数字图像处理之低通滤波器实现原理及方法（Matlab)

港控/mmm° 2022-07-15 10:10 289阅读 0赞

转自[http://blog.csdn.net/zhoufan900428/article/details/17194289][http_blog.csdn.net_zhoufan900428_article_details_17194289]

## 1.傅里叶变换与频域 ##

在之前的文中，我们已经进行过一些基本的图像处理。比如，使用低通滤波可以将图像模糊，也有些许降噪的作用。这些都是在空间域内进行的滤波处理，这个处理主要是依靠卷积来进行计算的。首先，从连续的一维卷积入手，如下所示。

![20131207233154609][]

将上式进行傅里叶变换，可以得到如下结果。

![20131207233404484][]

从这个式子，我们可以得到一个重要的结论。也就是，函数![20131207233545890][]与![20131207233549078][]卷积的傅里叶变换所得到的结果，是函数![20131207233545890][]与![20131207233549078][]的傅里叶变换![20131207233707609][]与![20131207233711093][]的乘积。再将其总结得简单易懂一些，有如下结论。

![20131207233820687][]

在将其扩展到二维的形况下，假设尺寸为MxN的图像，如下关系是成立的。

![20131207234129421][]

其实到这，基本的原理就明了的。我们所看到的图像，均为空间域内的表现形式，我们无法辨识出频域内的图像。要进行频域内的滤波器处理，首先就需要进行傅里叶变换，然后直接进行滤波处理，最后再用反傅里叶变换倒回到空间域内。

到此，已经可以开始空间域内的滤波处理了。但是，还有一点需要注意的地方。使用某个一维信号来举例子，一维信号的傅里叶变换是以2π为周期的函数。所以，我们常常使用的范围\[-π,π\]来表示这个信号的傅里叶变换，如下所示。

![20131208002637921][]

这样做的好处是，靠近0的成分就是低频，靠近-π与π的成分就表示高频。而对于图像而言，在Matlab中，我们使用fft2()这个函数来求取图像的傅里叶变换。

**\[plain\]** [view plain][] [copy][view plain]

[![在CODE上查看代码片][CODE]][CODE_CODE] [![派生到我的代码片][ico_fork.svg]][ico_fork.svg 1]

1.  g = fft2(f);

上面这个代码求取的，其实是 范围\[0,π\]内的傅里叶变换。为了方便理解，下图画出了本行代码所求取的图像的傅里叶变换的范围（右）和与其等效的一维傅里叶变换的范围（左）。

![20131208010819218][]

很显然，这并不是希望的范围，下面这个代码可以求取\[0,2π\]内的傅里叶变换。

**\[plain\]** [view plain][] [copy][view plain]

[![在CODE上查看代码片][CODE]][CODE_CODE] [![派生到我的代码片][ico_fork.svg]][ico_fork.svg 1]

1.  P = 2\*M; 
2.  Q = 2\*N; 
3.  F = fft2(f,P,Q);

下图画出了本行代码所求取的图像的傅里叶变换的范围（右）和与其等效的一维傅里叶变换的范围 （左） 。所得到的图像F(u,v)的尺寸为PxQ。

![20131208010832250][]

我们需要对其移动一下，如下图所示，我们需要的是粉色范围的区域。

![20131208010841359][]

下面，从数学上分析一下，如何获得这个部分的频谱。对于傅里叶变换，有如下性质。

![20131208005527218][]

这个特性称为平移特性，粉色部分的频谱，将![20131208010924031][]带入上式，我们可以得到如下式子。

![20131208011000093][]

为次，我们已经得到了粉色范围的频谱。越靠近傅里叶频谱图像中间的成分，代表了低频成分。其Matlab代码如下所示。

**\[plain\]** [view plain][] [copy][view plain]

[![在CODE上查看代码片][CODE]][CODE_CODE] [![派生到我的代码片][ico_fork.svg]][ico_fork.svg 1]

1.  \[M,N\] = size(f); 
2.  P = 2\*M; 
3.  Q = 2\*N; 
4.  fc = zeros(M,N); 
5.  
6.  for x = 1:1:M 
7.   for y = 1:1:N 
8.   fc(x,y) = f(x,y) \* (-1)^(x+y); 
9.   end 
10. end 
11. 
12. F = fft2(fc,P,Q);

代码所得到的结果，如下图所示。

![20131208120205656][]

接下来，我们总结一下频域滤波的步骤：

①：先将图像做频域内的水平移动，然后求原图像f(x,y)的DFT，得到其图像的傅里叶谱F(u,v)。

![20131208012143937][]

②：与频域滤波器做乘积，![20131208011652640][]

③：求取G(u,v)的IDFT，然后再将图像做频域内的水平移动（移动回去），其结果可能存在寄生的虚数，此时忽略即可。

![20131208012311875][]

④：这里使用ifft2函数进行IDFT变换，得到的图像的尺寸为PxQ。切取左上角的MxN的图像，就能得到结果了。

##  2.低通滤波器 ##

###  2.1理想的低通滤波器 ###

![20131208115855906][]

其中，D0表示通带的半径。D(u,v)的计算方式也就是两点间的距离，很简单就能得到。

![20131208120015265][]

使用低通滤波器所得到的结果如下所示。低通滤波器滤除了高频成分，所以使得图像模糊。由于理想低通滤波器的过度特性过于急峻，所以会产生了振铃现象。

![20131208120705156][]

![20131208120717859][]

###  2.2巴特沃斯低通滤波器 ###

![20131208121123250][]

同样的，D0表示通带的半径，n表示的是巴特沃斯滤波器的次数。随着次数的增加，振铃现象会越来越明显。

### ![20131208123312281][] ###

![20131208123323953][]

###  2.3高斯低通滤波器 ###

![20131208123519812][]

D0表示通带的半径。高斯滤波器的过度特性非常平坦，因此是不会产生振铃现象的。

![20131208123036031][]

![20131208123055046][]

###  3.实现代码 ###

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 <td style="border:none; margin:0px; padding:0px; vertical-align:top"> 
 <div style="border-left-width:1px; border-left-style:solid; border-left-color:rgb(222,226,227); overflow-x:auto; overflow-y:hidden"> 
 <pre style="word-wrap:normal; color:rgb(51,51,51); border:none; font-family:Menlo,'Liberation Mono',Consolas,'Courier New','andale mono','lucida console',monospace; margin-top:0px; margin-bottom:0px; padding:10px 0px; line-height:16px!important; background-color:rgb(255,255,255)"> 
 
 <div style="padding:0px 10px">
 
 
 close 
 
 
 all
 
 
 ;
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 clear 
 
 
 all
 
 
 ;
 
 
 </div>
 
 
 <div style="padding:0px 10px">

</div>
 
 
 <div style="padding:0px 10px">
 
 
 %% ---------Butterworth Lowpass Filters (Fre. Domain)------------
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 f 
 
 
 = 
 
 
 imread
 
 
 (
 
 
 'characters_test_pattern.tif'
 
 
 );
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 f 
 
 
 = 
 
 
 mat2gray
 
 
 (
 
 
 f
 
 
 ,[
 
 
 0 
 
 
 255
 
 
 ]);
 
 
 </div>
 
 
 <div style="padding:0px 10px">

</div>
 
 
 <div style="padding:0px 10px">
 
 
 [
 
 
 M
 
 
 ,
 
 
 N
 
 
 ] 
 
 
 = 
 
 
 size
 
 
 (
 
 
 f
 
 
 );
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 P 
 
 
 = 
 
 
 2
 
 
 *
 
 
 M
 
 
 ;
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 Q 
 
 
 = 
 
 
 2
 
 
 *
 
 
 N
 
 
 ;
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 fc 
 
 
 = 
 
 
 zeros
 
 
 (
 
 
 M
 
 
 ,
 
 
 N
 
 
 );
 
 
 </div>
 
 
 <div style="padding:0px 10px">

</div>
 
 
 <div style="padding:0px 10px">
 
 
 F 
 
 
 = 
 
 
 fft2
 
 
 (
 
 
 fc
 
 
 ,
 
 
 P
 
 
 ,
 
 
 Q
 
 
 );
 
 
 </div>
 
 
 <div style="padding:0px 10px">

</div>
 
 
 <div style="padding:0px 10px">
 
 
 H_1 
 
 
 = 
 
 
 zeros
 
 
 (
 
 
 P
 
 
 ,
 
 
 Q
 
 
 );
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 H_2 
 
 
 = 
 
 
 zeros
 
 
 (
 
 
 P
 
 
 ,
 
 
 Q
 
 
 );
 
 
 </div>
 
 
 <div style="padding:0px 10px">

</div>
 
 
 <div style="padding:0px 10px">
 
 
 for 
 
 
 x 
 
 
 = 
 
 
 (
 
 
 -
 
 
 P
 
 
 /
 
 
 2
 
 
 ):
 
 
 1
 
 
 :(
 
 
 P
 
 
 /
 
 
 2
 
 
 )
 
 
 -
 
 
 1
 
 
 </div>
 
 
 <div style="padding:0px 10px"> 
 
 
 for 
 
 
 y 
 
 
 = 
 
 
 (
 
 
 -
 
 
 Q
 
 
 /
 
 
 2
 
 
 ):
 
 
 1
 
 
 :(
 
 
 Q
 
 
 /
 
 
 2
 
 
 )
 
 
 -
 
 
 1
 
 
 </div>
 
 
 <div style="padding:0px 10px"> 
 
 
 D 
 
 
 = 
 
 
 (
 
 
 x^
 
 
 2 
 
 
 + 
 
 
 y^
 
 
 2
 
 
 )^
 
 
 (
 
 
 0.5
 
 
 );
 
 
 </div>
 
 
 <div style="padding:0px 10px"> 
 
 
 D_0 
 
 
 = 
 
 
 100
 
 
 ;
 
 
 </div>
 
 
 <div style="padding:0px 10px"> 
 
 
 H_1
 
 
 (
 
 
 x
 
 
 +
 
 
 (
 
 
 P
 
 
 /
 
 
 2
 
 
 )
 
 
 +
 
 
 1
 
 
 ,
 
 
 y
 
 
 +
 
 
 (
 
 
 Q
 
 
 /
 
 
 2
 
 
 )
 
 
 +
 
 
 1
 
 
 ) 
 
 
 = 
 
 
 1
 
 
 /
 
 
 (
 
 
 1
 
 
 +
 
 
 (
 
 
 D
 
 
 /
 
 
 D_0
 
 
 )^
 
 
 2
 
 
 ); 
 
 
 </div>
 
 
 <div style="padding:0px 10px"> 
 
 
 H_2
 
 
 (
 
 
 x
 
 
 +
 
 
 (
 
 
 P
 
 
 /
 
 
 2
 
 
 )
 
 
 +
 
 
 1
 
 
 ,
 
 
 y
 
 
 +
 
 
 (
 
 
 Q
 
 
 /
 
 
 2
 
 
 )
 
 
 +
 
 
 1
 
 
 ) 
 
 
 = 
 
 
 1
 
 
 /
 
 
 (
 
 
 1
 
 
 +
 
 
 (
 
 
 D
 
 
 /
 
 
 D_0
 
 
 )^
 
 
 6
 
 
 );
 
 
 </div>
 
 
 <div style="padding:0px 10px"> 
 
 
 end
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 end
 
 
 </div>
 
 
 <div style="padding:0px 10px">

</div>
 
 
 <div style="padding:0px 10px">

</div>
 
 
 <div style="padding:0px 10px">
 
 
 G_1 
 
 
 = 
 
 
 H_1 
 
 
 .* 
 
 
 F
 
 
 ;
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 G_2 
 
 
 = 
 
 
 H_2 
 
 
 .* 
 
 
 F
 
 
 ;
 
 
 </div>
 
 
 <div style="padding:0px 10px">

</div>
 
 
 <div style="padding:0px 10px">
 
 
 g_1 
 
 
 = 
 
 
 real
 
 
 (
 
 
 ifft2
 
 
 (
 
 
 G_1
 
 
 ));
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 g_1 
 
 
 = 
 
 
 g_1
 
 
 (
 
 
 1
 
 
 :
 
 
 1
 
 
 :
 
 
 M
 
 
 ,
 
 
 1
 
 
 :
 
 
 1
 
 
 :
 
 
 N
 
 
 );
 
 
 </div>
 
 
 <div style="padding:0px 10px">

</div>
 
 
 <div style="padding:0px 10px">
 
 
 g_2 
 
 
 = 
 
 
 real
 
 
 (
 
 
 ifft2
 
 
 (
 
 
 G_2
 
 
 ));
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 g_2 
 
 
 = 
 
 
 g_2
 
 
 (
 
 
 1
 
 
 :
 
 
 1
 
 
 :
 
 
 M
 
 
 ,
 
 
 1
 
 
 :
 
 
 1
 
 
 :
 
 
 N
 
 
 ); 
 
 
 </div>
 
 
 <div style="padding:0px 10px">

</div>
 
 
 <div style="padding:0px 10px">
 
 
 for 
 
 
 x 
 
 
 = 
 
 
 1
 
 
 :
 
 
 1
 
 
 :
 
 
 M
 
 
 </div>
 
 
 <div style="padding:0px 10px"> 
 
 
 for 
 
 
 y 
 
 
 = 
 
 
 1
 
 
 :
 
 
 1
 
 
 :
 
 
 N
 
 
 </div>
 
 
 <div style="padding:0px 10px"> 
 
 
 g_1
 
 
 (
 
 
 x
 
 
 ,
 
 
 y
 
 
 ) 
 
 
 = 
 
 
 g_1
 
 
 (
 
 
 x
 
 
 ,
 
 
 y
 
 
 ) 
 
 
 * 
 
 
 (
 
 
 -
 
 
 1
 
 
 )^
 
 
 (
 
 
 x
 
 
 +
 
 
 y
 
 
 );
 
 
 </div>
 
 
 <div style="padding:0px 10px"> 
 
 
 g_2
 
 
 (
 
 
 x
 
 
 ,
 
 
 y
 
 
 ) 
 
 
 = 
 
 
 g_2
 
 
 (
 
 
 x
 
 
 ,
 
 
 y
 
 
 ) 
 
 
 * 
 
 
 (
 
 
 -
 
 
 1
 
 
 )^
 
 
 (
 
 
 x
 
 
 +
 
 
 y
 
 
 );
 
 
 </div>
 
 
 <div style="padding:0px 10px"> 
 
 
 end
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 end
 
 
 </div>
 
 
 <div style="padding:0px 10px">

</div>
 
 
 <div style="padding:0px 10px">
 
 
 %% -----show-------
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 figure
 
 
 ();
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 subplot
 
 
 (
 
 
 1
 
 
 ,
 
 
 2
 
 
 ,
 
 
 1
 
 
 );
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 imshow
 
 
 (
 
 
 f
 
 
 ,[
 
 
 0 
 
 
 1
 
 
 ]);
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 xlabel
 
 
 (
 
 
 'a).Original Image'
 
 
 );
 
 
 </div>
 
 
 <div style="padding:0px 10px">

</div>
 
 
 <div style="padding:0px 10px">
 
 
 figure
 
 
 ();
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 subplot
 
 
 (
 
 
 1
 
 
 ,
 
 
 2
 
 
 ,
 
 
 1
 
 
 );
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 imshow
 
 
 (
 
 
 H_1
 
 
 ,[
 
 
 0 
 
 
 1
 
 
 ]);
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 xlabel
 
 
 (
 
 
 'c)Butterworth Lowpass (D_{0}=100,n=1)'
 
 
 );
 
 
 </div>
 
 
 <div style="padding:0px 10px">

</div>
 
 
 <div style="padding:0px 10px">
 
 
 subplot
 
 
 (
 
 
 1
 
 
 ,
 
 
 2
 
 
 ,
 
 
 2
 
 
 );
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 h 
 
 
 = 
 
 
 mesh
 
 
 (
 
 
 1
 
 
 :
 
 
 20
 
 
 :
 
 
 P
 
 
 ,
 
 
 1
 
 
 :
 
 
 20
 
 
 :
 
 
 Q
 
 
 ,
 
 
 H_1
 
 
 (
 
 
 1
 
 
 :
 
 
 20
 
 
 :
 
 
 P
 
 
 ,
 
 
 1
 
 
 :
 
 
 20
 
 
 :
 
 
 Q
 
 
 ));
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 set
 
 
 (
 
 
 h
 
 
 ,
 
 
 'EdgeColor'
 
 
 ,
 
 
 'k'
 
 
 );
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 axis
 
 
 ([
 
 
 0 
 
 
 P 
 
 
 0 
 
 
 Q 
 
 
 0 
 
 
 1
 
 
 ]);
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 xlabel
 
 
 (
 
 
 'u'
 
 
 );
 
 
 ylabel
 
 
 (
 
 
 'v'
 
 
 );
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 zlabel
 
 
 (
 
 
 '|H(u,v)|'
 
 
 );
 
 
 </div>
 
 
 <div style="padding:0px 10px">

</div>
 
 
 <div style="padding:0px 10px">
 
 
 figure
 
 
 ();
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 subplot
 
 
 (
 
 
 1
 
 
 ,
 
 
 2
 
 
 ,
 
 
 1
 
 
 );
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 imshow
 
 
 (
 
 
 log
 
 
 (
 
 
 1 
 
 
 + 
 
 
 abs
 
 
 (
 
 
 G_1
 
 
 )),[ 
 
 
 ]);
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 xlabel
 
 
 (
 
 
 'd).Result of filtering using c'
 
 
 );
 
 
 </div>
 
 
 <div style="padding:0px 10px">

</div>
 
 
 <div style="padding:0px 10px">
 
 
 figure
 
 
 ();
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 subplot
 
 
 (
 
 
 1
 
 
 ,
 
 
 2
 
 
 ,
 
 
 1
 
 
 );
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 imshow
 
 
 (
 
 
 H_2
 
 
 ,[
 
 
 0 
 
 
 1
 
 
 ]);
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 xlabel
 
 
 (
 
 
 'f).Butterworth Lowpass (D_{0}=100,n=3)'
 
 
 );
 
 
 </div>
 
 
 <div style="padding:0px 10px">

</div>
 
 
 <div style="padding:0px 10px">
 
 
 subplot
 
 
 (
 
 
 1
 
 
 ,
 
 
 2
 
 
 ,
 
 
 2
 
 
 );
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 h 
 
 
 = 
 
 
 mesh
 
 
 (
 
 
 1
 
 
 :
 
 
 20
 
 
 :
 
 
 P
 
 
 ,
 
 
 1
 
 
 :
 
 
 20
 
 
 :
 
 
 Q
 
 
 ,
 
 
 H_2
 
 
 (
 
 
 1
 
 
 :
 
 
 20
 
 
 :
 
 
 P
 
 
 ,
 
 
 1
 
 
 :
 
 
 20
 
 
 :
 
 
 Q
 
 
 ));
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 set
 
 
 (
 
 
 h
 
 
 ,
 
 
 'EdgeColor'
 
 
 ,
 
 
 'k'
 
 
 );
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 axis
 
 
 ([
 
 
 0 
 
 
 P 
 
 
 0 
 
 
 Q 
 
 
 0 
 
 
 1
 
 
 ]);
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 xlabel
 
 
 (
 
 
 'u'
 
 
 );
 
 
 ylabel
 
 
 (
 
 
 'v'
 
 
 );
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 zlabel
 
 
 (
 
 
 '|H(u,v)|'
 
 
 );
 
 
 </div>
 
 
 <div style="padding:0px 10px">

</div>
 
 
 <div style="padding:0px 10px">
 
 
 figure
 
 
 ();
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 subplot
 
 
 (
 
 
 1
 
 
 ,
 
 
 2
 
 
 ,
 
 
 1
 
 
 );
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 imshow
 
 
 (
 
 
 log
 
 
 (
 
 
 1 
 
 
 + 
 
 
 abs
 
 
 (
 
 
 G_2
 
 
 )),[ 
 
 
 ]);
 
 
 </div>
 
 
 <div style="padding:0px 10px">
 
 
 xlabel
 
 
 (
 
 
 'g).Result of filtering using e'
 
 
 );
 
 
 </div>
 
 
 <div style="padding:0px 10px">

##### [ 来自CODE的代码片][_CODE]  Butterworth\_Lowpass\_Filters.m #####

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 <td style="border:none; margin:0px; padding:0px; vertical-align:top"> 
 <div style="border-left-width:1px; border-left-style:solid; border-left-color:rgb(222,226,227); overflow-x:auto; overflow-y:hidden"> 
 <pre style="word-wrap:normal; color:rgb(51,51,51); border:none; font-family:Menlo,'Liberation Mono',Consolas,'Courier New','andale mono','lucida console',monospace; margin-top:0px; margin-bottom:0px; padding:10px 0px; line-height:16px!important; background-color:rgb(255,255,255)"> 
 
 <div style="padding:0px 10px">
 
 
 close 
 
 
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 clear 
 
 
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 <div style="padding:0px 10px">
 
 
 clc
 
 
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 <div style="padding:0px 10px">
 
 
 %% ---------Gaussian Lowpass Filters (Fre. Domain)------------
 
 
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 figure
 
 
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 subplot
 
 
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 xlabel
 
 
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 <div style="padding:0px 10px">

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 <div style="padding:0px 10px">
 
 
 subplot
 
 
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 <div style="padding:0px 10px">
 
 
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 subplot
 
 
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 h 
 
 
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 <div style="padding:0px 10px">
 
 
 set
 
 
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 <div style="padding:0px 10px">
 
 
 axis
 
 
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 <div style="padding:0px 10px">
 
 
 xlabel
 
 
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 ylabel
 
 
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 <div style="padding:0px 10px">
 
 
 zlabel
 
 
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 <div style="padding:0px 10px">
 
 
 figure
 
 
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 <div style="padding:0px 10px">
 
 
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 <div style="padding:0px 10px">

##### [ 来自CODE的代码片][_CODE]  Gaussian\_Lowpass\_Filters.m #####

<table style="border-collapse:collapse; border:none; margin:0px; padding:0px; table-layout:fixed"> 
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