Fourier transforms are often used since convolutions in the frequency
space become multiplications (see Section 2.3.3). Combining
Equation 2.1 and Equation 2.26 the
original image
is obtained by division

if the transformed PSF is non-zero and the noise

Another group of image restoration algorithms use iterative methods to
obtain a solution which is consistent with the data.
The maximum entropy method uses either the entropy :

in the optimizing procedure (Frieden 1972). It tends to enhance sharp features but a solution may depend on the initial guess and therefore not be unique.

A different scheme was introduced by Lucy (1974) who uses a correction
term based on the ratio between the image and the guess. A first
guess
must be specified (e.g. a constant) to start the
iteration. The first step in the iteration performs a convolution
with the PSF :

The second step computes a correction factor based on this frame and the original image :

The procedure repeats these two steps until the corrections are sufficiently small. After Equation 2.30 is computed the iteration continues with Equation 2.29. This scheme reaches a stable solution very quickly (i.e. 3-5 steps) and is little affected by noise. This makes it very useful for low signal-to-noise data. A photographic picture of a galaxy is used to illustrate this technique (see Figure 2.11). A fit to the stellar image was used to define the PSF.