Linearity Theorem
F { af1(x,y) + bf2(x,y)} = a F{f1(x,y)} + b F{f2(x,y)} (2.4)
The transform of the sum of two functions is simply the sum of
their individual transforms. The Fourier transform is a linear
operator or system.
Similarity Theorem
F {f(ax,by)} = F(u/a,v/b)/ab where F(u,v) = F {f(x,y)} (2.5)
Scale changes in the image domain results in an inverse scale
change in the frequency domain along with a change in the overall
amplitude of the spectrum.
Shift Theorem
F {f(x-a,y-b)} = F(u,v) exp [-j(ua+vb)] (2.6)
Translation of patterns in the image merely introduces a linear
phase shift in the frequency domain. The magnitude is invariant to
translation.
Parseval's Theorem
fJ/F(u,v) 12 du dv = If (x,y)12 dx dy (2.7)
The total energy in the images plane is exactly equal to the
energy in the frequency domain.
Convolution Theorem
F{f(x,y) g(x,y)} = ff F(u,v)F(uo-u,vo-V) du dv
(2.8)