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Maximum Entropy Reconstruction

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Encyclopedia of Biophysics

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Maximum entropy (MaxEnt) reconstruction is a technique for computing frequency spectra from time series data, e.g., a free induction decay. It is based on the principle of information entropy introduced by Claude Shannon (1948). The MaxEnt spectrum contains the smallest amount of information consistent with the experimental data. In contrast to the discrete Fourier transform (DFT), MaxEnt references the measured data indirectly, enabling it to handle nonuniformly sampled (NUS, also called sparse sampling) data. Thus an important application of MaxEnt is the computation of multidimensional NMR spectra from NUS data.

Mathematically MaxEnt reconstruction is formulated as

$$ \operatorname{Maximize}\ \mathrm{S}\left(\mathbf{f}\right)\ \mathrm{subject}\ \mathrm{to}\ \mathrm{C}\left(\mathbf{d},\mathbf{m}\right)\le {\mathrm{C}}_0 $$
(1)

where f is the frequency spectrum (of arbitrary dimension), d is the measured data, and m is the inverse discrete Fourier transform of f. S(f) is the entropy,...

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References

  • Daniell GJ, Hore PJ (1989) Maximum entropy and NMR – a new approach. J Magn Reson 84:515–536

    CAS  Google Scholar 

  • Ding K, Gronenborn AM (2002) Novel 2D triple-resonance NMR experiments for sequential resonance assignments of proteins. J Magn Reson 156:262–268

    Article  CAS  PubMed  Google Scholar 

  • Donoho DL, Johnstone IM, Stern AS, Hoch JC (1990) Does the maximum entropy method improve sensitivity? Proc Natl Acad Sci U S A 87:5066–5068

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Hoch JC, Stern AS (1996) NMR data processing. Wiley-Liss, New York

    Google Scholar 

  • Hoch JC, Stern AS, Donoho DL, Johnstone IM (1990) Maximum entropy reconstruction of complex (phase-sensitive) spectra. J Magn Reson 86:236–246

    CAS  Google Scholar 

  • Hyberts SG 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

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Kim S, Szyperski T (2003) GFT NMR, a new approach to rapidly obtain precise high-dimensional NMR spectral information. J Am Chem Soc 125:1385–1393

    Article  CAS  PubMed  Google Scholar 

  • Kupče E, Freeman R (2003) Projection-reconstruction of three-dimensional NMR spectra. J Am Chem Soc 125:13958–13959

    Article  CAS  PubMed  Google Scholar 

  • Mobli M, Stern AS, Hoch JC (2006) Spectral reconstruction methods in fast NMR: reduced dimensionality, random sampling and maximum entropy. J Magn Reson 182:96–105

    Article  CAS  Google Scholar 

  • Paramasivam S et al (2012) Enhanced sensitivity by nonuniform sampling enables multidimensional MAS NMR spectroscopy of protein assemblies. J Phys Chem B 116:7416–7427. https://doi.org/10.1021/Jp3032786

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  • Schmieder P, Stern AS, Wagner G, Hoch JC (1997) Quantification of maximum entropy reconstructions. J Magn Reson 125:332–339

    Article  CAS  PubMed  Google Scholar 

  • Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27:379–423

    Article  Google Scholar 

  • Skilling J, Bryan R (1984) Maximum entropy image reconstruction: general algorithm. Mon Not R Astron Soc 211:111–124

    Article  Google Scholar 

  • Stern AS, Li KB, Hoch JC (2002) Modern spectrum analysis in multidimensional NMR spectroscopy: comparison of linear-prediction extrapolation and maximum-entropy reconstruction. J Am Chem Soc 124:1982–1993

    Article  CAS  PubMed  Google Scholar 

  • Stern AS, Donoho DL, Hoch JC (2007) NMR data processing using iterative thresholding and minimum l1-norm reconstruction. J Magn Reson 188:295–300. https://doi.org/10.1016/j.jmr.2007.07.008

    Article  PubMed  PubMed Central  CAS  Google Scholar 

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Author information

Authors and Affiliations

  1. Molecular Biology and Biophysics, UConn Health, Farmington, CT, USA

    Jeffrey C. Hoch

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  1. Jeffrey C. Hoch
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Correspondence to Jeffrey C. Hoch .

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Editors and Affiliations

  1. University Leicester MRC Centre, Leicester, United Kingdom

    Gordon Roberts

  2. Dept Biochemistry, University of Oxford, Oxford, United Kingdom

    Anthony Watts

Section Editor information

  1. No affiliation provided

    Mitsu Ikura

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© 2018 European Biophysical Societies' Association (EBSA)

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Hoch, J.C. (2018). Maximum Entropy Reconstruction. In: Roberts, G., Watts, A. (eds) Encyclopedia of Biophysics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35943-9_337-1

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  • DOI: https://doi.org/10.1007/978-3-642-35943-9_337-1

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  • Print ISBN: 978-3-642-35943-9

  • Online ISBN: 978-3-642-35943-9

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