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

Question

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
This isn't a low-pass filter output.
So, what is the issue with my code?

Answer 1

You didn't correctly copy the formula in the high pass filter:
uv = u.^2+v.^2;
D = uv.^0.5;
frac = Dh./D;
p = 2*n;
denom = frac.^p;
k = 1./(1+denom); --> you forgot the "1+"
New output:

Answer 2

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