数字图像处理(频域处理)ppt课件

数字图像处理基础频域处理Kunming University of Science and Technology内 容v一维离散傅立叶变换及其反变换v二维离散傅立叶变换及其反变换v傅里叶变换的性质v频域滤波 图像平滑 图像锐化Kunming University of Science and TechnologyJean Baptiste Joseph Fourierv法国数学家傅立叶法国数学家傅立叶(生于生于1768年年)在在1822年出版的年出版的热分析理论热分析理论一书中一书中指出指出:任何周期函数都可以表达为不同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频率的正弦和或余弦和的形式,即傅立即傅立叶级数。叶级数。v20世纪世纪50年代后期年代后期,快速傅立叶变换快速傅立叶变换算法出现算法出现,得到了广泛的应用。得到了广泛的应用。Kunming University of Science and TechnologyThe 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 Image Processing(2002)Kunming University of Science and Technology一维离散傅立叶变换及其反变换v离散函数f(x)(其中x=0,1,2,M-1)的傅立叶变换:v傅立叶变换是可逆的，F(u)的反变换的反变换:12/0()()Mjux MxF ufx e1,.,2,1,0Mu12/01()()Mjux MufxF u eM1,.,2,1,0MxKunming University of Science and Technology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。sincosjej12/0101()()1()cos(2/)sin(2/)Mjux MuMuf xF u eMF uux Mjux MMKunming University of Science and Technology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。Kunming University of Science and Technology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-1.1010)/(2),(),(MxNyNvyMuxjeyxfvuFKunming University of Science and TechnologyDFT&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 Processing(2002)DFTKunming University of Science and TechnologyDFT&ImagesImages taken from Gonzalez&Woods,Digital Image Processing(2002)Kunming University of Science and TechnologyDFT&ImagesImages taken from Gonzalez&Woods,Digital Image Processing(2002)Kunming University of Science and TechnologyDFT&Images(cont)Images taken from Gonzalez&Woods,Digital Image Processing(2002)DFTScanning electron microscope image of an integrated circuit magnified 2500 timesFourier spectrum of the imageKunming University of Science and TechnologyDFT&Images(cont)Images taken from Gonzalez&Woods,Digital Image Processing(2002)Kunming University of Science and TechnologyDFT&Images(cont)Images taken from Gonzalez&Woods,Digital Image Processing(2002)Kunming University of Science and Technology二维 DFT反变换二维离散傅立叶逆变换:for x=0,1,2M-1 and y=0,1,2N-11010)/(2),(1),(MuNvNvyMuxjevuFMNyxfKunming University of Science and Technology二维 DFT反变换(cont)v(u,v)=(0,0)位置的傅里叶变换值为位置的傅里叶变换值为1100(0,0)(,)MNxyFf x y即图像即图像f(x,y)各像素灰度级的和，各像素灰度级的和，F(0,0)/MN即即为图像灰度的均值。为图像灰度的均值。F(0,0)称为频率谱的直流分称为频率谱的直流分量量(系数系数)，其它其它F(u,v)值称为交流分量值称为交流分量(交流系数交流系数)。Kunming University of Science and Technology二维DFT的极坐标表示 频率谱 相位谱 功率谱 ),(),()(vujevuFuF),(),(),(22vuIvuRvuF),(),(arctan),(vuRvuIvu),(),(),(22vuIvuRvuPKunming University of Science and Technology二维傅里叶变换的性质v周期性周期性 傅里叶级数傅里叶级数(DFS)有周期性有周期性MN，反变换也是周期性，反变换也是周期性的。的。DFT 是其中的一个周期。是其中的一个周期。1010)/(2),(),(MxNyNvyMuxjeyxfvuF1010)/(2),(1),(MuNvNvyMuxjevuFMNyxfKunming University of Science and Technology二维DFT傅里叶变换的性质v 平移特性平移特性)/(20000)/(20000),(),(),(),(NyvMuxjNyvMxujevuFyyxxfvvuuFeyxf当当u0=M/2,v0=N/2时时002(/)()(,)(,)(,)(1)ju x Mv y Njx yx yf x y ef x y ef x y通常在变换前用通常在变换前用(-1)x+y 乘以输入图像函数，实现频域乘以输入图像函数，实现频域中心化变换：中心化变换：(,)(1)(,)22xyMNfx yF uvKunming University of Science and TechnologyKunming University of Science and Technology频率域的基本性质频率域的基本性质 频率域的基本性质：频率域的基本性质：频域的中心邻域对应图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变化较快的部分（如：物体的边缘、线条等）。（如：物体的边缘、线条等）。Matlab示例：示例：example4_0.mKunming University of Science and Technology频域滤波v卷积定理：卷积定理：v空间域的乘法对应频域卷积空间域的乘法对应频域卷积Kunming University of Science and Technology频域滤波的折叠误差干扰v使用DFT进行滤波操作，图像及其变换均视为周期性的。v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。v通过使用零填充函数的方法避免折叠误差干扰。vMatlab示例：example4_1.mKunming University of Science and Technology频域滤波的步骤1.计算图像的离散傅立叶变换F(u,v)2.用滤波函数 H(u,v)乘以F(u,v)3.计算乘积的离散傅立叶反变换Images taken from Gonzalez&Woods,Digital Image Processing(2002)Kunming University of Science and Technology一些基本的频域滤波器Images taken from Gonzalez&Woods,Digital Image Processing(2002)Low Pass FilterHigh Pass FilterKunming University of Science and Technology低通滤波器滤波结果Images taken from Gonzalez&Woods,Digital Image Processing(2002)Kunming University of Science and Technology高通滤波器滤波结果Images taken from Gonzalez&Woods,Digital Image Processing(2002)Kunming University of Science and Technology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。Kunming University of Science and Technology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)Kunming University of Science and Technology理想低通滤波器(cont)理想低通滤波器的传输函数:其中D(u,v)为:00),(if 0),(if 1),(DvuDDvuDvuH2/122)2/()2/(),(NvMuvuDKunming University of Science and Technology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)Kunming University of Science and Technology理想低通滤波器(cont)Images taken from Gonzalez&Woods,Digital Image Processing(2002)Kunming University of Science and Technology理想低通滤波器(cont)Images taken from Gonzalez&Woods,Digital Image Processing(2002)Kunming University of Science and Technology理想低通滤波器(cont)Images taken from Gonzalez&Woods,Digital Image Processing(2002)OriginalimageResult of filtering with ideal low pass filter of radius 5Result of filtering with ideal low pass filter of radius 30Result of filtering with ideal low pass filter of radius 230Result of filtering with ideal low pass filter of radius 80Result of filtering with ideal low pass filter of radius 15Kunming University of Science and Technology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 5Kunming University of Science and Technology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 15Kunming University of Science and TechnologyButterworth 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)nDvuDvuH20/),(11),(Kunming University of Science and TechnologyButterworth 低通滤波器(cont)Images taken from Gonzalez&Woods,Digital Image Processing(2002)OriginalimageResult of filtering with Butterworth filter of order 2 and cutoff radius 5Result of filtering with Butterworth filter of order 2 and cutoff radius 30Result of filtering with Butterworth filter of order 2 and cutoff radius 230Result of filtering with Butterworth filter of order 2 and cutoff radius 80Result of filtering with Butterworth filter of order 2 and cutoff radius 15Kunming University of Science and TechnologyButterworth 低通滤波器(cont)Images taken from Gonzalez&Woods,Digital Image Processing(2002)OriginalimageResult of filtering with Butterworth filter of order 2 and cutoff radius 5Kunming University of Science and TechnologyButterworth 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 15Kunming University of Science and TechnologyGaussian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)2022/),(),(DvuDevuHKunming University of Science and TechnologyGaussian低通滤波器(cont)Images taken from Gonzalez&Woods,Digital Image Processing(2002)OriginalimageResult of filtering with Gaussian filter with cutoff radius 5Result of filtering with Gaussian filter with cutoff radius 30Result of filtering with Gaussian filter with cutoff radius 230Result of filtering with Gaussian filter with cutoff radius 85Result of filtering with Gaussian filter with cutoff radius 15Kunming University of Science and TechnologyGaussian低通滤波器Matlab示例vexample4_4.mKunming University of Science and Technology低通滤波器比较 Result of filtering with ideal low pass filter of radius 15Result of filtering with Butterworth filter of order 2 and cutoff radius 15Result of filtering with Gaussian filter with cutoff radius 15Images taken from Gonzalez&Woods,Digital Image Processing(2002)Kunming University of Science and Technology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)Kunming University of Science and Technology低通滤波器的应用Images taken from Gonzalez&Woods,Digital Image Processing(2002)Kunming University of Science and Technology低通滤波器的应用(cont)Different lowpass Gaussian filters used to remove blemishes in a photographImages taken from Gonzalez&Woods,Digital Image Processing(2002)Kunming University of Science and Technology低通滤波器的应用(cont)Images taken from Gonzalez&Woods,Digital Image Processing(2002)Kunming University of Science and Technology低通滤波器的应用(cont)Original imageGaussian lowpass filterProcessed imageSpectrum of original imageImages taken from Gonzalez&Woods,Digital Image Processing(2002)Kunming University of Science and Technology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)Kunming University of Science and Technology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00),(if 1),(if 0),(DvuDDvuDvuHImages taken from Gonzalez&Woods,Digital Image Processing(2002)Kunming University of Science and Technology理想高通滤波器(cont)Results of ideal high pass filtering with D0=15Results of ideal high pass filtering with D0=30Results of ideal high pass filtering with D0=80Images taken from Gonzalez&Woods,Digital Image Processing(2002)Kunming University of Science and TechnologyButterworth 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nvuDDvuH20),(/11),(Images taken from Gonzalez&Woods,Digital Image Processing(2002)Kunming University of Science and TechnologyButterworth 高通滤波器(cont)Results of Butterworth high pass filtering of order 2 with D0=15Results of Butterworth high pass filtering of order 2 with D0=80Results of Butterworth high pass filtering of order 2 with D0=30Images taken from Gonzalez&Woods,Digital Image Processing(2002)Kunming University of Science and TechnologyGaussian 高通滤波器Gaussian 高通滤波器定义为:D0 为截至频率2022/),(1),(DvuDevuHImages taken from Gonzalez&Woods,Digital Image Processing(2002)Matlab中绘制三维线框图：example4_5.mKunming University of Science and TechnologyGaussian 高通滤波器(cont)Results of Gaussian high pass filtering with D0=15Results of Gaussian high pass filtering with D0=80Results of Gaussian high pass filtering with D0=30Images taken from Gonzalez&Woods,Digital Image Processing(2002)Kunming University of Science and TechnologyGaussian高通滤波器Matlab示例vexample4_7.mKunming University of Science and Technology高通滤波器比较Results of ideal high pass filtering with D0=15Images taken from Gonzalez&Woods,Digital Image Processing(2002)Kunming University of Science and Technology高通滤波器比较Results of Butterworth high pass filtering of order 2 with D0=15Images taken from Gonzalez&Woods,Digital Image Processing(2002)Kunming University of Science and Technology高通滤波器比较Results of Gaussian high pass filtering with D0=15Images taken from Gonzalez&Woods,Digital Image Processing(2002)Kunming University of Science and Technology高通滤波器比较Results of ideal high pass filtering with D0=15Results of Gaussian high pass filtering with D0=15Results of Butterworth high pass filtering of order 2 with D0=15Images taken from Gonzalez&Woods,Digital Image Processing(2002)Kunming University of Science and Technology高频强调滤波v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。v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(,)(,)hfehpHu vabHu vKunming University of Science and Technology高频强调滤波例子Original imageHighpass filtering resultHigh frequency emphasis resultAfter histogram equalisationImages taken from Gonzalez&Woods,Digital Image Processing(2002)Kunming University of Science and TechnologyHighpass Filtering ExampleKunming University of Science and TechnologyHighpass Filtering ExampleKunming University of Science and TechnologyHighpass Filtering ExampleKunming University of Science and TechnologyHighpass Filtering Example