Exact Polygonal Filtering: Using Green's Theorem and Clipping for Anti-Aliasing

This is about taking an analytic sample of the scene with an expression that includes and accounts for the choice of filter, instead of integrating some number of point samples of the scene within a pixel.

In this case, the filtering and the sampling of the scene are both wrapped into the operation of intersection of the square with polygons. The filtering and the sampling are happening during rasterization, not before or after.

Keep in mind a pixel is an image sample, which is different from taking one or many point-samples of the scene in order to compute the pixel color.

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?

You might be making some incorrect assumptions about what this article is describing. It’s not limited to a single polygon against a background.

Analytic integration is always superior to multisampling, assuming the same choice of filter, and as long as the analytic integration is correct. Your comment is making an assumption that the analytic integration is incorrect in the presence of multiple polygons. This isn’t true though, the article is using multiple polygons, though the demo is limited in multiple ways for simplicity, it doesn’t appear to handle any arbitrary situation.

The limitations of the demo (whether it handles overlapping polygons, stitched meshes, textures, etc.) does not have any bearing on the conceptual point that computing the pixel analytically is better than taking multiple point samples. GPUs use multisampling because it’s easy and finite to compute, not because it’s higher quality. Multisampling is lower quality than analytic, but it’s far, far easier to productize, and it’s good enough for most things (especially games).

I think the story is a lot more complicated. Talking about "the best possible output quality" is a big claim, and I have no reason to believe it can be achieved by mathematically simple techniques (ie linear convolution with a kernel). Quality is ultimately a function of human perception, which is complex and poorly understood, and optimizing for that is similarly not going to be easy.

The Mitchell-Netravali paper[1] correctly describes sampling as a tradeoff space. If you optimize for frequency response (brick wall rejection of aliasing) the impulse response is sinc and you get a lot of ringing. If you optimize for total rejection of aliasing while maintaining positive support, you get something that looks like a Gaussian impulse response, which is very smooth but blurry. And if you optimize for small spatial support and lack of ringing, you get a box filter, which lets some aliasing through.

Which is best, I think, depends on what you're filtering. For natural scenes, you can make an argument that the oblique projection approach of Rocha et al[2] is the optimal point in the tradeoff space. I tried it on text, though, and there were noticeable ringing artifacts; box filtering is definitely better quality to my eyes.

I like to think about antialiasing specific test images. The Siemens star is very sensitive in showing aliasing, but it also makes sense to look at a half-plane and a thin line, as they're more accurate models of real 2D scenes that people care about. It's hard to imagine doing better than a box filter for a half-plane; either you get ringing (which has the additional negative impact of clipping when the half-planes are at the gamut boundary of the display; not something you have to worry about with natural images) or blurriness. In particular, a tent filter is going to be softer but your eye won't pick up the reduction in aliasing, though it is certainly present in the frequency domain.

A thin line is a different story. With a box filter, you get basically a non antialiased line of single pixel thickness, just less alpha, and it's clearly possible to do better; a tent filter is going to look better.

But a thin line is just a linear combination of two half-planes. So if you accept that a box filter is better visual quality than a tent filter for a half-plane, and the other way around for a thin line, then the conclusion is that linear filtering is not the correct path to truly highest quality.

With the exception of thin lines, for most 2D scenes a box filter with antialiasing done in the correct color space is very close to the best quality - maybe the midwit meme applies, and it does make sense to model a pixel as a little square in that case. But I am interested in the question of how to truly achieve the best quality, and I don't think we really know the answer yet.

[1] https://www.cs.utexas.edu/~fussell/courses/cs384g-fall2013/l...

[2] https://www.inf.ufrgs.br/~eslgastal/SBS3/Rocha_Oliveira_Gast...

Arvo’s work is also using Green’s theorem, in much the same way this article is. Integrating the phong-exponent light reflections is a little bit insane though (and btw I’ve seen Arvo’s code for it.)

The problem you run into with non-constant polygon colors is that you’d have to integrate the product of two different functions here - the polygon color and the filter function. For anything real-world, this is almost certainly going to result in an expression that is not analytically integrable.