SHARPNESS,  the principles

 Two basic principles:
Let's do the following exercise: move away from a line of text. At one point, it is no more possible to read the text which is unclear, but we still can recognize some letters. We have reach the legible limit. Moving more away, we can't distinguish the shape of the letters, but a line between two paragraphs is still clear. When it comes to disappear, we have gone beyond our limit of vision. These two limits are the two basic principles of our visual perception.


Viewing an image:
The minimum distance of clear viewing is conventionally 25cm. At this distance we see the whole of a postcard.
At this distance, the circle of confusion is c = 0.001 x 250 = 0.25 mm. We can distinguish all bigger shapes, and all the smaller ones will appear confused.
Moreover, it is possible that a shape, a remote outline are much thinner. Our eye is able to see a dot, or a line which size, or the smallest width is s = 0.0004 x 250 = 0.1 mm.
Circle of confusion:
This principle corresponds to the first limit.
On a sheet of paper let's draw a square, a dick, a triangle, a cross, of similar size, and go away, progressively. There is a distance to which we can't recognize them one from the other, and we see only spots.
Depending on their color, the lighting, the nature of the background, the contrast, this distance varies. For a  given viewing distance, the smallest dimension of these shapes for which the outline is not discernable defines the circle of confusion.


 It is the principle  of the circle of confusion which is applied for printing images and pictures on newspapers, magazines, and posters. A group of dots looks diffused and give the impression of a even color.
 The ratio between the circle of confusion and the viewing distance is a fix angle which commonly used value is 0.001 radians, or 3,4 minutes of arc. to calculate the diameter c of the circle of confusion, multiply the distance D by the angle in radians:
c = 0.001 x D
Visual Acuity:
When going away further from the previous shapes, there is the point where we cannot discern the spots which are too tiny. We have pass over our limit of visual acuity. It corresponds to an angle of one minute, so a = 0.0003 for a perfect acuity, but as average, a = 0.0004 will be used.
The smallest visible detail has a size equal to [a x D]
There is a relation between the circle of confusion and the visual acuity.           c = 2.5 x a x D



Therefore, for an image to be very clear, it is necessary that the patterns of the print ( screen in printing, or grain in photographic paper) are smaller than the circle of confusion of the eye for one part, and for the other part, that the tiniest details are as small as visual acuity may discern.


At each step of the process to make an image, at the same time, resolution of the optical device, resolution of the considered backing, and size of the obtained circle of confusion have to be considered.
Resolving power of optical device
The lens of camera may deteriorate the basic resolution of the film or of the sensor if it is of insufficient quality. The main reasons are the problems of distortion of shapes, of light diffusion and diffraction, of focusing offset.
Note also that the circle of confusion decreases when the diaphragm gets smaller (higher aperture values). So it gives an improvement of image sharpness which the photographers and the automatic exposure systems take in account by setting as preference the f5,6 to f16 values.
Mathematically, the resolving power is the inverse of the resolution.


The photographers have perfected various techniques to evaluate and to compare the resolution of lenses. the two main ones are the line patterns and the Modulation Transfer Function.
The resolving power is measured by searching to discriminate on a pattern alternating white and black lines. It is expressed in pairs of lines, (one white and one black) bu unit of lenght, usually pl/mm, or which is the same, in number of cycles per mm, cy/mm ( acycle being also a pair of lines). 100pl/mm is a good result in 24x36mm negatives.
The resolving powers of lenses are different depending on size of cameras.
Imager Systems:
Depending on size used, the size of the circle of confusion acceptable for the imager has different values.


The circle of confusion is linked to the resolution r of the imager by the formula:
c = 1,25 r
Resolution of film
The resolution of a film is:
r = 0.015mm for films 100 to 400 ISO
The degree of contrast and sharpness of the borders, quality named acutance, has influence on the resolution.
The resolving power is:
80 to 50 /mm for films 100 to 400 ISO 


Resolution of digital sensor
The pixels are normally square shape, and their dimension may be calculated:
i = U / I
I : number of pixels on the width
U : width of sensor
The resolution is: r = 2 i = 2 U/I
Circle of confusion of films:
The formula c = 1,25 r is applicable.
The following value of circle of confusion is traditionally used for calculations of depth of field and hyperfocal.of 24x36mm (35mm film)
c = 0,036 mm



Circle of confusion of sensors
The formula c = 1,25 r becomes now:
c = 2,5 U/I
Being the great number of sensors and pixel size formats, there is no standard values of circle of confusion
See the page photoscope for further details on the calculations of sensor size, pixel format, etc...


Resolution of photo prints and printed paper
On an enlargement, we move further to get the whole of it, usually to a distance of twice its diagonal but we go closer again, until a distance of 25cm to catch some details, and the image looks clear if the printing screen is lower than 0,25mm and if all tiniest details have size as small as 0,1mm.
This is why larger is the enlargement, better must be the definition of the sensor. To get it, a higher format may be necessary.
The circle of confusion c of the imager of size UxV becomes c' on the enlargement of size XxY; enlargement ratio is:
P = X/U = Y/V
The circle of confusion of the enlargement is:
 c' = c . P
To optimize the prints, see the page applications.


In the case of a print from film, the resolution of the photographic paper is much better than the eye acuity. As far as the printing device is of sufficient quality, there is no concern of its actual resolution. Note that the digital images can be printed on photographic paper, so producing the best printing quality achievable.
In the case of a printed image, we know that the professional printing equipment have particular systems. But what about the ink jet printers used with our computer? All depends on the definition N, normally expressed in ppi as pixels per inch. Conversion in mm is: i = 25,4 / N
The circle of confusion of the print is:
c'' = 1,25 r = 2,5 i = 63,5 / N
so to preserve the circle of confusion of the enlargement, it needs:
N = 63,5 / c P