Maximum entropy techniques problem

The mathematical basis for the exponential series method is Eq. (5.3), the use of which has recently been criticized by Phillips and Lyke.(19) Based on their analysis of the one-sided Laplace transform of model excited-state distribution functions, it is concluded that a small, finite series of decay constants cannot be used to represent a continuous distribution. Livesey and Brouchon(20) described a method of analysis using pulse fluorometry which determines a distribution using a maximum entropy method. Similarly to Phillips and Lyke, they viewed the determination of the distribution function as a problem related to the inversion of the Laplace transform of the distribution function convoluted with the excitation pulse. Since Laplace transform inversion is very sensitive to errors in experimental data,(21) physically and nonphysically realistic distributions can result from the same data. The latter technique provides for the exclusion of nonrealistic trial solutions and the determination of a physically realistic solution. These authors noted that this technique should be easily extendable to data from phase-modulation fluorometry. 

In this technique, a spectrum is generated to have maximum entropy (in the information theory sense)138 subject to being consistent with the observed Fid,136- 139 This inverse problem is solved iteratively. At each stage, the inverse Fourier transform of the spectrum is taken as an estimate of the FID and... 

Therefore, novel techniques potentially applicable to solving crystal structures are under continuous testing and development. A recent collective monograph on the structure determination from powder diffraction data provides an excellent discussion of the problem and introduces different approaches that may be used in its solution. In this chapter, unconventional structure solution methods are only briefly reviewed most of them are still controversial and do not always work well with different kinds of compounds and data, although solutions of several complex structures have been demonstrated. Summarized below are the genetic algorithm, maximum entropy, maximum likelihood, and simulated annealing methods. 

This is a familiar problem which can be analyzed by using the " maximum entropy method",(1-4) a technique used extensively elsewhere in the field of image analysis. 

The direct demodulation algorithm provides a general approach to handle a large variety of image restoration or reconstruction problems. Computer simulations and analysis results for COS-B and CGRO 7-ray data show that in comparison with traditional techniques, e.g. maximum entropy method, cross-correlation deconvolution or likelihood approach, the direct demodulation method has high sensitivity, high resolution ability and capability to effectively reduce the effect of statistical fluctuations and noise in data and to simultaneously restore both the extended and discrete features in the object. 

Maximum entropy is similar in some respects to maximum likelihood, although the technique is perhaps a little more esoteric and requires more expensive hardware and software. Nevertheless, the proponents of maximum entropy argue that it is the best possible solution to a problem such as deconvolution, as probability calculus is employed in the calculation of the most likely (maximum entropy) solution. 

Recently, maximum entropy image enhancement techniques have been successfully applied to a variety of ill-conditioned problems, including the inversion of imaginary... 

Maximum entropy, as applied to image restoration, can be thought of as a particular case of a more general technique known as regularimtion. One approach to ill-posed problems, such as image restoration, is to find solutions that are consistent with the data, but which possess other desirable features. Maximum entropy is but one of many possible desirable features. Another possibility is a measure such as... 

The maximum entropy method was one of several techniques that were used to restore the Hubble images. However, all techniques were hampered by lack of complete knowledge of the point-spread function. Using a point-spread function that included errors (either because of noise in a measured point-spread function, or modeling imperfections in predicted point-spread function) further adds to the ill-posed nature of the restoration problem. Although the modulation transfer function tended not to be zero-valued (at least not as severely as in the radio interferometry case described below), this was of only small comfort when imaging weak objects, as the... 

Ideally an image restoration technique will deliver an image that is consistent with available data and constraints (e.g., positivity), and which is free of obvious artifacts. Any technique that achieves this should be taken seriously, regardless of whether it is based on an ad hoc procedure or justified by a formalism such as maximum entropy. It is the data, ultimately, that must drive a restoration process. In analyzing the solution to any ill-posed problem, it is important to differentiate between those characteristics dicatated by the data, and those that are dependent on the solution technique. Any physically implausible feature that is not required by the data should be ignored. 

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