# Depth of field

A macro photograph illustrating the effect of depth of field on a tilted object.

For many cameras, depth of field (DOF) is the distance between the nearest and the farthest objects that are in acceptably sharp focus in an image. The depth of field can be calculated based on focal length, distance to subject, the acceptable circle of confusion size, and aperture. A particular depth of field may be chosen for technical or artistic purposes. Limitations of depth of field can sometimes be overcome with various techniques/equipment.

## Factors affecting depth of field

Effect of aperture on blur and DOF. The points in focus (2) project points onto the image plane (5), but points at different distances (1 and 3) project blurred images, or circles of confusion. Decreasing the aperture size (4) reduces the size of the blur spots for points not in the focused plane, so that the blurring is imperceptible, and all points are within the DOF.

For Cameras that can only focus on one object distance at a time, Depth of field is the distance between the nearest and the farthest objects that are in acceptably sharp focus.[1] "Acceptably sharp focus" is defined using a property called the circle of confusion.

The depth of field can be determined by focal length, distance to subject, the acceptable circle of confusion size, and aperture.[2] The approximate depth of field can be given by:

${\displaystyle {\text{DOF}}\approx {\frac {2u^{2}Nc}{f^{2}}}}$

for a given circle of confusion, c, focal length, f, F-number, N, and distance to subject, u.[3][4]

As distance or the size of the acceptable circle of confusion increases, the depth of field increases; however, increasing the size of the aperture or increasing the focal length reduces the depth of field. Depth of Field changes linearly with F-number and circle of confusion, but changes in proportional to the square of the focal length and the distance to the subject. As a result, photos taken at extremely close range have a proportionally much smaller depth of field.

Sensor size affects DOF only in that changing the sensor size on a camera requires changing the focal length to get the same picture. It is the change in focal length that then affects the DOF.[5][6][7]

### Effect of lens aperture

Points in an image produce a blur spot shaped like the aperture. Custom shaped apertures and bright points of light can be used artistically to produce images like this

For a given subject framing and camera position, the DOF is controlled by the lens aperture diameter, which is usually specified as the f-number (the ratio of lens focal length to aperture diameter). Reducing the aperture diameter (increasing the f-number) increases the DOF because only the light travelling at shallower angles passes through the aperture. Because the angles are shallow, the light rays are within the acceptable circle of confusion for a greater distance.[8]

Motion pictures make only limited use of this control; to produce a consistent image quality from shot to shot, cinematographers usually choose a single aperture setting for interiors and another for exteriors, and adjust exposure through the use of camera filters or light levels. Aperture settings are adjusted more frequently in still photography, where variations in depth of field are used to produce a variety of special effects.

Aperture = f/1.4. DOF=0.8 cm
Aperture = f/4.0. DOF=2.2 cm
Aperture = f/22. DOF=12.4 cm
Depth of field for different values of aperture using 50 mm objective lens and full-frame DSLR camera. Focus point is on the first blocks column.[9]

### Effect of circle of confusion

Precise focus is only possible at an exact distance from the lens;[a] at that distance, a point object will produce a point image. Otherwise, a point object will produce a blur spot shaped like the aperture, typically and approximately a circle. When this circular spot is sufficiently small, it is visually indistinguishable from a point, and appears to be in focus. The diameter of the largest circle that is indistinguishable from a point is known as the acceptable circle of confusion, or informally, simply as the circle of confusion. Points that produce a blur spot smaller than this acceptable circle of confusion are considered acceptably sharp.

The acceptable circle of confusion depends on how the final image will be used. It is generally accepted to be 0.25 mm for an image viewed from 25 cm away.[10]

For 35 mm motion pictures, the image area on the film is roughly 22 mm by 16 mm. The limit of tolerable error was traditionally set at 0.05 mm (0.002 in) diameter, while for 16 mm film, where the size is about half as large, the tolerance is stricter, 0.025 mm (0.001 in).[11] More modern practice for 35 mm productions set the circle of confusion limit at 0.025 mm (0.001 in).[12]

### Camera movements

Scheimpflug principle

The plane of focus (POF) in object space is normally parallel to the image plane. However, tilting the lens relative to the sensor can rotate the POF.

When the POF is rotated, the near and far limits of DOF are no longer parallel; the DOF becomes wedge-shaped, with the apex of the wedge nearest the camera.[13][14]

In some cases, rotating the POF can better fit the DOF to the scene, and achieve the required sharpness at a smaller f-number. Alternatively, rotating the POF, in combination with a small f-number, can minimize the part of an image that is within the DOF.

## Object field calculation methods

Traditional depth-of-field formulas can be hard to use in practice. As an alternative, the same effective calculation can be done without regard to the focal length and f-number.[b] Moritz von Rohr and later Merklinger observe that the effective absolute aperture diameter can be used for similar formula in certain circumstances.[15]

Moreover, traditional depth-of-field formulas assume equal acceptable circles of confusion for near and far objects. Merklinger[c] suggested that distant objects often need to be much sharper to be clearly recognizable, whereas closer objects, being larger on the film, do not need to be so sharp.[15] The loss of detail in distant objects may be particularly noticeable with extreme enlargements. Achieving this additional sharpness in distant objects usually requires focusing beyond the hyperfocal distance, sometimes almost at infinity. For example, if photographing a cityscape with a traffic bollard in the foreground, this approach, termed the object field method by Merklinger, would recommend focusing very close to infinity, and stopping down to make the bollard sharp enough. With this approach, foreground objects cannot always be made perfectly sharp, but the loss of sharpness in near objects may be acceptable if recognizability of distant objects is paramount.

Other authors such as Ansel Adams have taken the opposite position, maintaining that slight unsharpness in foreground objects is usually more disturbing than slight unsharpness in distant parts of a scene.[16]

## Overcoming DOF limitations

Some methods and equipment allow altering the apparent DOF, and some even allow the DOF to be determined after the image is made. For example, Focus stacking combines multiple images focused on different planes, resulting in an image with a greater (or less, if so desired) apparent depth of field than any of the individual source images. Similarly, in order to reconstruct the 3-dimensional shape of an object, a depth map can be generated from multiple photographs with different depths of field. This method is called "shape from focus."[17]

Other technologies use a combination of lens design and post-processing: Wavefront coding is a method by which controlled aberrations are added to the optical system so that the focus and depth of field can be improved later in the process.[18]

The lens design can be changed even more: in colour apodization the lens is modified such that each colour channel has a different lens aperture. For example, the red channel may be f/2.4, green may be f/2.4, whilst the blue channel may be f/5.6. Therefore, the blue channel will have a greater depth of field than the other colours. The image processing identifies blurred regions in the red and green channels and in these regions copies the sharper edge data from the blue channel. The result is an image that combines the best features from the different f-numbers.[19]

At the extreme, a plenoptic camera captures 4D light field information about a scene, so the focus and depth of field can be altered after the photo is taken.

## Diffraction and DOF

Diffraction causes images to lose sharpness at high F-numbers, and hence limits the potential depth of field.[20] In general photography this is rarely an issue; because large f-numbers typically require long exposure times, motion blur may cause greater loss of sharpness than the loss from diffraction. However, diffraction is a greater issue in close-up photography, and the tradeoff between DOF and overall sharpness can become quite noticeable as photographers are trying to maximise depth of field with very small apertures.[21][22]

Hansma and Peterson have discussed determining the combined effects of defocus and diffraction using a root-square combination of the individual blur spots.[23][24] Hansma's approach determines the f-number that will give the maximum possible sharpness; Peterson's approach determines the minimum f-number that will give the desired sharpness in the final image, and yields a maximum depth of field for which the desired sharpness can be achieved.[d] In combination, the two methods can be regarded as giving a maximum and minimum f-number for a given situation, with the photographer free to choose any value within the range, as conditions (e.g., potential motion blur) permit. Gibson gives a similar discussion, additionally considering blurring effects of camera lens aberrations, enlarging lens diffraction and aberrations, the negative emulsion, and the printing paper.[20][e] Couzin gave a formula essentially the same as Hansma's for optimal f-number, but did not discuss its derivation.[25]

Hopkins,[26] Stokseth,[27] and Williams and Becklund[28] have discussed the combined effects using the modulation transfer function.[29][30]

## DOF scales

Detail from a lens set to f/11. The point half-way between the 1 m and 2 m marks, the DOF limits at f/11, represents the focus distance of approximately 1.33 m (the reciprocal of the mean of the reciprocals of 1 and 2 being 4/3).

DOF scale on Tessina focusing dial

Many lenses include scales that indicate the DOF for a given focus distance and f-number; the 35 mm lens in the image is typical. That lens includes distance scales in feet and meters; when a marked distance is set opposite the large white index mark, the focus is set to that distance. The DOF scale below the distance scales includes markings on either side of the index that correspond to f-numbers. When the lens is set to a given f-number, the DOF extends between the distances that align with the f-number markings.

Photographers can use the lens scales to work backwards from the desired depth of field to find the necessary focus distance and aperture.[31] For the 35 mm lens shown, if it were desired for the DOF to extend from 1 m to 2 m, focus would be set so that index mark was centered between the marks for those distances, and the aperture would be set to f/11.[f]

On a view camera, the focus and f-number can be obtained by measuring the depth of field and performing simple calculations. Some view cameras include DOF calculators that indicate focus and f-number without the need for any calculations by the photographer.[32][33]

## Hyperfocal distance

Zeiss Ikon Contessa with red marks for hyperfocal distance 20 ft at f/8

Minox LX camera with hyperfocal red dot

In optics and photography, hyperfocal distance is a distance beyond which all objects can be brought into an "acceptable" focus. As the hyperfocal distance is the focus distance giving the maximum depth of field, it is the most desirable distance to set the focus of a fixed-focus camera. The hyperfocal distance is entirely dependent upon what level of sharpness is considered to be acceptable.

The hyperfocal distance has a property called "consecutive depths of field", where a lens focused at an object whose distance is at the hyperfocal distance H will hold a depth of field from H/2 to infinity, if the lens is focused to H/2, the depth of field will extend from H/3 to H; if the lens is then focused to H/3, the depth of field will extend from H/4 to H/2, etc.

Thomas Sutton and George Dawson first wrote about hyperfocal distance (or "focal range") in 1867. Louis Derr in 1906 may have been the first to derive a formula for hyperfocal distance. Rudolf Kingslake wrote in 1951 about the two methods of measuring hyperfocal distance.

Some cameras have their hyperfocal distance marked on the focus dial. For example, on the Minox LX focusing dial there is a red dot between 2 m and infinity; when the lens is set at the red dot, that is, focused at the hyperfocal distance, the depth of field stretches from 2 m to infinity.

## Near:far distribution

The DOF beyond the subject is always greater than the DOF in front of the subject. When the subject is at the hyperfocal distance or beyond, the far DOF is infinite, so the ratio is 1:∞; as the subject distance decreases, near:far DOF ratio increases, approaching unity at high magnification. For large apertures at typical portrait distances, the ratio is still close to 1:1.

## DOF formulae

This section covers some additional formula for evaluating depth of field; however they are all subject to significant simplifying assumptions: for example, they assume paraxial (Gaussian). They are suitable for practical photography, lens designers would use significantly more complex ones.

### Focus and f-number from DOF limits

For given near and far DOF limits ${\displaystyle D_{\mathrm {N} }}$  and ${\displaystyle D_{\mathrm {F} }}$ , the required f-number is smallest when focus is set to

${\displaystyle s={\frac {2D_{\mathrm {N} }D_{\mathrm {F} }}{D_{\mathrm {N} }+D_{\mathrm {F} }}},}$

the harmonic mean of the near and far distances. In practice, this is equivalent to the arithmetic mean for shallow depths of field.[34] Sometimes, view camera users refer to the difference ${\displaystyle v_{\mathrm {N} }-v_{\mathrm {F} }}$  as the focus spread.[35]

### Foreground and background blur

If a subject is at distance ${\displaystyle s}$  and the foreground or background is at distance ${\displaystyle D}$ , let the distance between the subject and the foreground or background be indicated by

${\displaystyle x_{\mathrm {d} }=|D-s|.}$

The blur disk diameter ${\displaystyle b}$  of a detail at distance ${\displaystyle x_{\mathrm {d} }}$  from the subject can be expressed as a function of the subject magnification ${\displaystyle m_{\mathrm {s} }}$ , focal length ${\displaystyle f}$ , f-number ${\displaystyle N}$ , or alternatively the aperture ${\displaystyle d}$ , according to

${\displaystyle b={\frac {fm_{\mathrm {s} }}{N}}{\frac {x_{\mathrm {d} }}{s\pm x_{\mathrm {d} }}}=dm_{\mathrm {s} }{\frac {x_{\mathrm {d} }}{D}}.}$

The minus sign applies to a foreground object, and the plus sign applies to a background object.

The blur increases with the distance from the subject; when ${\displaystyle b}$  is less than the circule of confusion, the detail is within the depth of field.

## Notes

1. ^ strictly, at an exact distance from a plane
2. ^ notwithstanding that the f-number is derived from the focal length
3. ^ Englander describes a similar approach in his paper Apparent Depth of Field: Practical Use in Landscape Photography; Conrad discusses this approach, under Different Circles of Confusion for Near and Far Limits of Depth of Field, and The Object Field Method, in Depth of Field in Depth
4. ^ Peterson does not give a closed-form expression for the minimum f-number, though such an expression obtains from simple algebraic manipulation of his Equation 3
5. ^ The analytical section at the end of Gibson (1975) was originally published as "Magnification and Depth of Detail in Photomacrography" in the Journal of the Photographic Society of America, Vol. 26, No. 6, June 1960
6. ^ The focus distance to have the DOF extend between given near and far object distances is the harmonic mean of the object conjugates. Most helicoid-focused lenses are marked with image plane-to-subject distances,[citation needed] so the focus determined from the lens distance scale is not exactly the harmonic mean of the marked near and far distances.

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