Depth of field

In optics, particularly film and photography, the depth of field (DOF) is the distance in front of and beyond the subject that appears to be in focus. There is only one distance at which a subject is precisely in focus, but focus falls off gradually on either side of that distance, and there is a region in which the blurring is imperceptible under normal viewing conditions.

Definition of “acceptably sharp”

Several factors determine whether the objective misfocus becomes noticeable. Subject matter, movement, the distance of the subject from the camera, and the way in which the image is displayed all have an influence. However, the most important factor is the actual degree of misfocus in relation to the area of film exposed.

A point subject at the focused distance will produce a point image. A point nearer or farther away will produce a circular image. The diameter of the circle increases with distance from the point of focus; 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.

For a 35 mm motion picture, the image area on the negative is roughly 22 mm by 16 mm (0.87 in by 0.63 in). The limit of tolerable error is usually set at 0.05 mm (0.002 in) diameter. For 16 mm film, where the image area is smaller, the tolerance is stricter, 0.025 mm (0.001 in). Standard depth-of-field tables are constructed on this basis, although generally 35 mm productions set it at 0.025 mm (0.001 in). Note that the acceptable circle of confusion values for these formats are different because of the relative amount of magnification each format will need in order to be projected on a full-sized movie screen.

(A table for 35 mm still photography would be somewhat different since more of the film is used for each image and the amount of enlargement is usually much less.)

The image format size also will affect the depth of field. The larger the format size, the longer a lens will need to be to capture the same framing as a smaller format. In motion pictures, for example, a frame with a 12 degree horizontal field of view will require a 50 mm lens on 16 mm film, a 100 mm lens on 35 mm film, and a 250 mm lens on 65 mm film. Conversely, using the same focal length lens with each of these formats will yield a progressively wider image as the film format gets larger: a 50 mm lens has a horizontal field of view of 12 degrees on 16 mm film, 23.6 degrees on 35 mm film, and 55.6 degrees on 65 mm film. What this all means is that because the larger formats require longer lenses than the smaller ones, they will accordingly have a smaller depth of field. Therefore, compensations in exposure, framing, or subject distance need to be made in order to make one format look like it was filmed in another format.

f-number

For a given subject framing, the DOF is controlled by the lens f-number. Increasing the f-number (reducing the aperture diameter) increases the DOF; however, it also reduces the amount of light transmitted, and increases diffraction, placing a practical limit on the extent to which the aperture size may be reduced. 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.

Artistic considerations

Depth of field can be anywhere from a fraction of an inch to virtually infinite. For instance, a closeup of a person's face may have shallow DOF (with someone just behind that person visible but out of focus—common, for instance, in melodramas and horror films); a shot of rolling hills might have great DOF, with both the foreground and background in focus. A closeup still photograph might employ a very shallow DOF to isolate the subject from a distracting background.

Hyperfocal distance

The hyperfocal distance is the nearest focus distance at which the DOF extends to infinity; focusing the camera at the hyperfocal distance results in the largest possible depth of field for a given f-number. Focusing beyond the hyperfocal distance does not increase the far DOF (which already extends to infinity), but it does decrease the DOF in front of the subject, decreasing the total DOF. Some photographers refer to this as “wasting DOF”; however, see The object field method below. Focusing ahead of the hyperfocal distance increases the DOF ahead of the subject, but decreases DOF beyond the subject, including objects near infinity. Of course, this latter approach may be appropriate for images that do not extend to infinity.

The object field method

Traditional depth-of-field formulae and tables assume equal circles of confusion for near and far objects. Some authors, such as Merklinger (1992),^[1] have 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. 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.

Moritz von Rohr also used an object field method, but unlike Merklinger, he used the conventional criterion of a maximum circle of confusion diameter in the image plane, leading to unequal front and rear depths of field.

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; as the subject distance decreases, near:far DOF ratio increases, approaching unity at high magnification. The oft-cited “rule” that 1/3 of the DOF is in front of the subject and 2/3 is beyond is true only when the subject distance is 1/3 the hyperfocal distance.

Depth of field formulae

The basis of these formulae is given in the section Derivation of the DOF formulae.^[2]

Hyperfocal Distance

Let <math>f</math> be the lens focal length, <math>N</math> be the lens f-number, and <math>c</math> be the circle of confusion for a given image format. The hyperfocal distance <math>H</math> is given by

<math>H \approx \frac {f^2} {N c}</math>

Moderate-to-large distances

Let <math>s</math> be the distance at which the camera is focused (the “subject distance”). When <math>s</math> is large in comparison with the lens focal length, the distance <math>D_{\mathrm N}</math> from the camera to the near limit of DOF and the distance <math>D_{\mathrm F}</math> from the camera to the far limit of DOF are

<math>D_{\mathrm N} \approx \frac {H s} {H + s}</math>

<math>D_{\mathrm F} \approx \frac {H s} {H - s} \mbox{ for } s < H</math>

When the subject distance is the hyperfocal distance,

<math>D_{\mathrm F} = \infty</math>

<math>D_{\mathrm N} = \frac H 2</math>

The depth of field <math>D_{\mathrm F} - D_{\mathrm N}</math> is

<math>

\mathrm {DOF} \approx \frac {2 Hs^2} {H^2 - s^2} \mbox{ for } s < H </math>

For <math>s \ge H</math>, the far limit of DOF is at infinity and the DOF is infinite; of course, only objects at or beyond the near limit of DOF will be recorded with acceptable sharpness.

Substituting for <math>H</math> and rearranging, DOF can be expressed as

<math>\mathrm {DOF} \approx \frac {2 N c f^2 s^2} {f^4 - N^2 c^2 s^2}</math>

Thus, for a given image format, depth of field is determined by three factors: the focal length of the lens, the f-number of the lens opening (the aperture), and the camera-to-subject distance.

Close-up

When the subject distance <math>s</math> approaches the focal length, using the formulae given above can result in significant errors. For close-up work, the hyperfocal distance has little applicability, and it usually is more convenient to express DOF in terms of image magnification. Let <math>m</math> be the magnification; when the subject distance is small in comparison with the hyperfocal distance,

<math>\mathrm {DOF} \approx 2 N c \left ( \frac {m + 1} {m^2} \right ),</math>

so that for a given magnification, DOF is independent of focal length. Stated otherwise, for the same subject magnification, all focal lengths give approximately the same DOF. This statement is true only when the subject distance is small in comparison with the hyperfocal distance, however.

The discussion thus far has assumed a symmetrical lens for which the entrance and exit pupils coincide with the front and rear nodal planes, and for which the pupil magnification (the ratio of exit pupil diameter to that of the entrance pupil)^[3] is unity. Although this assumption usually is reasonable for large-format lenses, it often is invalid for medium- and small-format lenses.

When <math>s \ll H</math>, the DOF for an asymmetrical lens is

<math>\mathrm {DOF} \approx \frac {2 N c (1 + m/P)}{m^2},</math>

where <math>P</math> is the pupil magnification. When the pupil magnification is unity, this equation reduces to that for a symmetrical lens.

Except for close-up and macro photography, the effect of lens asymmetry is minimal. At unity magnification, however, the errors from neglecting the pupil magnification can be significant. Consider a telephoto lens with <math>P = 0.5</math> and a retrofocus wide-angle lens with <math>P = 2</math>, at <math>m = 1.0</math>. The asymmetrical-lens formula gives <math>\mathrm {DOF} = 6 N c</math> and <math>\mathrm {DOF} = 3 N c</math>, respectively. The symmetrical-lens formula gives <math>\mathrm {DOF} = 4 N c</math> in either case. The errors are −33% and 33%, respectively.

Focus and f-number

Not all images require that sharpness extend to infinity; for given near and far DOF limits <math>D_{\mathrm N}</math> and <math>D_{\mathrm F}</math>, the required f-number is smallest when focus is set to

<math>s = \frac {2 D_{\mathrm N} D_{\mathrm F} }

{D_{\mathrm N} + D_{\mathrm F} } </math>

When the subject distance is large in comparison with the lens focal length, the required f-number is

<math>N \approx \frac {f^2} {c}

\frac {D_{\mathrm F} - D_{\mathrm N} } {2 D_{\mathrm N} D_{\mathrm F} } </math>

In practice, these settings usually are determined on the image side of the lens, using measurements on the bed or rail with a view camera, or using lens DOF scales on manual-focus lenses for small- and medium-format cameras. If <math>v_{\mathrm N}</math> and <math>v_{\mathrm F}</math> are the image distances that correspond to the near and far limits of DOF, the required f-number is minimized when the image distance <math>v</math> is

<math>v \approx \frac { v_{\mathrm N} + v_{\mathrm F} } {2}

= v_{\mathrm F} + \frac { v_{\mathrm N} - v_{\mathrm F} } {2} </math>

In practical terms, focus is set to halfway between the near and far image distances. The required f-number is

<math>N \approx \frac { v_{\mathrm N} - v_{\mathrm F} } { 2 c }</math>

The image distances are measured from the camera's image plane to the lens's image nodal plane, which is not always easy to locate. In most cases, focus and f-number can be determined with sufficient accuracy using the approximate formulae above, which require only the difference between the near and far image distances; view camera users often refer to the difference <math>v_{\mathrm N} - v_{\mathrm F}</math> as the focus spread. Most lens DOF scales are based on the same concept.

Practical complications

The distance scales on most medium- and small-format lenses indicate distance from the camera's image plane. Most DOF formulae, including those in this article, use the object distance <math>s</math> from the lens's object nodal plane, which often is not easy to locate. Moreover, for many zoom lenses and internal-focusing non-zoom lenses, the location of the object nodal plane, as well as focal length, changes with subject distance. When the subject distance is large in comparison with the lens focal length, the exact location of the object nodal plane is not critical; the distance is essentially the same whether measured from the front of the lens, the image plane, or the actual nodal plane. The same is not true for close-up photography; at unity magnification, a slight error in the location of the object nodal plane can result in a DOF error greater than the errors from any approximations in the DOF equations.

The asymmetrical lens formulae require knowledge of the pupil magnification, which usually is not specified for medium- and small-format lenses. The pupil magnification can be estimated by looking into the front and rear of the lens and measuring the diameters of the apparent apertures, and computing the ratio (rear diameter divided by front diameter).^[4] However, for many zoom lenses and internal-focusing non-zoom lenses, the pupil magnification changes with subject distance, and several measurements may be required.

Limitations

Most DOF formulae, including those discussed in this article, employ several simplifications:

1. Paraxial (Gaussian) optics is assumed, and technically, the formulae are valid only for rays that are infinitessimally close to the lens axis. However, Gaussian optics usually is more than adequate for determining DOF, and non-paraxial formulae are sufficiently complex that requiring their use would make determination of DOF impractical in most cases.
2. Lens aberrations are ignored. Including the effects of aberrations is nearly impossible, because doing so requires knowledge of the specific lens design. Moreover, in well-designed lenses, most aberrations are well corrected, and at least near the optical axis, often are almost negligible when the lens is stopped down 2–3 steps from maximum aperture. Because lenses usually are stopped down at least to this point when DOF is of interest, ignoring aberrations usually is reasonable. Not all aberrations are reduced by stopping down, however, so actual sharpness may be slightly less than predicted by DOF formulae.
3. Diffraction is ignored. DOF formulae imply that any arbitrary DOF can be achieved by using a sufficiently large f-number. Because of diffraction, however, this isn't quite true. Once a lens is stopped down to where most aberrations are well corrected, stopping down further will decrease sharpness in the center of the field. At the DOF limits, however, further stopping down decreases the size of the defocus blur spot, and the overall sharpness may increase. Consequently, choosing an f-number sometimes involves a tradeoff between center and edge sharpness, although viewers typically prefer uniform sharpness to slightly greater center sharpness. The choice, of course, is subjective, and may depend upon the particular image. Eventually, the defocus blur spot becomes negligibly small, and further stopping down serves only to decrease sharpness even at DOF limits. Typically, diffraction at DOF limits becomes significant only at fairly large f-numbers; because large f-numbers typically require long exposure times, motion blur often causes greater loss of sharpness than does diffraction. Combined defocus and diffraction is discussed in Hansma (1996) and in Conrad's Depth of Field in Depth (PDF) and Jacobson's Photographic Lenses Tutorial.
4. Post-capture manipulation of the image is ignored. Sharpening via techniques such as deconvolution or unsharp mask can increase the DOF in the final image, particularly when the original image has a large DOF. Conversely, noise reduction can reduce the DOF.
5. For digital capture with color filter array sensors, demosaicing is ignored. Demosaicing alone would normally reduce the DOF, but the demosaicing algorithm used might also include sharpening.

The lens designer cannot restrict analysis to Gaussian optics and cannot ignore lens aberrations. However, the requirements of practical photography are less demanding than those of lens design, and despite the simplifications employed in development of most DOF formulae, these formulae have proven useful in determining camera settings that result in acceptably sharp pictures. It should be recognized that DOF limits are not hard boundaries between sharp and unsharp, and that there is little point in determining DOF limits to a precision of many significant figures.

DOF vs. format size

To a first approximation, DOF is inversely proportional to format size. More precisely, if photographs with the same final-image size are taken in two different camera formats at the same subject distance with the same field of view and f-number, the DOF is, to a first approximation, inversely proportional to the format size. Strictly speaking, this is true only when the subject distance is large in comparison with the focal length and small in comparison with the hyperfocal distance, for both formats, but it nonetheless is generally useful for comparing results obtained from different formats

To maintain the same field of view, the lens focal lengths must be in proportion to the format sizes. Assuming, for purposes of comparison, that the 4×5 format is four times the size of 35 mm format, if a 4×5 camera used a 300 mm lens, a 35 mm camera would need a 75 mm lens for the same field of view. For the same f-number, the image made with the 35 mm camera would have four times the DOF of the image made with the 4×5 camera.

In many cases, the DOF is fixed by the requirements of the desired image. For a given DOF and field of view, the required f-number is proportional to the format size. For example, if a 35 mm camera required f/11, a 4×5 camera would require f/45 to give the same DOF. For the same ISO speed, the exposure time on the 4×5 would be sixteen times as long; if the 35 camera required 1/250 second, the 4×5 camera would require 1/15 second. In windy conditions, the exposure time with the larger camera might allow motion blur. Adjusting the f-number to the camera format is equivalent to maintaining the same absolute aperture diameter.

The greater DOF with the smaller format can be either an advantage or a disadvantage, depending on the desired effect. For the same amount of foreground and background blur, a small-format camera requires a smaller f-number and allows a shorter exposure time than a large-format camera; however, many point-and-shoot digital cameras cannot provide a very shallow DOF. For example, a point-and-shoot digital camera with a 1/1.8″ sensor (7.18 mm × 5.32 mm) at a normal focal length and f/2.8 has the same DOF as a 35 mm camera with a normal lens at f/13.

In some cases, movements (tilt or swing) can be used with view cameras to better fit the DOF to the scene, and achieve the required sharpness at a smaller f-number. A few small-format cameras can employ the same principle by using tilt/shift lenses.

Photolithography

In semiconductor photolithography applications, depth of field is extremely important as integrated circuit layout features must be printed with high accuracy at extremely small size. The difficulty is that the wafer surface is not perfectly flat, but may vary by several micrometres. Even this small variation causes some distortion in the projected image, and results in unwanted variations in the resulting pattern. Thus photolithography engineers take extreme measures to maximize the optical depth of field of the photolithography equipment. To minimize this distortion further, chip makers like IBM are forced to use chemical mechanical polishing machines to make the wafer surface even flatter before lithographic patterning.

Ophthalmology and optometry

A person may sometimes experience better vision in daylight than at night because of an increased depth of field due to constriction of the pupil (i.e., miosis).

Increasing DOF by digital compositing

Digital image processing can increase the depth of field of a photograph by combining multiple images taken at different focus distances, or by using techniques such as wavefront coding or plenoptic cameras. Available programs for multi-shot DOF enhancement include Helicon Focus and CombineZ5. See the linked online article by Rik Littlefield.

Getting sufficient depth of field can be particularly challenging in macro photography. The images at right illustrate the increase in DOF that can be achieved by combining multiple exposures.

Derivation of the DOF formulae

DOF limits

A symmetrical lens is illustrated at right. The subject at distance <math>s</math> is in focus at image distance <math>v</math>. Point objects at distances <math>D_\mathrm F</math> and <math>D_\mathrm N</math> would be in focus at image distances <math>v_\mathrm F</math> and <math>v_\mathrm N</math>, respectively; at image distance <math>v</math>, they are imaged as blur spots. The depth of field is controlled by the aperture stop diameter <math>d</math>; when the blur spot diameter is equal to the acceptable circle of confusion <math>c</math>, the near and far limits of DOF are at <math>D_\mathrm N</math> and <math>D_\mathrm F</math>. From similar triangles,

<math>\frac {v_\mathrm N - v} {v_\mathrm N} = \frac c d</math>

<math>\frac {v- v_\mathrm F} {v_\mathrm F} = \frac c d</math>

It usually is more convenient to work with the lens f-number than the aperture diameter; the f-number <math>N</math> is related to the lens focal length <math>f</math> and the aperture diameter <math>d</math> by

<math>N = \frac f d\,;</math>

substituting into the previous equations and rearranging gives

<math>v_\mathrm N = \frac {fv} {f - Nc}</math>

<math>v_\mathrm F = \frac {fv} {f + Nc}</math>

The image distance <math>v</math> is related to an object distance <math>u</math> by the thin-lens equation

<math>\frac 1 u + \frac 1 v = \frac 1 f\,;</math>

substituting into the two previous equations and rearranging gives the near and far limits of DOF:

<math>D_{\mathrm N} = \frac {s f^2} {f^2 + N c ( s - f ) }</math>

<math>D_{\mathrm F} = \frac {s f^2} {f^2 - N c ( s - f ) }</math>

Hyperfocal distance

Setting the far limit of DOF <math>D_{\mathrm F}</math> to infinity and solving for the focus distance <math>s</math> gives

<math>s = H = \frac {f^2} {N c} + f,</math>

where <math>H</math> is the hyperfocal distance. Setting the subject distance to the hyperfocal distance and solving for the near limit of DOF gives

<math>D_{\mathrm N} = \frac {f^2 / ( N c ) + f} {2} = \frac {H}{2}</math>

For any practical value of <math>H</math>, the focal length is negligible in comparison, so that

<math>H \approx \frac {f^2} {N c}</math>

Substituting the approximate expression for hyperfocal distance into the formulae for the near and far limits of DOF gives

<math>D_{\mathrm N} = \frac {H s}{H + ( s - f )}</math>

<math>D_{\mathrm F} = \frac {H s}{H - ( s - f )}</math>

Combining, the depth of field <math>D_{\mathrm F} - D_{\mathrm N}</math> is

<math>

\mathrm {DOF} = \frac {2 H s (s - f )} {H^2 - ( s - f )^2} \mbox{ for } s < H </math>

Moderate-to-large distances

When the subject distance is large in comparison with the lens focal length,

<math>D_{\mathrm N} \approx \frac {H s} {H + s}</math>

<math>D_{\mathrm F} \approx \frac {H s} {H - s} \mbox{ for } s < H</math>

<math>

\mathrm {DOF} \approx \frac {2 H s^2} {H^2 - s^2} \mbox{ for } s < H </math>

For <math>s \ge H</math>, the far limit of DOF is at infinity and the DOF is infinite; of course, only objects at or beyond the near limit of DOF will be recorded with acceptable sharpness.

Close-up

When the subject distance <math>s</math> approaches the lens focal length, the focal length no longer is negligible, and the approximate formulae above cannot be used without introducing significant error. At close distances, the hyperfocal distance has little applicability, and it usually is more convenient to express DOF in terms of magnification. Substituting

<math>s = \frac {m + 1} {m} f</math>

and

<math>s - f = \frac {f} {m}</math>

into the formula for DOF and rearranging gives

<math>

\mathrm {DOF} = \frac {2 f ( m + 1 ) / m } { ( f m ) / ( N c ) - ( N c ) / ( f m ) } </math>

At the hyperfocal distance, the terms in the denominator are equal, and the DOF is infinite. As the subject distance decreases, so does the second term in the denominator; when <math>s \ll H</math>, the second term becomes small in comparison with the first, and

<math>\mathrm {DOF} \approx 2 N c \left ( \frac {m + 1} {m^2} \right ),</math>

so that for a given magnification, DOF is independent of focal length. Stated otherwise, for the same subject magnification, all focal lengths for a given image format give approximately the same DOF. This statement is true only when the subject distance is small in comparison with the hyperfocal distance, however. Multiplying the numerator and denominator of the exact formula by

<math>\frac {N c m} {f}</math>

gives

<math>\mathrm {DOF} = \frac

{2 N c \left ( m + 1 \right )} {m^2 - \left ( \frac {N c} {f} \right )^2} </math>

Decreasing the focal length <math>f</math> increases the second term in the denominator, decreasing the denominator and increasing the value of the right-hand side, so that a shorter focal length gives greater DOF. The effect of focal length is greatest near the hyperfocal distance, and decreases as subject distance is decreased. However, the near/far perspective will differ for different focal lengths, so the difference in DOF may not be readily apparent. When the subject distance is small in comparison with the hyperfocal distance, the effect of focal length is negligible, and, as noted above, the DOF essentially is independent of focal length.

Near:far DOF ratio

From the “exact” equations for near and far limits of DOF, the DOF in front of the subject is

<math>s - D_{\mathrm N} = \frac {Ncs(s - f)} {f^2 + Nc(s - f)}\,,</math>

and the DOF beyond the subject is

<math>D_{\mathrm F} - s = \frac {Ncs(s - f)} {f^2 - Nc(s - f)}</math>

The near:far DOF ratio is

<math>\frac {s - D_{\mathrm N}} {D_{\mathrm F} - s}

= \frac {f^2 - Nc(s - f)} {f^2 + Nc(s - f)}</math>

This ratio is always less than unity; at moderate-to-large subject distances, <math>f \ll s</math>, and

<math>\frac {s - D_{\mathrm N}} {D_{\mathrm F} - s}

\approx \frac {f^2 - Ncs} {f^2 + Ncs} = \frac {H - s} {H + s}</math>

When the subject is at the hyperfocal distance or beyond, the far DOF is infinite, and the near:far ratio is zero. It's commonly stated that approximately 1/3 of the DOF is in front of the subject and approximately 2/3 is beyond; however, this is true only when <math>s \approx H/3</math>.

At closer subject distances, it's often more convenient to express the DOF ratio in terms of the magnification

<math>m = \frac f {s - f}</math>

Substitution into the “exact” equation for DOF ratio gives

<math>\frac {s - D_{\mathrm N}} {D_{\mathrm F} - s}

= \frac {m - Nc/f} {m + Nc/f}</math>

As magnification increases, the near:far ratio approaches a limiting value of unity.

Focus and f-number

Not all images require that sharpness extend to infinity; the equations for the DOF limits can be combined to eliminate <math>Nc</math> and solve for the subject distance. For given near and far DOF limits <math>D_{\mathrm N}</math> and <math>D_{\mathrm F}</math>, the subject distance is

<math>s = \frac {2 D_{\mathrm N} D_{\mathrm F} }

{D_{\mathrm N} + D_{\mathrm F} } </math>

The equations for DOF limits also can be combined to eliminate <math>s</math> and solve for the required f-number, giving

<math>N = \frac {f^2} {c}

\frac {D_{\mathrm F} - D_{\mathrm N} } {D_{\mathrm F} ( D_{\mathrm N} - f ) + D_{\mathrm N} ( D_{\mathrm F} - f ) } </math>

When the subject distance is large in comparison with the lens focal length, this simplifies to

<math>N \approx \frac {f^2} {c}

\frac {D_{\mathrm F} - D_{\mathrm N} } {2 D_{\mathrm N} D_{\mathrm F} } </math>

Most discussions of DOF concentrate on the object side of the lens, but the formulae are simpler and the measurements usually easier to make on the image side. If <math>v_{\mathrm N}</math> and <math>v_{\mathrm F}</math> are the image distances that correspond to the near and far limits of DOF, the required f-number is minimum when the image distance <math>v</math> is

<math>v = \frac {2 v_{\mathrm N} v_{\mathrm F} }

{v_{\mathrm N} + v_{\mathrm F} } </math>

The required f-number is

<math>N = \frac {f^2} {c}

\frac { v_{\mathrm N} - v_{\mathrm F} }

{v_{\mathrm N} + v_{\mathrm F} } </math>

The image distances are measured from the camera's image plane to the lens's image nodal plane, which is not always easy to locate. In most cases, focus and f-number can be determined with sufficient accuracy using the approximate formulae

<math>v \approx \frac { v_{\mathrm N} + v_{\mathrm F} } {2}

= v_{\mathrm F} + \frac { v_{\mathrm N} - v_{\mathrm F} } {2} </math>

<math>N \approx \frac { v_{\mathrm N} - v_{\mathrm F} } { 2 c },</math>

which require only the difference between the near and far image distances; focus is simply set to halfway between the near and far distances. View camera users often refer to the difference <math>v_{\mathrm N} - v_{\mathrm F}</math> as the focus spread; it usually is measured on the bed or focusing rail. On manual-focus small- and medium-format lenses, the focus and f-number usually are determined using the lens DOF scales, which often are based on the two equations above.

For close-up photography, the f-number is more accurately determined using

<math>N \approx \frac {1} { 1 + m } \frac { v_{\mathrm N} - v_{\mathrm F} } { 2 c },</math>

where <math>m</math> is the magnification.

Asymmetrical lenses

The discussion thus far has assumed a symmetrical lens for which the entrance and exit pupils coincide with the object and image nodal planes, and for which the pupil magnification is unity. Although this assumption usually is reasonable for large-format lenses, it often is invalid for medium- and small-format lenses.

For an asymmetrical lens, the DOF ahead of the subject distance and the DOF beyond the subject distance are given by^[5]

<math>\mathrm {DOF_N} = \frac

{N c (1 + m/P)} {m^2 [ 1 + (N c ) / ( f m ) ] } </math>

<math>\mathrm {DOF_F} = \frac

{N c (1 + m/P)} {m^2 [ 1 - (N c )/ ( f m ) ] }, </math>

where <math>P</math> is the pupil magnification.

Combining gives the total DOF:

<math>\mathrm {DOF} = \frac {2 f ( 1/m + 1/P ) }

{ ( f m ) / ( N c ) - ( N c ) / ( f m ) } </math>

When <math>s \ll H</math>, the second term in the denominator becomes small in comparison with the first, and

<math>\mathrm {DOF} \approx \frac {2 N c (1 + m/P)}{m^2}</math>

When the pupil magnification is unity, the equations for asymmetrical lenses reduce to those given earlier for symmetrical lenses.

Effect of lens asymmetry

Except for close-up and macro photography, the effect of lens asymmetry is minimal. A slight rearrangement of the last equation gives

<math>\mathrm {DOF} \approx \frac {2 N c} {m}

\left ( \frac 1 m + \frac 1 P \right ) </math>

As magnification decreases, the <math>1/P</math> term becomes smaller in comparison with the <math>1/m</math> term, and eventually the effect of pupil magnification becomes negligible.

Notes

1. ^ Englander describes a similar approach in his paper Apparent Depth of Field: Practical Use in Landscape Photography. (PDF); 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 (PDF)
2. ^ Derivations of DOF formulae are given in many texts, including Larmore (1965) and Ray (2002). Complete derivations also are given in Conrad's Depth of Field in Depth (PDF) and van Walree's Derivation of the DOF equations.
3. ^ A well-illustrated discussion of pupils and pupil magnification that assumes minimal knowledge of optics and mathematics is given in Shipman (1977).
4. ^ The procedure for estimating pupil magnification is described in detail in Shipman (1977).
5. ^ This is discussed in Jacobson's Photographic Lenses Tutorial. and complete derivations are given in Conrad's Depth of Field in Depth (PDF) and van Walree's Derivation of the DOF quations.

References

• Hansma, Paul K. 1996. View Camera Focusing in Practice. Photo Techniques, March/April 1996, 54–57. Available as GIF images on the Large Format page.
• Larmore, Lewis. 1965. Introduction to Photographic Principles. 2nd ed. New York: Dover Publications, Inc.
• Merklinger, Harold M. 1992. The INs and OUTs of FOCUS. v. 1.0.3. Bedford, Nova Scotia: Seaboard Printing Limited. Version 1.03e available in PDF at http://www.trenholm.org/hmmerk/. ISBN 0-9695025-0-8
• Ray, Sidney F. 2002. Applied Photographic Optics. 3rd ed. Oxford: Focal Press. ISBN 0-240-51540-4
• Shipman, Carl. 1977. SLR Photographers Handbook. Tucson: H.P. Books. ISBN 0-912656-59-X

Further reading

• Hummel, Rob (editor). 2001. American Cinematographer Manual. 8th ed. Hollywood: ASC Press. ISBN 0-935578-15-3

See also

• Angle of view
• Bokeh
• Circle of confusion
• Deep focus
• Depth-of-field adapter
• Depth of focus
• Frazier lens (very deep DOF)
• Hyperfocal distance
• Perspective distortion
• Shallow focus

Wikipedia information about Depth of field This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Depth of field". It may not have been reviewed by professional editors (see full disclaimer). Help contribute to Americola and edit this article.