Rectangular Grid Representations

If our goal is to use a uniform grid as support for the visibilities (that is, each time bin corresponds to a pixel of the gridded UV plane), we have to choose a fine enough grid that all points are independent. The innermost UV circle must therefore be large enough that no bins overlap, for if they do, there is no one-to-one correspondence between the UV map and the time bins, and there is a poor representation of the UV circle. (This is the greatest weakness of the rectangular representation.) It is easy to calculate that for 25 time bins (column 3 in Table 1), the finest grid that has no overlapping UV points has a radius of 9 pixels for the innermost UV circle. The UV coverage is shown in Figure 1, and a zoomed version shows the innermost circles. The radius of the largest UV circle is 81 times the radius of the smallest circle, So for the binnings of column 3, the radius of the largest UV circles would have to be 729 pixels, respectively. If the visibilities are to be put into square arrays suitable for IDL's FFT, the arrays would have to be big enough to contain the largest UV circle. Using the next power of 2 exceeding $2 \times 729$, the arrays would be $2048 \times 2048$.

Clearly, these arrays would be very sparse, and careful management of memory would be important if we used this representation. It is probably not efficient to use the standard 2D FFT for such arrays. (In IDL, the FFT of a $2048 \times 2048$ complex array is computationally equivalent to about $3 \times 10^8$ floating point multiplications.) Another problem with this representation is that the inner circles are not rendered very faithfully by nearest-neighbor sampling of circles, and some kind of interpolation might be necessary.

Example of visibilities and FFT Map for a point source at (600",200")