Low pass gaussian filter with a specified cut off frequency

Question

I\'m playing around with hybrid images, and wanted to use a gaussian filter to low pass filter an image. However, to make hybrid images, 2 filters are supposed to be used on

Answer 1

I'll first answer regarding 1D and the rest will follow. It may look trivial, but bear with me for a while. Lets assume the following code:

t=linspace(0,20,2^10); %time vector in seconds
w=0.2; %in Hz
signal=cos(2*pi*w*t)+rand(1,length(t))-0.5; % signal in seconds
dt=t(2)-t(1) ;
N=length(signal);
df=1/(N*dt);    % the frequency resolution (df=1/max_T)
if mod(N,2)==0
    f_vec= df*((1:N)-1-N/2);  % for EVEN length vectors
else
    f_vec= df*((1:N)-0.5-N/2); 
end

So, we have created a noisy signal of a specific frequency. f_vec is the frequency vector that stretches from f =[-f_max,-f_max+df,...,0,...,f_max], with f_max=1/(2*dt). If we now design a 1D Gaussian filter (in the fourier space) as follows:

f_vec0=0;
sigma=1;
filter=exp( -(f_vec-f_vec0).^2./(2*sigma^2)); 

and then filtering in the fourier doamin:

f_signal=fftshift(fft(signal));
filt_signal=fftshift(ifft(f_signal.*filter));

So, from the filter we applied, sigma=1 means the the cut off frequency (that I decided is 1% of the filter's maximum (which is 1)) is approximately at 3 Hz:

 cutoff_freq=f_vec(find(filter>=0.01,1,'last'))

Taking this to 2D is trivial, just be careful with units. For images, there are pixels as the position unit, and 1/pixels as the spacial frequency. The fspecial function generates a 2D matrix of a predefined filter. The usage of fspecial is usually like this:

PSF = fspecial('gaussian',hsize,sigma);
Blurred = imfilter(Image,PSF,'symmetric','conv');

Using convolution is just like multiplying in the Fourier domain. The sigma in the Fourier domain is proportional to 1/sigma of the position domain, etc...