Perform convolution SCULIB_WTFN_REGRID_2
This is an image space implementation of fourier techniques. In Fourier terms the technique could be described as follows:-
[1] Do a 2-d discrete Fourier transform of the data points. The result is a repeating pattern made up of copies of the transform of the map as a continuous function, each copied displaced from its neighbours by 1/dx, where dx is the sample spacing of the input points (assumed equal in x and y). Different copies of the ’continuous map’ transforms will overlap (’alias’) if there is any power in the map at frequencies greater than 1/(2dx). It is not possible to unravel aliased spectra and this constraint leads to the Nyquist sampling criterion.
[2] We want to derive the ’continuous map’ so that map values on the new grid mesh can be derived. Do this by multiplying the transform of the data by a function that has zero value beyond a radius of 0.5/dx from the origin. This will get rid of all the repeats in the pattern and leave just the transform of the ’continuous map’.
[3] Do an inverse FT on the remaining transform for the points where you wish the resampled points to be (note: FFTs implicitly assume that the data being transformed DO repeat ad infinitum so we’d have to be careful when using them to do this).
The analogue of these process steps in image space is as follows:
[1] Nothing.
[2] Convolve the input data with the FT of the function used to isolate the ‘continuous map’ transform.
[3] Nothing.
If the method is done properly, the rebinned map is in fact the map on the new sample mesh that has the same FT as the continuous function going through the original sample points.
Convolution Functions:-
[Bessel] For good data and with no time constraint on reduction the best convolution function would be one whose FT is a flat-topped cylinder in frequency space, centred on the origin and of a radius such that frequencies to which the telescope is not sensitive are set to zero. Unfortunately, this function is a Bessel function, which extends to infinity along both axes and has significant power out to a large radius. To work correctly this would require an infinite map and infinite computing time. However, a truncated Bessel function should work well on a large map, except near the edges. Edge effects can be removed by pretending that the map extends further - of course, this only works if you know what data the pretend map area should contain, i.e. zeros. Another problem with a Bessel function arises from the fact that it does truncate the FT of the map sharply. If the data are good then there should be nothing but noise power at the truncation radius and the truncation of the FT should have no serious effect. However, if the data has spikes (power at all frequencies in the FT) or suffers from seeing effects such that the data as measured DO have power beyond the truncation radius, then this will cause ringing in the rebinned map.
[Gaussian] In fact, any function that is finite in frequency space will have infinite extent in image space (I think). As such they will all drag in some power from the aliased versions of the map transform and all suffer from edge effects and large compute times. Some are worse than others, however. For example, a Gaussian can have most of its power concentrated over a much smaller footprint than a Bessel function, so the convolution calculation will be much faster. It is also more robust in the presence of spikes and seeing problems because it does not truncate the map FT as sharply as the Bessel function - such effects give rise to smoother defects in the rebinned map though the defects will still BE there.