Radio emissions from the Sun are characterized by extremely rapid time variation. Although they are relatively strong, the signal to noise ratio becomes critical when we observe them with high resolution in space and time. To improve the S-N ratio, we have investigated the feasibility of an acousto-optical processing of interferometer images and applied it to the 160 MHz interferometer of the Nobeyama Solar Radio Observatory. We present here results of simulation experiments and preliminary solar observations with the new image processor.

In optical astronomy, images are highly degraded by atmospheric turbulence. The diameter of the seeing disk may vary from 0.2 arcseconds to 10 arcseconds or more according to weather conditions. A typical diameter is of the order of 2 arcseconds which is the angular resolution of a 6cm lens at the diffraction limit.

Large telescopes have been built for their light collecting capabilities, not for their resolving power which is, in general, of the order of 0.2 arcseconds, that is far from the diffraction limit.

Labeyrie’s (1970, 1976) speckle interferometry permits an astronomical object to be resolved to a limit approaching λ/D (where λ is the mid-band wavelength of the light accepted by a telescope having a pupil aperture of diameter D) in the presence of severe atmospheric seeing. As Dainty (1973) has pointed out, this resolution limit is virtually independent of the accuracy to which the telescope is figured – the seeing can actually improve a telescope’s resolution! Even though a true image of the object cannot usually be constructed by Labeyrie’s data reduction procedures, nevertheless the autocorrelation of (the distribution of brightness over) the object can always be reconstructed – provided the extent (width, apparent angular diameter) of the object does not exceed that of the isoplanatic patch (cf. Bates and Gough, 1975). The autocorrelation is useful because its extent is necessarily twice that of the object.

In image formation from coherence functions we start with a set of observed coherences (visibilities) R′ (uk, vk), measured at locations uk, vk in the u,v-plane:

We have heard so far at this meeting from the practical-minded men of radio astronomy. Faced with the problem: ‘my map has sidelobes and noise on it’, they promptly reach for the nearest available weapons – ‘cleaners, polishers and kneaders’. All these use varying amounts of subjective human judgement, as for example in ‘window cleaning’.

The problem of estimating radio sky brightness distributions from incomplete, and noisy, visibility measurements, such as those collected by a long baseline interferometer, has recently been attacked using nonlinear data-adaptive techniques such as the Maximum Entropy method (Abies (1974), Wernecke (1976), Wernecke and D’Addario (1977), Gull and Daniell (1978)) and the Maximum Likelihood method (Papadopoulos (1975)).

Two rival expressions for entropy have been used in maximum entropy image formation. Attention is drawn to the fact that both have been shown to be limiting cases of a general expression for the entropy of the photon distribution swept up during an observation. All measures of entropy are based on some physical model of the radiation mechanism. It is argued that knowledge of its character should be fully included in the image formation process.

A common problem in radio synthesis work is that of determining the brightness at N grid points in the map domain when there are only n<N independent interferometer measurements available. The missing (N-n) equations can in principle be replaced by an equivalent amount of information in the form of a priori knowledge about the brightness distribution. One way of doing this is to add an equation of the form

i.e. some function of the brightness distribution is maximized subject to the condition that the map be compatible with the n measurements. This automatically gives (N-n) new equations, leaving us in the pleasant situation of having as many equations as there are unknowns.

In radio astronomy it is often necessary to estimate a brightness distribution from a limited number of samples of its Fourier transform. The manifest requirement that the brightness distribution be everywhere positive imposes definite constraints on its Fourier transform which yield information about unmeasured Fourier components. Here we discuss the question: given the first n+1 values, p0, P1 … pn, of a uniformly sampled Fourier transform of a real positive function, what can we say about Fourier terms of higher order?

Radioastronomical observations are often incomplete in the sense that either the angular resolution desired or the signal-to-noise ratio required are not adequate. Attempts to fill in the missing information by either smoothing or deconvolving the observed data date back to Bracewell (1956, 1958). More recently the need for more sophisticated restoration procedures has increased since observations with radioastronomical interferometers frequently suffer from incomplete coverage of the u – v plane. Procedures like CLEAN (Högbom 1974, Schwarz 1978) and MEM (Wernecke 1977, Gull and Daniell 1978) have been developed to interpolate the unknown data in the u – v plane.

Study of a model data analysis situation with the help of computer experiments reveals that the Maximum Entropy Method of Spectral Analysis owes its popularity to a peak-sharpening property which is found to be a strong function of the level of background white noise present in the spectrum. In one limit, misleading results may be obtained by this technique. Some suggestions for a more fruitful use of the technique are made.

The mathematical background of the method CLEAN, as given by Schwarz (1978) is summarized. A method to derive source parameters (flux, position, extent) as a direct application of the theory is described. The ambiguity of the CLEAN method is discussed from a theoretical and practical point of view. Empirical results on the influence of the loop-gain are given. A new approach for the processing of maps with limited phase information is presented and a novel application of CLEAN for beam-switching observations (as an example for the case of a non-symmetric beam) is given.

The problem of determining the optimum number of components in fitting a Gaussian model to aperture synthesis data is considered. As a measure of the badness of the fitted model, we propose the use of Akaike’s Information Criterion (AIC).

An optimum solution to a deconvolution problem has to fulfil three general criteria: (a) an explicit recognition of the smoothing nature of convolution; (b) a statistical treatment of noise, e.g., using the least-squares criterion; and (c) requiring the solution to conform to all our prior knowledge about it. In the usual least-squares method, one minimises a variance of ‘residuals’, or the departures of the observed data from the values expected according to the recovered solution. However, this condition does not lead to a stable solution in the case of deconvolution, since the only stable solutions are those conforming to a criterion of ‘regularisation’ or smoothness (see, e.g., Tikhonov and Arsenin 1977). In our method, the stability is achieved by minimising the variance of the second-differences of the solution simultaneously with the fulfilment of the least-squares criterion. Such a procedure was first used by Phillips(1962). However, the solution thus obtained is still unsatisfactory since it usually does not conform to our a priori information. When we seek the brightness distribution of an object, the most frequent violation of our prior knowledge is that of positiveness. This motivated us to develop an Optimum Deconvolution Method (ODM) which constrains the solution to satisfy prior knowledge while retaining the features of least-squares and smoothness criteria.

The data measured by an interferometer define the original image through linear equations of constraint containing the Fourier coefficients. These constraints and the requirement that all image points show a non-negative flux define a problem frequently encountered in operations research and numerically tractable through the Simplex Algorithm. When the data are undersampled, a continuum of original images is compatible with the measured points. The advantage of the operations research method in this situation is that the continuum of restorations is generated rather than a single example. As a practical matter, it appears most useful to select from this continuum the maximum sharpness restoration – which like CLEAN uses reasonable a priori information to chose one of the possibilities – and to select for comparison the maximum entropy solution – which embodies diametrically opposed a priori information. The first may be considered the best restoration for an astronomical source, while the second indicates the uncertainty due to undersampling. When our astrophysical deductions depend strongly on the restored structure of a source, a comparison of alternative solutions is necessary to acquire confidence in the deductions. The maximum sharpness solutions also have the advantage that small phase errors in the measured complex points can be compensated. In principle, large phase errors can be compensated, but uniqueness may be lost, and existing numerical procedures for this case are inefficient.

The limiting sensitivity for conventional single-dish radio continuum observations of extended objects at wavelengths of a few cm is set mainly by weather; for much of the time a varying component of atmospheric attenuation, and hence thermal emission, causes deviations in the response of the receiver far in excess of any deviations due to receiver noise or receiver instabilities.

Radioastronomy deals with radiowaves emitted by different kinds of sources. These sources are very far from the receiving antenna; the distance R is very large compared with the wavelength. In acoustic imagery of noise sources the conditions are quite different.

It is a great pleasure for me to address this colloquium of radio astronomers on reconstruction from projections, especially since this opportunity allows me to combine an early interest in astronomy with my current work.

Some of you have been applying methods of reconstruction since the mid-1950’s. Radio astronomers formed a cohesive society then and the methods reached fruition and matured in a natural progression. Reconstruction from strip scans and then interferometer data has been so fundamental to the formation of images in radio astronomy that one may well say that the field could not have progressed without them.