I am trying to understand how and why the fourier transform is used in image processing / computer vision. Below is what I have gathered so far. Would my understanding of it be correct? If not, could somebody explain it to me in simple, plain english? Or, does anybody have anything to add to it? Last but not least, could somebody explain the "discrete fourier transform"?
The fourier transform decomposes an image into its sine and cosine components. Put simply, sine and cosine are waves starting at a minimum and maximum respectively. In the real world, we can't tell whether a wave that we observe started at a maximum or minimum point, and therefore we can't really distinguish between the two. Therefore, sine and cosine are simply referred to as sinusoids.
When applying the FT to an image, we transform it from its spatial domain into a "frequency domain", which in essence is the image represented in terms of its variation in colour and brightness over time (well, not time, but space. That is, over a number of pixels).
EDIT: Why would I use the Fourier Transform? And what are its benefits over other methods? For example, one application in literature is in shape recognition or noise elimination. In basic terms, how could one go about shape recognition using the FT?