg4-arg1(1))/(arg1(mx)-arg1(1))*(ncols-1);
t = 1 + (arg5-arg2(1))/(arg2(my)-arg2(1))*(nrows-1);
end
if any(<), error('Z must be at least 2-by-2.'); end
if ~isequal(size(s),size(t)),
error('XI and YI must be the same size.');
end
% Check for out of range values of s and set to 1
sout = find((s<1)|(s>ncols));
if length(sout)>0, s(sout) = ones(size(sout)); end
% Check for out of range values of t and set to 1
tout = find((t<1)|(t>nrows));
if length(tout)>0, t(tout) = ones(size(tout)); end
% Matrix element indexing
ndx = floor(t)+floor(s-1)*nrows;
% Compute intepolation parameters, check for boundary value.
if isempty(s), d = s; else d = find(s==ncols); end
s(:) = (s - floor(s));
if length(d)>0, s(d) = s(d)+1; ndx(d) = ndx(d)-nrows; end
% Compute intepolation parameters, check for boundary value.
if isempty(t), d = t; else d = find(t==nrows); end
t(:) = (t - floor(t));
if length(d)>0, t(d) = t(d)+1; ndx(d) = ndx(d)-1; end
d = [];
% Now interpolate, reuse u and v to save memory.
if nargin==5,
F =( arg3(ndx).*(1-t) + arg3(ndx+1).*t ).*(1-s) + ...
( arg3(ndx+nrows).*(1-t) + arg3(ndx+(nrows+1)).*t ).*s;
else
F =( arg1(ndx).*(1-t) + arg1(ndx+1).*t ).*(1-s) + ...
( arg1(ndx+nrows).*(1-t) + arg1(ndx+(nrows+1)).*t ).*s;
end
% Now set out of range values to NaN.
if length(sout)>0, F(sout) = NaN; end
if length(tout)>0, F(tout) = NaN; end
%------------------------------------------------------
function F = cubic(arg1,arg2,arg3,arg4,arg5)
%CUBIC 2-D bicubic data interpolation.
% CUBIC(...) is the same as LINEAR(....) except that it uses
% bicubic interpolation.
%
% This function needs about 7-8 times SIZE(XI) memory to be available.
%
% See also LINEAR.
% Clay M. Thompson 4-26-91, revised 7-3-91, 3-22-93 by CMT.
% Based on "Cubic Convolution Interpolation for Digital Image
% Processing", Robert G. Keys, IEEE Trans. on Acoustics, Speech, and
% Signal Processing, Vol. 29, No. 6, Dec. 1981, pp. 1153-1160.
if nargin==1, % cubic(z), Expand Z
= size(arg1);
s = 1:.5:ncols; sizs = size(s);
t = (1:.5:nrows)'; sizt = size(t);
s = s(ones(sizt),:);
t = t(:,ones(sizs));
elseif nargin==2, % cubic(z,n), Expand Z n times
= size(arg1);
ntimes = floor(arg2);
s = 1:1/(2^ntimes):ncols; sizs = size(s);
t = (1:1/(2^ntimes):nrows)'; sizt = size(t);
s = s(ones(sizt),:);
t = t(:,ones(sizs));
elseif nargin==3, % cubic(z,s,t), No X or Y specified.
= size(arg1);
s = arg2; t = arg3;
elseif nargin==4,
error('Wrong number of input arguments.');
elseif nargin==5, % cubic(x,y,z,s,t), X and Y specified.
= size(arg3);
mx = prod(size(arg1)); my = prod(size(arg2));
if any( ~= ) & ...
~isequal(size(arg1),size(arg2),size(arg3))
error('The lengths of the X and Y vectors must match Z.');
end
if any(<), error('Z must be at least 3-by-3.'); end
s = 1 + (arg4-arg1(1))/(arg1(mx)-arg1(1))*(ncols-1);
t = 1 + (arg5-arg2(1))/(arg2(my)-arg2(1))*(nrows-1);
end
if any(<), error('Z must be at least 3-by-3.'); end
if ~isequal(size(s),size(t)),
error('XI and YI must be the same size.');
end
% Check for out of range values of s and set to 1
sout = find((s<1)|(s>ncols));
if length(sout)>0, s(sout) = ones(size(sout)); end
% Check for out of range values of t and set to 1
tout = find((t<1)|(t>nrows));
if length(tout)>0, t(tout) = ones(size(tout)); end
% Matrix element indexing
ndx = floor(t)+floor(s-1)*(nrows+2);
% Compute intepolation parameters, check for boundary value.
if isempty(s), d = s; else d = find(s==ncols); end
s(:) = (s - floor(s));
if length(d)>0, s(d) = s(d)+1; ndx(d) = ndx(d)-nrows-2; end
% Compute intepolation parameters, check for boundary value.
if isempty(t), d = t; else d = find(t==nrows); end
t(:) = (t - floor(t));
if length(d)>0, t(d) = t(d)+1; ndx(d) = ndx(d)-1; end
d = [];
if nargin==5,
% Expand z so interpolation is valid at the boundaries.
zz = zeros(size(arg3)+2);
zz(1,2:ncols+1) = 3*arg3(1,:)-3*arg3(2,:)+arg3(3,:);
zz(2:nrows+1,2:ncols+1) = arg3;
zz(nrows+2,2:ncols+1) = 3*arg3(nrows,:)-3*arg3(nrows-1,:)+arg3(nrows-2,:);
zz(:,1) = 3*zz(:,2)-3*zz(:,3)+zz(:,4);
zz(:,ncols+2) = 3*zz(:,ncols+1)-3*zz(:,ncols)+zz(:,ncols-1);
nrows = nrows+2; ncols = ncols+2;
else
% Expand z so interpolation is valid at the boundaries.
zz = zeros(size(arg1)+2);
zz(1,2:ncols+1) = 3*arg1(1,:)-3*arg1(2,:)+arg1(3,:);
zz(2:nrows+1,2:ncols+1) = arg1;
zz(nrows+2,2:ncols+1) = 3*arg1(nrows,:)-3*arg1(nrows-1,:)+arg1(nrows-2,:);
zz(:,1) = 3*zz(:,2)-3*zz(:,3)+zz(:,4);
zz(:,ncols+2) = 3*zz(:,ncols+1)-3*zz(:,ncols)+zz(:,ncols-1);
nrows = nrows+2; ncols = ncols+2;
end
% Now interpolate using computationally efficient algorithm.
t0 = ((2-t).*t-1).*t;
t1 = (3*t-5).*t.*t+2;
t2 = ((4-3*t).*t+1).*t;
t(:) = (t-1).*t.*t;
F = ( zz(ndx).*t0 + zz(ndx+1).*t1 + zz(ndx+2).*t2 + zz(ndx+3).*t ) ...
.* (((2-s).*s-1).*s);
ndx(:) = ndx + nrows;
F(:)= F + ( zz(ndx).*t0 + zz(ndx+1).*t1 + zz(ndx+2).*t2 + zz(ndx+3).*t ) ...
.* ((3*s-5).*s.*s+2);
ndx(:) = ndx + nrows;
F(:)= F + ( zz(ndx).*t0 + zz(ndx+1).*t1 + zz(ndx+2).*t2 + zz(ndx+3).*t ) ...
.* (((4-3*s).*s+1).*s);
ndx(:) = ndx + nrows;
F(:)= F + ( zz(ndx).*t0 + zz(ndx+1).*t1 + zz(ndx+2).*t2 + zz(ndx+3).*t ) ...
.* ((s-1).*s.*s);
F(:) = F/4;
% Now set out of range values to NaN.
if length(sout)>0, F(sout) = NaN; end
if length(tout)>0, F(tout) = NaN; end
%------------------------------------------------------
function F = nearest(arg1,arg2,arg3,arg4,arg5)
%NEAREST 2-D Nearest neighbor interpolation.
% ZI = NEAREST(X,Y,Z,XI,YI) uses nearest neighbor interpolation to
% find ZI, the values of the underlying 2-D function in Z at the points
% in matrices XI and YI.Matrices X and Y specify the points at which
% the data Z is given.X and Y can also be vectors specifying the
% abscissae for the matrix Z as for MESHGRID. In both cases, X
% and Y must be equally spaced and monotonic.
%
% Values of NaN are returned in ZI for values of XI and YI that are
% outside of the range of X and Y.
%
% If XI and YI are vectors, NEAREST returns vector ZI containing
% the interpolated values at the corresponding points (XI,YI).
%
% ZI = NEAREST(Z,XI,YI) assumes X = 1:N and Y = 1:M, where
% = SIZE(Z).
%
% F = NEAREST(Z,NTIMES) returns the matrix Z expanded by interleaving
% interpolates between every element.NEAREST(Z) is the same as
% NEAREST(Z,1).
%
% See also INTERP2, LINEAR, CUBIC.
% Clay M. Thompson 4-26-91, revised 7-3-91 by CMT.
if nargin==1, % nearest(z), Expand Z
= size(arg1);
u = ones(2*nrows-1,1)*(1:.5:ncols);
v = (1:.5:nrows)'*ones(1,2*ncols-1);
elseif nargin==2, % nearest(z,n), Expand Z n times
= size(arg1);
ntimes = floor(arg2);
u = 1:1/(2^ntimes):ncols; sizu = size(u);
v = (1:1/(2^ntimes):nrows)'; sizv = size(v);
u = u(ones(sizv),:);
v = v(:,ones(sizu));
elseif nargin==3, % nearest(z,u,v)
= size(arg1);
u = arg2; v = arg3;
elseif nargin==4,
error('Wrong number of input arguments.');
elseif nargin==5, % nearest(x,y,z,u,v), X and Y specified.
= size(arg3);
mx = prod(size(arg1)); my = prod(size(arg2));
if any( ~= ) & (size(arg1)~=size(arg3) | ...
size(arg2)~=size(arg3)),
error('The lengths of the X and Y vectors must match Z.');
end
if all(>),
u = 1 + (arg4-arg1(1))/(arg1(mx)-arg1(1))*(ncols-1);
v = 1 + (arg5-arg2(1))/(arg2(my)-arg2(1))*(nrows-1);
else
u = 1 + (arg4-arg1(1));
v = 1 + (arg5-arg2(1));
end
end
if size(u)~=size(v), error('XI and YI must be the same size.'); end
% Check for out of range values of u and set to 1
uout = (u<.5)|(u>=ncols+.5);
nuout = sum(uout(:));
if any(uout(:)), u(uout) = ones(nuout,1); end
% Check for out of range values of v and set to 1
vout = (v<.5)|(v>=nrows+.5);
nvout = sum(vout(:));
if any(vout(:)), v(vout) = ones(nvout,1); end
% Interpolation parameters
s = (u - round(u));t = (v - round(v));
u = round(u); v = round(v);
% Now interpolate
ndx = v+(u-1)*nrows;
if nargin==5,
F = arg3(ndx);
else
F = arg1(ndx);
end
% Now set out of range values to NaN.
if any(uout(:)), F(uout) = NaN; end
if any(vout(:)), F(vout) = NaN; end
%----------------------------------------------------------
function F = spline2(varargin)
%2-D spline interpolation
% Determine abscissa vectors
varargin{1} = varargin{1}(1,:);
varargin{2} = varargin{2}(:,1).';
%
% Check for plaid data.
%
xi = varargin{4}; yi = varargin{5};
xxi = xi(1,:); yyi = yi(:,1);
if (size(xi,2)>1 & ~isequal(repmat(xxi,size(xi,1),1),xi)) | ...
(size(yi,1)>1 & ~isequal(repmat(yyi,1,size(yi,2)),yi)),
F = splncore(varargin(2:-1:1),varargin{3},varargin(5:-1:4));
else
F = splncore(varargin(2:-1:1),varargin{3},{yyi(:).' xxi},'gridded');
end
页:
[1]