Theory of Bicubic interpolation

For the bicubic code which is given in the this link https://www.mathworks.com/matlabcentral/answers/405846-bicubic-interpolation-direct-interpolation-formula-matlab-source-code, where can I get the expalnation of this code. The explanation which is given in the link https://en.wikipedia.org/wiki/Bicubic_interpolation

is not matching.

[KALYAN ACHARJYA]

https://ftp.cs.wisc.edu/debooron/bicubic.pdf

Answer 1

[Bruno Luong]

It has been answered here

https://www.mathworks.com/matlabcentral/answers/78116-algorithm-of-bicubic-interpolation

In 2D you do in one direction followedred by another.

%%
x = [-1:2];
y = [-1:2];
[X,Y] = meshgrid(x,y);
xq = rand;
yq = rand;
Z = rand(size(X));

Zq = interp2(x,y,Z,xq,yq,'bicubic')

% Check bicubic formula
Pl = [1.5,-2.5,0,1];
Pr = [-0.5,2.5,-4,2];
cubicp = @(x) (x<=1).*polyval(Pl,x) + (x>1 & x<2).*polyval(Pr,x);
cubic = @(x) cubicp(abs(x));
Zq = 0;
[~,i0] = histc(xq,x);
[~,j0] = histc(yq,y);
for i = i0-1:i0+2
    for j = j0-1:j0+2
        k = sub2ind(size(Z),j,i);
        Zq = Zq + cubic(X(k)-xq)*cubic(Y(k)-yq)*Z(k);
    end
end
Zq

[Shubha B.]

where is the theory for this? how to find the coefficeients. we get here single value for Zq. Is that correct?

[Bruno Luong]

The theory is in the Key's paper on the reference of the document of interp1 and interp2 and in the link II provide (coeffcient in eqt 4 do you read it?).

Apparently it's free access here

Mostly coefficients are computed to provide a smooth convolution kernel and compact support.

The example I give is formula for interpolation at a singe point (xq,yq).

For more points just loop on it. No relevant if you want to understand the formula.

[Shubha B.]

thank you sir.