MATLAB –图像处理中的巴特沃斯低通滤波器

在图像处理领域，巴特沃斯低通滤波器(BLPF)用于频域中的图像平滑。它可以消除数字图像中的高频噪声，并保留低频分量。订单BLPF的传递函数$n$被定义为-
$H(u, v)=\frac{1}{1+\left[D(u, v) / D_{0}\right]^{2 n}}$
在哪里，

• $D_{0}$是一个正常数。 BLPF通过的所有频率小于$D_{0}$值不衰减，并切断所有大于该值的频率。
• 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(例如理想低通滤波器(ILPF))，而是引入了从1到0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 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

在MATLAB中的实现：

% 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, [ ]);

输入图像–

输出：

注意： 1阶巴特沃斯滤波器没有振铃伪影。通常，在2阶滤波器中不会感觉到振铃，但是在高阶滤波器中，振铃可能会成为重要因素。对于特定的截止频率，振铃随着滤波器阶数的增加而增加。