Editing Circle of confusion (section)

==Circle of confusion diameter limit in photography==
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:

{{ordered list
|1 = Visual acuity. For most people, the closest comfortable viewing distance, termed the ''near distance for distinct vision'',{{sfn|Ray|2000|p=52}} is approximately 25&nbsp;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&nbsp;mm in the final image.

|2 = Viewing conditions. If the final image is viewed at approximately 25&nbsp;cm, a final-image CoC of 0.2&nbsp;mm often is appropriate. A comfortable viewing distance is also one at which the angle of view is approximately 60°;{{sfn|Ray|2000|p=52}} at a distance of 25&nbsp;cm, this corresponds to about 30&nbsp;cm, approximately the diagonal of an [[large format|8-inch&nbsp;× 10-inch image]] (for comparison, [[A4 paper]] is {{cvt|210|×|297|mm|in|1|order=flip|disp=comma}}; [[Letter (paper size)|US Letter paper]] is {{cvt|8.5|×|11|in|mm|0|disp=comma}}). It often may be reasonable to assume that, for whole-image viewing, a final image larger than 8&nbsp;in&nbsp;× 10&nbsp;in will be viewed at a distance correspondingly greater than 25&nbsp;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&nbsp;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&nbsp;mm original image is enlarged to 25&nbsp;cm (10&nbsp;inches), the enlargement factor is approximately 7, and the CoC for the original image is 0.2&nbsp;mm&nbsp;/&nbsp;7, or 0.029&nbsp;mm.
}}

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:

{{block indent|1= CoC (in mm) = (viewing distance (in cm) / 25&nbsp;cm ) / (desired final-image resolution in lp/mm for a 25&nbsp;cm viewing distance) / enlargement}}

For example, to support a final-image resolution equivalent to 5&nbsp;lp/mm for a 25&nbsp;cm viewing distance when the anticipated viewing distance is 50&nbsp;cm and the anticipated enlargement is 8:

{{block indent|1= CoC = (50 / 25) / 5 / 8 = 0.05&nbsp;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&nbsp;cm width, along with a conventional final-image CoC of 0.2&nbsp;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&nbsp;mm format (24&nbsp;mm&nbsp;×&nbsp;36&nbsp;mm, 43&nbsp;mm diagonal), a widely used CoC limit is {{mvar|d}}/1500, or 0.029&nbsp;mm for full-frame 35&nbsp;mm format, which corresponds to resolving 5 lines per millimeter on a print of 30&nbsp;cm diagonal. Values of 0.030&nbsp;mm and 0.033&nbsp;mm are also common for full-frame 35&nbsp;mm format.

Criteria relating CoC to the lens focal length have also been used. Kodak recommended 2 minutes of arc (the [[Snellen chart|Snellen]] criterion of 30&nbsp;cycles/degree for normal vision) for critical viewing, yielding a CoC of about {{mvar|f}}/1720, where {{mvar|f}} is the lens focal length.{{sfn|Kodak|1972|p=5}} For a 50&nbsp;mm lens on full-frame 35&nbsp;mm format, the corresponding CoC is 0.0291&nbsp;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):

{{block indent|1= Viewing distance = focal length of taking lens × enlargement}}

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 {{mvar|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&nbsp;in&nbsp;× 10&nbsp;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.

===Circle of confusion diameter limit based on ''d''/1500===
{| class="wikitable sortable plainrowheaders"
|+ CoC diameter based on {{mvar|d}}/1500 for several image formats
|-
! scope="col" | [[Film format|Image Format]]
! scope="col" | Format class
! scope="col" | Frame size<ref>The frame size is an average of cameras that take photographs of this format. For example, not all 6×7 cameras take frames that are exactly {{val|56|u=mm}}&nbsp;× {{val|69|u=mm}}. Check with the specifications of a particular camera if this level of exactness is needed.</ref>
! scope="col" | CoC
|-
! scope="row | [[Image sensor format#Table of sensor formats and sizes|1" sensor (Nikon 1, Sony RX10, Sony RX100)]]
| rowspan=7 | Small format
| 8.8&nbsp;mm × 13.2&nbsp;mm
| 0.011&nbsp;mm
|-
! scope="row | [[Four Thirds System]]
| 13.5&nbsp;mm × 18&nbsp;mm
| 0.015&nbsp;mm
|-
! scope="row | [[Advanced Photo System|APS-C]]<ref>"[[APS-C]]" is a common format for digital SLRs. Dimensions vary slightly among different manufacturers; for example, Canon’s APS-C format is nominally {{val|15.0|u=mm}}&nbsp;× {{val|22.5|u=mm}}, while Nikon’s [[Nikon DX format|DX format]] is nominally {{val|16|u=mm}}&nbsp;× {{val|24|u=mm}}. Exact dimensions sometimes vary slightly among models with the same nominal format from a given manufacturer.</ref>
| 15.0&nbsp;mm × 22.5&nbsp;mm
| 0.018&nbsp;mm
|-
! scope="row | [[Advanced Photo System|APS-C Canon]]
| 14.8&nbsp;mm × 22.2&nbsp;mm
| 0.018&nbsp;mm
|-
! scope="row | [[Advanced Photo System|APS-C Nikon/Pentax/Sony]]
| 15.7&nbsp;mm × 23.6&nbsp;mm
| 0.019&nbsp;mm
|-
! scope="row | [[Advanced Photo System|APS-H Canon]]
| 19.0&nbsp;mm × 28.7&nbsp;mm
| 0.023&nbsp;mm
|-
! scope="row | [[135 film|35 mm]]
| 24&nbsp;mm × 36&nbsp;mm
| 0.029&nbsp;mm
|-
! scope="row | 645 (6×4.5)
| rowspan=6 | [[Medium format (film)|Medium Format]]
| 56&nbsp;mm × 42&nbsp;mm
| 0.047&nbsp;mm
|-
! scope="row | 6×6
| 56&nbsp;mm × 56&nbsp;mm
| 0.053&nbsp;mm
|-
! scope="row | 6×7
| 56&nbsp;mm × 69&nbsp;mm
| 0.059&nbsp;mm
|-
! scope="row | 6×9
| 56&nbsp;mm × 84&nbsp;mm
| 0.067&nbsp;mm
|-
! scope="row | 6×12
| 56&nbsp;mm × 112&nbsp;mm
| 0.083&nbsp;mm
|-
! scope="row | 6×17
| 56&nbsp;mm × 168&nbsp;mm
| 0.12&nbsp;mm
|-
! scope="row | 4×5
| rowspan=3 | [[large format|Large Format]]
| 102&nbsp;mm × 127&nbsp;mm
| 0.11&nbsp;mm
|-
! scope="row | 5×7
| 127&nbsp;mm × 178&nbsp;mm
| 0.15&nbsp;mm
|-
! scope="row | 8×10
| 203&nbsp;mm × 254&nbsp;mm
| 0.22&nbsp;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#Close-up 2|Depth of field]] article that

<math display=block>\mathrm {DoF} = \frac
{2 N c \left ( m + 1 \right )}
{m^2 - \left ( \frac {N c} {f} \right )^2} \,,
</math>

where {{mvar|N}} is the lens f-number, {{mvar|c}} is the CoC, {{mvar|m}} is the magnification, and {{mvar|f}} is the lens focal length. Because the f-number and CoC occur only as the product {{mvar|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&nbsp;mm, and the actual conditions require a CoC of 0.025&nbsp;mm, the CoC must be decreased by a factor of {{nowrap|1=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===
[[File:Circle of confusion calculation diagram.svg|thumb|400px|right|Lens and ray diagram for calculating the circle of confusion diameter {{mvar|c}} for an out-of-focus subject at distance {{math|''S''<sub>2</sub>}} when the camera is focused at {{math|''S''<sub>1</sub>}}. The auxiliary blur circle {{mvar|C}} in the object plane (dashed line) makes the calculation easier.]]
[[File:Long Short Focus 1866.jpg|thumb|400px|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 {{mvar|C}}, in the focused object plane at distance {{math|''S''<sub>1</sub>}}, is an unfocused virtual image of the object at distance {{math|''S''<sub>2</sub>}} as shown in the diagram. It depends only on these distances and the aperture diameter {{mvar|A}}, via similar triangles, independent of the lens focal length:

<math display=block> C = A {|S_2 - S_1| \over S_2} \,.</math>

The circle of confusion in the image plane is obtained by multiplying by magnification {{mvar|m}}:

<math display=block> c = C m \,,</math>

where the magnification {{mvar|m}} is given by the ratio of focus distances:

<math display=block> m = {f_1 \over S_1} \,.</math>

Using the lens equation we can solve for the auxiliary variable {{math|''f''<sub>1</sub>}}:

<math display=block> {1 \over f} = {1 \over f_1} + {1 \over S_1} \,,</math>

which yields

<math display=block> f_1 = {f S_1 \over S_1 - f} \,,</math>

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

<math display=block> m = {f \over S_1 - f} \,,</math>

which gives the final result:

<math display=block> c = A {|S_2 - S_1| \over S_2} {f \over S_1 - f} \,.</math>

This can optionally be expressed in terms of the [[f-number]] {{math|1= ''N'' = ''f/A''}} as:

<math display=block> c = {|S_2 - S_1| \over S_2} {f^2 \over N(S_1 - f)} \,.</math>

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 {{mvar|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 {{mvar|A}} is the [[entrance pupil]] diameter, the subject distances are measured from the entrance pupil, and the magnification is known:

<math display=block> c = A m {|S_2 - S_1| \over S_2} \,.</math>

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:

<math display=block> c = {f A \over S_2} = {f^2 \over N S_2} \,.</math>

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

<math display=block> c = {f A \over S_1 - f} = {f^2 \over N(S_1 - f)} \,.</math>

If the {{mvar|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.