how can i have image c and d ?

i try this but it doesnt result exact images

figure; imshow(log(1+S),[]); title('Spectrum');
% Define the center of the notch reject filter
[M, N] = size(I);
cx = M/2; cy = N/2;

% Define the radius and width of the notch reject filter
r = 30; % radius
w = 10; % width

% Compute the Butterworth notch reject filter
n = 4; % order
B = zeros(M,N);
for i = 1:M
    for j = 1:N
        d = sqrt((i-cx)^2 + (j-cy)^2);
        if d >= r-w/2 && d <= r+w/2
            B(i,j) = 1 / (1 + (d/r)^(2*n));
        else
            B(i,j) = 1;
        end
    end
end

% Apply the notch reject filter to the Fourier transform of the image
F_filtered = F_shifted .* B;
S_filtered = abs(F_filtered);
figure; imshow(log(1+S_filtered),[]); title('Filtered Spectrum');

% Compute the inverse Fourier transform of the filtered image
I_filtered = real(ifft2(ifftshift(F_filtered)));
figure; imshow(I_filtered,[]); title('Filtered Image')

[Image Analyst]

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Answer 1

[Abhishek]

Hi @hs,

It appears you are attempting to implement selective filtering using a Butterworth notch filter but are obtaining incorrect results.

The issue lies in the code for the filter which does not implement a notch filter. This can be confirmed by plotting the filter in image form.

I = imread('moire.png');
% Define the center of the notch reject filter
[M, N] = size(I);
cx = M/2; cy = N/2;
% Define the radius and width of the notch reject filter
r = 30; % radius
w = 10; % width
% Compute the Butterworth notch reject filter
n = 4; % order
B = zeros(M,N);

for i = 1:M
    for j = 1:N
        d = sqrt((i-cx)^2 + (j-cy)^2);
        if d >= r-w/2 && d <= r+w/2
            B(i,j) = 1 / (1 + (d/r)^(2*n));
        else
            B(i,j) = 1;
        end
    end
end
% Filter in image form
figure; imshow(B, []); title('Filter')

The correct implementation of Butterworth notch filter is shown below:

I = imread('moire.png');
% Define the center of the notch reject filter
[M, N] = size(I);
cx = M/2; cy = N/2;
% Define the radius and width of the notch reject filter
r = 15; % radius
% Compute the Butterworth notch reject filter
n = 4; % order
B = ones(M,N);

notch_centers = [-42.5, 25; 42.5, 25; -85, 25; 85, 25];
for i = 1:M
    for j = 1:N
        H = 1;
        for k = 1:size(notch_centers,1)
            % u-coordinate of the k-th notch center
            u_k = notch_centers(k,1);
            % v-coordinate of the k-th notch center
            v_k = notch_centers(k,2);
            
            % Distance to (u_k, v_k)
            Dk = sqrt((i - cx - u_k)^2 + (j - cy - v_k)^2);
            % Distance to symmetric notch (-u_k, -v_k)
            Dk_neg = sqrt((i - cx + u_k)^2 + (j - cy + v_k)^2);
            
            % Butterworth response for both notches
            H = H * 1/(1 + (r/Dk)^(2*n)) * 1/(1 + (r/Dk_neg)^(2*n));
        end
        B(i, j) = H;
    end
end
% Filter in image form
figure; imshow(B, []); title('Filter')

Visualizing the filter in image form confirms that it is a notch filer. Here, the notch centers are specified manually for demonstration but can be also calculated automatically using thresholding technique. With the correct filter in place, proceeding with remaining steps should yield the correct results.