MATLAB – Ideal Highpass Filter in Image Processing

In the field of Image Processing, Ideal Highpass Filter (IHPF) is used for image sharpening in the frequency domain. Image Sharpening is a technique to enhance the fine details and highlight the edges in a digital image. It removes low-frequency components from an image and preserves high-frequency components.

This ideal highpass filter is the reverse operation of the ideal lowpass filter. It can be determined using the following relation-  $H_{H P}(u, v)=1-H_{L P}(u, v)$
where, $H_{H P}(u, v)$ is the transfer function of the highpass filter and $H_{L P}(u, v)$ is the transfer function of the corresponding lowpass filter.

The transfer function of the IHPF can be specified by the function-
 $H(u, v)=\left\{\begin{array}{ll}0 & D(u, v) \leq D_{0} \\ 1 & D(u, v)>D_{0}\end{array}\right.$
Where,

  • D_{0} is a positive constant. IHPF passes all the frequencies outside of a circle of radius D_{0} from the origin without attenuation and cuts off all the frequencies within the circle.
  • This D_{0} is the transition point between H(u, v) = 1 and H(u, v) = 0, so this is termed as cutoff frequency.
  • 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 Cut-off Frequency D_{0}
Step 5: Designing filter: Ideal High 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 | Ideal High 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 Cut-off Frequency  
D0 = 10; % one can change this value accordingly
  
% Designing filter
u = 0:(M-1);
idx = find(u>M/2);
u(idx) = u(idx)-M;
v = 0:(N-1);
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);
  
% Comparing with the cut-off frequency and 
% determining the filtering mask
H = double(D > D0);
  
% 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:




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