Custom 2D surface fitting issue

I am trying to develop some code to fit a surface. However, the values I recieve back from Matlab's fit function are horrific and non-sensical. Despite it being, in my view, an easy thing to approximate.

The surface is a summation of sine and cosine modes (here just 1 of each) and yet the surface cannot see to be recovered by the fit function.

%% 2D surface generation
clear all ; clc; close all
N=128;
L=10;
M=1; %number of modes
kmax=N/2;
kx=(randi(kmax,1,M)-kmax/2);%random wave numbers
ky=(randi(kmax,1,M)-kmax/2);%random wave numbers
x=L*(0:N-1)/N;
y=x';
V=zeros(N,N);
test = zeros(N,M);
A=rand(M,M)*2-1; %random amplitudes
B=rand(M,M)*2-1;
for i=1:M
    for j=1:M
      V=V+A(i,j)*sin((2*pi/L)*kx(i)*x+(2*pi/L)*ky(j)*y)+B(i,j)*cos((2*pi/L)*kx(i)*x+(2*pi/L)*ky(j)*y);
    end
end
figure(1)
imagesc(x,y,V)

%% Fitting
strval = [];
coeffval = {};
fac = ['(2*pi/',num2str(L),')*'];
for i=1:M
    for j=1:M
    charMi = num2str(i);
    charMj = num2str(j);
    if i==1 & j==1
    form = ['A_',charMi,charMj,'*sin(',fac,'x*','wx_',charMi,'+ ',fac,'y*','wy_',charMj,') + B_',charMi,charMj,'*cos(',fac,'x*','wx_',charMi,'+ ',fac,'y*','wy_',charMj,')'];
    else
    form = [' + A_',charMi,charMj,'*sin(',fac,'x*','wx_',charMi,'+ ',fac,'y*','wy_',charMj,') + B_',charMi,charMj,'*cos(',fac,'x*','wx_',charMi,'+ ',fac,'y*','wy_',charMj,')'];
    end
    strval = strcat(strval,form);
    coeffs = {['A_',charMi], ['A_',charMj], ['B_',charMi], ['B_',charMj], ['wx_',charMi], ['wx_',charMj], ['wy_',charMi], ['wy_',charMj]};
    coeffval = [coeffval,coeffs];
    end
end
optsL = [-ones(2*M*M,1)',zeros(2*M,1)'];
optsH = [ones(2*M*M,1)',kmax.*ones(2*M,1)'];
optsS = [zeros(2*M*M,1)',(kmax/2).*ones(2*M,1)'];

%% Fit: 'untitled fit 1'.
[xData, yData, zData] = prepareSurfaceData( x, y, V );

% Set up fittype and options.
ft = fittype( strval, 'independent', {'x', 'y'}, 'dependent', 'z' );
opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
opts.Display = 'off';
opts.Lower = optsL;
opts.MaxIter = 5000;
opts.MaxFunEvals = 5000;
opts.Robust = 'LAR';
opts.StartPoint = optsS;
opts.Upper = optsH;
opts.Normal = 'on';

% Fit model to data.
[fitresult, gof] = fit( [xData, yData], zData, ft, opts );

% Plot fit with data.
figure(2, 'Name', 'untitled fit 1' );
h = plot( fitresult, [xData, yData], zData );
legend( h, 'untitled fit 1', 'V vs. x, y', 'Location', 'NorthEast' );
% Label axes
xlabel x
ylabel y
zlabel V
grid on
view( 9.7, 45.2 );
fitresult

Any help would be greatly appreciated.

Thanks in advance :)

Answer 1

[Raag]

Hi Nick,

As per my understanding, a possible cause of the issue is that fitting both amplitudes and wavenumbers for a sum of 2D sine and cosine modes leads to an overcomplicated and ill-conditioned problem, especially when the wavenumbers are already known. In such cases, it is both more robust and efficient to fit only the amplitudes, while keeping the wavenumbers fixed.

If the wavenumbers that create the surface are known, the amplitudes can be found out using MATLAB's 'fit' function.This makes the fitting easier and more accurate.

Suppose you have a surface defined as:

V = Asin(kxx + kyy) + Bcos(kxx + kyy); 
%where A, B, kx, and ky are constants, and x, y are grid coordinates.

To fit for amplitudes A and B (with known kx and ky):

% Prepare data 
xData = X(:); 
yData = Y(:); 
zData = V(:); 
 
% Define the fit type with fixed wavenumbers 
ft = fittype('A*sin((2*pi/L)*kx*x + (2*pi/L)*ky*y) + B*cos((2*pi/L)*kx*x + (2*pi/L)*ky*y)', ... 
    'independent', {'x', 'y'}, 'dependent', 'z', ... 
    'problem', {'kx', 'ky', 'L'}); 
 
% Fit the model 
[fitresult, gof] = fit([xData, yData], zData, ft, 'problem', {3, 2, 10}, 'StartPoint', [1, 1]); 
 
% Display results 
disp(fitresult); 

By specifying kx, ky, and L as known values (the wavenumbers and domain size), the fit only estimates A and B. This approach is accurate and avoids the instability of trying to fit for wavenumbers.

For a better understanding of the above solution, refer to the following MATLAB documentations:

1. fittype: https://www.mathworks.com/help/curvefit/fittype.html

2. fit: https://www.mathworks.com/help/curvefit/fit.html

3.non-linear least squares: https://www.mathworks.com/help/optim/nonlinear-least-squares-curve-fitting.html