Radio interferometry, Fourier transforms, and the guts of radio interferometry (part 2)

Today’s post is also from Dr Enno Middelberg and is the second part of two explaining in more detail about radio interferometry and the techniques used in producing the radio images in Radio Galaxy Zoo.

In a previous post I have explained how the similarity of the electric field at two antennas’ locations is related to the Fourier transform of the sky brightness. Unfortunately, we’re not quite there (yet). You may have heard about sine and cosine functions and know that they are one-dimensional. Images, and the sky brightness distribution, however, are two-dimensional. So how can we imagine a two-dimensional Fourier transform? In this case, we have to combine 2D waves with various frequencies, amplitudes, and orientations into one image. We can make a comparison with waves on a lake. Just like a sine or cosine wave, a water wave has an amplitude and a frequency, but in addition it also has an orientation, or a direction in which it travels. Now let us think of a few people sitting around a pond or lake. Everyone kneels down to generate waves which then propagate through the water. Let us further assume that the waves are not curved, but that the crests and valleys are parallel lines. Now all these waves, with properly chosen frequencies, amplitudes, and directions will propagate into the center of the pond, where the waves interfere. With just the right parameters, the interference pattern can be made to look like a 2D image. In a radio interferometer, every two telescopes make a measurement which represents the properties of such a wave, and all waves combined then can be turned into an image. Let me point out that the analogy with the lake is taking things a little bit too far: since the water waves keep moving across the lake, a potential image formed by their intererence will disappear quite quickly, but I hope you get the point about interfering 2D waves.

To illustrate this further I have made a little movie. Let us assume that the radio sky looks just like Jean Baptiste Joseph Fourier (top left panel in the movie). I have taken this image from Wikipedia, cropped it to 128×128 pixels, and calculated its Fourier transform. The Fourier transform is an image with the same dimensions, but the pixels indicate the amplitude, phase and frequency of 2D waves which, when combined, result in an image. Then I have taken an increasing number of pixels from this Fourier transform (which ones is indicated at the top right), calculated which 2D waves they represent (bottom right), and incrementally added them into an image (bottom left). At the beginning of the movie, when only few Fourier transform pixels are used, the reconstructed Mr. Fourier is barely recognizable, with 50 Fourier pixels added, one begins to identify a person, and with an increasing number of waves added, the image more and more resembles the input image. You should play it frame by frame, in particular at the beginning, when the changes in the reconstructed image are large. In radio interferometry, Mr. Fourier’s image is what we want (how does the sky look like?), but what we get is only the pixels shown in the upper right image. Each of these pixels, all by itself, provides information as illustrated in the bottom  right, but all together, they yield an image such as in the bottom left image. And the more pixels we measure, the more accurate the image becomes.

So in summary: a radio interferometer makes measurements of the similarity of the electric field at two locations, and the degree of similarity represents the Fourier transform of the sky radio brightness for the two antennas in that instant. Astronomers then reconstruct the sky brightness from all these measurements taken together – that’s also why the technique is called “synthesis imaging”, or “aperture synthesis”. And if you’ve kept reading until here without having your brain turn to mush – congratulations! This is typically the subject of lectures for advanced physics students. I’ve been learning about radio interferometry now for more than 15 years and am still discovering new and interesting bits.