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1、数字图像处理基础,频域处理,内 容,一维离散傅立叶变换及其反变换二维离散傅立叶变换及其反变换傅里叶变换的性质频域滤波图像平滑图像锐化,Jean Baptiste Joseph Fourier,法国数学家傅立叶(生于1768年)在1822年出版的热分析理论一书中指出:任何周期函数都可以表达为不同频率的正弦和或余弦和的形式,即傅立叶级数。20世纪50年代后期,快速傅立叶变换算法出现,得到了广泛的应用。,The Big Idea,=,任何周期函数都可以表达为不同频率的正弦和或余弦和的形式,即傅立叶级数。,Images taken from Gonzalez & Woods, Digital Imag
2、e Processing (2002),一维离散傅立叶变换及其反变换,离散函数f(x)(其中x=0,1,2,M-1)的傅立叶变换:傅立叶变换是可逆的，F(u)的反变换的反变换:,一维离散傅立叶变换及其反变换(cont),观察傅立叶反变换由欧拉公式则即，空域函数f(x)可表示为M个正弦(余弦)函数的累加，F(u)/M为对应频率分量的幅度(系数),因此，F(u) 覆盖的域(u值) 称为频率域。从另一角度,空域函数f(x)被表示为直流分量和交流分量F(0)/M对应的是直流分量的幅值,其余F(u)/M对应交流分量的幅值。,傅立叶变换的意义,傅立叶变换可以看作数学的棱镜,将函数基于频率分成不同成分.当我
3、们考虑光时,讨论它的光谱或频率谱。同样,傅立叶变换使我们能通过频率成分来分析一个函数。这是属于线性滤波核心的重要概念。,二维离散傅立叶变换Discrete Fourier Transform (DFT),一个图像尺寸为MN的函数f(x,y)的离散傅立叶变换F(u,v): (x = 0, 1, 2M-1 and y = 0,1,2N-1)for u = 0, 1, 2M-1 and v = 0, 1, 2N-1.,DFT & Images,二维图像的DFT结果可由图像各频率成分的频谱图展示,Images taken from Gonzalez & Woods, Digital Image Pro
4、cessing (2002),DFT,DFT & Images,Images taken from Gonzalez & Woods, Digital Image Processing (2002),DFT & Images,Images taken from Gonzalez & Woods, Digital Image Processing (2002),DFT & Images (cont),Images taken from Gonzalez & Woods, Digital Image Processing (2002),DFT,Scanning electron microscop
5、e image of an integrated circuit magnified 2500 times,Fourier spectrum of the image,DFT & Images (cont),Images taken from Gonzalez & Woods, Digital Image Processing (2002),DFT & Images (cont),Images taken from Gonzalez & Woods, Digital Image Processing (2002),二维 DFT反变换,二维离散傅立叶逆变换:for x = 0, 1, 2M-1
6、and y = 0, 1, 2N-1,二维 DFT反变换(cont),(u,v)=(0,0)位置的傅里叶变换值为,即图像f(x,y) 各像素灰度级的和，F(0,0) /MN即为图像灰度的均值。F(0,0) 称为频率谱的直流分量(系数)，其它F(u,v) 值称为交流分量(交流系数)。,二维DFT的极坐标表示,频率谱 相位谱 功率谱,二维傅里叶变换的性质,周期性 傅里叶级数(DFS)有周期性MN，反变换也是周期性的。DFT 是其中的一个周期。,二维DFT傅里叶变换的性质,平移特性,当u0=M/2, v0=N/2时,通常在变换前用(-1)x+y 乘以输入图像函数，实现频域中心化变换：,频率域的基本性
7、质,频率域的基本性质：频域的中心邻域对应图像中慢变化部分，较高的频率开始对应图像中变化较快的部分（如：物体的边缘、线条等）。Matlab示例：example4_0.m,频域滤波,卷积定理： 空间域的乘法对应频域卷积,频域滤波的折叠误差干扰,使用DFT进行滤波操作，图像及其变换均视为周期性的。若周期关于函数的非零部分持续时间很靠近，则对周期函数执行卷积运算会导致相邻周期间的干扰折叠误差干扰。通过使用零填充函数的方法避免折叠误差干扰。Matlab示例：example4_1.m,频域滤波的步骤,计算图像的离散傅立叶变换F(u,v)用滤波函数 H(u,v) 乘以F(u,v) 计算乘积的离散傅立叶反变换
8、,Images taken from Gonzalez & Woods, Digital Image Processing (2002),一些基本的频域滤波器,Images taken from Gonzalez & Woods, Digital Image Processing (2002),Low Pass Filter,High Pass Filter,低通滤波器滤波结果,Images taken from Gonzalez & Woods, Digital Image Processing (2002),高通滤波器滤波结果,Images taken from Gonzalez & Wo
9、ods, Digital Image Processing (2002),频域平滑滤波器,通过过滤掉频域的高频成分达到平滑目的滤波的基本模型:G(u,v) = H(u,v)F(u,v)其中 F(u,v) 是图像的傅立叶变换结果， H(u,v) 称为滤波器传输函数低通滤波器（Low pass filters） 仅保留低频分量而过滤掉高频分量。,理想低通滤波器,将频域中所有与原点的距离超过D0的高频分量截取掉改变距离D0将改变滤波器滤波效果,Images taken from Gonzalez & Woods, Digital Image Processing (2002),理想低通滤波器(con
10、t),理想低通滤波器的传输函数:其中D(u,v) 为:,理想低通滤波器(cont),左图为原始图像, 右图为其傅立叶变换结果，并在其上绘出半径分别为5, 15, 30, 80 和 230 的低通滤波器。,Images taken from Gonzalez & Woods, Digital Image Processing (2002),理想低通滤波器(cont),Images taken from Gonzalez & Woods, Digital Image Processing (2002),理想低通滤波器(cont),Images taken from Gonzalez & Woods
11、, Digital Image Processing (2002),理想低通滤波器(cont),Images taken from Gonzalez & Woods, Digital Image Processing (2002),Originalimage,Result of filtering with ideal low pass filter of radius 5,Result of filtering with ideal low pass filter of radius 30,Result of filtering with ideal low pass filter of r
12、adius 230,Result of filtering with ideal low pass filter of radius 80,Result of filtering with ideal low pass filter of radius 15,理想低通滤波器(cont),Images taken from Gonzalez & Woods, Digital Image Processing (2002),Result of filtering with ideal low pass filter of radius 5,理想低通滤波器(cont),Images taken fr
13、om Gonzalez & Woods, Digital Image Processing (2002),Result of filtering with ideal low pass filter of radius 15,Butterworth 低通滤波器,截至频率为D0 的n 阶Butterworth 低通滤波器定义为:,Images taken from Gonzalez & Woods, Digital Image Processing (2002),Butterworth 低通滤波器(cont),Images taken from Gonzalez & Woods, Digital
14、 Image Processing (2002),Originalimage,Result of filtering with Butterworth filter of order 2 and cutoff radius 5,Result of filtering with Butterworth filter of order 2 and cutoff radius 30,Result of filtering with Butterworth filter of order 2 and cutoff radius 230,Result of filtering with Butterwo
15、rth filter of order 2 and cutoff radius 80,Result of filtering with Butterworth filter of order 2 and cutoff radius 15,Butterworth 低通滤波器(cont),Images taken from Gonzalez & Woods, Digital Image Processing (2002),Originalimage,Result of filtering with Butterworth filter of order 2 and cutoff radius 5,
16、Butterworth 低通滤波器(cont),Images taken from Gonzalez & Woods, Digital Image Processing (2002),Result of filtering with Butterworth filter of order 2 and cutoff radius 15,Gaussian低通滤波器,Gaussian 低通滤波器的传输函数定义为:,Images taken from Gonzalez & Woods, Digital Image Processing (2002),Gaussian低通滤波器(cont),Images
17、 taken from Gonzalez & Woods, Digital Image Processing (2002),Originalimage,Result of filtering with Gaussian filter with cutoff radius 5,Result of filtering with Gaussian filter with cutoff radius 30,Result of filtering with Gaussian filter with cutoff radius 230,Result of filtering with Gaussian f
18、ilter with cutoff radius 85,Result of filtering with Gaussian filter with cutoff radius 15,Gaussian低通滤波器Matlab示例,example4_4.m,低通滤波器比较,Result of filtering with ideal low pass filter of radius 15,Result of filtering with Butterworth filter of order 2 and cutoff radius 15,Result of filtering with Gauss
19、ian filter with cutoff radius 15,Images taken from Gonzalez & Woods, Digital Image Processing (2002),低通滤波器的应用,使用高斯低通滤波器连接断裂的笔画,Images taken from Gonzalez & Woods, Digital Image Processing (2002),低通滤波器的应用,Images taken from Gonzalez & Woods, Digital Image Processing (2002),低通滤波器的应用(cont),Different low
20、pass Gaussian filters used to remove blemishes in a photograph,Images taken from Gonzalez & Woods, Digital Image Processing (2002),低通滤波器的应用(cont),Images taken from Gonzalez & Woods, Digital Image Processing (2002),低通滤波器的应用(cont),Original image,Gaussian lowpass filter,Processed image,Spectrum of orig
21、inal image,Images taken from Gonzalez & Woods, Digital Image Processing (2002),频域锐化,图像中的边缘等细节对应频域的高频分量高通滤波器（High pass filters） 仅保留高频分量，截取低频分量高通滤波器可通过低通滤波器获得:Hhp(u, v) = 1 Hlp(u, v),理想高通滤波器,理想高通滤波器定义为:与低通滤波器类似， D0 为截至频率,Images taken from Gonzalez & Woods, Digital Image Processing (2002),理想高通滤波器(cont)
22、,Results of ideal high pass filtering with D0 = 15,Results of ideal high pass filtering with D0 = 30,Results of ideal high pass filtering with D0 = 80,Images taken from Gonzalez & Woods, Digital Image Processing (2002),Butterworth 高通滤波器,Butterworth 高通滤波器定义为:其中， n为阶数， D0 为截至频率,Images taken from Gonza
23、lez & Woods, Digital Image Processing (2002),Butterworth 高通滤波器(cont),Results of Butterworth high pass filtering of order 2 with D0 = 15,Results of Butterworth high pass filtering of order 2 with D0 = 80,Results of Butterworth high pass filtering of order 2 with D0 = 30,Images taken from Gonzalez & W
24、oods, Digital Image Processing (2002),Gaussian 高通滤波器,Gaussian 高通滤波器定义为:D0 为截至频率,Images taken from Gonzalez & Woods, Digital Image Processing (2002),Matlab中绘制三维线框图：example4_5.m,Gaussian 高通滤波器(cont),Results of Gaussian high pass filtering with D0 = 15,Results of Gaussian high pass filtering with D0 =
25、80,Results of Gaussian high pass filtering with D0 = 30,Images taken from Gonzalez & Woods, Digital Image Processing (2002),Gaussian高通滤波器Matlab示例,example4_7.m,高通滤波器比较,Results of ideal high pass filtering with D0 = 15,Images taken from Gonzalez & Woods, Digital Image Processing (2002),高通滤波器比较,Results
26、 of Butterworth high pass filtering of order 2 with D0 = 15,Images taken from Gonzalez & Woods, Digital Image Processing (2002),高通滤波器比较,Results of Gaussian high pass filtering with D0 = 15,Images taken from Gonzalez & Woods, Digital Image Processing (2002),高通滤波器比较,Results of ideal high pass filterin
27、g with D0 = 15,Results of Gaussian high pass filtering with D0 = 15,Results of Butterworth high pass filtering of order 2 with D0 = 15,Images taken from Gonzalez & Woods, Digital Image Processing (2002),高频强调滤波,高通滤波去除了图像中的直流分量，图像失去大部分原图像中的背景色调。补偿方法：给高通滤波器加上一个偏移量并乘以一个大于1的常数高频强调滤波,高频强调滤波例子,Original image,Highpass filtering result,High frequency emphasis result,After histogram equalisation,Images taken from Gonzalez & Woods, Digital Image Processing (2002),Highpass Filtering Example,Highpass Filtering Example,Highpass Filtering Example,Highpass Filtering Example,
