如何在Matlab中实现低通Butterworth滤波器？

Question

从this answer，我知道如何创建一个高通巴特沃斯滤波器。我知道lowpasskernel = 1 - highpasskernel。

所以，我创建以下低通Butterworth滤波器电路，

function [out, kernel] = butterworth_lp(I, Dl, n) 
    height = size(I, 1); 
    width = size(I, 2); 

    [u, v] = meshgrid(-floor(width/2):floor(width/2)-1,-floor(height/2):floor(height/2)-1); 

    % lp_kernel = 1 - hp_kernel 
    kernel = 1 - butter_hp_kernel(u, v, Dl, n); 

    % fft the image 
    I_fft_shifted = fftshift(fft2(double(I))); 

    % apply lowpass filter 
    I_fft_shift_filtered = I_fft_shifted .* kernel; 

    % inverse FFT, get real components 
    out = real(ifft2(ifftshift(I_fft_shift_filtered))); 

    % normalize and cast 
    out = (out - min(out(:)))/(max(out(:)) - min(out(:))); 
    out = uint8(255*out); 

function k = butter_hp_kernel(u, v, Dh, n) 
    uv = u.^2+v.^2; 
    D = sqrt(uv); 
    frac = Dh./D; 
    p = 2*n; 
    denom = frac.^p; 
    k = 1./denom; 

输出

这不是一个低通滤波器的输出。

那么，我的代码有什么问题？

Answer 1

好。我已经跟随following formula (Page #8/48)解决了这个问题，

输出

源代码

butter_lp_kernel.m

function f = butter_lp_f(u, v, Dl, n) 
    uv = u.^2+v.^2; 
    Duv = sqrt(uv); 
    frac = Duv./Dl; 
    denom = frac.^(2*n); 
    f = 1./(1.+denom); 

function k = butter_lp_kernel(I, Dl, n) 
    Height = size(I,1); 
    Width = size(I,2); 

    [u, v] = meshgrid(... 
        -floor(Width/2) :floor(Width-1)/2, ... 
        -floor(Height/2): floor(Height-1)/2 ... 
       ); 

    k = butter_lp_f(u, v, Dl, n); 

ifftshow.m

function out = ifftshow(f) 
    f1 = abs(f); 
    fm = max(f1(:)); 
    out = f1/fm; 
end 

butterworth_lpf.m

function [out1, out2] = butterworth_lpf(I, Dl, n) 

    Kernel = butter_lp_kernel(I, Dl, n); 

    I_ffted_shifted = fftshift(fft2(I)); 

    I_ffted_shifted_filtered = I_ffted_shifted.*Kernel; 

    out1 = ifftshow(ifft2(I_ffted_shifted_filtered)); 

    out2 = ifft2(ifftshift(I_ffted_shifted_filtered)); 
end 

Answer 2

你没有公式正确复制的高通滤波器：

uv = u.^2+v.^2; 
D = uv.^0.5; 
frac = Dh./D; 
p = 2*n; 
denom = frac.^p; 
k = 1./(1+denom); --> you forgot the "1+" 

新的输出：