# How many printed dots are needed to represent a pixel with an inkjet printer?

Context

I have a digital picture with a size of x*y pixels. I'm trying to figure the maximum print size I could produce and still have an image which "look good" (relatively to human perception at a given distance).

So far, I have been using basic math and the notion of the human eye resolution to think about it.

For a screen, it's relatively straightforward :

1. Say that 1 pixel of my picture = 1 pixel on the screen (=3 sub-pixels RGB)
2. Compute physical distance between pixels on the screen
3. Compute the perceived angle function of the distance between screen and viewer
4. Conclusion : if the perceived angle is superior to the eye resolution, the eye can see two distinct pixels and the image doesn't look good. I agree that this criteria of "good looking" is purely mathematic and can be discussed at length. It's all I have got so far.

When it comes to printing, it gets more complicated, as an undefined number of dots are needed for one pixel. Moreover, complex algorithms are in play (halftone and arguably dithering, even if I tend to ignore the last for now).

Question

Is there a way to compute the number of ink dots needed to represent a pixel given the specs of the printer (brand+model) ? Is it a manufacturer "secret" ? Someone spoke of a native dpi specific to each printer (300, 600). In this context, the term dpi is confusing as this native dpi is the number of pixels - translated into ink dots - that the printer can represent on an inch of paper. Anyway, how is this conversion done ?

For example, the specs of my printer are 9600*2400 dpi (one dot = one drop of an unique ink). How can I translate 2400 dpi into something meaningful for my problem ? If the native dpi is 300, does it mean 1 pixel = 8*8 dots (2400/300=8) ?

## A simplified version.

There is a unit for offset commercial printing called lpi. This is defined by the real resolution of a laser printing on a negative film or plate.

Let us say the laser plate printer gives you 2400 dpi. If you need 256 tones of gray then you need a 16x16 square.

(this next image is only using an 8x8 square to simplify)

So the linear resolution is given by 2400/16=150; 150 lpi. On a 2400 dpi printer you can have 256 levels of gray resulting in 150 lpi.

The relation of ppi to dpi is given traditionally by this conversion: ppi = 2*lpi (to have good interpolation). So 150 lpi is like 300 ppi.

## Ink jet printers work differently.

The dots are random, the minimum size of this ones determines only whether you see the ink dot or not.

But what determines the file resolution is the real pixels you can actually see.

As this is a Photo forum think of the printer dot size as the grain of a photo, and the resolution as the sharpness of a lens. The 2 are not that related in modern inkjet printers.

This randomness is not given pixel by pixel, but... random. Each pixel can have a different number of ink dots, so the dots do not directly define the pixel size you can see.

Yes you can assume that you need a similar number of dots to form some shades of color. But using 300 ppi is a waste on a digital printer. Go for 200ppi or 150ppi, regardless of the printer model.

It is more important for the paper to have a small dot.

## So, what data really matter?

Contrary to popular belief you do not need a 300 ppi image for print on a digital media. 200 ppi is enough. It has more information than a normal magazine that only holds 150 ppi.

100ppi you can hardly see pixels on a print at, say, 50cm. So as a backup you use 200ppi.

If you double that viewing distance you need only half the ppi.

So here is a diagram (It is old and has an error: It should say ppi and not dpi)

This resolution can be adjusted by 2 factors. The real level of detail you need and other elements like text on the image.

You need more detail on an art gallery than on a billboard on a bus stop. And the text on the image, because a low resolution print with text is more noticeable than on a photo.

This graph is labeled mts. on the bottom for viewing distance of the printed material vs. ppi (not dots).

• The graph at the bottom is labeled "mts." Onnthe x axis. What is that? Aug 4, 2015 at 18:19
• Resolution is sharpness of lens: I don't like that, as a lens has bith a resolution and a sharpness, as 10% and 50% MFT. So why isn't sharpness the sharpness of the lens? Aug 4, 2015 at 18:24
• I aded a note about the mts. Thanks for theobservation. - Yes, the print resolution comparation can be resolution of a lens not sharpness, but I want to give just a quick comparation. Yes, a digital image can be high resolution but completly blurry or out of focus. But that is too many details for the example. Aug 4, 2015 at 18:43
• Offtopic. I am not sure resolution and sharpness of a lens are that diferent concepts. Interesting idea. I'll post it later as a question on the forum. Aug 4, 2015 at 18:46
• Viewing distance in some kind of unit? You might edit the image to change the dpi to ppi as well as label the x axis in a meaningful way. Aug 4, 2015 at 18:47

The variability of size and positioning precision of one "spit", as it's called, varies with printing technology and model lines over time.

As I recall, one particular printer I studied (and some generic lore mixed in) you have a certain volume of ink (a few picoliters) ejected with one action from the nozzle. Several such quantum spits can be performed rapidly to build up a larger drop in place. How that covers depends on whether it is dye or pigment, and what kind of solvent is used. In principle you can get several visible sizes/intensities of dot, but just a few. An extended sequence of spitting as the head moves will make an extended line though which does add to the size...but the ink wicks into the paper by about the same magnitude anyway.

Meanwhile, the way different colors mix varies. They can be translucent and/or mix while still wet so you get the mixes you expect. Or, one can opaquely block the other and you have to print different color dots side-by-side. The current tech for pigments are nano-particulate and are more dye-like in not blocking each other so much.

Given a set of particulars (ink properties) you can figure out how many distinct color and tone variations can be produced in a unit square of paper, say 1/300 of an inch on a side. That is rough, since the ink bleeds together and doesn't stay on a neat grid. Lighter shades stay localized, and you can look under a loupe and see for yourself that it is half toned, and you can also see how far apart the little lines are: that gives you a minimum resolution of how much area is needed for that shade, if you care to measure it.

For higher resolution, like 1400 per inch, you don't have many distinct shades so it must dither, using multiple pixels to average out a color. But it tries to keep edges sharp so that does mean something.

Effectively, the nominal stated resolution (like 300) gives most of the gamut in cells that size, with lighter colors still dithering out. But that might be out of date with newer print heads and smaller spits.

• thanks for your explanation about ink printing "engineering", even if it wasn't exactly what I was looking for :) Aug 17, 2015 at 20:42