Technique

Let an estimate of true intensity distribution on the sky be represented as the sum of two components:

where is the intensity of the point sources (mainly zero apart from -functions at the positions of designated sources) and is the intensity of the background. The corresponding estimates for the components which are observed are and where is the point spread function and is the convolution operator. The predicted total observed intensity is . We wish to obtain the ``best'' estimates for and and choose to maximize the quantity:

where is the observed intensity distribution, and the summation is performed over all pixels in the image. The second term is only evaluated over the background image and it has a regularizing effect. The quantity is a free parameter specified by the user and controls the strength of the regularization in comparison with the first term which is the logarithm of the likelihood. When is zero the objective function becomes the same as that of the standard Lucy-Richardson technique. takes the form of an entropy expression which may be calculated in two ways, relative to a flat background:

or as relative to a floating default :

Note that must be renormalized in calculating these entropy terms (Lucy 1994). The floating default is most conveniently chosen to be a smoothed version of the current estimate for the background. In practice the constant default works well for images in which the background is roughly flat such as those used for star cluster photometry. For cases where there is significant structure, such as stars superimposed on a galaxy, the floating default method works dramatically better and is essential for good results. Examples of both types of default are given below.

An iterative scheme has been developed to maximize the above objective function. It has also proved possible to accelerate this algorithm in a manner analogous to that which has been described for the standard Lucy-Richardson restoration scheme (Hook &Lucy 1993). The addition of the entropy term removes the automatic non-negativity conservation of the standard Lucy-Richardson method so care must be taken to avoid negative points which lead to gross instabilities. This may be done using the acceleration option which will act in these cases as a deceleration. The regularization of the background means that no ugly artifacts appear and the iterations may, and normally should, be allowed to continue as far as convergence.