What is the mathematical principle of two-dimensional cubic Spline interpolation ?

What is the mathematical principle of two-dimensional cubic Spline interpolation（the function: interp2(method＝spline)）in Matlab?

Can you provide a link or paper?

I know that in the case of one-dimensional data, Cubic function is required, but I don't understand two-dimensional data.

Answer 1

[Bruno Luong]

The 2D spline is tensorial of 1D splines, meaning it makes 1D is x, then 1D in y. If you do the other way around it give the same result.

[澳旋]

Thanks for your answer! But I still don't understand, if I interpolate the point (1.5,2.5), then I can get the interpolation on the four "known lines" around x = 1 and x = 2, and y = 2 and y = 3, respectively, how can I get the final interpolation at (1.5,2.5)?

[Bruno Luong]

No spline is "global" interpolation its is NOT local four point around is NOT enough (I though you know about 1D spline bit obviously youa e=are not).

Assume you grid poit are an ndgrid(1:10,1:10)

You start to perform 10 1D spline interpolations in x (first dimension) to compute

f(1.5,1) % consider f1(x) := f(x,1) as 1D spline then interpolate
f(1.5,2) % consider f2(x) := f(x,2)
...
f(1.5,10) % % consider f10(x) := f(x,10)

Then you use the 10 results of the first step, interpolate in to get

f(1.5,2.5) % consider g(y) := f(1.5,y) as 1D spline then interpolate

[Bruno Luong]

Code to illustrate

% Generate grid data (4 x 4)
p = 4;
x=1:p;
y=1:p;
[X,Y] = ndgrid(x,y);
Z = peaks(p);

% Generate fine query (grid) points
xq = 1:0.1:p;
yq = 1:0.1:p;
[XQ,YQ] = ndgrid(xq,yq);

% MATLAB spline 2D interpolation
ZQ = interp2(Y,X,Z,YQ,XQ,'spline');

[m,n] = size(Z);
G = zeros(size(XQ));
f = zeros(1,n);
for k=1:numel(XQ) % loop on the query points
    xq = XQ(k);
    yq = YQ(k);
    % tensorial interpolation in x
    for j=1:n
        f(j) = interp1(x, Z(:,j), xq, 'spline');
    end
    % then in y
    g = interp1(y, f, yq, 'spline');
    % store the result
    G(k) = g;
end

norm(ZQ(:)-G(:),'inf')

ans = 0

% Graphical Check
figure
subplot(2,1,1)
surf(ZQ)
title('interp2')
subplot(2,1,2)
surf(G)
title('tensorial interp1 (x/y)')

[澳旋]

I think I understand. Thank you!