2. Deconvolution equations

The observed CVs have well-known spatial frequencies (frequency pairs in the usual twodimensional case) and contain unknown complex-amplitude errors. The visibilities result from a Fourier integral over the intensity distribution in the image space. For practical purposes, the continuous intensity distribution has to be approximated by a discrete intensity grid. While it is not strictly necessary, it is nevertheless convenient, also for practical purposes, to consider a rectangular equidistant grid of constant-size intensity pixels.

Thus, we start with such grid and intensities z_jl at grid positions . Given an observed frequency pair (u_k,v_k), the observed CV (complex amplitude A_k) is given by

where f_k is a - generally known - telescope-dependent sensitivity factor which we shall put, for convenience, as equal to 1. is a complex error. The z_jl are the unknowns to be found from the observations. Thus, we call the righthand side of Eq. (1) the visibility model .

Another sensitivity factor S_k,jl is included to take into account the influence of the intensity distribution within the pixel jl on the observations. It is not known observationally but some assumption has to be made, at least implicitly. For a model, it is useful to assume the same distribution in all pixels: then S depends only on frequency, S=S_k. For example, S_k=1 if the intensities were concentrated in the mid-pixel positions . If, instead, one fourth of the intensity is put into each of the four pixel corners , then . A non-symmetric intensity distribution within the pixel would give rise to a complex S_k. If each pixel has constant surface brightness then

which is what we adopt in the following. The reason is that the influence of the pixel size chosen on the deconvolution result is diminished. In any case, S_k is near unity for small frequencies and/or small pixel size because, then, the exponential factor varies little within each pixel.

Note that each CV is a member of a DFT of the image but, unless the frequency pairs are given on a rectangular grid, each CV is a member of a different DFT. Interchanging the role of the two telescopes giving rise to a certain CV changes the sign of the frequencies. Both CVs are, of course, equivalent, and either may be used. Thus, A_-k = A_k^* and .

The CVs being a linear superposition of (weighted) intensities (which are the unknowns), an LSF is apt. The LSF weights can be understood in the sense that one expects to reproduce the corresponding term within an error that can be considered as a member of an (approximately) Gaussian distribution with variance . In principle, one can choose the freely, e.g. according to natural weighting, or to uniform weighting. It would, however, not be useful to choose them smaller than the actual measuring errors. In other words, the choice should be where of course the quadratic error expectation E is no more known than the actual error but can often be reasonably estimated. In MIM, the equations are normalized to a unit error expectation (0, 1), that is, zero average and unit variance. Also, the intensity units can be normalized to multiples of the point source detection limit

We keep the errors formally in the equations and minimize the absolute value of model minus observed CVs,

The result of differentiation in respect to z_nm is

with z^* = z/z₀





The system is to be solved in the sense that the residuals fit approximately into a (0, 1) distribution. The double sum is, apparently, a discrete convolution (DC) of the intensity map z^* with a dirty beam a, i.e., we have a straight deconvolution problem, not a Fourier deconvolution problem.

It is convenient to put . Then . Each term of the dirty map b is the real part of one (weighted) term of a DFT of the CVs. But the DM as a whole is the real part of such DFT only if the CVs are given on a rectangular frequency grid (and complemented by zeros where missing). The same holds for the DB. In this case, a and b would, except for the weighting, just be the DB and DM of the usual procedure. Of course, the pixels of the image z would have to be defined on a corresponding rectangular grid; in particular, the pixel number has to equal the number of (complemented) CVs, and each Fourier term has to have equal weight. To the contrary, our DM and DB, i.e., b and a, can be considered as a (weighted) direct Fourier transform (see Sramek & Schwab 1986) of the CVs and the corresponding unit numbers using both signs of the frequencies. Note that, inspite of its name, this transformation is not a Fourier transform in the usual sense. Unless certain conditions are met, not even a unique back transformation is possible.

We can thus freely choose a common pixel size for CM, DM, and DB, as well as the number of pixels. Of course, a reasonable pixel size is approximately given by the special problem under consideration. The central beam should not be undersampled while there is little advantage in oversampling it.

The size of the image need not be larger than what is needed to contain all expected intensity. The DB should be large enough to allow a full convolution with the CM in the range of the DM. It should be noted that none of the 3 maps is periodic or assumed to be so. Even if the CVs were given on an rectangular grid, there would be no periodicity.

The overall goodness of the fit can be checked by the sum of the squared residual visibilities. We find