Maximum entropy image restoration in astronomy

[Nature India Special Issue: Lighting the way in physics]

An individual radio antenna has an angular field of view on the sky– informally known as ‘the beam’ – which can result in a very poor angular resolution at long wavelengths. Only the combined effect of all the sources in the beam is measured. Martin Ryle won the 1974 Nobel Prize for Physics for a dramatic improvement called ‘aperture synthesis’ - work that was carried out in Cambridge, UK. Combining the signals from two antennas modifies the response to each source, depending on its position in the beam. It can be enhanced twofold if the path difference is an integer number of wavelengths, or become zero if there is an extra half wavelength. Physically, this is the same effect that Thomas Young used in 1801 to establish the wave nature of light in his famous two-slit experiment.

Mathematically, the sources in the beam are now multiplied by a sine/cosine wave – a series of stripes in the sky. This is nothing but a single measurement of the Fourier transform of the sky brightness. Clearly, many such pairs with different spacings and orientations are needed to build up enough information to unscramble – or deconvolve – all the sources and make an image. Ryle was clear that all the measurements should be made out to a maximum spacing, which defined the angular resolution.

Later, astronomers elsewhere – and even a few in Cambridge – started playing the dangerous game of ‘reconstructing’ the map of the emission in the sky with fewer measurements.

One such method on the horizon in 1982 was ‘maximum entropy’ (MEM). The basic idea is that there are many possible maps consistent with a limited set of measurements. The one that is ‘most likely’ can then be chosen. Thermodynamics tells us that a gas fills a container simply because that is the configuration that can occur in the largest number of ways, as measured by its entropy. This idea was then extended to decoding messages in Claude Shannon’s information theory.

The astronomer Jon Ables visited the Raman Research Institute (RRI) from Sydney in 1973 and showed us some startling successes using this approach. He generously shared his code with Rajendra Bhandari, who was intrigued but was also sceptical of reconstruction that almost seemed like resurrection. He soon came up with examples where the MEM did not work that well. It also did not help that the ‘entropy’ as a function of the intensity in the sky depended on whether you were in Cambridge ( - B ln B) or in Sydney (ln B)

Ramesh Narayan came to RRI in 1978 after completing PhD studies at India’s National Aerospace Laboratories. He had experience in Fourier transforms. One deconvolution in X-rays was already under his belt. Max Komesaroff was visiting from Sydney at that time and had a new take relating MEM to the positivity constraint. Their collaboration resulted in a much better mathematical understanding of the “ ln B “ MEM in one dimension and they shared some of the excitement with me. This triggered further conceptual and computational activity. Ronald Ekers visited later and brought insight into CLEAN – the algorithm of choice at that time for radio maps. Narayan and I began to develop our own codes for CLEAN and MEM in the realistic two dimensional case and tried out different entropy functions. We knew the algebraic form for the reconstruction which the two forms of MEM gave: a nonlinear function, either the reciprocal or the logarithm of a Fourier series. The breakthrough was Narayan’s graphical insight that such a nonlinear transformation automatically sharpened peaks and suppressed undesirable baseline ripples.

A lot of things fell into place once we were armed with this knowledge, but they had to be backed up with extensive simulations starting with a known image, throwing away some information and seeing if the MEM could recover it. We could understand when it worked and when it didn’t and how to control its behaviour. We poured cold water on information theory even in our title – ‘a practical non-information-theoretic approach’ to the MEM, since we had many more functions which worked just as well. Our paper was published in the newly founded Journal of Astrophysics and Astronomy of the Indian Academy of Sciences (J. Astrophys. Astr. 3, 419–450; 1982).

It helped a great deal that Narayan soon attended a meeting in Sydney on indirect imaging on his way to a fellowship at the California Institute of Technology (Cal Tech). We submitted a generalised version to this conference, including the polarisation of the radio waves. Our pragmatic and practical message won some supporters. In the conference summary, R.H.Bates from New Zealand joked that folklore had N=2 kinds of MEM, but Bangalore had pushed N to infinity!