[dahart]

There’s a picture of the exact operation in the article. Under “Filters”, the first row of 3 pictures has the caption “Box Filter”. The one on the right (with internal caption “Contribution (product of both)”) demonstrates the analytic box filter. The analytic box filter is computed by taking the intersection of the pixel boundary with all visible polygons that touch the pixel, and then summing the resulting colors weighted by their area. Note the polygon fragments also have to be non-overlapping, so if there are overlapping polygons, the hidden parts need to be first trimmed away using boolean clipping operations. This can all be fairly expensive to compute, depending on how many overlapping polygons touch the pixel.

[rsp1984]

OK, so reading a bit further this boils down to clipping the polygon to the pixel and then using the shoelace formula for finding the area? Why call it "box filter" then?

[dahart]

It’s very useful to point out that it’s a Box Filter because the article moves on to using other filters, and larger clipping regions than a single pixel. This is framing the operation in known signal processing terminology, because that’s what you need to do in order to fully understand very high quality rendering.

Dig a little further into the “bilinear filter” and “bicubic filter” that follow the box filter discussion. They are more interesting than the box filter because the contribution of a clipped polygon is not constant across the polygon fragment, unlike the box filter which is constant across each fragment. Integrating non-constant contribution is where Green’s Theorem comes in.

It’s also conceptually useful to understand the equivalence between box filtering with analytic computation and box filtering with multi-sample point sampling. It is the same mathematical convolution in both cases, but it expressed very very differently depending on how you sample & integrate.

[rsp1984]

OK, I get it now. Thanks for the explanation. I would just never in a lifetime call it a "filter". That's extremely poor naming.

If they called it a choice of basis or influence function, it would've been so much clearer.

[dahart]

Oh interesting, I hadn’t thought about it, but why does it seem like poor naming? I believe “filter” is totally standard in signal processing and has been for a long time, and that term does make sense to me in this case because what we’re trying to do is low-pass filter the signal, really. The filtering is achieve through convolution, and to your point I think there might be cases in which referring to the filter function as a basis function occurs.

I would think that conceptually that a basis function is different form a filter function because a basis function is usually about transforming a point in one space to some different space, and basis functions come in a set that’s the size of the dimensionality of the target space. Filters, even if you can think of the function as a sort of basis, aren’t meant for changing spaces or encoding & decoding against a different basis than the signal. Filters transform the signal but keep it in the same space it started from, and the filter is singular and might lose data.

May be better if I just link to what others say about filters than me trying to blabber on https://en.wikipedia.org/wiki/Filter_(signal_processing)