With interferometry, you're getting an incomplete sampling of the Fourier transform of the sky image, and if you just invert the samples, you get what we call a "dirty" image.
But you know your sampling of the Fourier plane exactly, since that's just a function of the projected baselines between every pair of telescopes during the observation, so you can create a "dirty beam" - now all you have to do is remove the effects of the dirty beam from the dirty image. Of course, that's a deconvolution problem, and given that you don't have all the information - you sampled it - it can never be exact.
But it can be very good! There are very sophisticated radio synthesis image deconvolution algorithms, including CLEAN and Maximum Entropy. For Maximum Entropy methods, you can apply a Bayesian prior on your images - most of the time, the prior we apply is a blank sky (seriously!) but if we have other constraints that we can use (e.g., the approximate size of the region with extended emission), Bayes tells us that we would be remiss not to use it.
If you look at this image [1] from Paper IV [2], we show the image results from different techniques on different observing days. Those are the inputs to what is the "consensus image" - you can check how close they all are to each other.
Does that make sense...?
[1] https://iopscience-event-horizon.s3.amazonaws.com/2041-8205/... [2] https://iopscience.iop.org/article/10.3847/2041-8213/ab0e85
This ted talk has a very basic explanation of how they constructed this image. I was curious if anyone with image interpolation experience could weigh in on the method. https://www.youtube.com/watch?v=P7n2rYt9wfU
When she first starts explaining their method around 8:00m in, I was initially very skeptical of this result because she said that they feed images of what we "think" a black hole should look like and use algorithms to compare the captured data with those images.
She then goes into explaining the measure they take to keep the resulting image from being biased by passing environmental images and images of other astronomical anomaly to make sure that those images return similar results.
But I can't for the life of me figure out how passing non-stellar imagery could return something similar. And if it does, why do we need to feed it an example of what we think it should look like at all?