Maximum Entropy

Based on ideas from information theory, one can perform image analysis and construct meaningful tomographic algorithms, algorithms. Suppose there is a source which generates a discrete set of independent messages with probabilities  Then the information associated with is defined logarithmically as

The entropy of the source is defined as the average information generated generated by the source and can be calculated as

When the source is the image, the probability can be replaced by the gray level for the th pixel and entropy can be redefined as

For natural systems, the organization of intensities over the image can be expected to follow the second law of thermodynamics namely,

This is the basis of the MAXENT algorithm. For interferometric images, one can view the pixel temperature as the information content and entropy built up using their magnitudes. In the absence of any constraint, the solution of the above optimization problem will  correspond to a constant temperature distribution, more generally a uniform histogram in terms of probabilities. Hence, the MAXENT algorithm is properly posed only along with the projections as constraints.

Requiring that the entropy of the system be a maximum along with the interferometric projections as constraints is known as the Maximum entropy optimization technique (MAXENT). It produces an unbiased solution and is maximally noncommittal about the unmeasured parameters. This technique is particularly attractive when the projection data is incomplete. The MAXENT algorithm is described below:

Consider a continuous function with condition and values pixels. In the present context, the entropy technique refers to the extermination of the

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Subject to a set of constraints. In constraints. In MAXENT the collected projection data and any other a priori information about the field to be reconstructed can be viewed as the constraints over which the entropy is to be maximized. A typical maximum entropy problem can be stated as:

Different schemes are available for optimizing a function over some constraints, for example the Lagrange multiplier technique. The MART algorithms have been shown to be equivalent to the maximum entropy algorithm in the literature. Hence the entropy algorithm has not been considered further in the present article.