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 is defined as-

Where,

• is a positive constant. BLPF passes all the frequencies less than value without attenuation and cuts off all the frequencies greater than it.
• This 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.
• is the Euclidean Distance from any point (u, v) to the origin of the frequency plane, i.e,

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 and cut-off frequency
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.