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.
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.
This principle corresponds to the
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
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
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
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
There is a relation between the circle
of confusion and the visual acuity. c
= 2.5 x a x
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
- Resolving power of optical
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
The resolving powers of lenses
are different depending on size of cameras.
Depending on size used, the size
of the circle of confusion acceptable for the imager has
The circle of confusion is linked
to the resolution r of the imager by the formula:
c = 1,25
Resolution of film
The resolution of a film is:
r = 0.015mm for films 100 to
The degree of contrast and sharpness
of the borders, quality named acutance, has influence on
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
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...
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
P = X/U
The circle of confusion of the
= c . P
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
The circle of confusion of the
c'' = 1,25 r = 2,5 i = 63,5
so to preserve the circle of confusion
of the enlargement, it needs:
63,5 / c P