# Deconvolution with maximum entropy algorithms

This discussion was lifted from Tim Cornwell's article in the NRAO imaging workshop (1988).

CLEAN approaches the deconvolution problem by using a procedure which selects a plausible image from the set of feasible images. This makes a noise analysis of CLEAN very difficult. The Maximum Entropy Method (MEM) is not procedural. The image selected is that which fits the data, to within the noise level, and also has maximum entropy. This has nothing to do with physical entropy, it is just something that when maximised, produces a positive image with a compressed range of pixel values. The latter aspect forces the MEM image to be smooth, and the positivity forces super-resolution on bright, isolated objects.

One general-purpose definition of entropy (championed by Gull and Skilling) is

where is the brightness of the kth pixel, and is some default image. An example might be a low-resolution image of the object. This allows a priori information to be incorporated into the problem. An alternate form, suggested by Cornwell (sometimes call the `maximum emptiness' criterion) is

This second form does not enforce positivity, and so can be used for Stokes parameters other than I.

The requirement that each visibility be fitted exactly by the model usually invalidates the positivity constraint. Therefore, data are incorporated with the constraint that the fit, , of the predicted visibility to that observed, be close to the expected value:

Maximising H subject to the constraint that be equal to its expected value leads to an image which fits the long spacings too well, and the zero and short spacings poorly. To remedy this, an added constraint can be added to the problem. Typically this is a flux constraint, which ensures that the flux density in the maximum entropy image is correct. That is, the flux density of the maximum entropy image,

is constrained to equal the known flux density for the source. This is quite a useful constraint. Although the maximum entropy task, described below, does not force you to give a flux constraint, you should attempt to, if at all possible.

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