June 22, 2014

The importance of ppi


I asked 20 people whether they are used to printing their digital photographs: 19 said they never do, preferring to look at them on a tablet or on a computer monitor.
Looking at your own photographs only through a display has several advantages, however it raises some concerns about quality. One first of all: is pixel density high enough to prevent the human eye to discern single pixels? 

The ppi (pixel per inch) is an intrinsic characteristic of a display, measuring the pixel density, i.e. the number of pixels that are contained in a 1-inch segment.
In ideal conditions, a display should have a pixel density greater than the resolution power of the human eye at the observation distance.
A healthy eye has a visual acuity of about 30 line pairs (lp), i.e. 60 lines, on a visual angle of 1 degree [1]. At a distance L a visual angle of 1o encompasses a circular area of diameter

tan( 0.5 o ) . L ~   (π /180) . L

so the required display pixel density can be easily calculated as a function of the observation distance L (in inch):

ppi > 60 . / (π /180)  . L) ~  3438 / L             (1)

The most demanding case corresponds to the closest viewing distance.
The closest viewing distance a healthy eye is able to focus (near-point) is usually about 25cm [2], i.e. about 10 inches. So the required ppi shall be:

ppi > 3438 / 10 ~ 344        (2)

A display with a resolution Nw x Nh and diagonal size D (inch) has a pixel density given by

Capture

If we plot, for different display types, the relationship between the diagonal size D and the ppi, we obtain a chart like this.

ppi-size-1

The chart highlights four clusters of devices, corresponding to different display types (phones/tablets, laptops, monitors, TVs).
The chart clearly shows that only the smaller devices (smartphones) have a ppi almost adequate for the closest observation distance (as a general rule, smaller the screen better is the ppi).
Fortunately, most people prefer to look at an image not so closely. Experiments [3] carried out with images of different dimensions have shown that the average viewing distance of a group of observers is, for images whose diagonal d is greater than 16”, the one which subtends an angle of 36o, that is

Lav = d / ( 2٠ tan(18 o) ) = 1.54 ٠ d           

i.e. about 1.5X the diagonal.
Therefore:

ppi >  2232 / d      (3)

If we plot the curve (3) in the previous chart, we discover that on many display types, when viewed from the average viewing distance, pixels cannot be discerned by the human eye.

ppi-size-2

In particular, the graph shows that at the average viewing distance (1.5X the diagonal):
  • HD (1920x1080) TVs and monitors are fairly fine;
  • on a 15.4” 1366x768 laptop pixels will be discernible, so it is much more advisable to go for a 1920x1080 resolution.
Another distance to be taken under consideration is the one for which the perspective of the scene depicted in the image is correctly perceived. Any image obeying to perspective projection, be it a photograph or a painting or a computer generated picture, is rendered by projecting the points of the three-dimensional scene on a plane through a centre of projection.
The viewing distance of a photographic image for which the perspective is correctly perceived is given by the focal length of the lens (referred to a 24x36mm sensor or film) multiplied by the enlarging factor of the image [4], i.e.:

Lp =  f ٠ d / 43.33    (4)


where    Lp is the viewing distance for a correct perspective perception
   d is the measure of the diagonal of the image
   f   is the focal length in mm, referred to a 24x36mm sensor or film 
   (both Lp and d are expressed in the same units).


It shall be noted that the above relation (4) is not restricted to photographs, but is applicable to other kind of images, as long as they obey to the rules of perspective. Many paintings made in central perspective provide clues allowing to deduce the correct viewing distance with a graphical procedure based on a few elements of the depicted scene and on sensible assumptions (e.g. the tiles of a floor are square) [5].
From (4) we obtain that at the average viewing distance (~1.5X the diagonal) the perspective of an image taken with a 67mm (quite close to the ‘normal’ focal length 50mm) is correctly perceived. Shorter focal lengths require closer viewing distance, more demanding in terms of pixel density, e.g.:


                 Lp = 0.80٠ d           for a 35mm lens
                 Lp = 0.65٠ d           for a 28mm lens
                 Lp = 0.46٠ d           for a 20mm lens


The required ppi can be easily calculated:


ppi >  148925 / ( d ٠ f  )  


and plotted in the above chart for three typical focal lengths: 20mm, 35mm, 85mm.



ppi-size-3-1


The graphs show that, from a viewing distance for which the perspective is correctly perceived, an image taken with a 35mm lens require at least a 4K resolution (3840x2160), already available on a few monitors and TVs. An image taken with a 20mm ultra-wide would require a display with an 8K resolution (7680x4320), not available yet apart from a few TV 8K prototypes (as of June 2014).

Laptops (except maybe the MacBook Pro) are totally unsuitable for the close viewing distances required by wide-angle lenses. Smart phones and tablets are unsuitable too, because they would require a viewing distance closer that the near-point (so, impossible to focus).

References

[1]  K.R. Huxlin, The Human Visual System, Focal Encyclopedia of Photography 4th edition, pp. 629-636
[2] The range of accommodation of the eye, Nuffield Foundation, 2011, http://www.nuffieldfoundation.org/practical-physics/range-accommodation-eye
[3]  Cooper, E. A., Piazza, E. A., & Banks, M. S. (2012). The perceptual basis of common photographic practice. Journal of Vision, 12(5):8, 1–14, http://www.journalofvision.org/content/12/5/8.full.pdf, doi:10.1167/12.5.8
[4] N. Salvaggio 2009, Basic Photographic Materials and Processes, 3rd edition, Focal Press, 2009, p.342
[5] Frantz M., Crannell A., Viewpoints: Mathematical Perspective and Fractal Geometry in Art, draft 2009, Chapter. 3 (Vanishing points and Viewpoints), http://sp.rpcs.org/faculty/BrickmanR/12/Course%20Documents/Viewpoints.pdf

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