How to make a parabolic interpolation?

Hello,

I have 3d graph as follow:

image1=imread('image.tif');
mu=[]
for i=1:30
    for j=1:30
        mu(i,j)=image1(i,j);
    end 
end 

        
fmu=fft2(mu);
fmu(1,1)=0;
fmu2=fmu.*conj(fmu);
ps=ifft2(fmu2);
z=ps(1,1);
ps=real(fftshift(ps/z));


x=-15:1:14
y=-15:1:14
i=1:1:30
j=1:1:30
Z=ps(i,j)
[X,Y] = meshgrid(x,y);
surf(X,Y,Z)

set(gcf,'color','w') % pour les bordures.
xlabel('x (pixels)','FontSize',12, 'FontName','Times New Roman')
ylabel('y (pixels)', 'FontSize',12,'FontName','Times New Roman')
zlabel('Autocorrelation function', 'FontSize',12,'FontName','Times New Roman')

I want to make a prabolic interpolation in the vicinity of the maximum using the following equation :

Z=1-(a*x²+2b*x*y+c*y²)

How to make a parabolic interpolation? and how to get the parameters a, b and c?

Thank you for help.

Answer 1

[Sulaymon Eshkabilov]

Hi,

An easy solution is to employ the toolbox: cftool of MATLAB.

1. Launch it: >> cftool
2. Select X data: X
3. Select Y data: Y
4. Select Z data Z
5. Fir model type: Polynomial
6. Degrees x: 2
7. Degrees y: 2
8. You will get the fit model

Good luck

[John D'Errico]

Actually, no. Selecting 2nd degree for both x and y there will NOT yield the model the OP has asked to get.

Answer 2

[Sulaymon Eshkabilov]

Hi,

An exact custom equation as asked in the question can be attained on step 5 Fit Model Type selection via Custom Equation (from drop down options) and manual entry of:

1 -( a*x^2 + 2*b*x*y+c*y^2)

Good luck.