It is hard to understand what this article actually advocates. All the images have the filters aligned with the input pixels, with no grid of output pixels overlaid, as one would need to actually scale or rotate the image. It seems like the article actually suggests that we should use a filter in order to display an unscaled image on screen, even adding a border of filter garbage from outside the confines of the original image.

We are told that the square filter depicted is bad, no actual reason is given.

If we actually add in some scaling and rotation then it is really bad, but then again if we scale it down really far, then so are all the other filters, horrible aliasing all the way round.

Conceptually a filter operation can be thought of as two steps, first we apply an input filter to produce an intermediate infinite resolution version of the input image, then for each output pixel we use an output filter to sample the intermediate version and produce a simple colour value. In practice of course there is no intermediate image, the input and output filters are combined to a single formula that deals with a finite amount of data.

The reason that this model is not often mentioned is that the output filter is commonly just sampling a single point, thus the combined filter and the input filter becomes the same thing. This is often a very poor choice, leading to uneven sampling distribution and the aforementioned aliasing. It is possible to mostly avoid these issues by picking the exact properties of the input filter to match the desired level of scaling, but that is not something I have generally seen applied outside of one specific context.

That context is trilinear filtering used in 3D graphics. This input filter produce an intermediate that is exactly as blurry as it needs to be to avoid the aliasing that results from heavy downscaling. It is still visibly not perfect, and therefore we also use an output filter called an anisotropic filter. Rather that a single point sample it picks multiple point samples in a circle, typically 16. I think there is an argument to be made that an integral over the pixel-shaped square would produce slightly better results. But an exact integral is really expensive to compute, and the circle complements the trilinear filtering better than a 4x4 grid of samples, leading to a more even sampling of the input in cases where a texture is viewed at a sharp angle.

So modern filtering in 3D games don't use any of the fancy filters you see in papers like this. Not because a modern video card couldn't, but because they don't solve the problems video games care about, like aliasing.

For offline filters, like the ones you would apply in an image processing program, I do think squares have some merit. But we have to think of it as two filter steps. One approach would be to simply render each input pixel as a square in the intermediate image, and then for each output pixel sample a square from this image. The input and the output squares are possibly a different size, and oriented differently, but with a bit of maths we can still compute the exact result in finite time. The resulting image is decent, with each input pixel contributing the same amount to the output image, but the blurriness is possibly uneven, which in some cases can look jarring.

An alternative is to use a bilinear input filter, while retaining sampling a square for the output. This is a bit blurrier, but the blur is much more even.

Okay, but what if we use a circular output filter? Or also use bilinear for output, throw in some bicubic, or sinc or some other thing? The real world result is that you are now staring at a bunch of similar images trying to deduce which one looks best, and they are all kind of the same. The only markable difference is that some of them are a bit blurrier, and the others artefact a bit more, and ultimately that tradeoff is the main concern when choosing filters.