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