## The Discrete Two Dimensional Fourier Transform

An image (such as the projection of a molecule) is defined as a two-dimensional (2D) array representing samples of the optical density distribution on a regular grid:

The coordinates are multiples of the sampling distance Ax = Ay; xt = iAx; yk = kAx. The Fourier transform is a mathematical representation of the image using sine waves or complex exponentials as basis functions. One can see the Fourier transform as an alternative representation of the information contained in the image. What singles out the Fourier transform, among many other expansions using sets of orthonormalized basis functions, is that it allows the influence of instrument aberrations and the presence of periodicities in the object to be readily analyzed. It also provides a key to an understanding of the relationship between the projections and the object they originate from, and to the principle underlying three-dimensional (3D) reconstruction.

The basis of the Fourier representation of an image is a set of ''elementary images''—the basis functions. In the case of the sine transform, to be used as an introduction here, these elementary images have sinusoidal density distribution:

eim\,k = eim(xi, yO = sin[2<MZx; + vmyt)] (A1.2)

Each of these images (examples in figure A1.1), represented by a discrete set of pixels at positions (xi, yk) of a square lattice, is characterized by its discrete spatial frequency components (ul, vm)—describing how many full waves fit into the frame in the horizontal (i ) and vertical (m) directions:

I m ui = —, I = 0, ..., I - 1; vm = —, m = 0, ..., K - 1 (A1.3) IAx KAy

Any discrete image {pik; i = 1,..., I; k = 1,..., K} can be represented by a sum of such elementary images, if one uses appropriate amplitudes am and phases 'lm. The amplitudes determine the weights of each elementary image, and the phases determine how much it has to be shifted relative to the normal position (first zero assumed at xt = yk = 0; note that in figure A1.1 this occurs in the upper left corner since a left-handed coordinate system is used):