Low-pass filter applied in frequency domain after fft2 and before ifft2

Part of my task is to filter an image in frequency domain. Filter has to be low-pass with cut-off frequency (K0) determined by user.

I found interesting code written by other user but I am not sure how it works. Could you please explain me step by step this approach? The author does not reply. Maybe you know better way? If yes, please tell me.

Fragment:

% Application of low pass filter in reconstruction
%Image dimensions 
[N,M] = size(I0grey); %[height, width]
%Sampling intervals 
dx = 1; 
dy = 1; 
%Characteristic wavelengths 
KX0 = (mod(1/2 + (0:(M-1))/M, 1) - 1/2); 
KX1 = KX0 * (2*pi/dx); 
KY0 = (mod(1/2 + (0:(N-1))/N, 1) - 1/2); 
KY1 = KY0 * (2*pi/dx); 
[KX,KY] = meshgrid(KX1,KY1); 
%Filter formulation 
lpf = (KX.*KX + KY.*KY < K0^2); 
%Filter Application 
rec = ifft2(lpf.*ff);

All my script:

%%PSIB 2015, Trabalho final, o problema 2

% This application  performs the filtering of an image in the frequency 
% domain. It applies a low-pass filter for multiplying a transfer function 
% suitable for 2D Fourier transform of the image, then inverse Fourier 
% transform. An image and a cut-off frequency are choosen by user.
% Results are compared to images obtained using spatial filtering
% operators.

%%Input
% Image
% Image file selection dialog box
[filename,pathname] = uigetfile('*.jpg;*.tif;*.png;*.gif','Select the image file');
%Reading a file and converting to greyscale
I0 = imread(fullfile(pathname, filename)); %MxNx3
I0grey = im2double(rgb2gray(I0)); %grey, MxN

% Cut-off frequency for low pass filter
prompt = 'Enter a cut-off frequency for a low-pass filter:';
dlg_title = 'Cut-off frequency';
num_lines = 1;
def = {'0.20'};
answer = inputdlg(prompt,dlg_title,num_lines,def);
K0 = str2double(answer);

% If loop that checks if K0 is correct (in range of frequencies)
ff = fft2(I0grey); % Take Fourier Transform 2D
while(1)
    if (K0 >= min(ff(:))) && (K0 <= max(ff(:)))
        break;
    else
        prompt = 'Entered cut-off frequency is incorrect, type another:';
        dlg_title = 'Cut-off frequency';
        num_lines = 1;
        def = {'0.20'};
        answer = inputdlg(prompt,dlg_title,num_lines,def);
        K0 = str2double(answer);
    end
end
%%Processing
% Fourier transform
ff = fft2(I0grey); % Take Fourier Transform 2D
F = 20*log(abs(fftshift(ff))); % Shift center; get log magnitude

% Application of low pass filter in reconstruction
%Image dimensions 
[N,M] = size(I0grey); %[height, width]
%Sampling intervals 
dx = 1; 
dy = 1; 
%Characteristic wavelengths 
KX0 = (mod(1/2 + (0:(M-1))/M, 1) - 1/2); 
KX1 = KX0 * (2*pi/dx); 
KY0 = (mod(1/2 + (0:(N-1))/N, 1) - 1/2); 
KY1 = KY0 * (2*pi/dx); 
[KX,KY] = meshgrid(KX1,KY1); 
%Filter formulation 
lpf = (KX.*KX + KY.*KY < K0^2); 
%Filter Application 
rec = ifft2(lpf.*ff);

%
lp = fir1(32,K0);  % Generate the lowpass filter (order, cut-off frequency)
lp_2D = ftrans2(lp);  % Convert to 2-dimensions
I_double = im2double(I0grey);
I_lowpass_rep = imfilter(I_double,lp_2D,'replicate');
I_lowpass_gray = mat2gray(I_lowpass_rep);

% Spatial filtering operators
% Construction of 2D filters
%with default parameters to not influence ?
h1 = fspecial('gaussian'); %gaussian low-pass filter
h2 = fspecial('log');

% Image filtering
I_f1 = imfilter(I0grey,h1,'replicate');
I_f2= imfilter(I0grey,h2,'replicate');
%%Results
%Graphics
figure(1)
% Plot original
subplot(2,3,1)
imshow(I0grey); 
xlabel('X','FontSize',14); 
ylabel('Y','FontSize',14);

%show FFT, magnitude
subplot(2,3,2)
mesh(F); colormap(hot); % Plot Fourier Transform as function
xlabel('Horizontal Frequency','FontSize',12); 
ylabel('Vertical Frequency','FontSize',12);

% Reconstruction after filtering in frequency domain
subplot(2,3,3)
imshow(rec);
xlabel('X','FontSize',14); 
ylabel('Y','FontSize',14);
title('Hand-made low-pass filter');

% Filtered image 1, FIR
subplot(2,3,4)
imshow(I_lowpass_gray);
xlabel('X','FontSize',14); 
ylabel('Y','FontSize',14);
title('FIR low-pass');

% Filtered image 2, fspecial Gaussian
subplot(2,3,5)
imshow(I_f1);
xlabel('X','FontSize',14); 
ylabel('Y','FontSize',14);
title('Gaussian');

% Filtered image 3, fspecial Laplacian of Gaussian
subplot(2,3,6)
imshow(I_f2);
xlabel('X','FontSize',14); 
ylabel('Y','FontSize',14);
title('Laplacian of Gaussian');

Thank you in advance.

Answer 1

[Image Analyst]

The equation

lpf = (KX.*KX + KY.*KY < K0^2);

Constructs a circle. See the FAQ for more info. The filter is true (1) inside the radius and false (0) outside it.

Then they multiply the circle mask by the Fourier transform by doing this: lpf.*ff. This retains all the low frequencies inside the circle and zeros out all high frequencies outside the circle. Then they inverse transform to get an image with only low frequencies in it:

%Filter Application 
rec = ifft2(lpf.*ff);

[Katarzyna Wieciorek]

And what is done before?