Circle of confusion
In optics, a circle of confusion 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 uses
Two important uses of this term and concept need to be distinguished:1. For describing the largest blur spot that is indistinguishable from a point. A lens can precisely focus objects at only one distance; objects at other distances are defocused. Defocused object points are imaged as blur spots rather than points; the greater the distance an object is from the plane of focus, the greater the size of the blur spot. Such a blur spot has the same shape as the lens aperture, but for simplicity, is usually treated as if it were circular. In practice, objects at considerably different distances from the camera can still appear sharp ; the range of object distances over which objects appear sharp is the depth of field. The common criterion for “acceptable sharpness” in the final image is that the blur spot be indistinguishable from a point.
2. For describing the blur spot achieved by a lens, at its best focus or more generally, Recognizing that real lenses do not focus all rays perfectly under even the best conditions, the term circle of least confusion is often used for the smallest blur spot a lens can make, for example by picking a best focus position that makes a good compromise between the varying effective focal lengths of different lens zones due to spherical or other aberrations. The term circle of confusion'' is applied more generally, to the size of the out-of-focus spot to which a lens images an object point. Diffraction effects from wave optics and the finite aperture of a lens determine the circle of least confusion; the more general usage of "circle of confusion" for out-of-focus points can be computed purely in terms of ray optics.
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 vary with application.
Circle of confusion diameter limit in photography
In photography, the circle of confusion diameter limit 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 or on the original image.With this definition, the CoC limit in the original image 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, 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 for critical viewing, giving, where f is the lens focal length. For a 50 mm lens on full-frame 35 mm format, this gave CoC ≈ 0.0291 mm. This criterion evidently assumed that a final image would be viewed at “perspective-correct” distance :
However, images seldom are viewed at the “correct” distance;
the viewer usually doesn't 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 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″×10″ 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 isn't 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.
Circle of confusion diameter limit based on ''d''/1500
Adjusting the circle of confusion diameter for a lens’s DoF scale
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 thatwhere 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, and vice versa. 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 ; 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.
Determining a circle of confusion diameter from the object field
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:
The circle of confusion in the image plane is obtained by multiplying by magnification m:
where the magnification m is given by the ratio of focus distances:
Using the lens equation we can solve for the auxiliary variable f1:
which yields
and express the magnification in terms of focused distance and focal length:
which gives the final result:
This can optionally be expressed in terms of the f-number N = f/A as:
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:
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:
And for the blur circle of an object at infinity when the focus distance is finite:
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.
History
Henry Coddington 1829
Before it was applied to photography, the concept of circle of confusion was applied to optical instruments such as telescopes. Coddington quantifies both a circle of least confusion and a least circle of confusion for a spherical reflecting surface.Society for the Diffusion of Useful Knowledge 1832
The applied it to third-order aberrations:T.H. 1866
Circle-of-confusion calculations: An early precursor to depth of field calculations is the 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. The formula he comes up with for what he terms "the indistinctness" is equivalent, in modern terms, tofor focal length, aperture diameter A, and subject distance S. But he does not invert this to find the S corresponding to a given c criterion, 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" .
Dallmeyer and Abney
, in an expanded re-publication of his father John Henry Dallmeyer's 1874 pamphlet On the Choice and Use of Photographic Lenses, says:This latter statement is clearly incorrect, or misstated, being off by a factor of focal distance. He goes on:
Numerically, 1/100 of an inch at 12 to 15 inches is closer to two minutes of arc. This choice of COC limit remains the most widely used even today. 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 to 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.