I'm a bit confused about how this works in practice — if you were to dense sample and return the average, then take a single sample per pixel of that, wouldn't you end up in the same place after just doing the dense sample?

Here's my understanding but please correct me if I'm wrong. Imagine there's a truth representation of a triangle behind one screen pixel in a discrete format (although it should be continuous because we define the triangle with the three vertices)

"truth" [2, 4, 8, 16, 32] With dense sampling, let's say it samples one for every two values, the result is: dense sample: [2, 8, 32] And with sparse sampling, eg. samples one out of five values, the result is: sparse sample: [8]

If you convert your sparse sample directly into a pixel, the value will be 8. But if you do dense sampling first, and then average the values into one pixel, you get (2 + 8 + 32) / 3 = 14. which is more close the truth representation of that pixel.