Circle of confusion

In optics, a circle of confusion, (also known as disk of confusion, circle of indistinctness, blur circle, etc.), 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.



Two uses
Two important uses of this term and concept need to be distinguished:

1. To calculate a camera's depth of field (&ldquo;DoF&rdquo;), one needs to know how large a circle of confusion can be considered to be an acceptable focus. The maximum acceptable diameter of such a circle of confusion is known as the maximum permissible circle of confusion, the circle of confusion diameter limit, or the circle of confusion criterion, but is often incorrectly called simply the circle of confusion.

2. Recognizing that real lenses do not focus all rays perfectly under even the best of conditions, the circle of confusion of a lens is a characterization of its optical spot. The term circle of least confusion is often used for the smallest optical 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. Diffraction effects from wave optics and the finite aperture of a lens can be included in the circle of least confusion, or the term can be applied in pure ray (geometric) optics.

In idealized ray optics, where rays are assumed to converge to a point when perfectly focused, the shape of a mis-focused spot from a lens with a circular aperture is a hard-edged disk of light (that is, a hockey-puck shape when intensity is plotted as a function of x and y coordinates in the focal plane). A more general circle of confusion has soft edges due to diffraction and aberrations, and may be non-circular due to the aperture (diaphragm) shape. So the diameter concept needs to be carefully defined to be meaningful. The diameter of the smallest circle that can contain 90% of the optical energy is a typical suitable definition for the diameter of a circle of confusion; in the case of the ideal hockey-puck shape, it gives an answer about 5% less than the actual diameter.

Basis for circle of confusion diameter limit
In photography, the circle of confusion diameter limit (&ldquo;CoC&rdquo;) is sometimes defined as the largest blur circle that will still be perceived by the human eye as a point when viewed at a distance of 25 cm (and variations thereon).

With this definition, the CoC in the original image depends on three factors:

 Visual acuity. For most people, the closest comfortable viewing distance, termed the near distance for distinct vision (Ray 2002, 216), is approximately 25 cm. At this distance, a person with good vision can usually distinguish an image resolution of 5 line pairs per millimeter (lp/mm), equivalent to a CoC of 0.2 mm in the final image.

Viewing conditions. If the final image is viewed at approximately 25 cm, a final-image CoC of 0.2 mm often is appropriate. A comfortable viewing distance is also one at which the angle of view is approximately 60&deg; (Ray 2002, 216); at a distance of 25 cm, this corresponds to about 30 cm, approximately the diagonal of an 8&times;10 inch image. It often may be reasonable to assume that, for whole-image viewing, an image larger than 8&times;10 will be viewed at a distance greater than 25 cm, for which a larger CoC may be acceptable.

Enlargement from the original image (the focal plane image on the film or image sensor) to the final image (print, usually). If an 8&times;10 inch original image is contact printed, there is no enlargement, and the CoC for the original image is the same as that in the final image. However, if the long dimension of a 35 mm image is enlarged to approximately 25 cm (10 inches), the enlargement is approximately 7&times;, and the CoC for the original image is 0.2 mm/7, or 0.029 mm. 

All three factors are accommodated with this formula:


 * CoC Diameter Limit (mm) = anticipated viewing distance (cm) / desired print resolution (lp/mm) for a 25 cm viewing distance / anticipated enlargement factor / 25

For example, to support a print resolution equivalent to 5 lp/mm for a 25 cm viewing distance when the anticipated viewing distance is 50 cm and the anticipated enlargement factor is 8:


 * CoC Diameter Limit = 50 / 5 / 8 / 25 = 0.05 mm

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.

Using the so-called &ldquo;Zeiss formula&rdquo; the circle of confusion is sometimes calculated as d/1730 where d is the diagonal measure of the original image (the camera format). For full-frame 35 mm format (24 mm &times; 36 mm, 43 mm diagonal) this comes out to be 0.024 mm. A more widely used CoC 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. For practical purposes, d /1730, a final-image CoC of 0.2 mm, and d /1500 give very similar results.

Angular criteria for CoC have also been used. Kodak (1972) recommended 2 minutes of arc (the Snellen criterion of 30 cycles/degree for normal vision) for critical viewing, giving CoC &asymp; $$f/1720$$, where $$f$$ is the lens focal length. For a 50 mm lens on full-frame 35 format, this gave CoC &asymp; 0.0291 mm. Angular criteria evidently assumed that a final image would be viewed at &ldquo;perspective-correct&rdquo; distance (i.e., the angle of view would be the same as that of the original image):


 * Viewing distance = focal length of taking lens &times; enlargement

However, images seldom are viewed at the &ldquo;correct&rdquo; distance; the viewer usually doesn't know the the focal length of the taking lens, and the &ldquo;correct&rdquo; distance may be uncomfortably short or long. Consequently, angular criteria have generally given way to a CoC fixed to the camera format.

The common values for CoC may not be applicable if reproduction or viewing conditions differ significantly from those assumed in determining those values. If the photograph will be magnified to a larger size, or viewed at a closer distance, then a smaller CoC will be required. If the photo is printed or displayed using a device, such as a computer monitor, that introduces additional blur or resolution limitation, then a larger CoC may be appropriate since the detectability of blur will be limited by the reproduction medium rather than by human vision; for example, an 8&Prime; &times; 10&Prime; image displayed on a CRT may have greater depth of field than an 8&Prime; &times; 10&Prime; print of the same photo, due to the CRT display having lower resolution; the CRT image is less sharp overall, and therefore it takes a greater misfocus for a region to appear blurred.

Depth of field formulae 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. The CoC is decreased by increasing the lens f-number, 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.

Calculating a circle of confusion diameter


To calculate the diameter of the circle of confusion in the focal plane for an out-of-focus subject, the easiest 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, 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:


 * $$ C = A \cdot {|S_2 - S_1| \over S_2}$$

The circle of confusion in the focal plane is obtained by multiplying by magnification m:


 * $$ c = C\cdot m$$

where the magnification m is given by the ratio of focus distances:


 * $$ m = {f_1 \over S_1}$$

Using the lens equation we can solve for the auxiliary variable f1:


 * $$ {1 \over f} = {1 \over f_1} + {1 \over S_1}$$
 * $$ f_1 = {f\cdot S_1 \over S_1 - f}$$

and express the magnification in terms of focused distance and focal length:


 * $$ m = {f \over S_1 - f}$$

which gives the final result:


 * $$ c = A \cdot {|S_2 - S_1| \over S_2}\cdot{f \over S_1 - f}$$

and which can optionally be expressed in terms of the f-number N = f/A as:


 * $$ c = {|S_2 - S_1| \over S_2}\cdot{f^2 \over N(S_1 - f)}$$

This formula is exact for a simple paraxial thin-lens system, 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:


 * $$ c = A \cdot m \cdot {|S_2 - S_1| \over S_2}$$

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:


 * $$ c = {f A \over S_2} = {f^2 \over N S_2}$$

And for the blur of an object at infinity when the focus distance is finite:


 * $$ c = {f A \over S_1 - f} = {f^2 \over N(S_1 - f)}$$

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.

Society for the Diffusion of Useful Knowledge 1838
Before it was applied to photography, the concept of circle of confusion was applied to optical instruments such as telescopes. The 1838 Natural Philosophy: With an Explanation of Scientific Terms, and an Index applied it to third-order aberrations:


 * "This spherical aberration produces an indistinctness of vision, by spreading out every mathematical point of the object into a small spot in its picture; which spots, by mixing with each other, confuse the whole. The diameter of this circle of confusion, at the focus of the central rays F, over which every point is spread, will be L K (fig. 17.); and when the aperture of the reflector is moderate it equals the cube of the aperture, divided by the square of the radius  (...): this circle is called the aberration of latitude."

T.H. 1866
Circle-of-confusion calculations: An early precursor to depth of field calculations is the 1866 calculation of a circle-of-confusion diameter from a subject distance, for a lens focused at infinity, in a one-page article "Long and Short Focus" by an anonymous T. H. (British Journal of Photography XIII p. 138; this article was pointed out by Moritz von Rohr in his 1899 book Photographische Objektive). The formula he comes up with for what he terms "the indistinctness" is equivalent, in modern terms, to


 * $$c = {f A \over S}$$

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 and Abney
Thomas R. Dallmeyer's 1892 expanded re-publication of his father John Henry Dallmeyer's 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:


 * "Thus every point in an object out of focus is represented in the picture by a disc, or circle of confusion, the size of which is proportionate to the aperture in relation to the focus of the lens employed. If a point in the object is 1/100 of an inch out of focus, it will be represented by a circle of confusion measuring but 1/100 part of the aperture of the lens."

This latter statement is clearly incorrect, or misstated, being off by a factor of focal distance (focal length). He goes on:


 * "and when the circles of confusion are sufficiently small the eye fails to see them as such; they are then seen as points only, and the picture appears sharp. At the ordinary distance of vision, of from twelve to fifteen inches, circles of confusion are seen as points, if the angle subtended by them does not exceed one minute of arc, or roughly, if  they do not exceed the 1/100 of an inch in diameter."

Numerically, 1/100 of an inch at 12 to 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. Sir Abney, in his 1881 A Treatise on Photography, 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.

Wall 1889
The common 1/100 inch COC limit has been applied to blur other than mis-focus blur. For example, Edward John Wall, in his 1889 A Dictionary of Photography for the Amateur and Professional Photographer, says:


 * To find how quickly a shutter must act to take an object in motion that there may be a circle of confusion less than 1/100in. in diameter, divide the distance of the object by 100 times the focus of the lens, and divide the rapidity of motion of object in inches per second by the results, when you have the longest duration of exposure in fraction of a second.

Trivia
The Circle of Confusion is a popular and often appropriate name for photography clubs.