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

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.

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.

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

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

Modulation transfer function

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

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(k x /2) sinc(k y /2) This picture is an optical determination of the Fourier Transform of the square function! x y f(x,y) F (2) {f(x,y)}

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