How Big Can You Print That Image? PPI Explained

[Jason Dunn]

Digital Media Thoughts reader <i>ctmagnus</i> asked a question about PPI in our forums and I thought a column on this topic would be useful for everyone. PPI is a confusing issue, but hopefully this column will clarify it for you. Here's the question <i>ctmagnus</i> asked, and it's certainly a common one:<br /><br /><i>"I have an image I'd like to have printed. It's 2272 pixels by 1704 pixels at 180dpi. How big can I get this printed before it starts looking bad?"</i><br /><br />First off, what is PPI you might ask? It stands for "pixels per inch". PPI is a physical measurement of how many dots are printed on a piece of paper. <!> Ignore what you know about DPI and screen resolution - PPI is a measurement of the physical world, where digital pixels get transformed into physical dots on the page. <b>The basic concept is this: the more pixels you can have per square inch on your page, the better the quality of your print will be.</b><br /><br /><span><b>The Painting Analogy...</b></span><br />Here's an analogy that might help this concept make sense: let's say you have one gallon of paint. If you have a wall that's 10 feet high and ten feet wide, you can give that wall several coats of paint and it will look great - full, lush, rich colour. But if you had to paint a wall that was 100 feet high and 100 feet wide, you'd have to spread your paint very thin to cover it all - the net result would be a poor quality paint job. The concept is exactly the same when it comes to printing your digital images. If you take a low resolution (640 x 480) image without many pixels (paint) and print it at a 3" x 2" wallet size (small wall), the quality will be acceptable. But if you took that same 640 x 480 image and tried to do an 8" x 10" print (a big wall), the quality would be poor because there's not enough pixels (paint) to cover it properly.<br /><br /><span><b>How Many Pixels Per Inch Do You Need?</b></span><br />You can't have a PPI value until you know what size of paper you're printing it on - all you have is the resolution (total number of pixels). PPI is an unknown until you pick a physical print size.<br /><br />In the scenario above, <i>ctmagnus</i> has 3.8 megapixels worth of image data (2272 x 1704 = 3,871,488). If you have a program such as <a href="http://www.picasa.net">Picasa</a>, it will give you the PPI value for each print size that you choose, and warn you when the PPI value is low. 300 PPI is ideal - anything above that is overkill. I was surprised to find, however, that I could go all the way down to 100 ppi on my 8.5" x 11" sample print image and it still looked quite good at 8.5 x 11 when viewed from a normal distance (holding the image at chest-height). At 50 PPI I could see significant image degradation. This will vary from image to image, and perhaps from printer to printer (I used a Canon i950 for my images), but 100 PPI is the "danger zone" that you don't want to go below. Some might argue that 150 PPI is the line never to cross - it depends on the eyes of the person looking at the image, and how closely they inspect it for flaws).<br /><br /><span><b>How To Calculate PPI</b></span><br />I taught a college class on PPI and printing a few weeks ago, and I spent some time working out the math (with the help of <a href="http://www.ehansberry.com">Ed Hansberry</a>). How do you calculate PPI? There's the quick way and the painful way. <br /><br />The quick way is to take the largest pixel value on the image (in this case, 2272) and divide it by the longest print side of the page size you're going to print it on. So if you were printing an 8" x 10", you'd divide 2272 by 10 and get 227 PPI. No quality problems with that PPI! At 11" x 14" you'd have 162 PPI. Still ok there, and at 16" x 20" you'd be at 113 PPI. This is pushing the barrier of acceptable quality, but I think it would still look decent - especially since the larger the image the further away people tend to view it.<br /><br />So what's the painful way of calculating PPI? If you're a math person, you won't find this painful, but I'm not, so I did find it moderately painful. :wink: The formula for the data above would look like this: (2272 x 1704) / (8.5 x 11) = x. In this instance, we'd get 41,406 and change. We'd then take the square root of that number and get a PPI value of 203. This is the most accurate method of calculating PPI (the quick method above gave us 227 PPI), but it's more involved. If your software gives you the PPI value, all the better, but if it doesn't you can safely use the quick method to crank out a PPI value before you waste valuable ink and paper.<br /><br />And there you have it. Go out and conquer the digital photo printing world with your new-found knowledge of PPI! :way to go:

[Doug Johnson]

Now if someone would be kind enough to explain Lines Per Inch...

[PJE]

Now if someone would be kind enough to explain Lines Per Inch...

Check out...

http://desktoppub.about.com/cs/intermediate/a/measure_lpi.htm

PJE

[jim s]

If you want to be really accurate, you should figure in the margins on the long version, so 8.5x11 might be more like 8x10.

[Jason Dunn]

If you want to be really accurate, you should figure in the margins on the long version, so 8.5x11 might be more like 8x10.

Depends on the printer - my i950 does full page prints, edge to edge (usually called borderless), so I was quoting the true size of the print. :-)

[sub_tex]

Great article!

That quick and dirty ppi calc is going to make my life a little easier.

[Gordo]

If the image size is not large enough for the size of print you really want; one trick I read about, but haven’t tried, is to resize the image larger in 10% increments. If I recall correctly the author did this step upsizing 20 times in 10% increments to produce a poster size image. I have not tried this approach but heard from a co-worker who tried that it and they indicated it worked quite well.

[craigf]

Technicallly, the formula given isn't exactly correct unless the ratio of length to width is the same for both the image and the print.

If you have an 1600x1200 image, the ratio is 4:3. That's great if you're making a 4x3 or 8x6 print. But, if you want a standard size, say 10x8 (which has a ratio of 5:4), you're going to have to crop the image and make it more squarish. That means you'll lose pixels off of your longest side, reducing your PPI and degrading your print quality.

To continue the example: If we were to print a 1600x1200 pixel image onto a 10" long piece of paper, the article would lead you to believe you'll get 160 ppi. However, the width of the image would be only 7.5 inches. How'd I get that? 1600 pixels over 10" = 160ppi. Now, divide 1200 pixels by 160 ppi to get 7.5 inches (just keep your units straight). So, in order to fill up a 10x8 image, we're going to have to use our short side: 1200 pixels divided by 8" = 150ppi. Multiply 150ppi by the 10" of our paper and you get only 1500 pixels. That means you'll lose 100 pixels off the long end of the image due to cropping.

So, while 150 ppi versus 160 ppi doesn't seem to make a lot of difference (and it won't most of the time), if you start moving to/from highly rectangular image sizes (e.g. 3:2 ratios), then the effect becomes more pronounced.

Anyway, I thought I'd mention this, since it isn't obvious unless you've been mangling, er, managing digital photos for a while and it's helpful to understand (especially when sending images off to online printers like Shutterfly and Ofoto).

[Jason Dunn]

Technicallly, the formula given isn't exactly correct unless the ratio of length to width is the same for both the image and the print.

That's true - I was taking into account the final pixels once the aspect ratio has been fixed, but you're right in that if someone uploads an image to Ofoto and crops it, they'll be losing some PPI.

That's one of the things I like about Picasa - it shows you the PPI value for the image as you change the print size, so there are no surprises.

[Pony99CA]

The quick way is to take the largest pixel value on the image (in this case, 2272) and divide it by the longest print side of the page size you're going to print it on. So if you were printing an 8" x 10", you'd divide 2272 by 10 and get 227 PPI. No quality problems with that PPI! At 11" x 14" you'd have 162 PPI. Still ok there, and at 16" x 20" you'd be at 113 PPI. This is pushing the barrier of acceptable quality, but I think it would still look decent - especially since the larger the image the further away people tend to view it.

So what's the painful way of calculating PPI? If you're a math person, you won't find this painful, but I'm not, so I did find it moderately painful. :wink: The formula for the data above would look like this: (2272 x 1704) / (8.5 x 11) = x. In this instance, we'd get 41,406 and change. We'd then take the square root of that number and get a PPI value of 203. This is the most accurate method of calculating PPI (the quick method above gave us 227 PPI), but it's more involved.
I am a math person (BS in Math and Computer Science from Michigan), and I don't see how the "complicated" calculation is "more accurate". What you're doing is (if I recall correctly) taking the geometric mean of the two ratios, which is creating an artifical number that doesn't reflect either dimension properly.

I haven't thought completely about it, but I believe the following is the most accurate way to do things.

Assume we're going to be printing the larger dimension of the image on the longer side of the paper (typically landscape mode unless you rotated the camera).

Take the larger image pixel count and divide by the larger print dimension. In the example above, 2272/11 = 206.55 ppi.

Take the smaller image pixel count and divide by the smaller print dimension. 1704/8.5 = 200.47 ppi in the above example.

Use the greater of the two pixels per inch values. That corresponds to the dimension that will print all the way to the edge; the other dimension will be letter-boxed (assuming truly round dots) because it will print at the same density as the other dimension.

Of course, to actually fill up that 8.5x11" paper, we'll have to resize the image to match the printer's specifications. (Some programs fit-to-page automatically, but resizing it yourself allows you to inspect the image before committing to paper.)

If you're going to print at 300 dpi, that's 2550x3300 pixels. Resize the number with the higher ppi to its corresponding size (2272 goes to 3300 in the example; 3300/2272 = 1.45 expansion ratio). The other dimension will be printed using 2475 dots and will use 8.25" of the paper.

Does that sound right?

The best algorithm for increasing the size of an image is left to the reader. :-D

Steve

[Jason Dunn]

I received an email from a fellow named Ken who tracked down my personal blog and sent me this message - I figured I'd post it here since he obviously didn't feel like it. ;-)

*************

I read a portion of your article that discusses PPI http://www.digitalmediathoughts.com/articles.php?action=expand,5812.

I do not wish to come off wrong or being a basher but some of the information in your article is in great error.
First of all DPI or dots per inch is exactly that "DOTS" per inch. PPI pixels per inch is exactly that Pixels (they are squares by the way). The number of these per inch is a standard formula that is used to define resolution.

In some systems dots is the method by which images are printed to the paper or medium, in others (not so common) Pixels are used.
Since some printing devices use pixels SQUARES they have a better fill factor than Dots - therefore a lower resolution PPI Printing device can look as good as a higher resolution DPI printing device. A square fills more area than a circle - overlay a circle on a floor tile where the diameter of the circle is no larger than the square. You can still see the square, if you swap the positions an overlay the tile on the circle - you see no circle only a square.
If one has an image that is 4 x 6 at 300 PPI, and wish to print it to paper - whether or not the medium prints Dots or pixels at 300 of those dots or pixels per inch - the image will be 4 x 6. Your monitor "paints- or prints to the screen" at a different resolution of Pixels or Dots per inch typically 72 or
96) the 4 x 6 image will look a different size (larger 3 to 4 inches of screen for each inch in relation to the print from the printing device at 300).

If some one has an image 2100 x 2100 Pixels at '300DPI' - it is really 300PPI (in most cases - cameras use pixels - some now use triangles) then any image manipulation they can have an image 4200 x 4200 at 150 Dpi this is why you see the enlargement or miniaturization of photos when they are printed to a printer. The DPI / PPI can best be looked at in terms of the spacing of the dots or pixels for a given image across a medium - if your printer had a variable resolution and could print at 10 DPI you could easily understand the concept, BUT yor printer resolution IS FIXED. Your monitor on the other hand may offer you other choices from 72 to 101.
The point that you do not know PPI until you have a printing medium is incorrect. I have an Image that is 2100 PIXELS by 2100 PIXELS the image is in PPI .. Pixels 300PPI.
What really is the difference between the two - fill factor, and since most printers print in Dots -you end up with with printing 2100 Pixels on the paper represented as dots at the resolution the printer was told to print it at (usually 300).
If your printer could print Pixels.. you would end up with only a better Fill Factor.. that is the only difference.

- Ken