Circle of confusion

In optics, a circle of confusion (CoC) is an optical spot caused by a cone of light rays from a lens not coming to a perfect focus when imaging a point source. It is also known as disk of confusion, circle of indistinctness, blur circle, or blur spot.
In photography, the circle of confusion is used to determine the depth of field, the part of an image that is acceptably sharp. A standard value of CoC is often associated with each image format, but the most appropriate value depends on visual acuity, viewing conditions, and the amount of enlargement. Usages in context include maximum permissible circle of confusion, circle of confusion diameter limit, and the circle of confusion criterion.
Real lenses do not focus all rays perfectly, so that even at best focus, a point is imaged as a spot rather than a point. The smallest such spot that a lens can produce is often referred to as the circle of least confusion.
Two important uses of this term and concept need to be distinguished:
In idealized ray optics, where rays are assumed to converge to a point when perfectly focused, the shape of a defocus blur spot from a lens with a circular aperture is a hard-edged circle of light. A more general blur spot has soft edges due to diffraction and aberrations, and may be non-circular due to the aperture shape. Therefore, the diameter concept needs to be carefully defined in order to be meaningful. Suitable definitions often use the concept of encircled energy, the fraction of the total optical energy of the spot that is within the specified diameter. Values of the fraction (e.g., 80%, 90%) vary with application.
In photography, the circle of confusion diameter limit (CoC limit or CoC criterion) is often defined as the largest blur spot that will still be perceived by the human eye as a point, when viewed on a final image from a standard viewing distance. The CoC limit can be specified on a final image (e.g. a print) or on the original image (on film or image sensor).
With this definition, the CoC limit in the original image (the image on the film or electronic sensor) can be set based on several factors:
The common values for CoC limit may not be applicable if reproduction or viewing conditions differ significantly from those assumed in determining those values. If the original image will be given greater enlargement, or viewed at a closer distance, then a smaller CoC will be required. All three factors above are accommodated with this formula:
For example, to support a final-image resolution equivalent to 5 lp/mm for a 25 cm viewing distance when the anticipated viewing distance is 50 cm and the anticipated enlargement is 8:
Since the final-image size is not usually known at the time of taking a photograph, it is common to assume a standard size such as 25 cm width, along with a conventional final-image CoC of 0.2 mm, which is 1/1250 of the image width. Conventions in terms of the diagonal measure are also commonly used. The DoF computed using these conventions will need to be adjusted if the original image is cropped before enlarging to the final image size, or if the size and viewing assumptions are altered.
For full-frame 35 mm format (24 mm × 36 mm, 43 mm diagonal), a widely used CoC limit is d/1500, or 0.029 mm for full-frame 35 mm format, which corresponds to resolving 5 lines per millimeter on a print of 30 cm diagonal. Values of 0.030 mm and 0.033 mm are also common for full-frame 35 mm format.
Criteria relating CoC to the lens focal length have also been used. Kodak recommended 2 minutes of arc (the Snellen criterion of 30 cycles/degree for normal vision) for critical viewing, yielding a CoC of about f/1720, where f is the lens focal length. For a 50 mm lens on full-frame 35 mm format, the corresponding CoC is 0.0291 mm. This criterion evidently assumed that a final image would be viewed at perspective-correct distance (i.e., the angle of view would be the same as that of the original image):
However, images seldom are viewed at the so-called 'correct' distance; the viewer usually does not know the focal length of the taking lens, and the "correct" distance may be uncomfortably short or long. Consequently, criteria based on lens focal length have generally given way to criteria (such as d/1500) related to the camera format.
If an image is viewed on a low-resolution display medium such as a computer monitor, the detectability of blur will be limited by the display medium rather than by human vision. For example, the optical blur will be more difficult to detect in an 8 in × 10 in image displayed on a computer monitor than in an 8×10 print of the same original image viewed at the same distance. If the image is to be viewed only on a low-resolution device, a larger CoC may be appropriate; however, if the image may also be viewed in a high-resolution medium such as a print, the criteria discussed above will govern.
Depth of field formulas derived from geometrical optics imply that any arbitrary DoF can be achieved by using a sufficiently small CoC. Because of diffraction, however, this is not quite true. Using a smaller CoC requires increasing the lens f-number to achieve the same DoF, and if the lens is stopped down sufficiently far, the reduction in defocus blur is offset by the increased blur from diffraction. See the Depth of field article for a more detailed discussion.
The f-number determined from a lens DoF scale can be adjusted to reflect a CoC different from the one on which the DoF scale is based. It is shown in the Depth of field article that
D o F = 2 N c ( m + 1 ) m 2 − ( N c f ) 2 , {\displaystyle \mathrm {DoF} ={\frac {2Nc\left(m+1\right)}{m^{2}-\left({\frac {Nc}{f}}\right)^{2}}}\,,}
where N is the lens f-number, c is the CoC, m is the magnification, and f is the lens focal length. Because the f-number and CoC occur only as the product Nc, an increase in one is equivalent to a corresponding decrease in the other. For example, if it is known that a lens DoF scale is based on a CoC of 0.035 mm, and the actual conditions require a CoC of 0.025 mm, the CoC must be decreased by a factor of 0.035 / 0.025 = 1.4; this can be accomplished by increasing the f-number determined from the DoF scale by the same factor, or about 1 stop, so the lens can simply be closed down 1 stop from the value indicated on the scale.
The same approach can usually be used with a DoF calculator on a view camera.
To calculate the diameter of the circle of confusion in the image plane for an out-of-focus subject, one method is to first calculate the diameter of the blur circle in a virtual image in the object plane, which is simply done using similar triangles, and then multiply by the magnification of the system, which is calculated with the help of the lens equation.
The blur circle, of diameter C, in the focused object plane at distance S1, is an unfocused virtual image of the object at distance S2 as shown in the diagram. It depends only on these distances and the aperture diameter A, via similar triangles, independent of the lens focal length:
C = A | S 2 − S 1 | S 2 . {\displaystyle C=A{|S_{2}-S_{1}| \over S_{2}}\,.}
The circle of confusion in the image plane is obtained by multiplying by magnification m:
c = C m , {\displaystyle c=Cm\,,}
where the magnification m is given by the ratio of focus distances:
m = f 1 S 1 . {\displaystyle m={f_{1} \over S_{1}}\,.}
Using the lens equation we can solve for the auxiliary variable f1:
1 f = 1 f 1 + 1 S 1 , {\displaystyle {1 \over f}={1 \over f_{1}}+{1 \over S_{1}}\,,}
which yields
f 1 = f S 1 S 1 − f , {\displaystyle f_{1}={fS_{1} \over S_{1}-f}\,,}
and express the magnification in terms of focused distance and focal length:
m = f S 1 − f , {\displaystyle m={f \over S_{1}-f}\,,}
which gives the final result:
c = A | S 2 − S 1 | S 2 f S 1 − f . {\displaystyle c=A{|S_{2}-S_{1}| \over S_{2}}{f \over S_{1}-f}\,.}
This can optionally be expressed in terms of the f-number N = f/A as:
c = | S 2 − S 1 | S 2 f 2 N ( S 1 − f ) . {\displaystyle c={|S_{2}-S_{1}| \over S_{2}}{f^{2} \over N(S_{1}-f)}\,.}
This formula is exact for a simple paraxial thin lens or a symmetrical lens, in which the entrance pupil and exit pupil are both of diameter A. More complex lens designs with a non-unity pupil magnification will need a more complex analysis, as addressed in depth of field.
More generally, this approach leads to an exact paraxial result for all optical systems if A is the entrance pupil diameter, the subject distances are measured from the entrance pupil, and the magnification is known:
c = A m | S 2 − S 1 | S 2 . {\displaystyle c=Am{|S_{2}-S_{1}| \over S_{2}}\,.}
If either the focus distance or the out-of-focus subject distance is infinite, the equations can be evaluated in the limit. For infinite focus distance:
c = f A S 2 = f 2 N S 2 . {\displaystyle c={fA \over S_{2}}={f^{2} \over NS_{2}}\,.}
And for the blur circle of an object at infinity when the focus distance is finite:
c = f A S 1 − f = f 2 N ( S 1 − f ) . {\displaystyle c={fA \over S_{1}-f}={f^{2} \over N(S_{1}-f)}\,.}
If the c value is fixed as a circle of confusion diameter limit, either of these can be solved for subject distance to get the hyperfocal distance, with approximately equivalent results.
Before it was applied to photography, the concept of circle of confusion was applied to optical instruments such as telescopes. Coddington (1829, p. 54) quantifies both a circle of least confusion and a least circle of confusion for a spherical reflecting surface.
The Society for the Diffusion of Useful Knowledge (1832, p. 11) applied it to third-order aberrations:
Circle-of-confusion calculations: An early precursor to depth of field calculations is the TH (1866, p. 138) calculation of a circle-of-confusion diameter from a subject distance, for a lens focused at infinity; this article was pointed out by von Rohr (1899). The formula he comes up with for what he terms "the indistinctness" is equivalent, in modern terms, to
c = f A S {\displaystyle c={fA \over S}}
for focal length f, aperture diameter A, and subject distance S. But he does not invert this to find the S corresponding to a given c criterion (i.e. he does not solve for the hyperfocal distance), nor does he consider focusing at any other distance than infinity.
He finally observes "long-focus lenses have usually a larger aperture than short ones, and on this account have less depth of focus" [his italic emphasis].
Dallmeyer (1892, p. 24), in an expanded re-publication of his father John Henry Dallmeyer's 1874 (Dallmeyer 1874) pamphlet On the Choice and Use of Photographic Lenses (in material that is not in the 1874 edition and appears to have been added from a paper by J.H.D. "On the Use of Diaphragms or Stops" of unknown date), says:
This latter statement is clearly incorrect, or misstated, being off by a factor of focal distance (focal length). He goes on:
Numerically, 1/100 inch at 12–15 inches is closer to two minutes of arc. This choice of CoC limit remains (for a large print) the most widely used even today. Abney (1881, pp. 207–08) takes a similar approach based on a visual acuity of one minute of arc, and chooses a circle of confusion of 0.025 cm for viewing at 40–50 cm, essentially making the same factor-of-two error in metric units. It is unclear whether Abney or Dallmeyer was earlier to set the CoC standard thereby.
The common 1/100 inch CoC limit has been applied to blur other than defocus blur. For example, Wall (1889, p. 92) says: