matlablowpass-filterbutterworth

How can I implement a low-pass Butterworth filter in Matlab?


From this answer, I know how to create a High-pass Butterworth filter.

From this video, I know that, lowpasskernel = 1 - highpasskernel.

So, I created the following Low-pass Butterworth Filter,

function [out, kernel] = butterworth_lp(I, Dl, n)
    height = size(I, 1);
    width  = size(I, 2);

    [u, v] = meshgrid(-floor(width/2):floor(width/2)-1,-floor(height/2):floor(height/2)-1);

    % lp_kernel = 1 - hp_kernel
    kernel = 1 - butter_hp_kernel(u, v, Dl, n);

    % fft the image
    I_fft_shifted = fftshift(fft2(double(I)));

    % apply lowpass filter
    I_fft_shift_filtered = I_fft_shifted .* kernel;

    % inverse FFT, get real components
    out = real(ifft2(ifftshift(I_fft_shift_filtered)));

    % normalize and cast
    out = (out - min(out(:))) / (max(out(:)) - min(out(:)));
    out = uint8(255*out);

function k = butter_hp_kernel(u, v, Dh, n)
    uv =  u.^2+v.^2;
    D = sqrt(uv);
    frac = Dh./D;
    p = 2*n;
    denom =  frac.^p;
    k = 1./denom;

Output

enter image description here

This isn't a low-pass filter output.

So, what is the issue with my code?


Solution

  • Okay. I have solved the problem by following the following formula (Page #8/48),

    enter image description here

    Output

    enter image description here

    Source Code

    butter_lp_kernel.m

    function f = butter_lp_f(u, v, Dl, n)
        uv = u.^2+v.^2;
        Duv = sqrt(uv);
        frac = Duv./Dl;
        denom = frac.^(2*n);
        f = 1./(1.+denom);
    
    function k = butter_lp_kernel(I, Dl, n) 
        Height = size(I,1); 
        Width = size(I,2); 
    
        [u, v] = meshgrid( ...
                        -floor(Width/2) :floor(Width-1)/2, ...
                        -floor(Height/2): floor(Height-1)/2 ...
                     ); 
    
        k = butter_lp_f(u, v, Dl, n);
    

    ifftshow.m

    function out = ifftshow(f)
        f1 = abs(f);
        fm = max(f1(:));
        out = f1/fm;
    end
    

    butterworth_lpf.m

    function [out1, out2] = butterworth_lpf(I, Dl, n)
    
        Kernel = butter_lp_kernel(I, Dl, n);
    
        I_ffted_shifted = fftshift(fft2(I));
    
        I_ffted_shifted_filtered = I_ffted_shifted.*Kernel;
    
        out1 = ifftshow(ifft2(I_ffted_shifted_filtered));
    
        out2 = ifft2(ifftshift(I_ffted_shifted_filtered));
    end