Ideal Low Pass Filter Concept in MATLAB

Please help me understand the following MATLAB code for Ideal Low pass filter. I am unable to understand the Part2 in the below code. Please explain me why we are doing like this.

       %IDEAL LOW-PASS FILTER

      %Part 1
              function idealfilter(X,P) % X is the input image and P is the cut-off freq
              f=imread(X);  % reading an image X
              [M,N]=size(f); % Saving the the rows of X in M and columns in N
              F=fft2(double(f)); % Taking Fourier transform to the input image
      %Part 2 % I don't understand this part
              u=0:(M-1);
              v=0:(N-1);
              idx=find(u>M/2);
              u(idx)=u(idx)-M;
              idy=find(v>N/2);
              v(idy)=v(idy)-N;
              [V,U]=meshgrid(v,u);
              D=sqrt(U.^2+V.^2);

      %Part 3
              H=double(D<=P);       % Comparing with the cut-off frequency 
              G=H.*F;               % Convolution with the Fourier transformed image
              g=real(ifft2(double(G))); % Inverse Fourier transform
              imshow(f),figure,imshow(g,[ ]); % Displaying input and output image
              end

I tried to run each commands in Part2 individually for M= 8 and N=8. I get

u=0:(M-1); ==> u = 0 1 2 3 4 5 6 7

v=0:(N-1); ==> v = 0 1 2 3 4 5 6 7

idx=find(u>M/2); ==> idx = 6 7 8

u(idx)=u(idx)-M; ==> 0 1 2 3 4 -3 -2 -1

idy=find(v>N/2); ==> idy = 6 7 8

v(idy)=v(idy)-N; ==> 0 1 2 3 4 -3 -2 -1

[V,U]=meshgrid(v,u); ==>

V=

     0     1     2     3     4    -3    -2    -1
     0     1     2     3     4    -3    -2    -1
     0     1     2     3     4    -3    -2    -1
     0     1     2     3     4    -3    -2    -1
     0     1     2     3     4    -3    -2    -1
     0     1     2     3     4    -3    -2    -1
     0     1     2     3     4    -3    -2    -1
     0     1     2     3     4    -3    -2    -1

U =

     0     0     0     0     0     0     0     0
     1     1     1     1     1     1     1     1
     2     2     2     2     2     2     2     2
     3     3     3     3     3     3     3     3
     4     4     4     4     4     4     4     4
    -3    -3    -3    -3    -3    -3    -3    -3
    -2    -2    -2    -2    -2    -2    -2    -2

Please help me understand this program.. If possible explain this with an example.

Answer 1

[Hari]

Hi Premnath,

I understand that you want to uderstand the Part 2 of the code for an Ideal Low-Pass Filter, focusing on how it manipulates frequency indices and sets up the frequency domain.

Here is the explantion of the important steps of the code you have provided:

Frequency Indexing:

This part of the code generates frequency index vectors u and v for the image dimensions. It then adjusts indices greater than half the size to negative values, effectively centering the zero frequency.

u = 0:(M-1); v = 0:(N-1);
u(u > M/2) = u(u > M/2) - M;
v(v > N/2) = v(v > N/2) - N;

Meshgrid Creation:

Here, the code uses “meshgrid” to create a 2D grid of frequency coordinates. This grid is used to map out the frequency space, allowing for further calculations.

[V, U] = meshgrid(v, u);

Distance Calculation:

This section calculates the Euclidean distance D from the origin for each point in the frequency domain. This distance helps in defining the boundary of the filter.

D = sqrt(U.^2 + V.^2);

Example Explanation:

For an 8x8 matrix, this code shifts the frequency indices so that they range symmetrically around zero, like turning u into [0 1 2 3 4 -3 -2 -1], which is crucial for centering.

Purpose in Filtering:

By setting up D, the code allows you to apply a circular filter, retaining frequencies within a specified cutoff P and attenuating others, effectively filtering the image.

Refer to the documentation of “meshgrid” function to know more about its usage:

• https://www.mathworks.com/help/matlab/ref/meshgrid.html

Hope this helps!