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 (“CoC”) 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. Properly, this is the maximum permissible circle of confusion, the circle of confusion diameter limit, or the circle of confusion criterion, but is often informally called simply the circle of confusion.

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.

The depth of field is the region where the CoC is less than the resolution of the human eye (or of the display medium).

Two uses[link]

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 (Ray 2000, 50); the range of object distances over which objects appear sharp is the depth of field (“DoF”). The common criterion for “acceptable sharpness” in the final image (e.g., print, projection screen, or electronic display) 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 (Ray 2002, 89), 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 can be included in the circle of least confusion;^[1] the more general circle of confusion for out-of-focus points is often computed in terms of pure ray (geometric) optics.^[2]

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 (Stokseth 1969, 1317; Merklinger 1992, 45–46), and may be non-circular due to the aperture shape. So the diameter concept needs to be carefully defined 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.

Circle of confusion diameter limit in photography[link]

In photography, the circle of confusion diameter limit (“CoC”) for the final image is often defined as the largest blur spot that will still be perceived by the human eye as a point.

With this definition, the CoC in the original image (the image on the film or electronic sensor) depends on three factors:

1. Visual acuity. For most people, the closest comfortable viewing distance, termed the near distance for distinct vision (Ray 2000, 52), 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.
2. 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° (Ray 2000, 52); at a distance of 25 cm, this corresponds to about 30 cm, approximately the diagonal of an 8″×10″ image. It often may be reasonable to assume that, for whole-image viewing, a final image larger than 8″×10″ will be viewed at a distance correspondingly greater than 25 cm, and for which a larger CoC may be acceptable; the original-image CoC is then the same as that determined from the standard final-image size and viewing distance. But if the larger final image will be viewed at the normal distance of 25 cm, a smaller original-image CoC will be needed to provide acceptable sharpness.
3. Enlargement from the original image to the final image. If there is no enlargement (e.g., a contact print of an 8×10 original image), the CoC for the original image is the same as that in the final image. But if, for example, the long dimension of a 35 mm original image is enlarged to 25 cm (10 inches), the enlargement is approximately 7×, and the CoC for the original image is 0.2 mm / 7, or 0.029 mm.

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 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:

CoC (mm) = viewing distance (cm) / desired final-image resolution (lp/mm) for a 25 cm viewing distance / enlargement / 25

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:

CoC = 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 “Zeiss formula”, 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 × 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.

Criteria relating CoC to the lens focal length have also been used. Kodak (1972, 5) recommended 2 minutes of arc (the Snellen criterion of 30 cycles/degree for normal vision) for critical viewing, giving CoC ≈ f /1720, 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 (i.e., the angle of view would be the same as that of the original image):

Viewing distance = focal length of taking lens × enlargement

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 (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″×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.

[edit] Circle of confusion diameter limit based on d/1500

Adjusting the circle of confusion diameter for a lens’s DoF scale[link]

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm {DoF} = \frac {2 N c \left ( m + 1 \right )} {m^2 - \left ( \frac {N c} {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, 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 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.

Determining a circle of confusion diameter from the object field[link]

Lens and ray diagram for calculating the circle of confusion diameter c for an out-of-focus subject at distance S₂ when the camera is focused at S₁. The auxiliary blur circle C in the object plane (dashed line) makes the calculation easier.

An early calculation of CoC diameter (“indistinctness”) by “T.H.” in 1866.

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:

Failed to parse (Missing texvc executable; please see math/README to configure.): C = A {|S_2 - S_1| \over S_2} \,.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): c = C m \,,

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

Failed to parse (Missing texvc executable; please see math/README to configure.): m = {f_1 \over S_1} \,.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): {1 \over f} = {1 \over f_1} + {1 \over S_1} \,,

which yields

Failed to parse (Missing texvc executable; please see math/README to configure.): f_1 = {f S_1 \over S_1 - f} \,.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): m = {f \over S_1 - f} \,,

which gives the final result:

Failed to parse (Missing texvc executable; please see math/README to configure.): 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:

Failed to parse (Missing texvc executable; please see math/README to configure.): 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:

Failed to parse (Missing texvc executable; please see math/README to configure.): c = A m {|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:

Failed to parse (Missing texvc executable; please see math/README to configure.): c = {f A \over S_2} = {f^2 \over N S_2} \,.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): 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.

History[link]

Henry Coddington 1829[link]

Before it was applied to photography, the concept of circle of confusion was applied to optical instruments such as telescopes. Coddington (1829, 54) quantifies both a circle of least confusion and a least circle of confusion for a spherical reflecting surface.

"This we may consider as the nearest approach to a simple focus, and term the circle of least confusion."

Society for the Diffusion of Useful Knowledge 1832[link]

The Society for the Diffusion of Useful Knowledge (1832, 11) 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[link]

Circle-of-confusion calculations: An early precursor to depth of field calculations is the T.H. (1866, 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

Failed to parse (Missing texvc executable; please see math/README to configure.): c = {f A \over S}

for focal length Failed to parse (Missing texvc executable; please see math/README to configure.): 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[link]

T Dallmeyer (1892, 24), in an 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. Abney (1881, 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 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[link]

The common 1/100 inch COC limit has been applied to blur other than defocus blur. For example, Wall (1889, 92) 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."

See also[link]

Photography portal

• Airy disk
• Astigmatism
• Bokeh
• Chromatic aberration
• Focal cloud
• Point spread function

Notes[link]

References[link]

• Abney, Sir William de Wiveleslie. 1881. A Treatise on Photography. London: Longmans, Green and Co.
• Coddington, Henry. 1829. A Treatise on the Reflexion and Refraction of light: being Part I. of a System of Optics. Cambridge: J. Smith.
• Dallmeyer, John Henry. 1874. On the Choice and Use of Photographic Lenses. New York: E. and H.T. Anthony and Co.
• Dallmeyer, Thomas R. 1892. On the Choice and Use of Photographic Lenses. London: J. Pitcher.
• Eastman Kodak Company. 1972. Optical Formulas and Their Application, Kodak Publication No. AA-26, Rev. 11-72-BX. Rochester, New York: Eastman Kodak Company.
• Kodak. See Eastman Kodak Company.
• Merklinger, Harold M. 1992. The INs and OUTs of FOCUS: An Alternative Way to Estimate Depth-of-Field and Sharpness in the Photographic Image. v. 1.0.3. Bedford, Nova Scotia: Seaboard Printing Limited. ISBN 0-9695025-0-8. Version 1.03e available in PDF at http://www.trenholm.org/hmmerk/.
• Ray, Sidney F. 2000. The geometry of image formation. In The Manual of Photography: Photographic and Digital Imaging, 9th ed. Ed. Ralph E. Jacobson, Sidney F. Ray, Geoffrey G. Atteridge, and Norman R. Axford. Oxford: Focal Press. ISBN 0-240-51574-9
• Ray, Sidney F. 2002. Applied Photographic Optics, 3rd ed. Oxford: Focal Press. ISBN 0-240-51540-4
• Society for the Diffusion of Useful Knowledge. 1832. Natural Philosophy: With an Explanation of Scientific Terms, and an Index. London: Baldwin and Cradock, Paternoster-Row.
• Stokseth, Per A. 1969. Properties of a Defocused Optical System. Journal of the Optical Society of America 59:10, Oct. 1969.
• T.H. [pseud.]. 1866. “Long and Short Focus”. British Journal of Photography 13.
• von Rohr, Moritz. 1899. Photographische Objektiv. Berlin: Verlag Julius Springer.
• Wall, Edward John. 1889. A Dictionary of Photography for the Amateur and Professional Photographer. New York: E. and H.T. Anthony and Co.

External links[link]

• Circles of Confusion for Digital Cameras DOFMaster
• Depth of field and circle of confusion
• Depth of Field in Depth (PDF) Includes discussion of circle of confusion criteria
• What Is the “Circle of Confusion” ƒ/Calc

Thomas J. "Tom" Rowe (b. November 4, 1950, Lewiston, Maine - d. January 17, 2004, Portland, Maine) was the bass player and a singer in the folk trios Schooner Fare and Turkey Hollow. He was noted for playing a model XL2 Steinberger bass guitar. Born to Charles "Bud" Rowe, and his wife, Thoma, he had two brothers named Russ and Chuck, and a sister named Shirley. He went to school in Auburn, Maine and graduated from Edward Little High School as part of the class of 1969. He majored in Music Education at Gorham State College, now the University of Southern Maine, though never graduated. In 1975, Rowe joined the Maine folk/rock group Devonsquare. Three months after joining, the band split into two separate groups, and he went along with Steve and Chuck Romanoff, to form Schooner Fare. As part of the band, Tom sang and played electric bass, pennywhistle, and sometimes guitar. He also wrote a number of songs for the band, which included "Salt Water Farm", "Big House, Middle House, Back House, Barn", "John Cook", "The Royal Tar", and "Way Down Below."

In 1991 Rowe released a solo album titled "Whistle Up a Storm," much of which he recorded with his son, Dave Rowe at the controls. In 1992 Tom and son formed Rowe by Rowe. After six years as a duo, Rowe by Rowe added Rowe's longtime friend Denny Breau, to form Turkey Hollow Consort; later called simply Turkey Hollow. He was diagnosed with throat cancer in November of 2003, and died on January 17, 2004, due to complications from the cancer treatment. Schooner Fare continues to perform: the Romanoff brothers decided against replacing him, and perform now as a duo. Turkey Hollow continued together for about two more years, but subsequently disbanded. Since then Dave Rowe continues to perform as soloist, with his trio, and with The Squid Jiggers.

External links and references[link]

• "Tom Rowe" (obituary). Outer Green Records web site. 2005-03-05. http://www.outergreen.com/tomrowe.html. Retrieved 2008-02-29. 

This article about a United States singer is a stub. You can help Wikipedia by expanding it. v t e

Gary Robinson
Nationality	American
Education	Bard College, Courant Institute[1]
Occupation	Computer programmer
Employer	Emergent Music LLC[1]
Known for	SpamBayes, SpamAssassin, Recommendation engine, Collaborative filtering
Title	Chief Technology officer[1]
Website
GaryRobinson.net

Gary Robinson is an American software engineer notable for his mathematical algorithms to fight spam.^[2]

Fighting spam with algorithms[link]

In 2003, Robinson published an article in Linux Journal which discussed mathematical approaches for fighting spam which led to work along with Tim Peters on the SpamBayes project which began in 2002.^[3][4] The SpamBayes approach was notable because it assigned scores to both spam and ham (useful emails) and used an algorithm to guess intelligently whether an incoming email was spam; the scoring system enabled the program to return a value of unsure if both the spam and ham scores were high.^[5] His mathematical approaches were also used in projects such as SpamAssassin.^[6][7] SpamAssassin combined a Bayesian statistical approach using a text-classifier rule-based approach to whittle down spam.^[7][8] Spamming is the abuse of electronic messaging systems to send unsolicited, undesired bulk messages, according to David Anderson, who quoted Robinson's approaches in 2006.^[9] Robinson commented in Linux Journal on how fighting spam was a collaborative effort:

The approach described here truly has been a distributed effort in the best open-source tradition. Paul Graham, an author of books on Lisp, suggested an approach to filtering spam in his on-line article, “A Plan for Spam”. I took his approach for generating probabilities associated with words, altered it slightly and proposed a Bayesian calculation for dealing with words that hadn't appeared very often … an approach based on the chi-square distribution for combining the individual word probabilities into a combined probability (actually a pair of probabilities—see below) representing an e-mail. Finally, Tim Peters of the Spambayes Project proposed a way of generating a particularly useful spamminess indicator based on the combined probabilities. All along the way the work was guided by ongoing testing of embodiments written in Python by Tim Peters for Spambayes and in C by Greg Louis of the Bogofilter Project. The testing was done by a number of people involved with those projects. — Gary Robinson, 2003.^[4]

Online music[link]

In 2010, Robinson was the chief technology officer at FlyFi, an online music service owned by Maine-based^[10] Emergent Discovery which uses his anti-spam programming techniques along with collaborative filtering technologies to help make music recommendations to web users.^[11][12] His blog Gary Robinson's Rants has been quoted by others in the computer and online music industries^[11] and cited by academic papers.^[9][13][14] Robinson helped develop recommendation engine technology which applies high-power mathematical techniques using software algorithms to have a computer guess intelligently about what a consumer might like.^[15] For example, if a consumer likes music by artists such as the Beach Boys, Bob Dylan and the Talking Heads, the computer software will match these preferences with a much larger dataset of other consumers who also like those three artists but which cumulatively has much greater musical knowledge than the single consumer. Accordingly, the computer will find music that the user might like but hasn't been exposed to, and therefore hopefully offer intelligent recommendations. But the mathematics behind such comparisons can become quite complex and involved. Robinson studied mathematics at Bard College and graduated in 1979 and studied further at the Courant Institute of New York University.^[1] In the 1980s, Robinson worked on an entrepreneurial start-up dating service called 212-Romance which used similar computer algorithms to match singles romantically.^[16] The New York City-based voice mail dating service created community-based automated recommendations and used collaborative filtering technologies which Robinson developed further in other capacities.

References[link]

External links[link]

• Gary Robinson's Rants blog