Maximum entropy reconstruction

Papoular, R.J. and Gillon, B. (1990) Maximum entropy reconstruction of spin density maps in crystals from polarized neutron diffraction data, Europhys. Lett., 13(5), 429 134. 

Maximum entropy reconstruction of spin densities involving non-uniform prior... 

Unfortunately, there is great scope for confusion, as two distinct techniques include the phrase maximum entropy in their names. The first technique, due to Burg,135 uses the autocorrelation coefficients of the time series signal, and is effectively an alternative means of calculating linear prediction coefficients. It has become known as the maximum-entropy method (MEM). The second technique, which is more directly rooted in information theory, estimates a spectrum with the maximum entropy (i.e. assumes the least about its form) consistent with the measured FID. This second technique has become known as maximum-entropy reconstruction (MaxEnt). The two methods will be discussed only briefly here. Further details can be found in references 24, 99, 136 and 137. Note that Laue et a/.136 describe the MaxEnt technique although they refer to it as MEM. 

Maximum entropy reconstruction [32] is claimed to remrn a maximally noncommittal solution. It calculates a small set of proposed S Fi,F maps that are compatible with the measured projections within the experimental errors, and selects the one with the least information content. For this reason it suppresses all noise and artefacts in the reconstruction and is therefore prone to be misleading. In another terminology, it rejects false positives but is likely to return false negatives . This particular feature suggests that the maximum entropy solution could prove to be a useful starting point for more sophisticated statistical programs. 

Rovnyak D, Frueh DP, Sastry M, Sun ZYJ, Stem AS, Hoch JC, Wagner G (2004) Accelerated acquisition of high resolution triple-resonance spectra using non-uniform sampling and maximum entropy reconstruction. J Magn Reson 170 15-21... 

Hyberts SG, Takeuchi K, Wagner G (2010) Poisson-gap sampling and forward maximum entropy reconstruction for enhancing the resolution and sensitivity of protein NMR data. J Am Chem Soc 132 2145-2147... 

Mehdi M, Alan SS, Jeffrey CH (2006) Spectral Reconstmction Methods in Fast NMR Reduced Dimensionality, Random Sampling, and Maximum Entropy Reconstruction. J Magn Reson 192 96-105... 

The Forward Maximum Entropy Reconstruction Relative to Other Procedures. 127... 

Hyberts SG, Heffron GJ, Tarragona NG, Solanky K, Edmonds KA, Luithardt H, Fejzo J, Chorev M, Aktas H, Colson K et al (2007) Ultrahigh-resolution (1)H-(13)C HSQC spectra of metabolite mixtures using nonlinear sampling and forward maximum entropy reconstruction. J Am Chem Soc 129 5108-5116... 

Jones DH, Opella SJ (2006) Application of maximum entropy reconstruction to PISEMA spectra. J Magn Reson 179(1) 105-113... 

To make the raw experimental data interpretable they must be converted into a frequency-domain spectrum. The classical, and commonest, method is discrete Fourier transformation. This is a linear operation the information content of the data is unchanged. Alternative methods such as maximum entropy reconstruction and linear prediction are nonlinear, changing the information content. In Fourier processing a weighting function is usually applied to the time-domain data before transformation a DC correction based on the last portion of the data may also be performed. After weighting and any... 

Although linear prediction can be used to extract spectral data directly from a FID ( parametric LP ), it is commonly used to extrapolate the experimental time-domain data, which are then weighted and transformed as normal. Maximum entropy reconstruction, in contrast, seeks to fit the experimental FID with a model function that contains the minimum amount of information consistent with fitting experiment to within the estimated noise level. The criterion of minimum information corresponds to the maximum Shannon informational entropy S(p), which for a probability distribution p is defined as... 

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