6 ms·

A Pixel Is Not a Little Square (1995) [pdf]

30 points by justin_ 1y ago

turtleyacht 1y ago
See also: Pixel is a unit of length and area - https://news.ycombinator.com/item?id=43769478 https://news.ycombinator.com/item?id=43769478 - 1 hour ago (11 points, 20 comments)
mkl 1y ago
Never really convinced me; I've done lots of graphics stuff, and I find thinking about pixels as squares works fine. Under magnification, LCD pixels are usually square blocks of rectangular RGB segments (OLED and phone screens can be stranger geometry), and camera sensors are usually made of square (ish) pixel sensor blocks in a Bayer colour array pattern. They're not point sources or point samples, they emit or sense light over an area. Maybe I'm missing something. Lots of past discussions: https://news.ycombinator.com/item?id=35076487 https://news.ycombinator.com/item?id=35076487 74 points, 2 years ago, 69 comments https://news.ycombinator.com/item?id=26950455 https://news.ycombinator.com/item?id=26950455 81 points, 4 years ago,70 comments https://news.ycombinator.com/item?id=20535984 https://news.ycombinator.com/item?id=20535984 143 points, 6 years ago, 79 comments https://news.ycombinator.com/item?id=8614159 https://news.ycombinator.com/item?id=8614159 118 points, 10 years ago, 64 comments https://news.ycombinator.com/item?id=1472175 https://news.ycombinator.com/item?id=1472175 46 points, 15 years ago, 20 comments
codeflo 1y ago
This classic article is wrong, BTW, there's no nicer way to put it. It applies the wrong theory. It was already wrong in 1995 when monitors where CRTs, and it's way wrong in 2025 in the LCD/OLED era where pixels are truly discrete. Audio samples are point samples (usually). This is nice, because there's a whole theory on how to upsample point samples without loss of information. But more importantly, this theory works because it matches how your playback hardware functions (for both analog and digital reasons that I won't go into). Pixels, however, are actually displayed by the hardware as little physical rectangles. Take a magnifying glass and check. Treating them as points is a bad approximation that can only result in unnecessarily blurry images. I have no idea why this article is quoted so often. Maybe "everybody is doing it wrong" is just a popular article genre. Maybe not everyone is familiar enough with sampling theory to know exactly why it works in audio (to see why those reasons don't apply to graphics).
- p0w3n3d 1y ago
  I think you're wrong. According to my knowledge the pixels on CRT were rectangular and were throwing their color on neighbouring pixels. Graphics created for CRT were shown nicely on those screens, had much better visuals than displayed on LCD/LED and were antialiases by default (i.e. by the display technology)
  - badmintonbaseba 1y ago
    Pixels displayed on CRT displays are not squares, but they are not infinitely small dots either. They are much less well-defined blobs, that even overlap with each other. There is also the complication of composite video signals, where you can't treat pixels as linearly independent components.
    - codeflo 1y ago
      Good PC monitors (especially in 1995) displayed pixels as almost perfect discrete squares. It was home consoles on televisions where people get the idea from that all CRTs were blurry.
      - p0w3n3d 1y ago
        Let me guess: when game creators were thinking about result visuals were they considering that everyone would have this good pc monitor with exact square pixel, or were they taking into account possible distortions that would occur on average CRT? Also: people playing retro nowadays use shaders to emulate CRT https://youtube.com/shorts/W_ZI3w9CYnI https://youtube.com/shorts/W_ZI3w9CYnI
        codeflo 1y ago
        As I said, >> It was home consoles on televisions where people get the idea from that all CRTs were blurry. The video you linked shows a PS1 game, which proves my point. It's possible you're too young to remember the big difference between a CRT TV and a CRT monitor. Monitors really did show discrete pixels (which was important for tiny text in applications to be readable), while TVs were blurry messes.
captainmuon 1y ago
Except pixels are little squares. Sure, if you look under a microscope, they have funny shapes, but they are always laid out in a rectangular grid. I've never seen any system where the logical pixels are staggered like a hex grid, for example. No matter how the actual light emitters are arranged, the abstraction offered to the programmer is a rectangular grid. If you light up pixels in a row, you get a line - a long thin rectangle - and not a chain of blobs. If you light them up diagnoally, you get a jagged line. For me that is proof that they squares - at least close enough to squares. Heck even on old displays that don't have a square pixel ratio they are squished squares ;-). And you have to treat them like little squares if you want to understand antialiasing, or why you sometimes have to add (0.5, 0.5) to get sharp lines. (And a counterpoint: The signal-theoretical view that they are point samples is useful if you want to understand the role of gamma in anti-aliasing, or if you want to do things like superresolution with RGB-sub-pixels.)
- mkl 1y ago
  There are some screen types with variations on the geometry, like some sub-pixels shared between logical pixels. E.g. Samsung's diamond pixel https://global.samsungdisplay.com/29043/ https://global.samsungdisplay.com/29043/, Apple watch https://imgur.com/GkKjjwy https://imgur.com/GkKjjwy. They are still programmed as squares, but the light isn't emitted exactly like that (still coming from discrete areas, not points). See also https://www.reddit.com/r/apple/comments/9fp1ty/did_you_ever_looked_at_apple_screens_under_a/ https://www.reddit.com/r/apple/comments/9fp1ty/did_you_ever_....
- ChrisMarshallNY 1y ago
  I don't remember the manufacturer (may have been Fuji[0]), but someone made a camera sensor that was laid out around a 45-degree angle. [0] https://en.wikipedia.org/wiki/Super_CCD https://en.wikipedia.org/wiki/Super_CCD
  - IAmBroom 1y ago
    There's also a hex-pattern camera sensor out there. It claimed to have better resolution without increased chip-printing cost (N pixels/area produced better effective visual resolution), but never took off.
roflmaostc 1y ago
Mathematically speaking the paper is correct. I think it actually depends what you define as "pixel". Sure, the pixel on your screen emits light on a tiny square into space. And sure, a sensor pixel measures the intensity on a tiny square. But let's say I calculate something like: # samples from 0, 0.1, ..., 1 x = range(0, 1, 11) # evaluate the sin function at each point y = sin.(x) Then each pixel (or entry in the array) is not a tiny square. It represents the value of sin at this specific location. A real pixelated detector would have integrated sin from `y[u] = int_{u}^{u + 0.1} sin(x) dx` which is entirely different from the point wise evaluation before. So for me that's the main difference to understand.
gitroom 1y ago
i think i've argued with friends over this exact thing - like, once you zoom in, does it even matter what shape the pixel is or is it just about how we use it? you think treating pixels as points or little squares actually changes decisions when making art or code
mordae 1y ago
There's pixels and pixels. Screen pixels are (nowadays) usually three vertical rectangles that occupy a square spot on the grid that forms the screen. This is sometimes exploited for sub-pixel font smoothing purposes. Digital photography pixels are reconstructed from sensors that perceive cone of incoming light of certain frequency band, arranged in a Bayer grid. Rendered 3D scene pixels are point samples unless they approximate cones via sampling neighborhood of the pixel center. In any case, Nyquist will tear your head off and spit into your neck hole as soon as you come close to any kind of pixel. Square or point.
GrantMoyer 1y ago
People get caught up on display technology, but how pixels are displayed on a screen is irrelevant. From a typical viewing distance and with imperfect human lenses, a point impulse and a little square are barely distingishable. The important part is that thinking about pixels as little squares instead of points makes all the math you do with them harder for no benefit. Consider the Direct3D rasterization rules[1], which offset each sample point by 0.5 on each axis to sample "at the pixel center". Why are the "pixel centers" even at half-integer coordinates in the first place? Because if thinking of pixels as little squares, it's tempting to align the "corners" with integer coordinates like graph paper. If instead the specifiers had thought of pixels as lattice of sample points, it would have been natural to align the sample points with integer coordinates. "Little square" pixels resulted in an unneeded complication to sampling, an extra translation by a fractional distance, so now every use of the API for pixel perfect rendering must apply the inverse transform. [1]: https://learn.microsoft.com/en-us/windows/win32/direct3d11/d3d10-graphics-programming-guide-rasterizer-stage-rules https://learn.microsoft.com/en-us/windows/win32/direct3d11/d...
- leguminous 1y ago
  The half pixel offset makes sense, though. If you have two textures, you want the edges to align, not the centers of the pixels. See, for example: https://bartwronski.com/2021/02/15/bilinear-down-upsampling-pixel-grids-and-that-half-pixel-offset/ https://bartwronski.com/2021/02/15/bilinear-down-upsampling-... Implementations of resizing based on aligning pixel centers resulted in slight shifts, which caused a lot of trouble.
- IAmBroom 1y ago
  > People get caught up on display technology, but how pixels are displayed on a screen is irrelevant. To a user, usually. To a home entertainment customer, never (even if they wouldn't really notice!). To an optical engineer like myself, never true.
gomijacogeo 1y ago
The paper would be a lot less infamous if the title had more accurately been "A Texel is Not a Little Square".

30 points by justin_ 1y ago

See also: Pixel is a unit of length and area - https://news.ycombinator.com/item?id=43769478 https://news.ycombinator.com/item?id=43769478 - 1 hour ago (11 points, 20 comments)

mkl 1y ago

Never really convinced me; I've done lots of graphics stuff, and I find thinking about pixels as squares works fine. Under magnification, LCD pixels are usually square blocks of rectangular RGB segments (OLED and phone screens can be stranger geometry), and camera sensors are usually made of square (ish) pixel sensor blocks in a Bayer colour array pattern. They're not point sources or point samples, they emit or sense light over an area. Maybe I'm missing something. Lots of past discussions: https://news.ycombinator.com/item?id=35076487 https://news.ycombinator.com/item?id=35076487 74 points, 2 years ago, 69 comments https://news.ycombinator.com/item?id=26950455 https://news.ycombinator.com/item?id=26950455 81 points, 4 years ago,70 comments https://news.ycombinator.com/item?id=20535984 https://news.ycombinator.com/item?id=20535984 143 points, 6 years ago, 79 comments https://news.ycombinator.com/item?id=8614159 https://news.ycombinator.com/item?id=8614159 118 points, 10 years ago, 64 comments https://news.ycombinator.com/item?id=1472175 https://news.ycombinator.com/item?id=1472175 46 points, 15 years ago, 20 comments

codeflo 1y ago

This classic article is wrong, BTW, there's no nicer way to put it. It applies the wrong theory. It was already wrong in 1995 when monitors where CRTs, and it's way wrong in 2025 in the LCD/OLED era where pixels are truly discrete. Audio samples are point samples (usually). This is nice, because there's a whole theory on how to upsample point samples without loss of information. But more importantly, this theory works because it matches how your playback hardware functions (for both analog and digital reasons that I won't go into). Pixels, however, are actually displayed by the hardware as little physical rectangles. Take a magnifying glass and check. Treating them as points is a bad approximation that can only result in unnecessarily blurry images. I have no idea why this article is quoted so often. Maybe "everybody is doing it wrong" is just a popular article genre. Maybe not everyone is familiar enough with sampling theory to know exactly why it works in audio (to see why those reasons don't apply to graphics).

p0w3n3d 1y ago
I think you're wrong. According to my knowledge the pixels on CRT were rectangular and were throwing their color on neighbouring pixels. Graphics created for CRT were shown nicely on those screens, had much better visuals than displayed on LCD/LED and were antialiases by default (i.e. by the display technology)
- badmintonbaseba 1y ago
  Pixels displayed on CRT displays are not squares, but they are not infinitely small dots either. They are much less well-defined blobs, that even overlap with each other. There is also the complication of composite video signals, where you can't treat pixels as linearly independent components.
  - codeflo 1y ago
    Good PC monitors (especially in 1995) displayed pixels as almost perfect discrete squares. It was home consoles on televisions where people get the idea from that all CRTs were blurry.
    - p0w3n3d 1y ago
      Let me guess: when game creators were thinking about result visuals were they considering that everyone would have this good pc monitor with exact square pixel, or were they taking into account possible distortions that would occur on average CRT? Also: people playing retro nowadays use shaders to emulate CRT https://youtube.com/shorts/W_ZI3w9CYnI https://youtube.com/shorts/W_ZI3w9CYnI
      - codeflo 1y ago
        As I said, >> It was home consoles on televisions where people get the idea from that all CRTs were blurry. The video you linked shows a PS1 game, which proves my point. It's possible you're too young to remember the big difference between a CRT TV and a CRT monitor. Monitors really did show discrete pixels (which was important for tiny text in applications to be readable), while TVs were blurry messes.

captainmuon 1y ago

Except pixels are little squares. Sure, if you look under a microscope, they have funny shapes, but they are always laid out in a rectangular grid. I've never seen any system where the logical pixels are staggered like a hex grid, for example. No matter how the actual light emitters are arranged, the abstraction offered to the programmer is a rectangular grid. If you light up pixels in a row, you get a line - a long thin rectangle - and not a chain of blobs. If you light them up diagnoally, you get a jagged line. For me that is proof that they squares - at least close enough to squares. Heck even on old displays that don't have a square pixel ratio they are squished squares ;-). And you have to treat them like little squares if you want to understand antialiasing, or why you sometimes have to add (0.5, 0.5) to get sharp lines. (And a counterpoint: The signal-theoretical view that they are point samples is useful if you want to understand the role of gamma in anti-aliasing, or if you want to do things like superresolution with RGB-sub-pixels.)

mkl 1y ago
There are some screen types with variations on the geometry, like some sub-pixels shared between logical pixels. E.g. Samsung's diamond pixel https://global.samsungdisplay.com/29043/ https://global.samsungdisplay.com/29043/, Apple watch https://imgur.com/GkKjjwy https://imgur.com/GkKjjwy. They are still programmed as squares, but the light isn't emitted exactly like that (still coming from discrete areas, not points). See also https://www.reddit.com/r/apple/comments/9fp1ty/did_you_ever_looked_at_apple_screens_under_a/ https://www.reddit.com/r/apple/comments/9fp1ty/did_you_ever_....
ChrisMarshallNY 1y ago
I don't remember the manufacturer (may have been Fuji[0]), but someone made a camera sensor that was laid out around a 45-degree angle. [0] https://en.wikipedia.org/wiki/Super_CCD https://en.wikipedia.org/wiki/Super_CCD
- IAmBroom 1y ago
  There's also a hex-pattern camera sensor out there. It claimed to have better resolution without increased chip-printing cost (N pixels/area produced better effective visual resolution), but never took off.

roflmaostc 1y ago

Mathematically speaking the paper is correct. I think it actually depends what you define as "pixel". Sure, the pixel on your screen emits light on a tiny square into space. And sure, a sensor pixel measures the intensity on a tiny square. But let's say I calculate something like: # samples from 0, 0.1, ..., 1 x = range(0, 1, 11) # evaluate the sin function at each point y = sin.(x) Then each pixel (or entry in the array) is not a tiny square. It represents the value of sin at this specific location. A real pixelated detector would have integrated sin from `y[u] = int_{u}^{u + 0.1} sin(x) dx` which is entirely different from the point wise evaluation before. So for me that's the main difference to understand.

gitroom 1y ago

i think i've argued with friends over this exact thing - like, once you zoom in, does it even matter what shape the pixel is or is it just about how we use it? you think treating pixels as points or little squares actually changes decisions when making art or code

mordae 1y ago

There's pixels and pixels. Screen pixels are (nowadays) usually three vertical rectangles that occupy a square spot on the grid that forms the screen. This is sometimes exploited for sub-pixel font smoothing purposes. Digital photography pixels are reconstructed from sensors that perceive cone of incoming light of certain frequency band, arranged in a Bayer grid. Rendered 3D scene pixels are point samples unless they approximate cones via sampling neighborhood of the pixel center. In any case, Nyquist will tear your head off and spit into your neck hole as soon as you come close to any kind of pixel. Square or point.

GrantMoyer 1y ago

People get caught up on display technology, but how pixels are displayed on a screen is irrelevant. From a typical viewing distance and with imperfect human lenses, a point impulse and a little square are barely distingishable. The important part is that thinking about pixels as little squares instead of points makes all the math you do with them harder for no benefit. Consider the Direct3D rasterization rules[1], which offset each sample point by 0.5 on each axis to sample "at the pixel center". Why are the "pixel centers" even at half-integer coordinates in the first place? Because if thinking of pixels as little squares, it's tempting to align the "corners" with integer coordinates like graph paper. If instead the specifiers had thought of pixels as lattice of sample points, it would have been natural to align the sample points with integer coordinates. "Little square" pixels resulted in an unneeded complication to sampling, an extra translation by a fractional distance, so now every use of the API for pixel perfect rendering must apply the inverse transform. [1]: https://learn.microsoft.com/en-us/windows/win32/direct3d11/d3d10-graphics-programming-guide-rasterizer-stage-rules https://learn.microsoft.com/en-us/windows/win32/direct3d11/d...

leguminous 1y ago
The half pixel offset makes sense, though. If you have two textures, you want the edges to align, not the centers of the pixels. See, for example: https://bartwronski.com/2021/02/15/bilinear-down-upsampling-pixel-grids-and-that-half-pixel-offset/ https://bartwronski.com/2021/02/15/bilinear-down-upsampling-... Implementations of resizing based on aligning pixel centers resulted in slight shifts, which caused a lot of trouble.
IAmBroom 1y ago
> People get caught up on display technology, but how pixels are displayed on a screen is irrelevant. To a user, usually. To a home entertainment customer, never (even if they wouldn't really notice!). To an optical engineer like myself, never true.

gomijacogeo 1y ago

The paper would be a lot less infamous if the title had more accurately been "A Texel is Not a Little Square".