Open In App

MATLAB – Butterworth Lowpass Filter in Image Processing

Improve
Improve
Like Article
Like
Save
Share
Report
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:
% 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, [ ]);

                    
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.

Last Updated : 10 May, 2020
Like Article
Save Article
Previous
Next
Share your thoughts in the comments
Similar Reads