Maximum Entropy and the Nearly Black Object

• Published in: Journal of the Royal Statistical Society. Series B, Methodological Methodological, 1992, Vol.54 (1), p.41-81
• Main Authors: Donoho, David L., Johnstone, Iain M., Hoch, Jeffrey C., Stern, Alan S.
• Format: Article
• Language: English
• Subjects: Astronomical objects; Data smoothing; diffraction‐limited imaging; Entropy; Estimators; Exact sciences and technology; Fourier transformations; inverse problems; Least squares; magnetic resonance spectroscopy; Mathematical methods in physics; Minimax; minimax decision theory; non‐linear recovery; Nuclear magnetic resonance; Physics; positivity constraints; Signal noise; Spectroscopy; superresolution
• ISSN: 0035-9246; 1369-7412; 2517-6161; 1467-9868
• DOI: 10.1111/j.2517-6161.1992.tb01864.x

Maximum entropy (ME) inversion is a non-linear inversion technique for inverse problems where the object to be recovered is known to be positive. It has been applied in areas ranging from radio astronomy to various forms of spectroscopy, sometimes with dramatic success. In some cases, ME has attained an order of magnitude finer resolution and/or an order of magnitude smaller noise level than that obtainable by standard linear methods. The dramatic successes all seem to occur in cases where the object to be recovered is 'nearly black': essentially zero in the vast majority of samples. We show that near-blackness is required, both for signal-to-noise enhancements and for superresolution. However, other methods--in particular, minimum l1-norm reconstruction--may exploit near-blackness to an even greater extent.