Metric-dimensioned resolution

[Stanley]

Hi everyone,

I have been thinking about resolution. As we all know, the benchmark resolution for photographs in publishing is 300 ppi (pixels per inch).

A problem is that things get so darned confusing when we are discussing other matters using metric sizes (centimeters, millimeters, micrometers, what have you) and also referring to resolution using the inch/pound system.

So the thought occurred to me to convert 300 ppi to pixels per centimeter (let’s call it px/cm). That’s easy to do: divide 25.4 into 300, and you get a bit more than 118.1. Well, that’s not even an integer, but even if it were, the number 118 is not convenient to use, so let’s change that to the nearest round number, which is 120.

Thus, 120 px/cm is very close to 300 ppi, just slightly higher resolution. And that’s fine. After all, there’s nothing magical about 300 ppi. At some time in the past it was determined to use the standard for publication to be approximately 300 ppi, so it was decided to use exactly 300 ppi. But if instead the number used were, let’s say 301, or 305, or 298, any such number like that would be indistinguishable from 300.

In the same way, there’s scant difference between 118.1 and 120 (120 px/cm is exactly equivalent to 304.8 ppi).

So there we have it: Using metric notation, the standard resolution for publishing should be 120 px/cm.

But wait – there’s more.

Easy arithmetic shows that 120 px/cm equals 12 px/mm (that is, pixels per millimeter).

And we can go one step further, noting that 12 pixels per millimeter equals 12 pixels per 1000 micrometers. (For those who may not know it, a micrometer is the formal name for a micron, and its symbol is μm. The diameter of a strand of human hair may be as narrow as 10 micrometers, and if so, you could lie 100 such strands next to each other in a millimeter).

OK, let’s do one more division and see that 12 pixels per 1000 micrometers equals 3 pixels per 250 micrometers.

Now, 250 micrometers – in other words, one quarter of a millimeter – is a very special number. I will quote from a magnificently excellent web site devoted to measurement. First, here is the link to that web site:

http://www.unc.edu/~rowlett/units/

This site refers to a Q (just the capital letter), and defines a Q as “a metric unit of distance equal to exactly 0.25 millimeter ... used by typographers and page designers in Japan, in Germany, and in other countries in preference to the traditional point … This unit is also spelled kyu.”

A Q is a most useful measurement. It is very fine, yet sufficiently coarse as to make rough calculations on small objects.

So, let’s recapitulate what we have. Working in the metric system, we have the following equations serving as a benchmark for publishing-quality resolution:

120 px/cm = 12 px/mm = 3 px/Q

Stanley

[lauriek]

I got to the end thinking this sounded good but the final number is too small. 3px per 1/4mm is not 300dpi, you converted to mm instead of cm at the start - 25.4mm = 1" not 25.4cm so I think you end up with 30px/Q? Which sounds more reasonable to me...

[puzzledpaul]

Seem to recollect reading somewhere that 400ppi is used in Japan?

[rjlittlefield]

I think the final number is OK. There's a glitch near the beginning, where it says "divide 25.4 into 300, and you get a bit more than 118.1". Of course the proper calculation is "divide 2.54 into 300".

For some reason, my head is just abuzz with thoughts sparked by this thread. I can't resist sharing some of them...

1. Much as I love the metric system, it does have the minor downside that very different units have names that sound very much alike. You don't often see people writing "feet" when they mean "inches", but it's surprisingly common to see people writing "cm" when they mean "mm". Even though Stanley is obviously passionate about the topic of measurement, we've seen this type of error in several of his postings already. Sure, they're just typos (unless you rely on the result...), but to my eye they undercut the argument of how superior the metric system is.

2. The ppi/dpi nomenclature obviously originated with inch-using people. It's odd, really, that this convention has become so widespread even in metric areas. I just checked with www.canon.de, for example, and was surprised to find that their German-language brochures quote resolutions like "9.600 x 2.400 dpi". Now of course a scientifically trained but geographically naive American would think at first that this is absurdly low resolution, expressed to absurdly high precision (three digits after the decimal separator). But knowing that this is German, we can quickly figure out that what it really means is 9,600 x 2,400 dpi (using the One True Notation of "," for a group separator*). I can't help wondering if the devices are really manufactured to a dpi specification, or if they actually come out to nice even numbers in metric and the dpi specification is just a close translation.

3. The use of ppi/dpi is not universal. AdobePress, for example, notes that

Res. One other method of discussing resolution uses the term res. Res is simply the number of pixels per millimeter, and it's a great deal more common in Europe than in the metrically challenged United States. People usually talk about res when they're discussing capture resolution on drum scanners. For example, a file scanned at res 12 is scanned at 12 sample points (pixels) per millimeter—which is 120 sample points per centimeter, or—in common usage—304.8 sample points per inch (see Table 3-2).

4. Although 300 ppi is one commonly accepted standard for what's needed to look good, there's wide variation in just what that means. At least one prominent writer has done studies showing that 600 ppi is notably better, and if I recall correctly, that was for viewing at 20 inches(!).

5. All of the above thoughts come together in the order form for my favorite large-print supplier:

Printing resolution and optimum files size for Lightjet: 304.8 ppi at output size (res 12). The Lightjet has an excellent hardware interpolator and can produce great prints with files down to 180 ppi at output size.

In fact, when I sent them a test file that had black/white/black/white pixels at 12 pixels per mm, what I got back was not clearly different from uniform gray. Doing the same thing with black/black/white/white etc. produced sharp bars. Obviously the effective resolution is someplace between 12 and 6 pixels per mm, but I'm not sure where. Don't care either, since the prints look great.

6. The unit of Q is unfamiliar to me (except for the various math and engineering uses documented at Wikipedia). Now that I've heard of it, I'm appalled. Typographical "points" are 1/72 inch (see the Wikipedia article, a fascinating read). This Q thingie not only mixes its paradigms (a metric unit, divided by 4), but manages to be different by a factor of only about 4/3 from the thing that it's used in preference to. Good grief!

Whew -- glad I got that off my chest. Now I can go back to something productive, like lunch.

--Rik

* Footnote: Yes, the comment about One True Notation for group separators is tongue-in-cheek. See http://en.wikipedia.org/wiki/Decimal_separator for a map showing how evenly split the world is on this usage.

[Stanley]

Hi Laurie,

Thank you for your response.

Sorry, I don't see where you get that. But I want to make sure that I don't make a mistake.

We agree with the following, right?

120 px/cm = 12 px/mm (I am dividing by 10)

So next we can write this:

120 px/cm = 12 px/mm = 12 px/1000 μm (since 1 mm = 1000 μm)

Next I will divide the expression on the right by 4, as such:

12 px/mm = 3 px/250 μm

By definition, however, 250 μm = 1 Q.

And thus we have,

120 px/cm = 12 px/mm = 3 px/Q

Isn't this correct?

Stanley

[ChrisR]

Isn't this correct?

Yes it is, but not one I'd care to remember!
Don't we have enough units already?

Pipe comes, for instance, in 1/2 inch and 3/4" sizes, measured internally, and adapts to 15mm, and 22 "mil", copper, which is called tube, not pipe, measured externally. But to a real gas man, those are simply "ten light pipe" "fifteen light tube" and so on. I believe that people whose life is bound up in their own little world, love to use their own measurement systems to confuse the heck out of everyone else.

In the British woodyard of today
"8 x 4 x 12"
means a sheet about 8 feet by 4 feet by 12mm thick
and
"4 x 2 x 3"
Is about four inches by two inches by 3 metres long.

If you asked for a "4 x 4 x 6" you'd obviously be just plain stupid

[Stanley]

Hi Rik,

And thank you for your response. It crossed mine at the same time that I was responding to Laurie.

It is, as always, most interesting, also with helpful links.

I will carefully read and consider what you have to say.

Stanley

[rjlittlefield]

If you asked for a "4 x 4 x 6" you'd obviously be just plain stupid

This reminds me of the old story about the guy who took his bicycle to Alaska but left his chain behind. So he walks into the local hardware store and asks to buy 6 feet of chain. The clerk says "No problem. How big do you want the links?" The bicyclist conjures up a memory of his old chain and replies, "Oh, about 2 inches." The clerk says that'll be a special order, which seems odd to the bicyclist, but he says to go ahead. Two weeks later the chain arrives. Turns out that in Alaska they measure chain links by the diameter of the rod.

(Now where is that dropping-anvil-on-foot emoticon, anyway?)

--Rik