数字图像处理冈萨雷斯N09学习教案

1、会计学1数字图像处理冈萨雷斯数字图像处理冈萨雷斯N09第一页，共29页。 Given a 1-d sequence f(x) , x 0,1,2,3,.NFourier transformImage transform no. 21 , , 1 , 0/j2exp)(1)()(10NuNuxxfNuFxfNxF Fourier transform is periodic with 2Inverse Fourier transform1 , , 1 , 0/j2exp)()()(101NxNuxuFxfuFNuF The variable u is called frequency variab

2、le.using Eulers formula:uxuxux2sin j2cosj2 exp 第2页/共29页第二页，共29页。 How is the Fourier transform of a sequence sk related to the Fourier transform of the continuous signalImage transform no. 3 Continuous-time Fourier transform第3页/共29页第三页，共29页。Image transform no. 5第4页/共29页第四页，共29页。Image transform no. 4第

3、5页/共29页第五页，共29页。Image transform no. 6第6页/共29页第六页，共29页。Image transform no. 7第7页/共29页第七页，共29页。 Given a 2-d matrix of image samples: f(x,y), x,yZ2Fourier transformImage transform no. 8 10101 , , 1 , 0,/ )(j2exp),( 1),(NxNyNvuNvyuxyxfNvuF Fourier transform is 2-periodic both in u and v.Inverse Fourier t

4、ransform 10101 , , 1 , 0,/ )(j2exp),( 1),(NuNvNyxNvyuxvuFNyxf 第8页/共29页第八页，共29页。 How is the Fourier transform of sm,n related to the Fourier transform of the continuous signalImage transform no. 9 Continuous-space 2-d Fourier transform第9页/共29页第九页，共29页。Image transform no. 10 xyf ()x,y 第10页/共29页第十页，共29

5、页。Image transform no. 11第11页/共29页第十一页，共29页。Image transform no. 12第12页/共29页第十二页，共29页。Image transform no. 13第13页/共29页第十三页，共29页。Image transform no. 14第14页/共29页第十四页，共29页。Image transform no. 15第15页/共29页第十五页，共29页。Image transform no. 16第16页/共29页第十六页，共29页。Image transform no. 17第17页/共29页第十七页，共29页。Image trans

6、form no. 18第18页/共29页第十八页，共29页。Image transform no. 19第19页/共29页第十九页，共29页。the discrete Fourior transform can be expressed in separate form, thus can be obtained in two steps by successive applications of 1D FFT. (So is its inverse) Image transform no. 201 , , 1 , 0/j2exp),(1),(10NvNvyyxfNNvxFNy 1 , , 1

7、 , 0,/xj2exp),(1),(10NvuNuvxFNvuFNx How x or y?第20页/共29页第二十页，共29页。 the translation properties of the Fourier transform pair areImage transform no. 21),(/ )(j2exp),(0000vvuuFNyvxuyxf and, (a shift in f(x,y) does not affect the magnitude of its FT)/ )(j2exp),(),(0000NvyuxvuFyyxxf Note an import applic

8、ation here, to move the origin to the center,) 2/, 2/() 1)(,()(NvNuFyxfyx第21页/共29页第二十一页，共29页。 The discrete Fourier transform and its inverse are periodic with period N, Image transform no. 22),(),(),(),(NvNuFNvuFvNuFvuF If f(x,y) is real, the Fourier transform also exhibits conjugate Symmetry:),(),(

9、*vuFvuF Or, more interestingly, ),( ),( vuFvuF which means that the magnitude of the transform is centered on origin,calculating half of the transform can determine whole. 第22页/共29页第二十二页，共29页。 polar coordinatesImage transform no. 23sin,cos,sin,coswvwuryrx then f (x, y) and F(u, v) become f (r, ) and

10、 F(w, ) , respectively),(),(00wFrf That is rotating f(x,y) by angle rotates the F(u,v) by the same angle, vice versa 第23页/共29页第二十三页，共29页。 From the definition of the Fourier transformImage transform no. 24),(),(),(),(2121yxfyxfyxfyxfFFF in general,),(),(),(),(2121yxfyxfyxfyxfFFF 第24页/共29页第二十四页，共29页。I

11、mage transform no. 25For two scalars a and b, ),(),(vuaFyxaf bvauFabbyaxf,1),( 第25页/共29页第二十五页，共29页。Image transform no. 26The average value of 2D discrete function is defined as, 10102),( 1),(NxNyyxfNyxf From 2D Fourier transform definition, when u=0 and v=0, 1010),(1)0 , 0(NxNyyxfNFTherefore, )0 , 0(1),