數(shù)字圖像處理第四章課件

1、Chapter 4 Image Enhancement in the Frequency Domain第四章 頻率域中的圖像增強(qiáng) 下面的函數(shù)曲線是上面4條函數(shù)曲線之和。Fourier在1807年認(rèn)為：周期函數(shù)可以表示為周期函數(shù)可以表示為Sines和和Cosine的加權(quán)和。的加權(quán)和。這在當(dāng)時(shí)倍受懷疑。Chapter 4 Image Enhancement in the Frequency Domain4.1 Fourier變換入門與頻率域dueuFxfdxexfuFuxjuxj22)()()()(10/21, 2 , 1 , 0)(1)(MxMuxjMuforexfMuF10/21, 2 , 1

2、 , 0)()(MuMuxjMxforexFxf10/2sin/2cos)(1)(MxMuxjMuxxfMuFsincosjejChapter 4 Image Enhancement in the Frequency Domain)()(tan)()()()()()(2/122)(uRuIacuuIuRuFeuFuFuj)()()()(222uIuRuFuPFourier譜or功率譜：幅度or頻率譜相角or位譜Chapter 4 Image Enhancement in the Frequency Domain2D DFT and Its Inverse10)/(2101, 2 , 1 , 0

3、1, 2 , 1 , 0),(),(MxNvyMuxjNyNvMuforeyxfvuF10)/(2101, 2 , 1 , 01, 2 , 1 , 0),(1),(MuNvyMuxjNvNyMxforevuFMNyxf空間域和頻率域之間的采樣步距關(guān)系：yNvxMu11Chapter 4 Image Enhancement in the Frequency Domain(a)在512*512黑圖中的20*40白塊(b)居中的富氏頻譜，經(jīng)log(1+p)變換 橫向相當(dāng)是圖4.2d, 縱向是圖4.2bChapter 4 Image Enhancement in the Frequency Domai

4、n 正變換是原信號(hào)f(x)在各基向量上的投影投影（矢量內(nèi)積），得到各系數(shù)F(u)； 反變換是由各基向量的F(u)加權(quán)和加權(quán)和，得到原信號(hào)f(x)。Chapter 4 Image Enhancement in the Frequency DomainChapter 4 Image Enhancement in the Frequency DomainChapter 4 Image Enhancement in the Frequency DomainChapter 4 Image Enhancement in the Frequency Domain頻域中的濾波損壞的集成電路掃描電鏡圖像Chap

5、ter 4 Image Enhancement in the Frequency Domain頻率域?yàn)V波的基本步驟關(guān)鍵步驟：),(*),(),(),(),(),(),(1yxfyxhvuGFyxgvuFvuHvuGChapter 4 Image Enhancement in the Frequency Domain卷積定理),(*),(),(),(),(),(),(*),(vuFvuHyxhyxfvuFvuHyxhyxf兩個(gè)空間函數(shù)的卷積可以通過計(jì)算兩個(gè)傅利葉變換函數(shù)的乘積的逆變換得到。兩個(gè)空間函數(shù)的卷積的傅利葉變換恰好等于兩個(gè)函數(shù)的傅利葉變換的乘積。Chapter 4 Image Enhan

6、cement in the Frequency Domain4.4圖b中央的直流分量被置零，然后被反變換H(1,1)=0, G(u,v)=H(u,v)F(u,v) Chapter 4 Image Enhancement in the Frequency Domain一個(gè)2D低通濾波函數(shù)及其效果（上）一個(gè)2D高通濾波函數(shù)及其效果（下）Chapter 4 Image Enhancement in the Frequency Domain在高通濾波器中加入一個(gè)常數(shù)后的效果（直流分量不為零）Chapter 4 Image Enhancement in the Frequency Domain頻率域高斯

7、低通濾波器頻率域高斯高通濾波器對(duì)應(yīng)的空間域低通濾波器對(duì)應(yīng)的空間域高通濾波器Chapter 4 Image Enhancement in the Frequency Domain4.3 頻率域平滑濾波器理想的低通濾波器函數(shù)Chapter 4 Image Enhancement in the Frequency DomainuvMuNvvuPvuP1010),(/ ),(100頻譜能量的百分比：例例：Chapter 4 Image Enhancement in the Frequency Domain原圖(100%)5/250半徑，低通92%15/250半徑低通94.6%30/250半徑低通96.

8、4%80/250半徑低通98%230/250半徑，低通99.5%例例(續(xù)續(xù)1)：Chapter 4 Image Enhancement in the Frequency Domain例例(續(xù)續(xù)2)：頻域理想低通濾波器(左上)所對(duì)應(yīng)的空間域?yàn)V波器，有振鈴效應(yīng)振鈴效應(yīng)d=b(x,y)*c(x,y)d圖的對(duì)角剖面Chapter 4 Image Enhancement in the Frequency DomainButterworth Lowpass Filters (BLPF)nDvuDvuH20/ ),(11),(不是理想的濾波器，避免振鈴效應(yīng)D0為截止頻率Chapter 4 Image Enh

9、ancement in the Frequency DomainILPFBLPF(n=2)Chapter 4 Image Enhancement in the Frequency DomainBLPF當(dāng)n4時(shí)開始有明顯振鈴?fù)ǔ?，BLPF在n=2時(shí)能折中折中有效的低通與可容忍的振鈴Chapter 4 Image Enhancement in the Frequency DomainGaussian低通濾波器(GLPF)222/ ),(),(vuDevuH無振鈴效應(yīng)Chapter 4 Image Enhancement in the Frequency DomainBLPFGLPFChapter

10、4 Image Enhancement in the Frequency DomainGLPF應(yīng)用實(shí)例：Chapter 4 Image Enhancement in the Frequency Domain其它LPF實(shí)例：過濾眼角魚尾紋Chapter 4 Image Enhancement in the Frequency DomainNOAA衛(wèi)星在墨西哥灣和佛州的雷達(dá)輻射圖像：低通濾波是一種消除掃描線的粗糙方法Chapter 4 Image Enhancement in the Frequency Domain4.4 Sharpening(銳化) Frequency Domain Filte

11、rsIHPFBHPFGHPFChapter 4 Image Enhancement in the Frequency Domain上述3種濾波器的空間響應(yīng)：Chapter 4 Image Enhancement in the Frequency Domain不同截止頻率參數(shù)的IHPF所得效果：00),(1),(0),(DvuifDDvuifDvuHChapter 4 Image Enhancement in the Frequency Domain不同截止頻率參數(shù)的BHPF所得效果：nvuDDvuH20),(/11),(Chapter 4 Image Enhancement in the Fr

12、equency Domain不同截止頻率參數(shù)D0的GHPF所得效果：2022/ ),(1),(DvuDevuHChapter 4 Image Enhancement in the Frequency Domain頻域中的Laplacian22)2/()2/(),(NvMuvuH原點(diǎn)(M/2,N/2)有峰值0),()2/()2/(),(2212vuFNvMuFyxf),()2/()2/(1),(),(),(2212vuFNvMuFyxfyxfyxg 負(fù)負(fù)得正，實(shí)際上是原圖f加上高通圖f”，邊緣得到增強(qiáng)Chapter 4 Image Enhancement in the Frequency Dom

13、ain頻域中的Laplacian22)2/()2/(),(NvMuvuH頻域：空間域：上式的圖形/圖像4角高頻為負(fù)系數(shù)中點(diǎn)有峰值0剖面的實(shí)際值下降快Chapter 4 Image Enhancement in the Frequency Domain),()2/()2/(1),(),(),(2212vuFNvMuFyxfyxfyxgChapter 4 Image Enhancement in the Frequency Domain),(),(),(yxfyxfyxflphp),(),(),(yxfyxAfyxflphbUnsharp Masking: (鈍化掩膜）High-boost fil

14、tering: (提升濾波),(),(),() 1(),(yxfyxfyxfAyxflphb),(),() 1(),(yxfyxfAyxfhphbwhere A=1Unsharp Masking, High-boost filtering,High-frequency Emphasis FilteringChapter 4 Image Enhancement in the Frequency DomainHigh-boost filtering),() 1(),(vuHAvuHhphbA=1),(1),(vuHvuHlphpChapter 4 Image Enhancement in the

15、Frequency Domain),(),(vubHavuHhphfeHigh-Frequency Emphasis FilteringBHPFHist-EqualizationHfeF(a=0.5 b=2.0)Chapter 4 Image Enhancement in the Frequency Domain4.5 Homomorphic Filteringf(x,y)=i(x,y)r(x,y) 照度分布i(x,y)屬低頻，物體反射r(x,y)屬高頻，高頻應(yīng)增強(qiáng)Chapter 4 Image Enhancement in the Frequency Domain同態(tài)濾波同態(tài)濾波增強(qiáng)實(shí)例：C

16、hapter 4 Image Enhancement in the Frequency Domain位移：位移：如果 F(u,v) f(x,y)則 F(u-M/2, v-N/2) (-1)(x+y)f(x,y)4.6 Implementation of FT4.6.1 2D FT的特性的特性1423F0,04321F0,0變換后重排后Chapter 4 Image Enhancement in the Frequency DomainF(u-M/2, v-N/2) (-1)(x+y)f(x,y)Chapter 4 Image Enhancement in the Frequency Domai

17、n比例尺與旋轉(zhuǎn) af(x,y)aF(u,v) f(ax,by) F(u/a,v/b)/|ab|但是，如果f(x,y)與圖框一起放大，F(xiàn)(u,v)不變。f(x,y)旋轉(zhuǎn)角， F(u,v)也在同方向旋轉(zhuǎn)角Chapter 4 Image Enhancement in the Frequency Domain線性組合k1 f(x,y) + k2 g(x,y) k1 F(u,v) + k2 G(u,v) 但是,通常：f(x,y)g(x,y) F(u,v)G(u,v)Chapter 4 Image Enhancement in the Frequency DomainPeriodicity and con

18、jugate symmetry周期性F(u,v)=F(u+M,v)=F(u,v+N)=F(u+M,v+N)f(x,y)=f(x+M,y)=f(x,y+N)=f(x+m,y+N)共軛對(duì)稱 (共軛: z=a+jb, z*=a-jb )F(u,v)=F*(-u,-v)|F(u,v)|=|F(-u,-v)|Chapter 4 Image Enhancement in the Frequency DomainSeparability可分離性，2D1D:vyNjNyMxxuMjeeyxfMNvuF210102),(11),(10)/(210),(1),(MxNvyMuxjNyeyxfMNvuF102),(

19、1),(MxxuMjeyxfMyuFvyNjNyeyuFNvuF210),(1),(Chapter 4 Image Enhancement in the Frequency Domaina)f(x,y); b)F(u,y);c)F(u,v) 一次2D FT 可由兩次1D FT完成，O(n2)2O(n)Chapter 4 Image Enhancement in the Frequency Domain利用正向FT做反向FT則該反向FT的運(yùn)算式與正向相同對(duì)計(jì)算結(jié)果f*(x)求共軛后乘M即得f(x)MuxjMueuFMxfM/210*)(1)(110/21, 2 , 1 , 0)(1)(MxMux

20、jMuforexfMuF10/21, 2 , 1 , 0)()(MuMuxjMxforexFxf正向FT:反向FT:對(duì)反向FT兩邊取共軛, 除以M得：Chapter 4 Image Enhancement in the Frequency Domain周期信號(hào)引起卷積混疊的現(xiàn)象：j圖實(shí)際上是由e圖的梯形復(fù)制平移后疊加Chapter 4 Image Enhancement in the Frequency Domain為避免卷積混疊需要加大信號(hào)周期：Chapter 4 Image Enhancement in the Frequency Domainf*h，設(shè)f與h為大小相同的正方形；Chapt

21、er 4 Image Enhancement in the Frequency Domain理想低通濾波器卷積實(shí)現(xiàn)中的擴(kuò)充實(shí)例Chapter 4 Image Enhancement in the Frequency DomainConvolution & Correlation Theorems Convolution1010),(),(1),(*),(MmNnnymxhnmfMNyxhyxf),(),(),(*),(),(),(),(*),(yxhyxfvuHvuFvuHvuFyxhyxf卷積定理：卷積定理：Chapter 4 Image Enhancement in the Fre

22、quency DomainCorrelation1010),(),(1),(),(MmNnnymxhnmfMNyxhyxf相關(guān)定理相關(guān)定理:),(),(),(),(),(),(),(),(*yxhyxfvuHvuFvuHvuFyxhyxfChapter 4 Image Enhancement in the Frequency Domain相關(guān)匹配的實(shí)現(xiàn)舉例：擴(kuò)充：Chapter 4 Image Enhancement in the Frequency Domain表4.1、2D-FFT的一些特性平均值頻率譜相位角位移變換對(duì)Chapter 4 Image Enhancement in the Frequency Domain表4.1、2D-FFT的一些特性(續(xù)1）共軛對(duì)稱分配率變比例周期性可分性旋轉(zhuǎn)Chapter 4 Image Enhancement in the Frequency Domain表4.1、2D-FFT的一些特性(續(xù)2）卷積相關(guān)卷積定理卷積定理相關(guān)定理Chapter 4 Image Enhancement in the Frequency Domain表4.1、2D-FFT的一些特性(續(xù)3）一些有用的變換對(duì)：脈沖1高斯高斯Cha