Wikipedia:Reference desk/Archives/Mathematics/2022 October 5

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October 5

Are square pixel widths 0.5*2^(minus 0.5) times eye resolution enough?

I can see how <0.5 can help in some cases cause Nyquist but is there really less information in the diagonal direction? Sagittarian Milky Way (talk) 05:27, 5 October 2022 (UTC)

The two questions are not entirely clear. Enough for what? Generally, smaller pixels (so that the representation of an object takes up more pixels) give a better image quality and may make previously illegible text legible – smaller is better. But if one needs to see whether a 3-pixel black shape on an otherwise white field is a ▙, ▟, ▜ , or ▛, one needs pixels whose angular diameter is not too small – larger is better.

How do you want to characterize the amount of information? The fovea centralis of a normal human eye is isotropic. From an information-theoretic point of view, two multi-pixel shapes of the same size in a field whose individual pixels are iid carry the same amount of information; their orientation is irrelevant.  --Lambiam 08:08, 5 October 2022 (UTC)

No the content stays the same size in millimeters and the naked eye-visible information increases the smaller the pixels are then stops increasing. Are those black and white "zebra patterns" fading into gray or weird pie centers the state-of-the-art for this? Where they have skinny 000000-colored rectangles or pie slices and skinny FFFFFF-colored ones alternating. So can there be any benefit for square pixels of side length 2*sqrt2 times smaller than Snellen chart resolution instead of just 2 times smaller? While an infinitely thin line through the diagonal of square pixels would have only 0.707 times as many pixels per inch as an orthogonal line you can also have diagonal lines of alternating colors only 0.707 pixel widths apart. So let's say 20/20 eyeballs are looking (60 seconds of arc pixels, I can read 20/15 lines very quickly but anyway). I'm wondering if there's anything you could display on a monitor that looks better at 1 arcsecond pixels than 29 arcseconds. And would this 20/20 eyeball need 21 arcseconds or maybe there's some orientation of un/badly-antialiased line pairs or some other image where 29 arcseconds is not enough but 21.22 is overkill. Is 25 enough? 27? 23? 28? 26? Sagittarian Milky Way (talk) 16:44, 5 October 2022 (UTC)

The question involves converting analog to digital, which is its own subfield. I recommend using existing standards rather than trying to roll your own. For example the Library of Congress standard for digitized print is 400 ppi, but it would depend on what type of image you're talking about. With images having fine regular patterns, such as newspaper photos, you may run into issues with aliasing, and you'd need to either use much higher resolution than the eye or some kind of anti-aliasing filter. People have been dealing with this issue for decades and have discovered most of the unexpected complication complication that arise, so it's probably not an area where you should reinvent the wheel. --RDBury (talk) 13:25, 5 October 2022 (UTC)

Is “0.5*2^(minus 0.5)” a roundabout say of saying 2**(-3/2) = 1/√8, or something else? —Tamfang (talk) 00:17, 13 October 2022 (UTC)

Some thoughts about balls

Suppose we have a box with one red ball and one blue ball. Then we draw a ball by chance. If we draw a red ball, we add two new red balls. If we draw a blue ball, we add a red and a blue ball. How does the system evolve? 2A02:908:424:9D60:142:5C74:A95B:A7D1 (talk) 16:36, 5 October 2022 (UTC)

We have two random sequence $R_{i}$ and $B_{i}$, representing the number of red balls and blue balls at time $i$. At time $0$ they both have the one-point distribution given by $R_{0}=B_{0}=1.$ They are not independent; together they satisfy the invariant $R_{i}+B_{i}=i+2.$ Moreover, at each round, either we do not draw a blue ball and then we do not add a blue ball, or we draw a blue ball and then we add a blue ball. So the number of blue balls does not change: $B_{i}=1$ for all $i.$ Therefore $R_{i}=i+1.$  --Lambiam 17:25, 5 October 2022 (UTC)

One can simplify the drawing process description: no matter what ball we draw, we add a red one to it and put them both back into the box. To simplify things even further: we do not draw anything but just put the additional red ball into the box on each turn. :-) --CiaPan (talk) 20:38, 12 October 2022 (UTC)