[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

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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             (
             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
             );
            
     
             end
            

             end
            


            


            

             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
             );
            
    
             end
            

             end
            


            

             %% -----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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             close 
             all
             ;
            

             clear 
             all
             ;
            


            

             %% ---------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
             );
            
    
             end
            

             end
            


            

             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
             ));
            
     
             end
            

             end
            


            

             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
             );
            
    
             end
            

             end
            


            


            

             %% -----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

 
       <a target=_blank id="L1" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L1" rel="#L1" style="color: rgb(102, 102, 102); text-decoration: none;">  1</a>
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             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
             );
            
    
             end
            

             end
            


            

             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
            
     
             end
            

             end
            


            

             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
             );
            
    
             end
            

             end
            

             %% -----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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             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
             );
            
    
             end
            

             end
            


            

             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
             ));   
            
     
             end
            

             end
            


            

             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
             );
            
    
             end
            

             end
            


            

             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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             1
             :
             M
            
    
             for 
             y 
             = 
             1
             :
             1
             :
             N
            
        
             fc
             (
             x
             ,
             y
             ) 
             = 
             f
             (
             x
             ,
             y
             ) 
             * 
             (
             -
             1
             )^
             (
             x
             +
             y
             );
            
    
             end
            

             end
            


            

             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
             ));
            
     
             end
            

             end
            


            

             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
             );
            
    
             end
            

             end
            


            


            

             %% ---------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
             );   
            
     
             end
            

             end
            


            

             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
             );
            
    
             end
            

             end
            


            

             %% -----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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<a target=_blank id="L28" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L28" rel="#L28" style="color: rgb(102, 102, 102); text-decoration: none;"> 28</a>
<a target=_blank id="L29" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L29" rel="#L29" style="color: rgb(102, 102, 102); text-decoration: none;"> 29</a>
<a target=_blank id="L30" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L30" rel="#L30" style="color: rgb(102, 102, 102); text-decoration: none;"> 30</a>
<a target=_blank id="L31" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L31" rel="#L31" style="color: rgb(102, 102, 102); text-decoration: none;"> 31</a>
<a target=_blank id="L32" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L32" rel="#L32" style="color: rgb(102, 102, 102); text-decoration: none;"> 32</a>
<a target=_blank id="L33" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L33" rel="#L33" style="color: rgb(102, 102, 102); text-decoration: none;"> 33</a>
<a target=_blank id="L34" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L34" rel="#L34" style="color: rgb(102, 102, 102); text-decoration: none;"> 34</a>
<a target=_blank id="L35" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L35" rel="#L35" style="color: rgb(102, 102, 102); text-decoration: none;"> 35</a>
<a target=_blank id="L36" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L36" rel="#L36" style="color: rgb(102, 102, 102); text-decoration: none;"> 36</a>
<a target=_blank id="L37" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L37" rel="#L37" style="color: rgb(102, 102, 102); text-decoration: none;"> 37</a>
<a target=_blank id="L38" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L38" rel="#L38" style="color: rgb(102, 102, 102); text-decoration: none;"> 38</a>
<a target=_blank id="L39" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L39" rel="#L39" style="color: rgb(102, 102, 102); text-decoration: none;"> 39</a>
<a target=_blank id="L40" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L40" rel="#L40" style="color: rgb(102, 102, 102); text-decoration: none;"> 40</a>
<a target=_blank id="L41" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L41" rel="#L41" style="color: rgb(102, 102, 102); text-decoration: none;"> 41</a>
<a target=_blank id="L42" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L42" rel="#L42" style="color: rgb(102, 102, 102); text-decoration: none;"> 42</a>
<a target=_blank id="L43" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L43" rel="#L43" style="color: rgb(102, 102, 102); text-decoration: none;"> 43</a>
<a target=_blank id="L44" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L44" rel="#L44" style="color: rgb(102, 102, 102); text-decoration: none;"> 44</a>
<a target=_blank id="L45" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L45" rel="#L45" style="color: rgb(102, 102, 102); text-decoration: none;"> 45</a>
<a target=_blank id="L46" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L46" rel="#L46" style="color: rgb(102, 102, 102); text-decoration: none;"> 46</a>
<a target=_blank id="L47" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L47" rel="#L47" style="color: rgb(102, 102, 102); text-decoration: none;"> 47</a>
<a target=_blank id="L48" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L48" rel="#L48" style="color: rgb(102, 102, 102); text-decoration: none;"> 48</a>
<a target=_blank id="L49" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L49" rel="#L49" style="color: rgb(102, 102, 102); text-decoration: none;"> 49</a>
<a target=_blank id="L50" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L50" rel="#L50" style="color: rgb(102, 102, 102); text-decoration: none;"> 50</a>
<a target=_blank id="L51" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L51" rel="#L51" style="color: rgb(102, 102, 102); text-decoration: none;"> 51</a>
<a target=_blank id="L52" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L52" rel="#L52" style="color: rgb(102, 102, 102); text-decoration: none;"> 52</a>
<a target=_blank id="L53" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L53" rel="#L53" style="color: rgb(102, 102, 102); text-decoration: none;"> 53</a>
<a target=_blank id="L54" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L54" rel="#L54" style="color: rgb(102, 102, 102); text-decoration: none;"> 54</a>
<a target=_blank id="L55" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L55" rel="#L55" style="color: rgb(102, 102, 102); text-decoration: none;"> 55</a>
<a target=_blank id="L56" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L56" rel="#L56" style="color: rgb(102, 102, 102); text-decoration: none;"> 56</a>
<a target=_blank id="L57" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L57" rel="#L57" style="color: rgb(102, 102, 102); text-decoration: none;"> 57</a>
<a target=_blank id="L58" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L58" rel="#L58" style="color: rgb(102, 102, 102); text-decoration: none;"> 58</a>
<a target=_blank id="L59" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L59" rel="#L59" style="color: rgb(102, 102, 102); text-decoration: none;"> 59</a>
<a target=_blank id="L60" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L60" rel="#L60" style="color: rgb(102, 102, 102); text-decoration: none;"> 60</a>
<a target=_blank id="L61" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L61" rel="#L61" style="color: rgb(102, 102, 102); text-decoration: none;"> 61</a>
<a target=_blank id="L62" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L62" rel="#L62" style="color: rgb(102, 102, 102); text-decoration: none;"> 62</a>
<a target=_blank id="L63" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L63" rel="#L63" style="color: rgb(102, 102, 102); text-decoration: none;"> 63</a>
<a target=_blank id="L64" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L64" rel="#L64" style="color: rgb(102, 102, 102); text-decoration: none;"> 64</a>
<a target=_blank id="L65" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L65" rel="#L65" style="color: rgb(102, 102, 102); text-decoration: none;"> 65</a>
<a target=_blank id="L66" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L66" rel="#L66" style="color: rgb(102, 102, 102); text-decoration: none;"> 66</a>
<a target=_blank id="L67" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L67" rel="#L67" style="color: rgb(102, 102, 102); text-decoration: none;"> 67</a>
<a target=_blank id="L68" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L68" rel="#L68" style="color: rgb(102, 102, 102); text-decoration: none;"> 68</a>
<a target=_blank id="L69" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L69" rel="#L69" style="color: rgb(102, 102, 102); text-decoration: none;"> 69</a>
<a target=_blank id="L70" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L70" rel="#L70" style="color: rgb(102, 102, 102); text-decoration: none;"> 70</a>
<a target=_blank id="L71" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L71" rel="#L71" style="color: rgb(102, 102, 102); text-decoration: none;"> 71</a>
<a target=_blank id="L72" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L72" rel="#L72" style="color: rgb(102, 102, 102); text-decoration: none;"> 72</a>
<a target=_blank id="L73" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L73" rel="#L73" style="color: rgb(102, 102, 102); 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>
<a target=_blank id="L79" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L79" rel="#L79" style="color: rgb(102, 102, 102); text-decoration: none;"> 79</a>
<a target=_blank id="L80" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L80" rel="#L80" style="color: rgb(102, 102, 102); text-decoration: none;"> 80</a>
<a target=_blank id="L81" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L81" rel="#L81" style="color: rgb(102, 102, 102); text-decoration: none;"> 81</a>
<a target=_blank id="L82" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L82" rel="#L82" style="color: rgb(102, 102, 102); text-decoration: none;"> 82</a>
<a target=_blank id="L83" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L83" rel="#L83" style="color: rgb(102, 102, 102); text-decoration: none;"> 83</a>
<a target=_blank id="L84" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L84" rel="#L84" style="color: rgb(102, 102, 102); text-decoration: none;"> 84</a>
<a target=_blank id="L85" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L85" rel="#L85" style="color: rgb(102, 102, 102); text-decoration: none;"> 85</a>
<a target=_blank id="L86" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L86" rel="#L86" style="color: rgb(102, 102, 102); text-decoration: none;"> 86</a>
<a target=_blank id="L87" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L87" rel="#L87" style="color: rgb(102, 102, 102); text-decoration: none;"> 87</a>
<a target=_blank id="L88" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L88" rel="#L88" style="color: rgb(102, 102, 102); text-decoration: none;"> 88</a>
<a target=_blank id="L89" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L89" rel="#L89" style="color: rgb(102, 102, 102); text-decoration: none;"> 89</a>
<a target=_blank id="L90" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L90" rel="#L90" style="color: rgb(102, 102, 102); text-decoration: none;"> 90</a>
<a target=_blank id="L91" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L91" rel="#L91" style="color: rgb(102, 102, 102); text-decoration: none;"> 91</a>
<a target=_blank id="L92" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L92" rel="#L92" style="color: rgb(102, 102, 102); text-decoration: none;"> 92</a>
<a target=_blank id="L93" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L93" rel="#L93" style="color: rgb(102, 102, 102); text-decoration: none;"> 93</a>
<a target=_blank id="L94" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L94" rel="#L94" style="color: rgb(102, 102, 102); text-decoration: none;"> 94</a>
<a target=_blank id="L95" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L95" rel="#L95" style="color: rgb(102, 102, 102); text-decoration: none;"> 95</a>
<a target=_blank id="L96" href="http://blog.csdn.net/zhoufan900428/article/details/17201293#L96" rel="#L96" style="color: rgb(102, 102, 102); text-decoration: none;"> 96</a> 
           
             close 
             all
             ;
            

             clear 
             all
             ;
            

             clc
             ;
            


            

             %% ---------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
             );
            
    
             end
            

             end
            


            

             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
             );
            
     
             end
            

             end
            


            

             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
             );
            
    
             end
            

             end
            


            

             %% -----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'
             );
            
 
          
 
     