The 2D Fourier TransformDepartamento de Mtem225;tica 二维傅里叶变换省matem和天加225;
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1、The 2D Fourier TransformF (2)f(x,y) = F(kx,ky) = f(x,y) exp-i(kxx+kyy) dx dyIf f(x,y) = fx(x) fy(y), then the 2D FT splits into two 1D FTs. But this doesnt always happen.F (2)f(x,y)xyf(x,y)The Fourier transform in 2 dimensionsThe Fourier transform can act in any number of dimensions,It is separable
2、and the order does not matter. g(x,y)x,yg(x,y)eikyyeikxxdxdy g(x,y)x,y g(x,y)x g(x,y)yCentral Slice TheoremThe equivalence of the zero-frequency rule in 2D is the central slice theorem.orSo a slice of the 2-D FT that passes through the origin corresponds to the 1 D FT of the projection in real space
3、. g(x,y)x,ykx0g(x,y)eikyyeikxxdxdykx0 g(x,y)x,ykx0eikyydyg(x,y)dxkx0FilteringWe can change the information content in the image by manipulating the information in reciprocal space.Weighting function in k-space.FilteringWe can also emphasis the high frequency components.Weighting function in k-space.
4、Modulation transfer function i(x,y) o(x,y) PSF(x,y) noise c c c cI(kx,ky)O(kx,ky) MTF(kx,ky) noiseThe 2D Fourier TransformF (2)f(x,y) = F(kx,ky) = f(x,y) exp-i(kxx+kyy) dx dyIf f(x,y) = fx(x) fy(y), then the 2D FT splits into two 1D FTs. But this doesnt always happen.F (2)f(x,y)xyf(x,y)A 2D Fourier
5、Transform: a square functionConsider a square function in the xy plane: f(x,y) = rect(x) rect(y)The 2D Fourier Transform splits into the product of two 1D Fourier Transforms: F f(x,y) = sinc(kx/2) sinc(ky/2)This picture is an optical determinationof the Fourier Transform of the square function!xyf(x,y)F (2)f(x,y)Fourier Transform Magnitude and Phase Pictures reconstructedusing the spectral phaseof the other pictureThe phase of the Fourier transform (spectral phase) is much more important than the magnitude in reconstructing an image.RickLindaMagF LindaPhaseF RickMagF Rick PhaseF Linda
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