2D Fourier transform

A 2D Fourier transform, the adaptation of the Fourier transform to two-dimensional functions (over bounded domains), yields a two-dimensional complex function capable of reconstructing the original function, or in the non-complex (i.e., real number) version, one capturing some of the function's characteristics. As with the 1D Fourier transform, the reconstruction consists of applying the corresponding inverse Fourier transform. Some useful manipulations of the original function are easily carried out in its transformed form. For example, a 2D spatial Fourier transform (over space dimensions) of image data (essentially a sample of values of a 2D function) makes it straightforward to derive that of an approximation of the image, useful for compression of image files as well as analysis, such as image recognition. The 2D transform and its inverse are also key to aperture synthesis, the retrieval of images from diffraction interference patterns.