cat vision

Sharpness is, really, an illusion. It doesn't represent anything about the world, it's just an indication that the limits of resolution of the visual system have been met. In that sense, it's relative. I've thought about sharpness a lot, starting in graduate school, when I first wondered whether, having learned the quantitative difference in visual spatial resolution between cat and human, a cat could see stars. My first thought was no - stars are so small, if blurred they can't be seen; then I recognized that human acuity is nothing special (a sort of Copernican principle for vision), and that sure, cats should be able to see stars just as humans can. But they would look different, wouldn't they? Blurrier? No. To see blur implies you have the acuity to see what is missing. So then, they would look larger? No, for the same reason - if a star appears as a disk, that implies its edges are seen separately, which implies acuity to separate them. So to a cat, whose acuity is almost an order of magnitude worse than a human, stars must also appear as points. How to make sense of this?

This gets at a more general sort of paradox about visual resolution. Lower acuity isn't the same as blur, not at all. Acuity is an ability or a capacity; blur is a state or an affordance. A certain acuity enables you to see a certain amount of blur - that is the relationship. But we easily confuse the two by trying to represent the effects of acuity as blur. This is a common demonstration: illustrate the spatial resolution of the visual field as round window with a focused center and increasing blur towards its boundaries. This kind of demonstration is useful in that it shows what is lost in peripheral vision (in terms only of 1st-order resolution) relative to central vision. But it is harmful in that it conflates blur with this relative difference in visibility. Because really, no matter what the resolution is, there is finer content that cannot be seen.  We can think of this kind of demonstration, of comparing resolution at different visual field locations in terms of blur, an 'isometric' demonstration, since space is kept constant or symmetric over the whole field, though apparent sharpness falsely appears to change. This demonstration doesn't violate our intuitions about space - space seems, and is, symmetric to translations across the visual field - though it does fool us regarding blur.

Another way of demonstrating the same variation in resolution across the visual field is to reverse this relationship; that is, with an isoambylic representation of the field. This representation would have equal sharpness everywhere, but would vary metrically across the field, giving something like a fish-eye lens view of the scene. For some reason, even though the isoambylic representation is just as 'fair' as the isometric, its distortions are more disturbing. Maybe it's because the spatial asymmetry is unfamiliar, whereas blur asymmetries are more familiar.

So now we go back to cat vision. Imagine that you and a talking, scientifically interested cat, are discussing the topic at hand, and wondering how to explain to one another the differences in your spatial acuity. I think it's time for a dialogue!

Tacitus: So, here we are.
Otho: True.
Tacitus: We're supposed to demonstrate to one another the differences in our visual fields, in terms of spatial acuity. How do you think we might do that?
Otho: Well, for starters, let's use pictures.
Tacitus: That's kind of a given.
Otho: Good. Here are two copies of a scene. The one on the left represents your acuity: you see that in the center, the image is sharper, and it gets blurrier as you go out towards the edges.
Tacitus: I do see that. Nice and sharp in the center, blurrier toward the edges.
Otho: If you stand right here, and look at the center of the picture, you shouldn't be able to tell that there's any blur, because the blur is matched to your acuity. What do you see?
Tacitus: It's just as you say. Interesting!
Otho: Good. Now, this picture, on the right, represents my acuity. It's similar in that in its center, it's sharper, getting blurrier towards the edges.
Tacitus: I see that, but...
Otho: But what?
Tacitus: But it looks just like my picture. I can't see a difference. Maybe.. it's not quite as strong a trend, from the center outward, but I can barely tell.
Otho: Well, the difference is obvious to me. It's because my acuity is so much better than yours, all around.
Tacitus: Well, then this isn't fair. Why can you see so clearly the difference between our visual fields, while I can't see it at all? I feel left out.
Otho: Hm.
Tacitus: Here, let's try this. Instead of using blur to represent acuity, let's change the size of the images. We'll transform the images so that the acuity limit, which is just a measure of distance within the visual field, will be a fixed distance.
Otho: So that means that when acuity is high, the image will be relatively magnified, since you're taking a small distance in the visual field and stretching it to, let's say, one centimeter. And when acuity is low, the image will be compressed, since you're taking a big distance in the field and squeezing it into that same centimeter.
Tacitus: Exactly!
Otho: The images will look funny, though.
Tacitus: Well, the funny-ness will be our explanatory tool. We should both be able to notice changes in size, right? I can see a spot a centimeter across from this distance, and so can you.
Otho: It does seem fair.
Tacitus: Okay, here we go.
Otho: Wow! My visual field is so big! And look at the distortion, it's like a fish-eye lens! Why is your field so small?
Tacitus: Didn't you just explain it to me?
Otho: I know, I was just surprised.
Tacitus: And mine also looks like a fish-eye lens, just a bit less extreme. Yours is interesting, I can see so many details there that I can't see in mine. I didn't realize you could see such small things!
Otho: I'm sure you did, you just haven't realized it in a visual sense.
Tacitus: Well here it is.
Otho: Can I go back to bed now? It's 3am.
Tacitus: Go ahead and try. We'll see how it goes.