Depth of Field and Resolution

The depth of field refers to that section of a scene that appears in focus in an image of the scene. Being in focus means that details are discernible on the surfaces of the scene captured in the image. It is generally observed in images that the details of some minimum size are uniformly discernible for all the surfaces in the scene between two distances from the lens, u_n and u_f. Rather than a point of focus, there is a field of uniform focus. The distance between these two points u_f – u_n is the depth of field. This level of detail in the image is not available from surfaces at distances shorter than the near extent of the depth of field u_n nor at distances longer than the far extent of the depth of field u_f.

The ideal focusing model of a lens of focal length f under geometric optics uses the focusing equation

1/u_s + 1/v_s = 1/f

to describe the relationship between the distance u_s between a lens and a subject in a scene and the distance v_s behind the lens at which subject can be captured in focus, that is, with its smallest detail. This model of ideal behavior allows the smallest detail in the scene, a point, to focus as a point in the image. A surface at a distance u_f farther from the lens is in focus at a distance v_f closer to the lens, and a surface at a distance u_n nearer to the lens is in focus at a distance v_n farther from the lens. When distance to the surface gets very large, the term 1/u_s on the left of the focusing equation becomes very small and can be neglected. In this case the resulting equation tells us that the surface is in focus at v_s = f. A surface at a very large distance from the lens is in focus at the focal length f of the lens.

u_f > u_s > u_n > 0 < v_f < v_s < v_n

The focusing equation has the geometric property that a focal plane at v_s, that is, a plane of points all of which are in focus and that cuts through the lens axis at v_s, corresponds to a plane of points in the scene that cuts through the lens axis at u_s.

Consider points on either side of the actual focal plane at v_s that are not in the focal plane. A focusing diagram for these points shows that a point v_f in front of the actual focal plane expands in size from its point in focus in front of the lens to a wide blur as it crosses the actual focal plane. Similarly, a point v_n behind the actual focal plane is a wide blur as it crosses the actual focal plane on its way to becoming its point in focus behind the actual focal plane.

Based on the focusing equation alone, one should expect that there is only a single plane passing through the scene in which all points appear in focus in the image. Clearly (no pun intended), this is not the case.

In reality, imaging systems have a limited capacity to capture small details. Details are differences in tone or color. They give rise to contrast. The easiest details to see are those that are black on a white field or white on a black field. The smallest detail that a system can capture is characterized by the distance across the detail–its width. In an isotropic, two-dimensional image, the smallest details can be thought of as circles characterized by their diameters. The width or the diameter of these minimum discernible details is called the resolving power of an image system. It can be reported as its reciprocal, the resolution of an imaging system. The resolution is the number of recurrent details per distance, that is, the number of cycles per distance. This reporting scheme is used because originally resolution was measured by visually differentiating series of fine black lines of the same width and spacing on a white field and because now resolution is measured by numerically decomposing the image of the edge of a black surface in a white field into oscillating components each of which is characterized by a spatial frequency, that is, their number of cycles per distance.

The depth of field is created by the limitation on the minimum size of detail that is discernible at the focal plane. It is a phenomenon that is produced by the limited resolution of an imaging system. The physical model that produces depth of field, that is, a field of uniform focus, starts with the model for ideal focusing behavior and adds a constraint limiting the minimum size of discernible detail. While the model of ideal behavior predicts that any detail–no matter how small–can be brought into perfect focus by an ideal lens and aperture, the model of real behavior predicts that details of a certain size and smaller are brought into real focus (if you are willing to allow call it focus because it is the best the imaging system can do) as blurs of some small diameter. The limitation affects not only the points in the focal plane that would ideally be in focus, but it also affects the points that would ideally be in focus near the focal plane. The constraint on this range of points creates the field of uniform focus.

Due to limited resolution, a real imaging system cannot bring a single plane in the scene into focus. In the focusing diagram, one must add a bracket in the focal plane around the point in focus to illustrate that the point must take on some minimum diameter. The points within some distance from the focal plane also appear in the image with a diameter that matches that minimum diameter produced by the point in the focal plane. One can imagine a box or a cylinder extending out of the focal plane to illustrate with its width or its diameter which points produce a detail with the same minimum diameter as the point in the focal plane. The height of this box or the diameter of this cylinder represents the size of the smallest detail available to the imaging system and, thus, to the points that are at or near focus. All the points that fall within this box or this cylinder, while not in focus in the ideal sense, have the same level of detail in the image of the scene. This constraining feature creates a field of uniform focus in the image.

The lens, the photosensitive array, and the camera support all contribute to the limited resolution of an imaging systems. The limited resolution of the photosensitive array might be the easiest constraint to visualize. The array is always placed at the actual focal plane, and the size of the photosensitive grains in an emulsion or the spacing of photosensitive pixels in a solid-state device are easily recognized as the minimum sizes that can discern detail.

In addition, it should be recognized that lenses also have a limited resolution, This is obvious when viewing the image of a small and distant subject on the ground glass of a large format view camera. No matter how carefully focused the subject, it appears as a blur and not a point. The size of the subject in the image gets smaller when using better quality lenses, but there is always a minimum size. The capacity of a real lens to focus is always limited.

Internal and external camera support also contributes to the limited resolution of imaging system by reducing the motion of the system. Everyone has certainly experienced a loss of visual resolution while trying to read the lettering on a page while walking or in a moving car. Loss of resolution can appear in an image due to mirror recoil or hand-holding the camera.

While the focal length of the lens and the diameter of the aperture are parameters that describe the imaging system, the lens and the aperture are part of the model of ideal focusing behavior that creates a single point of focus as well as part of the model of real focusing behavior that creates a field of uniform focus. The only difference between these two models is the constraint on the minimum size of discernible detail. When this minimum size is allowed in theory (in the ideal case) to go away or to go to zero, the field of uniform focus becomes a single point of focus. Changing the values of the focal length of the lens and the diameter of the aperture does not produce nor eliminate the field of uniform focus in the way that the minimum discernible size does.

Expressions for the near and far extents of this uniform field of focus, u_n and u_f, respectively, and its depth of field u_f – u_n can be derived for an arbitrary resolution of the imaging system by taking the the points v_n and v_f to be such that the diameter of the unfocused images at these points is that same as the diameter d_c of the smallest available detail. That is to say, the distance between the focusing lines that travel between the ends of the aperture and cross the focal plane at v_s for each of the points v_n and v_f is equal to d_c, the diameter of the smallest detail. The resulting equations describe the field of uniform focus for a thin, symmetrical lens with geometric optics.

Near Extent = u_n = h u_s / [h + (u_s – f)]

Far Extent = u_f = h u_s / [h – (u_s – f)]

Depth of Field = u_f – u_n = 2 h u_s (u_s – f) / [h² – (u_s – f)²]

where h = f² / A d_c

While the depth of field and the distances to its near and far extents are produced by the resolution of the selected imaging system, the derivation and the calculation of the these quantities depends on the focal length of the lens f, the diameter of the aperture D_a (in terms of the aperture number A = f/D_a), and the focusing distance u_s. Focusing at or beyond a particular distance (which depends upon these parameters) called the hyperfocal distance u_h,

Hyperfocal Distance = u_h = h + f

the far extent of the field of uniform focus and, with it, the depth of field become very large. In mathematical parlance, the far extent and the depth of field become infinite. In this case the field of uniform focus extends as far as one can see, and there are no surfaces in the apparent distance within the image that are not in uniform focus.

These equations can be used to create an expression for the fraction of the depth of field in front of the focusing point.

Fraction of Depth of Field in Front of Focus

= (u_s – u_n) / (u_f – u_n) = (u_h – u_s) / 2h

This fraction tells us that the position of the focusing point within the field of uniform focus depends on the focusing distance relative to the hyperfocal distance. The fraction varies smoothly from 0 when focusing close to the camera to 1/2 when focusing at the hyperfocal distance. Because the fraction varies linearly with focusing distance and definitely does not take on a constant value, the popular one-third rule for the focusing point at a constant, relative location in the depth of field is unequivocally invalid.

These equations can be approximated under the generally valid assumption that distances are much greater than the focal length, so that the focal length can be neglected in each difference (u – f) that appears in these equations, that is, u >> f, so (u – f) ≈ u.

Near Extent = u_n ≈ h u_s / (h + u_s)

Far Extent = u_f ≈ h u_s / (h – u_s)

Depth of Field = u_f – u_n ≈ 2 h u_s² / (h² – u_s²)

Fraction of Depth of Field in Front of Focus

= (u_s – u_n) / (u_f – u_n) ≈ (h – u_s) / 2 h

The reciprocals of these equations in terms of the reciprocals of the focusing distance, near extent, far extent, and hyperfocal distance are simple linear equations that can be solved together (simultaneously) to produce equations for approximate values of the hyperfocal distance (and, thus, the aperture number) and the focusing distance from given near and far extents of a field of uniform focus.

Hyperfocal Distance = u_h ≈ h ≈ 2 u_n u_f / (u_f – u_n)

Focusing Distance = u_s ≈ 2 u_n u_f / (u_f + u_n)

Using the focusing equation to replace each of the distances in front of the lens with its counterpart behind the lens, produces similar equations for approximate values of the hyperfocal distance (and, thus, the aperture number) and the focusing distance behind the lens from given distances behind the lens.

Hyperfocal Distance = u_h ≈ h ≈ 2 v_n v_f / (v_f – v_n)

Focusing Distance behind the Lens = v_s ≈ 2 v_n v_f / (v_f + v_n)

The resolution of the imaging system is determined by the combined resolutions of the lens, the emulsion or solid-state photosensitive array, and the system support. resolutions are combined by adding their reciprocals, that is, by adding the diameters of their smallest details, their resolving powers. The result is that the resolution of the system is always smaller than the resolutions of the individual components, that is, a system is able to discern fewer cycles per distance than any of its components. Typical resolutions used in the calculation of the near and far extents of the uniform field of focus and its depth of field are 33 cycles/mm (a 0.030 mm diameter) and 40 cycles/mm (a 0.025 mm diameter).

The smallest discernible detail produced by an imaging system is often called the circle of confusion, the circle of indistinctness, or the blur circle. These names are somewhat misleading because details of this size are just visible. It is the smaller details that are confused, indistinct, or blurred. Of course, being a limit or a boundary between two distinct situations, the glass can be half full or half empty. Irrespective of which label is given to the smallest discernible detail, it is incorrect to call the quantity d_c used in the calculations of the hyperfocal distance and of the depth of field a circle of anything. A circle is not distance; it is geometric figure that is characterized by a distance called its diameter.

Depth of field, the limiting effect of resolution on focus, is a phenomenon of geometric or particle optics. As distances get much smaller than those that characterize resolution, light begins to act as waves that interact with surfaces, edges of surfaces, and each other producing visible phenomenon such as diffraction rings. A photographer, after considering the formula for the depth of field, might want to reduce the aperture (increase the aperture number) to increase the depth of field. Unfortunately,the light waves interacting with relatively small apertures (at relatively large aperture numbers) add unappealing diffraction patterns to the resulting images.