MATLAB – Butterworth Lowpass Filter in Image Processing


In the field of Image Processing, Butterworth Lowpass Filter (BLPF) is used for image smoothing in the frequency domain. It removes high-frequency noise from a digital image and preserves low-frequency components. The transfer function of BLPF of order n is defined as-
H(u, v)=\frac{1}{1+\left[D(u, v) / D_{0}\right]^{2 n}}
Where,

  • D_{0} is a positive constant. BLPF passes all the frequencies less than D_{0} value without attenuation and cuts off all the frequencies greater than it.
  • This D_{0} is the transition point between H(u, v) = 1 and H(u, v) = 0, so this is termed as cutoff frequency. But instead of making a sharp cut-off (like, Ideal Lowpass Filter (ILPF)), it introduces a smooth transition from 1 to 0 to reduce ringing artifacts.
  • D(u, v) is the Euclidean Distance from any point (u, v) to the origin of the frequency plane, i.e, $D(u, v)=\sqrt{\left(u^{2}+v^{2}\right)}$

Approach:
Step 1: Input – Read an image
Step 2: Saving the size of the input image in pixels
Step 3: Get the Fourier Transform of the input_image
Step 4: Assign the order n and cut-off frequency D_{0}
Step 5: Designing filter: Butterworth Low Pass Filter
Step 6: Convolution between the Fourier Transformed input image and the filtering mask
Step 7: Take Inverse Fourier Transform of the convoluted image
Step 8: Display the resultant image as output

Implementation in MATLAB:

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% MATLAB Code | Butterworth Low Pass Filter
      
% Reading input image : input_image
input_image = imread('[name of input image file].[file format]');
  
% Saving the size of the input_image in pixels-
% M : no of rows (height of the image)
% N : no of columns (width of the image)
[M, N] = size(input_image);
  
% Getting Fourier Transform of the input_image
% using MATLAB library function fft2 (2D fast fourier transform)
FT_img = fft2(double(input_image));
  
% Assign the order value
n = 2; % one can change this value accordingly
  
% Assign Cut-off Frequency
D0 = 20; % one can change this value accordingly
  
% Designing filter
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;
  
% MATLAB library function meshgrid(v, u) returns 
% 2D grid which contains the coordinates of vectors 
% v and u. Matrix V with each row is a copy of v 
% and matrix U with each column is a copy of u 
[V, U] = meshgrid(v, u);
  
% Calculating Euclidean Distance
D = sqrt(U.^2 + V.^2);
  
% determining the filtering mask
H = 1./(1 + (D./D0).^(2*n));
  
% Convolution between the Fourier Transformed 
% image and the mask
G = H.*FT_img;
  
% Getting the resultant image by Inverse Fourier Transform 
% of the convoluted image using MATLAB library function  
% ifft2 (2D inverse fast fourier transform)   
output_image = real(ifft2(double(G))); 
    
% Displaying Input Image and Output Image 
subplot(2, 1, 1), imshow(input_image), 
subplot(2, 1, 2), imshow(output_image, [ ]);

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Input Image –

Output:

Note: A Butterworth filter of order 1 has no ringing artifact. Generally ringing is imperceptible in filters of order 2. But it can become a significant factor in filters of a higher order. For a specific cut-off frequency, ringing increases with an increase in the filter order.

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