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 the 2 images being combined with different cut off frequencies.

Does fspecial() allow us to specify cut off frequencies when we employ it to make a gaussian filter? (I know that we can specify the filter size and sigma and that there is some relation between sigma and cut off frequency). If we can only specify cut off frequencies using sigma, then what sigma would I need to set to get a cut off frequency of say 0.2 Hz.

I don't have much experience with this but I expect you're on the right track with using sigma to specify a cutoff frequency. Bear in mind that cutoff frequency corresponds to the point where the input signal is attenuated by a certain amount, so you have to define what attenuation (usually 3dB) the cutoff frequency corresponds to before you can identify what point you're solving for. — Salain
fspecial creates a predefined 2D matrix filter, from what you wrote I understand that you want to filter a 1D signal (0.2 Hz, meaning you have a time vector or a frequency vector). So are you filtering a vector or a matrix ? — bla
@nate I'm filtering a jpg file which is a matrix..How would i specify a frequency for that..? — Siddhartha
@Siddhartha These are spacial frequencies, fx and fy. If you want a good reference on the matter, I recommend "Introduction to Fourier optics" by J.W. Goodman. I'll try to give you an appropriate answer below... — bla
What is the meaning of 0.2Hz in image processing ? Hz is time related and I don't see any time information in a fixed image. — bokan

bla bla · Accepted Answer · 2012-09-09T21:33:11

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...

Low pass gaussian filter with a specified cut off frequency

1 Answers