Digitised images are an approximate representation of the real world. As mentioned previously, they are subject to sampling, quantisation, windowing and noise. Normally, they are represented in a two dimensional spatial domain, i.e. x and y, or .

However, they may equally be represented in a spatial frequency domain, and . Transformation from one domain to another is based commonly on Fourier analysis. Usually, this is devloped in one dimension for example, in the analysis of speech or music where is a time varying signal; here we consider briefly the extension to two dimensions.

Image transforms are widely used in filtering, data compression, image description, e.g. extraction of frequency domain characteristics for classification of texture, image restoration and low level processing. Low level image processing operations transform generally from one image to another image; this might be the first stage in a computer vision system. Considering first the equivalence of the time and frequency domains, this can be described by the Fourier transformation, where

Figure 1: Examples of one dimensional signals and their corresponding transforms

and are continuous aperiodic signals of time, t, and radial frequency, , respectively. In image processing terms, transformation between the spatial and spatial frequency domains is expressed,

and

where describes the intensity function of an image and is it's frequency spectrum in terms of the variables , which corresponds to a frequency variable along the x-axis and , which corresponds to a frequency variable along the y-axis. In a digitised image, the incoming signal is sampled at discrete intervals in space. Assuming a square image of dimension, N, the corresponding transformational equations are,

and

The 2-dimensional Fourier transform of an digital image is thus essentially a Fourier series representation of a 2-dimensional field. Within a digital image processing system, the computation of these equations requires operations but this can be reduced substantially by successive passes of the 1-dimensional FFT to .