splin3d - spline gridded 3d interpolation
This function computes a 3d tensor spline s which interpolates the (xi,yj,zk,vijk) points, ie, we have s(xi,yj,zk)=vijk for all i=1,..,nx, j=1,..,ny and k=1,..,nz. The resulting spline s is defined by tl which consists in a B-spline-tensor representation of s. The evaluation of s at some points must be done by the interp3d function (to compute s and its first derivatives) or by the bsplin3val function (to compute an arbitrary derivative of s) . Several kind of splines may be computed by selecting the order of the spline in each direction order=[kx,ky,kz] .
This function works under the conditions:
nx, ny, nz >= 3
2 <= kx < nx
2 <= ky < ny
2 <= kz < nz
an error being issued when they are not respected.
// example 1
func = "v=cos(2*%pi*x).*sin(2*%pi*y).*cos(2*%pi*z)";
deff("v=f(x,y,z)",func);
n = 10; // n x n x n interpolation points
x = linspace(0,1,n); y=x; z=x; // interpolation grid
[X,Y,Z] = ndgrid(x,y,z);
V = f(X,Y,Z);
tl = splin3d(x,y,z,V,[5 5 5]);
m = 10000;
// compute an approximated error
xp = grand(m,1,"def"); yp = grand(m,1,"def"); zp = grand(m,1,"def");
vp_exact = f(xp,yp,zp);
vp_interp = interp3d(xp,yp,zp, tl);
er = max(abs(vp_exact - vp_interp))
// now retry with n=20 and see the error
// example 2 (see linear_interpn help page which have the
// same example with trilinear interpolation)
getf("SCI/demos/interp/interp_demo.sci")
func = "v=(x-0.5).^2 + (y-0.5).^3 + (z-0.5).^2";
deff("v=f(x,y,z)",func);
n = 5;
x = linspace(0,1,n); y=x; z=x;
[X,Y,Z] = ndgrid(x,y,z);
V = f(X,Y,Z);
tl = splin3d(x,y,z,V);
// compute (and display) the 3d spline interpolant on some slices
m = 41;
dir = ["z=" "z=" "z=" "x=" "y="];
val = [ 0.1 0.5 0.9 0.5 0.5];
ebox = [0 1 0 1 0 1];
XF=[]; YF=[]; ZF=[]; VF=[];
for i = 1:length(val)
[Xm,Xp,Ym,Yp,Zm,Zp] = slice_parallelepiped(dir(i), val(i), ebox, m, m, m);
Vm = interp3d(Xm,Ym,Zm, tl);
[xf,yf,zf,vf] = nf3dq(Xm,Ym,Zm,Vm,1);
XF = [XF xf]; YF = [YF yf]; ZF = [ZF zf]; VF = [VF vf];
Vp = interp3d(Xp,Yp,Zp, tl);
[xf,yf,zf,vf] = nf3dq(Xp,Yp,Zp,Vp,1);
XF = [XF xf]; YF = [YF yf]; ZF = [ZF zf]; VF = [VF vf];
end
nb_col = 128;
vmin = min(VF); vmax = max(VF);
color = dsearch(VF,linspace(vmin,vmax,nb_col+1));
xset("colormap",jetcolormap(nb_col));
xbasc(); xset("hidden3d",xget("background"));
colorbar(vmin,vmax)
plot3d(XF, YF, list(ZF,color), flag=[-1 6 4])
xtitle("3d spline interpolation of "+func)
xselect()
linear_interpn , interp3d , bsplin3val ,