MATLAB –图像处理中的巴特沃斯高通滤波器

在图像处理领域，巴特沃斯高通滤波器(BHPF)用于频域中的图像锐化。图像锐化是一种增强精细细节并突出显示数字图像边缘的技术。它从图像中去除了低频成分，并保留了高频成分。

此Butterworth高通滤波器是Butterworth低通滤波器的反向操作。可以使用以下关系式确定： $$H_{H P}(u, v)=1-H_{L P}(u, v)$$在哪里， $$H_{H P}(u, v)$$是高通滤波器的传递函数， $$H_{L P}(u, v)$$是相应的低通滤波器的传递函数。

订单BHPF的传递函数$n$被定义为-
$H(u, v)=\frac{1}{1+\left[D_{0} / D(u, v)\right]^{2 n}}$
在哪里，

• $D_{0}$是一个正常数。 BHPF通过所有大于$D_{0}$值不衰减，并切断所有小于该值的频率。
• 这$D_{0}$是H(u，v)= 1和H(u，v)= 0之间的过渡点，因此称为截止频率。但是，它没有进行陡峭的截止(例如理想高通滤波器(IHPF))，而是引入了从0到1的平滑过渡以减少振铃失真。
• $D(u, v)$是从任意点(u，v)到频率平面原点的欧几里得距离，即$$D(u, v)=\sqrt{\left(u^{2}+v^{2}\right)}$$

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 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

在MATLAB中的实现：

% MATLAB Code | Butterworth 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 the order value
n = 2; % one can change this value accordingly
  
% Assign Cut-off Frequency
D0 = 10; % 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 + (D0./D).^(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, [ ]);

输入图像–

输出：

注意： BHPF的结果比IHPF平滑得多。在这里，即使对于最小截止频率值，边界也不会失真得多。使用BHPF可以更平滑地过渡到更高的截止频率值。