The 2D Fourier Transform

1. f(x,y) y x The 2D Fourier Transform F(2){f(x,y)} = F(kx,ky) = f(x,y) exp[-i(kxx+kyy)] dx dy Iff(x,y) = fx(x) fy(y), then the 2D FT splits into two 1D FT's. But this doesn’t always happen. F (2){f(x,y)}

2. The Fourier transform in 2 dimensions The Fourier transform can act in any number of dimensions, It is separable and the order does not matter.

3. Central Slice Theorem The equivalence of the zero-frequency rule in 2D is the central slice theorem. or So a slice of the 2-D FT that passes through the origin corresponds to the 1 D FT of the projection in real space.

4. Filtering We can change the information content in the image by manipulating the information in reciprocal space. Weighting function in k-space.

5. Filtering We can also emphasis the high frequency components. Weighting function in k-space.

6. Modulation transfer function

7. f(x,y) y x The 2D Fourier Transform F (2){f(x,y)} = F(kx,ky) = f(x,y) exp[-i(kxx+kyy)] dx dy Iff(x,y) = fx(x) fy(y), then the 2D FT splits into two 1D FT's. But this doesn’t always happen. F (2){f(x,y)}

8. f(x,y) y x A 2D Fourier Transform: a square function Consider 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 determination of the Fourier Transform of the square function! F (2){f(x,y)}

9. Rick Linda Fourier Transform Magnitude and Phase Pictures reconstructed using the spectral phase of the other picture Mag{F [Linda]} Phase{F [Rick]} Mag{F [Rick]} Phase{F {Linda]} The phase of the Fourier transform (spectral phase) is much more important than the magnitude in reconstructing an image.