# Fourier series approximations to images

Images are invariably finite signals. Given any image, it is possible to construct a periodic image by just tiling a plane with the image. Thus, there is again a close relationship between a periodic image and a finite one.We have seen sinusoidal images, so it follows that it ought to be possible to construct a Fourier series representation of an image. The only hard part is that images have a two-dimensional domain, and thus are finite in two distinct dimensions. The sinusoidal images that we saw correspondingly could have a vertical frequency, a horizontal frequency, or both.

Suppose that the domain of an image is [*a*, *b*] ×
[*c*, *d*] ⊂ *Reals ×
Reals*. Let *p _{H}* =

*b − a*, and

*p*=

_{V}*d− c*represent the horizontal and vertical "periods" for the equivalent periodic image. For constructing a Fourier series representation, we can define the horizontal and vertical fundamental frequencies as follows:

*= 2π /*

_{H}*p*

_{H}*= 2π /*

_{V}*p*

_{V}The Fourier series representation of *Image*: [*a*, *b*]
× [*c*, *d*] →
*Intensity* is

For convenience, we have included the constant term *A*_{0,0}
in the summation, so we assume that Φ _{0}
= φ _{0} = 0 (recall that cos(0) = 1).