Some great online references are in the matlab code. Whenever I post links at the top of an answer, the answer gets flagged as spam, but then they encourage you to post reference links? Oh well. Here's some code to help with this problem:
% Online references for FFT's
% https://www.gaussianwaves.com/2015/11/interpreting-fft-results-complex-dft-frequency-bins-and-fftshift/
% https://blogs.uoregon.edu/seis/wiki/unpacking-the-matlab-fft/
dx_m = 0.5; % [m]
dy_m = 0.5; % [m]
% generate fractal texture for this spatial frame
MaxLevel = 6; % size of image is 2^MaxLevel+1
seed = 8675309; % seed enables repeatability
H = 0.5; % Hurst parameters a values between 0 and 1
FractalImage = midpoint(MaxLevel,H,seed);
N = 2.0^MaxLevel;
% spatial coordinates
X1D = dx_m.*[-N/2:N/2];
Y1D = dy_m.*[-N/2:N/2];
% visualize spatial data
figure('Color','white');
subplot(2,1,1)
imagesc(X1D,Y1D,FractalImage,[-3 3]);
title({'Fractional Brownian Motion';['Hurst =' num2str(H) ...
' Fractal Dimension =' num2str(3-H)]},'fontsize',12);
axis equal
axis tight
colormap(bone);
colorbar
set(gca,'fontweight','bold');
xlabel('X [m]'); ylabel('Y [m]');
% now to show the power spectrum
% with frequency space grid
ny = N;
nx = N;
dfy = 1/(ny*dy_m);
dfx = 1/(nx*dx_m);
fy = (-0.5/dy_m:dfy:(0.5/dy_m-1/(ny*dy_m)));
fx = (-0.5/dx_m:dfx:(0.5/dx_m-1/(nx*dx_m)));
FFT_FractalImage = fft2(FractalImage);
subplot(2,1,2)
imagesc(fx,fy,20*log10(abs(fftshift(FFT_FractalImage))));
axis equal; axis tight; colormap(bone); colorbar
set(gca,'fontweight','bold');
xlabel('Frequency [1/m]'); ylabel('Frequency [1/m]');
title('FFT2D Output','fontsize',12);
To run the code, you can download the midpoint function here (thank you if you do!):
