Circle of confusion

Encyclopedia

In optics

, a

s from a lens not coming to a perfect focus

when imaging a point source. It is also known as

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.

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.

, 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:

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:

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.

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,

0.2 mm, and

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

, 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):

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.

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 ; 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.

The blur circle, of diameter C, in the focused object plane at distance S

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 f

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.

In optics, particularly as it relates to film and photography, depth of field is the distance between the nearest and farthest objects in a scene that appear acceptably sharp in an image...

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

for 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 (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].

'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:

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

Optics

Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

, a

**circle of confusion**is an optical spot caused by a cone of light rayRay (optics)

In optics, a ray is an idealized narrow beam of light. Rays are used to model the propagation of light through an optical system, by dividing the real light field up into discrete rays that can be computationally propagated through the system by the techniques of ray tracing. This allows even very...

s from a lens not coming to a perfect focus

Focus (optics)

In geometrical optics, a focus, also called an image point, is the point where light rays originating from a point on the object converge. Although the focus is conceptually a point, physically the focus has a spatial extent, called the blur circle. This non-ideal focusing may be caused by...

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

Film format

A film format is a technical definition of a set of standard characteristics regarding image capture on photographic film, for either stills or movies. It can also apply to projected film, either slides or movies. The primary characteristic of a film format is its size and shape.In the case of...

, but the most appropriate value depends on visual acuity

Visual acuity

Visual acuity is acuteness or clearness of vision, which is dependent on the sharpness of the retinal focus within the eye and the sensitivity of the interpretative faculty of the brain....

, 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.

## Two uses

Two important uses of this term and concept need to be distinguished:- 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 fieldDepth of fieldIn optics, particularly as it relates to film and photography, depth of field is the distance between the nearest and farthest objects in a scene that appear acceptably sharp in an image...

(“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. - For describing the blur spot achieved at a lens’s best focus. Recognizing that real lenses do not focus all rays perfectly under even the best conditions, the circle of confusion of a lens is a characterization of that blur spot. 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 lengthFocal lengthThe focal length of an optical system is a measure of how strongly the system converges or diverges light. For an optical system in air, it is the distance over which initially collimated rays are brought to a focus...

s of different lens zones due to spherical or other aberrationsAberration in optical systemsAberrations are departures of the performance of an optical system from the predictions of paraxial optics. Aberration leads to blurring of the image produced by an image-forming optical system. It occurs when light from one point of an object after transmission through the system does not converge...

. DiffractionDiffractionDiffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...

effects from wave optics and the finite apertureApertureIn optics, an aperture is a hole or an opening through which light travels. More specifically, the aperture of an optical system is the opening that determines the cone angle of a bundle of rays that come to a focus in the image plane. The aperture determines how collimated the admitted rays are,...

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 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

Encircled energy

The optics term encircled energy refers to a measure of concentration of energy in an optical image, or projected laser at a given range. If a single star is brought to its sharpest focus by a lens giving the smallest image possible with that given lens , calculation of the encircled energy of the...

, 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

In photographyPhotography

Photography is the art, science and practice of creating durable images by recording light or other electromagnetic radiation, either electronically by means of an image sensor or chemically by means of a light-sensitive material such as photographic film...

, 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:

- 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 resolutionImage resolutionImage resolution is an umbrella term that describes the detail an image holds. The term applies to raster digital images, film images, and other types of images. Higher resolution means more image detail....

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° (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. - Enlargement from the original image to the final image.

If there is no enlargement (e.g., a contact print of an 8×10Large formatLarge format refers to any imaging format of 4×5 inches or larger. Large format is larger than "medium format", the 6×6 cm or 6×9 cm size of Hasselblad, Rollei, Kowa, Pentax etc cameras , and much larger than the 24×36 mm frame of 35 mm format.The main advantage...

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

Zeiss formula

In photographic optics, the Zeiss formula is a supposed formula for computing a circle of confusion criterion for depth of field calculations...

”, 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 of0.2 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

Snellen chart

A Snellen chart is an eye chart used by eye care professionals and others to measure visual acuity. Snellen charts are named after the Dutch ophthalmologist Herman Snellen who developed the chart during 1862...

criterion of

30 cycles/degree for normal vision) 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 (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

Geometrical optics

Geometrical optics, or ray optics, describes light propagation in terms of "rays". The "ray" in geometric optics is an abstraction, or "instrument", which can be used to approximately model how light will propagate. Light rays are defined to propagate in a rectilinear path as far as they travel in...

imply that any

arbitrary DoF can be achieved by using a sufficiently small CoC. Because

of diffraction

Diffraction

Diffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...

, however, this isn't quite true. Using a smaller CoC

requires increasing the lens f-number

F-number

In optics, the f-number of an optical system expresses the diameter of the entrance pupil in terms of the focal length of the lens; in simpler terms, the f-number is the focal length divided by the "effective" aperture diameter...

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

Depth of field

In optics, particularly as it relates to film and photography, depth of field is the distance between the nearest and farthest objects in a scene that appear acceptably sharp in an image...

article for a more

detailed discussion.

### Circle of confusion diameter limit based on d/1500

Image Format Film format A film format is a technical definition of a set of standard characteristics regarding image capture on photographic film, for either stills or movies. It can also apply to projected film, either slides or movies. The primary characteristic of a film format is its size and shape.In the case of... |
Frame size | CoC |
---|---|---|

Small Format | ||

Four Thirds System Four Thirds System The Four Thirds system is a standard created by Olympus and Kodak for digital single-lens reflex camera design and development.The system provides a standard that, with digital cameras and lenses available from multiple manufacturers, allows for the interchange of lenses and bodies from different... |
13.5 mm × 18 mm | 0.015 mm |

APS-C Advanced Photo System Advanced Photo System is a film format for still photography first produced in 1996. It was marketed by Eastman Kodak under the brand name Advantix, by FujiFilm under the name Nexia, by AgfaPhoto under the name Futura and by Konica as Centuria.- Design :The film is 24 mm wide, and has three... |
15.0 mm × 22.5 mm | 0.018 mm |

APS-C Canon Advanced Photo System Advanced Photo System is a film format for still photography first produced in 1996. It was marketed by Eastman Kodak under the brand name Advantix, by FujiFilm under the name Nexia, by AgfaPhoto under the name Futura and by Konica as Centuria.- Design :The film is 24 mm wide, and has three... |
14.8 mm × 22.2 mm | 0.018 mm |

APS-C Nikon/Pentax/Sony Advanced Photo System Advanced Photo System is a film format for still photography first produced in 1996. It was marketed by Eastman Kodak under the brand name Advantix, by FujiFilm under the name Nexia, by AgfaPhoto under the name Futura and by Konica as Centuria.- Design :The film is 24 mm wide, and has three... |
15.7 mm × 23.6 mm | 0.019 mm |

APS-H Canon Advanced Photo System |
19.0 mm × 28.7 mm | 0.023 mm |

35 mm 135 film The term 135 was introduced by Kodak in 1934 as a designation for cartridge film wide, specifically for still photography. It quickly grew in popularity, surpassing 120 film by the late 1960s to become the most popular photographic film format... |
24 mm × 36 mm | 0.029 mm |

Medium Format | ||

645 (6×4.5) | 56 mm × 42 mm | 0.047 mm |

6×6 | 56 mm × 56 mm | 0.053 mm |

6×7 | 56 mm × 69 mm | 0.059 mm |

6×9 | 56 mm × 84 mm | 0.067 mm |

6×12 | 56 mm × 112 mm | 0.083 mm |

6×17 | 56 mm × 168 mm | 0.12 mm |

Large Format Large format Large format refers to any imaging format of 4×5 inches or larger. Large format is larger than "medium format", the 6×6 cm or 6×9 cm size of Hasselblad, Rollei, Kowa, Pentax etc cameras , and much larger than the 24×36 mm frame of 35 mm format.The main advantage... |
||

4×5 | 102 mm × 127 mm | 0.11 mm |

5×7 | 127 mm × 178 mm | 0.15 mm |

8×10 | 203 mm × 254 mm | 0.22 mm |

### 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 S

_{1}, is an unfocused virtual image of the object at distance S_{2}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 f

_{1}: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

F-number

In optics, the f-number of an optical system expresses the diameter of the entrance pupil in terms of the focal length of the lens; in simpler terms, the f-number is the focal length divided by the "effective" aperture diameter...

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

Depth of field

In optics, particularly as it relates to film and photography, depth of field is the distance between the nearest and farthest objects in a scene that appear acceptably sharp in an image...

.

More generally, this approach leads to an exact paraxial result for all optical systems if A is the entrance pupil

Entrance pupil

In an optical system, the entrance pupil is the optical image of the physical aperture stop, as 'seen' through the front of the lens system. The corresponding image of the aperture as seen through the back of the lens system is called the exit 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

Hyperfocal distance

In optics and photography, hyperfocal distance is a distance beyond which all objects can be brought into an "acceptable" focus. There are two commonly used definitions of hyperfocal distance, leading to values that differ only slightly:...

, with approximately equivalent results.

### Henry Coddington 1829

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

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

Circle-of-confusion calculations: An early precursor to depth of fieldDepth 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

for 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 (i.e. he does not solve for the hyperfocal distance

Hyperfocal distance

In optics and photography, hyperfocal distance is a distance beyond which all objects can be brought into an "acceptable" focus. There are two commonly used definitions of hyperfocal distance, leading to values that differ only slightly:...

), 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

T Dallmeyer (1892, 24), in an expanded re-publication of his father John Henry DallmeyerJohn Henry Dallmeyer

John Henry Dallmeyer , Anglo-German optician, was born at Loxten, Westphalia, the son of a landowner.On leaving school at the age of sixteen he was apprenticed to an Osnabrück optician, and in 1851 he came to London, where he obtained work with an optician, W Hewitt, who shortly afterwards, with...

'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

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

- Airy disk
- Astigmatism
- BokehBokehIn photography, bokeh is the blur, or the aesthetic quality of the blur, in out-of-focus areas of an image, or "the way the lens renders out-of-focus points of light."...
- Chromatic aberration
- Point spread functionPoint spread functionThe point spread function describes the response of an imaging system to a point source or point object. A more general term for the PSF is a system's impulse response, the PSF being the impulse response of a focused optical system. The PSF in many contexts can be thought of as the extended blob...

## External links

- 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