Maximum Entropy Method

A Deconvolution Algorithm (sometimes abbreviated MEM) which functions by minimizing a smoothness function (``Entropy'') in an image. Maximum entropy is also called the All-Poles Model or Autoregressive Model. For images with more than a million pixels, maximum entropy is faster than the CLEAN Algorithm.

MEM is commonly employed in astronomical synthesis imaging. In this application, the resolution depends on the signal-to-noise ratio, which must be specified. Therefore, resolution is image dependent and varies across the map. MEM is also biased, since the ensemble average of the estimated noise is Nonzero. However, this bias is much smaller than the Noise for pixels with a ${\rm SNR}\gg 1$ . It can yield super-resolution, which can usually be trusted to an order of magnitude in Solid Angle.

Several definitions of ``Entropy'' normalized to the flux in the image are

$\displaystyle H_1$	$\textstyle \equiv$	$\displaystyle \sum_k \ln\left({I_k\over M_k}\right)$	(1)
$\displaystyle H_2$	$\textstyle \equiv$	$\displaystyle -\sum_k I_k\ln\left({I_k\over M_ke}\right),$	(2)

where

is a ``default image'' and

is the smoothed image. Some unnormalized entropy measures (Cornwell 1982, p. 3) are given by

$\displaystyle H_1$	$\textstyle \equiv$	$\displaystyle -\sum f_i\ln(f_i)$	(3)
$\displaystyle H_2$	$\textstyle \equiv$	$\displaystyle \sum \ln(f_i)$	(4)
$\displaystyle H_3$	$\textstyle \equiv$	$\displaystyle -\sum {1\over\ln(f_i)}$	(5)
$\displaystyle H_4$	$\textstyle \equiv$	$\displaystyle -\sum {1\over [\ln(f_i)]^2}$	(6)
$\displaystyle H_5$	$\textstyle \equiv$	$\displaystyle \sum\sqrt{\ln(f_i)}\,.$	(7)

See also CLEAN Algorithm, Deconvolution, LUCY

References

Cornwell, T. J. ``Can CLEAN be Improved?'' VLA Scientific Memorandum No. 141, March 1982.

Cornwell, T. and Braun, R. ``Deconvolution.'' Ch. 8 in Synthesis Imaging in Radio Astronomy: Third NRAO Summer School, 1988 (Ed. R. A. Perley, F. R. Schwab, and A. H. Bridle). San Francisco, CA: Astronomical Society of the Pacific, pp. 167-183, 1989.

Christiansen, W. N. and Högbom, J. A. Radiotelescopes, 2nd ed. Cambridge, England: Cambridge University Press, pp. 217-218, 1985.

Narayan, R. and Nityananda, R. ``Maximum Entropy Restoration in Astronomy.'' Ann. Rev. Astron. Astrophys. 24, 127-170, 1986.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Power Spectrum Estimation by the Maximum Entropy (All Poles) Method'' and ``Maximum Entropy Image Restoration.'' §13.7 and 18.7 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 565-569 and 809-817, 1992.

Thompson, A. R.; Moran, J. M.; and Swenson, G. W. Jr. §3.2 in Interferometry and Synthesis in Radio Astronomy. New York: Wiley, pp. 349-352, 1986.