[数字图像处理]频域滤波(2)--高通滤波器，带阻滤波器与陷波滤波器
2015-06-18 17:31
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1.高通滤波器
首先，对一副图像进行如下二维傅里叶变换。
我们将u=0和v=0带上式，我们可以得到如下式子。
根据上式，可以到F(0,0)的值是非常大的。这里，我们将F(0,0)称为直流分量，直流分量比其他的成分要大好几个数量级。所以，这也就是傅里叶谱为什么需要使用对数变换才能看清楚的原因。
这里，对于高通滤波器而言，由于直流分量被衰减，所以，所得到的图像的动态范围是非常狭窄的，也就造成了图像偏灰。进一步而言，保持直流（DC）分量，对别的部分进行增幅，可以增强图像的细节。这样的滤波器称为锐化滤波器。这一小节主要介绍高通滤波器与锐化滤波器。
1.1理想高通滤波器
这里的D0是滤波器的阻带半径，而D(u,v)是点到滤波器中央的距离。理想高通的滤波器的振幅特性如下所示。
用这个滤波器对图像进行处理，可得到如下所示的结果。我们可以看到，与理想的低通滤波器一样，所得到的图像有很明显的振铃现象。结果图像从视觉上来看，有些偏暗，这是因为图像的直流分量被滤掉的原因。
1.2巴特沃斯高通滤波器
同样的，巴特沃斯高通滤波器也可以通过改变次数n，对过度特性进行调整。过大的n会造成振铃现象。
1.3高斯高通滤波器
高斯滤波器的过度特性很好，所以不会发生振铃现象。
1.4 上述三种滤波器的实现Matlab代码
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%% ---------Butterworth Highpass Filters (Fre. Domain)------------f = imread('characters_test_pattern.tif');f = mat2gray(f,[0 255]);
[M,N] = size(f);P = 2*M;Q = 2*N;fc = zeros(M,N);
for x = 1:1:M for y = 1:1:N fc(x,y) = f(x,y) * (-1)^(x+y); endend
F = fft2(fc,P,Q);
H_1 = zeros(P,Q);H_2 = zeros(P,Q);
for x = (-P/2):1:(P/2)-1 for y = (-Q/2):1:(Q/2)-1 D = (x^2 + y^2)^(0.5); D_0 = 100; H_1(x+(P/2)+1,y+(Q/2)+1) = 1/(1+(D_0/D)^2); H_2(x+(P/2)+1,y+(Q/2)+1) = 1/(1+(D_0/D)^6); endend
G_1 = H_1 .* F;G_2 = H_2 .* F;
g_1 = real(ifft2(G_1));g_1 = g_1(1:1:M,1:1:N);
g_2 = real(ifft2(G_2));g_2 = g_2(1:1:M,1:1:N);
for x = 1:1:M for y = 1:1:N g_1(x,y) = g_1(x,y) * (-1)^(x+y); g_2(x,y) = g_2(x,y) * (-1)^(x+y); endend
%% -----show-------figure();subplot(1,2,1);imshow(f,[0 1]);xlabel('a).Original Image');
subplot(1,2,2);imshow(log(1 + abs(F)),[ ]);xlabel('b).Fourier spectrum of a');
figure();subplot(1,2,1);imshow(H_1,[0 1]);xlabel('c)Butterworth Lowpass (D_{0}=100,n=1)');
subplot(1,2,2);h = mesh(1:20:P,1:20:Q,H_1(1:20:P,1:20:Q));set(h,'EdgeColor','k');axis([0 P 0 Q 0 1]);xlabel('u');ylabel('v');zlabel('|H(u,v)|');
figure();subplot(1,2,1);imshow(log(1 + abs(G_1)),[ ]);xlabel('d).Result of filtering using c');
subplot(1,2,2);imshow(g_1,[0 1]);xlabel('e).Result image');
figure();subplot(1,2,1);imshow(H_2,[0 1]);xlabel('f).Butterworth Lowpass (D_{0}=100,n=3)');
subplot(1,2,2);h = mesh(1:20:P,1:20:Q,H_2(1:20:P,1:20:Q));set(h,'EdgeColor','k');axis([0 P 0 Q 0 1]);xlabel('u');ylabel('v');zlabel('|H(u,v)|');
figure();subplot(1,2,1);imshow(log(1 + abs(G_2)),[ ]);xlabel('g).Result of filtering using e');
subplot(1,2,2);imshow(g_2,[0 1]);xlabel('h).Result image'); |
来自CODE的代码片
Butterworth_Highpass_Filters.m
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%% ---------Gaussian Highpass Filters (Fre. Domain)------------f = imread('characters_test_pattern.tif');f = mat2gray(f,[0 255]);
[M,N] = size(f);P = 2*M;Q = 2*N;fc = zeros(M,N);
for x = 1:1:M for y = 1:1:N fc(x,y) = f(x,y) * (-1)^(x+y); endend
F = fft2(fc,P,Q);
H_1 = zeros(P,Q);H_2 = zeros(P,Q);
for x = (-P/2):1:(P/2)-1 for y = (-Q/2):1:(Q/2)-1 D = (x^2 + y^2)^(0.5); D_0 = 60; H_1(x+(P/2)+1,y+(Q/2)+1) = 1 - exp(-(D*D)/(2*D_0*D_0)); D_0 = 160; H_2(x+(P/2)+1,y+(Q/2)+1) = 1 - exp(-(D*D)/(2*D_0*D_0)); endend
G_1 = H_1 .* F;G_2 = H_2 .* F;
g_1 = real(ifft2(G_1));g_1 = g_1(1:1:M,1:1:N);
g_2 = real(ifft2(G_2));g_2 = g_2(1:1:M,1:1:N);
for x = 1:1:M for y = 1:1:N g_1(x,y) = g_1(x,y) * (-1)^(x+y); g_2(x,y) = g_2(x,y) * (-1)^(x+y); endend
%% -----show-------figure();subplot(1,2,1);imshow(f,[0 1]);xlabel('a).Original Image');
subplot(1,2,2);imshow(log(1 + abs(F)),[ ]);xlabel('b).Fourier spectrum of a');
figure();subplot(1,2,1);imshow(H_1,[0 1]);xlabel('c)Gaussian Highpass (D_{0}=60)');
subplot(1,2,2);h = mesh(1:20:P,1:20:Q,H_1(1:20:P,1:20:Q));set(h,'EdgeColor','k');axis([0 P 0 Q 0 1]);xlabel('u');ylabel('v');zlabel('|H(u,v)|');
figure();subplot(1,2,1);imshow(log(1 + abs(G_1)),[ ]);xlabel('d).Result of filtering using c');
subplot(1,2,2);imshow(g_1,[0 1]);xlabel('e).Result image');
figure();subplot(1,2,1);imshow(H_2,[0 1]);xlabel('f).Gaussian Highpass (D_{0}=160)');
subplot(1,2,2);h = mesh(1:20:P,1:20:Q,H_2(1:20:P,1:20:Q));set(h,'EdgeColor','k');axis([0 P 0 Q 0 1]);xlabel('u');ylabel('v');zlabel('|H(u,v)|');
figure();subplot(1,2,1);imshow(log(1 + abs(G_2)),[ ]);xlabel('g).Result of filtering using e');
subplot(1,2,2);imshow(g_2,[0 1]);xlabel('h).Result image'); |
来自CODE的代码片
Gaussian_Highpass_Filters.m
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text-decoration: none;"> 97</a> | close all;clear all;
%% ---------Ideal Highpass Filters (Fre. Domain)------------f = imread('characters_test_pattern.tif');f = mat2gray(f,[0 255]);
[M,N] = size(f);P = 2*M;Q = 2*N;fc = zeros(M,N);
for x = 1:1:M for y = 1:1:N fc(x,y) = f(x,y) * (-1)^(x+y); endend
F = fft2(fc,P,Q);
H_1 = ones(P,Q);H_2 = ones(P,Q);
for x = (-P/2):1:(P/2)-1 for y = (-Q/2):1:(Q/2)-1 D = (x^2 + y^2)^(0.5); if(D <= 60) H_1(x+(P/2)+1,y+(Q/2)+1) = 0; end if(D <= 160) H_2(x+(P/2)+1,y+(Q/2)+1) = 0; end endend
G_1 = H_1 .* F;G_2 = H_2 .* F;
g_1 = real(ifft2(G_1));g_1 = g_1(1:1:M,1:1:N);
g_2 = real(ifft2(G_2));g_2 = g_2(1:1:M,1:1:N);
for x = 1:1:M for y = 1:1:N g_1(x,y) = g_1(x,y) * (-1)^(x+y); g_2(x,y) = g_2(x,y) * (-1)^(x+y); endend%% -----show-------figure();subplot(1,2,1);imshow(f,[0 1]);xlabel('a).Original Image');
subplot(1,2,2);imshow(log(1 + abs(F)),[ ]);xlabel('b).Fourier spectrum of a');
figure();subplot(1,2,1);imshow(H_1,[0 1]);xlabel('c).Ideal Highpass filter(D=60)');
subplot(1,2,2);h = mesh(1:20:P,1:20:Q,H_1(1:20:P,1:20:Q));set(h,'EdgeColor','k');axis([0 P 0 Q 0 1]);xlabel('u');ylabel('v');zlabel('|H(u,v)|');
figure();subplot(1,2,1);imshow(log(1 + abs(G_1)),[ ]);xlabel('d).Result of filtering using c');
subplot(1,2,2);imshow(g_1,[0 1]);xlabel('e).Result image');
figure();subplot(1,2,1);imshow(H_2,[0 1]);xlabel('f).Ideal Highpass filter(D=160)');
subplot(1,2,2);h = mesh(1:20:P,1:20:Q,H_2(1:20:P,1:20:Q));set(h,'EdgeColor','k');axis([0 P 0 Q 0 1]);xlabel('u');ylabel('v');zlabel('|H(u,v)|');
figure();subplot(1,2,1);imshow(log(1 + abs(G_2)),[ ]);xlabel('g).Result of filtering using e');
subplot(1,2,2);imshow(g_2,[0 1]);xlabel('h).Result image'); |
来自CODE的代码片
Ideal_Highpass.m
1.5 锐化滤波器
按照之前所说的，锐化滤波器是将傅里叶谱的直流分量保留，然后将其余的成分增幅。使用锐化滤波器，可以对图像的细节进行增强，使得细节凸显出来。锐化滤波器的表达式如下所示。
其实上式的目的很明显，就是先将原图的傅里叶谱
保留下来，然后叠加上高通滤波器的结果
，所得到的图像就是锐化后的图像了。这里为了调整锐化程度，引入了两个变量
与
。
可以调整直流分量的衰减程度，
可以调整高频分量的增幅程度。
下面是代码。
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text-decoration: none;"> 73</a> <a target=_blank id="L74" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L74" rel="#L74" style="color: rgb(102, 102, 102); text-decoration: none;"> 74</a> <a target=_blank id="L75" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L75" rel="#L75" style="color: rgb(102, 102, 102); text-decoration: none;"> 75</a> <a target=_blank id="L76" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L76" rel="#L76" style="color: rgb(102, 102, 102); text-decoration: none;"> 76</a> <a target=_blank id="L77" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L77" rel="#L77" style="color: rgb(102, 102, 102); text-decoration: none;"> 77</a> <a target=_blank id="L78" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L78" rel="#L78" style="color: rgb(102, 102, 102); text-decoration: none;"> 78</a> | close all;clear all;clc;
%% ---------The High-Fre-Emphasis Filters (Fre. Domain)------------f = imread('blurry_moon.tif');f = mat2gray(f,[0 255]);
[M,N] = size(f); P = 2*M;Q = 2*N;fc = zeros(M,N);
for x = 1:1:M for y = 1:1:N fc(x,y) = f(x,y) * (-1)^(x+y); endend
F = fft2(fc,P,Q);
H_HP = zeros(P,Q);
for x = (-P/2):1:(P/2)-1 for y = (-Q/2):1:(Q/2)-1 D = (x^2 + y^2)^(0.5); D_0 = 80; H_HP(x+(P/2)+1,y+(Q/2)+1) = 1 - exp(-(D*D)/(2*D_0*D_0)); endend
G_HP = H_HP .* F;
G_HFE = (1 + 1.1 * H_HP) .* F;
g_1 = real(ifft2(G_HP));g_1 = g_1(1:1:M,1:1:N);
g_2 = real(ifft2(G_HFE));g_2 = g_2(1:1:M,1:1:N);
for x = 1:1:M for y = 1:1:N g_1(x,y) = g_1(x,y) * (-1)^(x+y); g_2(x,y) = g_2(x,y) * (-1)^(x+y); endend
g = histeq(g_2);
%g_1 = mat2gray(g_1);%% -----show-------figure();subplot(1,2,1);imshow(f,[0 1]);xlabel('a).Original Image');
subplot(1,2,2);imshow(log(1 + abs(F)),[ ]);xlabel('b).Fourier spectrum of a');
figure();subplot(1,2,1);imshow(g_1,[0 1]);xlabel('c).Result image of High-pass Filter');
subplot(1,2,2);imshow(log(1 + abs(G_HP)),[ ]);xlabel('d).Result of filtering using e');
figure();subplot(1,2,1);imshow(g_2,[0 1]);xlabel('e).Result image of High-Fre-Emphasis Filter');
subplot(1,2,2);imshow(log(1 + abs(G_HFE)),[ ]);xlabel('f).Result of filtering using e'); |
来自CODE的代码片
High-Fre-Emphasis_Filters.m
2.带阻滤波器
同样的，带阻滤波器也有三种特性。高斯、巴特沃斯和理想，三种类型，其数学表达式如下所示。
其带通滤波器可以使用上面的表格转化而得。
带阻滤波器可以用于去除周期性噪声，为了体现带阻滤波器的特性，我们先对一幅图像增加很严重的噪声。
在原图的傅里叶谱上添加了几个很明显的亮点。在对其做IDFT，可以看到，原图被严重的周期噪声污染了。此时，我们可以使用带阻滤波器，可以有很好的去噪效果。为了避免振铃现象，选择使用如下所示巴特沃斯带阻滤波器，所用滤波器的次数为2次。使用空间域的操作，要去除这种噪声基本是不可能的，这也是频域内的操作的优点。
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text-decoration: none;"> 97</a> <a target=_blank id="L98" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L98" rel="#L98" style="color: rgb(102, 102, 102); text-decoration: none;"> 98</a> <a target=_blank id="L99" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L99" rel="#L99" style="color: rgb(102, 102, 102); text-decoration: none;"> 99</a> <a target=_blank id="L100" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L100" rel="#L100" style="color: rgb(102, 102, 102); text-decoration: none;"> 100</a> | close all;clear all;clc;%% ---------------------Add Noise-------------------------f = imread('left.tif');f = mat2gray(f,[0 255]);
[M,N] = size(f);P = 2*M;Q = 2*N;fc = zeros(M,N);
for x = 1:1:M for y = 1:1:N fc(x,y) = f(x,y) * (-1)^(x+y); endend
F = fft2(fc,P,Q);
H_NP = ones(P,Q);
for x = (-P/2):1:(P/2)-1 for y = (-Q/2):1:(Q/2)-1 D = 2; v_k = -200; u_k = 150; D_k = ((x+u_k)^2 + (y+v_k)^2)^(0.5); H_NP(x+(P/2)+1,y+(Q/2)+1) = H_NP(x+(P/2)+1,y+(Q/2)+1) * 1/(1+(D/D_k)^4); D_k = ((x-u_k)^2 + (y-v_k)^2)^(0.5); H_NP(x+(P/2)+1,y+(Q/2)+1) = H_NP(x+(P/2)+1,y+(Q/2)+1) * 1/(1+(D/D_k)^4); v_k = 200; u_k = 150; D_k = ((x+u_k)^2 + (y+v_k)^2)^(0.5); H_NP(x+(P/2)+1,y+(Q/2)+1) = H_NP(x+(P/2)+1,y+(Q/2)+1) * 1/(1+(D/D_k)^4); D_k = ((x-u_k)^2 + (y-v_k)^2)^(0.5); H_NP(x+(P/2)+1,y+(Q/2)+1) = H_NP(x+(P/2)+1,y+(Q/2)+1) * 1/(1+(D/D_k)^4); v_k = 0; u_k = 250; D_k = ((x+u_k)^2 + (y+v_k)^2)^(0.5); H_NP(x+(P/2)+1,y+(Q/2)+1) = H_NP(x+(P/2)+1,y+(Q/2)+1) * 1/(1+(D/D_k)^4); D_k = ((x-u_k)^2 + (y-v_k)^2)^(0.5); H_NP(x+(P/2)+1,y+(Q/2)+1) = H_NP(x+(P/2)+1,y+(Q/2)+1) * 1/(1+(D/D_k)^4); v_k = 250; u_k = 0; D_k = ((x+u_k)^2 + (y+v_k)^2)^(0.5); H_NP(x+(P/2)+1,y+(Q/2)+1) = H_NP(x+(P/2)+1,y+(Q/2)+1) * 1/(1+(D/D_k)^4); D_k = ((x-u_k)^2 + (y-v_k)^2)^(0.5); H_NP(x+(P/2)+1,y+(Q/2)+1) = H_NP(x+(P/2)+1,y+(Q/2)+1) * 1/(1+(D/D_k)^4); H_NP(x+(P/2)+1,y+(Q/2)+1) = 1 + 700*(1 - H_NP(x+(P/2)+1,y+(Q/2)+1)); endend
G_Noise = F .* H_NP;
g_noise = real(ifft2(G_Noise));g_noise = g_noise(1:1:M,1:1:N);
for x = 1:1:M for y = 1:1:N g_noise(x,y) = g_noise(x,y) * (-1)^(x+y); endend
%% ---------Bondpass Filters (Fre. Domain)------------H_1 = ones(P,Q);
for x = (-P/2):1:(P/2)-1 for y = (-Q/2):1:(Q/2)-1 D = (x^2 + y^2)^(0.5); D_0 = 250; W = 30; H_1(x+(P/2)+1,y+(Q/2)+1) = 1/(1+((D*W)/((D*D) - (D_0*D_0)))^6); endend
G_1 = H_1 .* G_Noise;
g_1 = real(ifft2(G_1));g_1 = g_1(1:1:M,1:1:N);
for x = 1:1:M for y = 1:1:N g_1(x,y) = g_1(x,y) * (-1)^(x+y); endend
%% -----show-------close all;
figure();subplot(1,2,1);imshow(f,[0 1]);xlabel('a).Original Image');
subplot(1,2,2);imshow(log(1 + abs(F)),[ ]);xlabel('b).Fourier spectrum of a');
figure();subplot(1,2,2);imshow(log(1 + abs(G_Noise)),[ ]);xlabel('c).Fourier spectrum of b');
subplot(1,2,1);imshow(g_noise,[0 1]);xlabel('b).Result of add noise');
figure();subplot(1,2,1);imshow(H_1,[0 1]);xlabel('d).Butterworth Bandpass filter(D=355 W=40 n =2)');
subplot(1,2,2);h = mesh(1:20:Q,1:20:P,H_1(1:20:P,1:20:Q));set(h,'EdgeColor','k');axis([0 Q 0 P 0 1]);xlabel('u');ylabel('v');zlabel('|H(u,v)|');
figure();subplot(1,2,2);imshow(log(1 + abs(G_1)),[ ]);xlabel('e).Fourier spectrum of f');
subplot(1,2,1);imshow(g_1,[0 1]);xlabel('f).Result of denoise'); |
来自CODE的代码片
Butterworth_Boundpass.m
3.陷波滤波器(Notch Filter)
陷波滤波器也用于去除周期噪声，虽然带阻滤波器也能可以去除周期噪声，但是带阻滤波器对噪声以外的成分也有衰减。而陷波滤波器主要对，某个点进行衰减，对其余的成分不损失。使用下图将更容易理解。
从空间域内看，图像存在着周期性噪声。其傅里叶频谱中，也能看到噪声的所在之处(这里我用红圈标注出来了，他们不是数据的一部分)。我们如果使用带阻滤波器的话，是非常麻烦的，也会使得图像有损失。陷波滤波器，能够直接对噪声处进行衰减，可以有很好的去噪效果。
其表达式如下所示，陷波滤波器可以通过对高通滤波器的中心，进行位移就可以得到了。
这里，由于傅里叶的周期性，傅里叶频谱上不可能单独存在一个点的噪声，必定是关于远点对称的一个噪声对。这里的
与
需要去除的噪声点，其求取的方式如下所示。
针对于上图，我们设计如下滤波器，去进行去噪。
（图片下标错了，后续更新改过来！）
所得到的结果，如下所示。噪声已经被去除了，画质得到了很大的改善。
（图片下标错了，后续更新改过来！）
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%% ---------Butterworth Notch filter (Fre. Domain)------------f = imread('car_75DPI_Moire.tif');f = mat2gray(f,[0 255]);
[M,N] = size(f);P = 2*M;Q = 2*N;fc = zeros(M,N);
for x = 1:1:M for y = 1:1:N fc(x,y) = f(x,y) * (-1)^(x+y); endend
F = fft2(fc,P,Q);
H_NF = ones(P,Q);
for x = (-P/2):1:(P/2)-1 for y = (-Q/2):1:(Q/2)-1 D = 30; v_k = 59; u_k = 77; D_k = ((x+u_k)^2 + (y+v_k)^2)^(0.5); H_NF(x+(P/2)+1,y+(Q/2)+1) = H_NF(x+(P/2)+1,y+(Q/2)+1) * 1/(1+(D/D_k)^4); D_k = ((x-u_k)^2 + (y-v_k)^2)^(0.5); H_NF(x+(P/2)+1,y+(Q/2)+1) = H_NF(x+(P/2)+1,y+(Q/2)+1) * 1/(1+(D/D_k)^4); v_k = 59; u_k = 159; D_k = ((x+u_k)^2 + (y+v_k)^2)^(0.5); H_NF(x+(P/2)+1,y+(Q/2)+1) = H_NF(x+(P/2)+1,y+(Q/2)+1) * 1/(1+(D/D_k)^4); D_k = ((x-u_k)^2 + (y-v_k)^2)^(0.5); H_NF(x+(P/2)+1,y+(Q/2)+1) = H_NF(x+(P/2)+1,y+(Q/2)+1) * 1/(1+(D/D_k)^4); v_k = -54; u_k = 84; D_k = ((x+u_k)^2 + (y+v_k)^2)^(0.5); H_NF(x+(P/2)+1,y+(Q/2)+1) = H_NF(x+(P/2)+1,y+(Q/2)+1) * 1/(1+(D/D_k)^4); D_k = ((x-u_k)^2 + (y-v_k)^2)^(0.5); H_NF(x+(P/2)+1,y+(Q/2)+1) = H_NF(x+(P/2)+1,y+(Q/2)+1) * 1/(1+(D/D_k)^4); v_k = -54; u_k = 167; D_k = ((x+u_k)^2 + (y+v_k)^2)^(0.5); H_NF(x+(P/2)+1,y+(Q/2)+1) = H_NF(x+(P/2)+1,y+(Q/2)+1) * 1/(1+(D/D_k)^4); D_k = ((x-u_k)^2 + (y-v_k)^2)^(0.5); H_NF(x+(P/2)+1,y+(Q/2)+1) = H_NF(x+(P/2)+1,y+(Q/2)+1) * 1/(1+(D/D_k)^4); endend
G_1 = H_NF .* F;
g_1 = real(ifft2(G_1));g_1 = g_1(1:1:M,1:1:N);
for x = 1:1:M for y = 1:1:N g_1(x,y) = g_1(x,y) * (-1)^(x+y); endend
%% -----show-------close all;
figure();subplot(1,2,1);imshow(f,[0 1]);xlabel('a).Original Image');
subplot(1,2,2);imshow(log(1 + abs(F)),[ ]);xlabel('b).Fourier spectrum of a');
figure();subplot(1,2,1);imshow(H_NF,[0 1]);xlabel('c).Butterworth Notch filter(D=30 n=2)');
subplot(1,2,2);h = mesh(1:10:Q,1:10:P,H_NF(1:10:P,1:10:Q));set(h,'EdgeColor','k');axis([0 Q 0 P 0 1]);xlabel('u');ylabel('v');zlabel('|H(u,v)|');
figure();subplot(1,2,2);imshow(log(1 + abs(G_1)),[ ]);xlabel('e).Fourier spectrum of d');
subplot(1,2,1);imshow(g_1,[0 1]);xlabel('d).Result image'); |
来自CODE的代码片
Notch_Filtering.m
原文发于博客：http://blog.csdn.net/thnh169/
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