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**Session 107 - Radio Astronomy and VLBI Instruments.**

*Display session, Thursday, January 16*

*Metropolitan Ballroom, *

## [107.09] A Maximum Entropy Method for the Ill-Posed Inversion Problem

*R. G. Lyon (CESDIS - NASA/GSFC), J. M. Hollis, J. E. Dorband (NASA/GSFC)*
Estimation of the true source emission signature from an
observed data set corrupted by the detection process is a
classical inverse problem arising in both imaging and
spectroscopic observations. The data are generally distorted
due to the finite spatial, spectral and/or temporal response
of the instrument. Furthermore, noise can be introduced into
the data via the detection process, and in some cases the data
can be partially corrupted, saturated or even lost. If the
number of observed data points is greater than or equal to the
number of independent variables, the inverse problem is
well-posed and can be uniquely solved if a feasible solution
exists. In principle, maximum likelihood estimation (MLE)
methods can be applied to problems of this type; in practice,
numerical problems usually arise due to ill-conditioning. If
the number of observed data points is less than the number of
independent variables due to missing data, the problem is ill-
posed and generally an infinite set of MLE solutions exist. We
develop a maximum entropy method with MLE constraints suitable
for both the ill- and well-posed cases. For the ill-posed case
the method chooses out of the infinite set of MLE solutions
that which has maximal entropy, i.e. the least informative
solution, thereby mitigating against an over interpretation of
the data. The MLE constraints also insure that, for the well-
posed case, the solution must be the unique MLE solution if a
feasible solution exists. This insures continuity across the
boundary between the ill- and well-posed cases. This work
suggests the possibility that maximum likelihood estimation
theory may be interpreted as a subset of maximum entropy
theory. Spectroscopic and imaging examples are shown as
demonstrations.

If you would like more information about this abstract, please follow the link to cesdis.gsfc.nasa.gov/~lyon. This link was provided by the author. When you follow it, you will leave the the Web space for this meeting; to return, you should use the Back button on your browser.The author(s) of this abstract have provided an email address for comments about the abstract: lyon@nibbles.gsfc.nasa.gov

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