The Fourier transform of the product of two images is the

convolution of their associated individual transforms. Also the

Fourier transform of the convolution of two images is the product of

the individual transforms.

Correlation Theorem


 Rfg(x,y) = fff(x,y) f(x-xo,y-yo) dxo dyo (2.9)


 The correlation is very similar to the convolution except that

neither function is inverted.


Autocorrelation (Wiener-Khintchine) Theorem


 Off(u,v) = F {Rff(x,y)} (2.10)


 This special case of the convolution theorem shows that the

autocorrelation and the power spectral density are Fourier transform

pairs.

Fourier integral Theorem


 f(x,y) = F-1{ F{f(x,y)}} (2.11)


 f(-x,-y) =F {F {f(x,y)}}


 Successive transformation and inverse transformation yield that

function again. If the Fourier transform is applied twice

successively, the result is the original image inverted and perverted.

 It is also useful to define here the impulse function. Also known

as the Dirac delta function, it describes a function which is infinite

at the origin, zero elsewhere, and contains a volume equal to unity.

One definition of the Dirac delta function is