How many pixels do you need in an image? Sounds a simple enough question and the inexorable megapixel race doesn't seem to have ended, in much the same way that the PC processor race marched on unabated for several decades.
Of course, megahertz envy ultimately resulted in people not caring a whit anymore. A 4GHz whizbang processor didn't make a blind bit of difference compared to a 2GHz popzoom processor when it came to finishing writing a Christmas thank you letter to Auntie Marge. That said, you try loading a 50MP image into a PC with an integrated graphics card from five years ago with 1Gb of main system RAM and, well, you'd better have plenty of spare time on your hands!
And so the principle is very similar to the photographic world and the ever-ramping up of the MP count for all those pixel peepers. So let's start with the premise of the actual use of the resultant image - why are you actually taking a photo? That then drives the second question - what are you going to do with it?
If you are shooting for a client, then they will have in mind a single, or more likely, a range of applications for which they want to make use of your images. Anything from Instagram, to photobook, to canvas, to full page ad, to billboard. Each of these is displayed/printed at a different size, on different media, with different print techniques.
So how many pixels do you need? The answer, which might seem counter-intuitive at first, is as many as you need to see. There are two closely related aspects to this:
1. Viewing Distance: as a rough rule of thumb, the viewing distance should be 1.5 to 2 times the diagonal length. This then gives an estimate of how far away you should be for the human eye to focus on a whole image. The bigger an image gets, the further away you need to be (doh!).
2. Pixels per Inch (ppi): for the viewing distance above, how many pixels do you need to fool the eye into believing you are showing smooth continuous tones? The short answer is at least 3438/viewing distance.
A 6x4 photo? You view it about 12" away: 3438/12=286.5. Hence the recommendation for 300ppi.
An A1 canvas? Maybe 4ft away: 3438/48=71.6. Hence the recommendation for 72ppi.
So where does the magical 3438 number come from? Well, this is based on the visual acuity of a "normal" human eye where 1 arc minute (0.000290888 radians) of angle is how much resolution the eye can see. With some High School trig (remember SOHCAHTOA?!) we calculate
1/ppi = 2 x Viewing Distance x tan(0.000290888/2)
1/ppi = Viewing Distance x tan(0.000290888)
ppi = 3438/Viewing Distance
As long as you keep your ppi above this value you should be fine at this viewing distance. If you have an image (such as a photo montage) where people are likely to look at parts of the image or where people may simply want to look at it more closely (possibly a landscape), then you'll need to up the ppi. It's also a timely reminder that, for an A1 canvas, you are only looking at a minimum 4MP requirement (although you might want it to be higher). Resolution, therefore, provides latitude in your photographic workflow - latitude to crop-in, rotate and in any other way you see fit, lose pixels during post-production.
Image courtesy of OpenClipart-Vectors
(Original inspired by having this problem and some great StackExchangers)