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62 2 Image formation

(a) (b)

Figure 2.20 Regular and zoom lens depth of field indicators.

focused image is formed z i can be expressed as

1 1 1
+ = , (2.96)
z o z i f
where f is called the focal length of the lens. If we let z o →∞, i.e., we adjust the lens (move
the image plane) so that objects at infinity are in focus, we get z i = f, which is why we can
think of a lens of focal length f as being equivalent (to a first approximation) to a pinhole a
distance f from the focal plane (Figure 2.10), whose field of view is given by (2.60).
If the focal plane is moved away from its proper in-focus setting of z i (e.g., by twisting
the focus ring on the lens), objects at z o are no longer in focus, as shown by the gray plane in
Figure 2.19. The amount of mis-focus is measured by the circle of confusion c (shown as short
7
thick blue line segments on the gray plane). The equation for the circle of confusion can be
derived using similar triangles; it depends on the distance of travel in the focal plane Δz i
relative to the original focus distance z i and the diameter of the aperture d (see Exercise 2.4).
The allowable depth variation in the scene that limits the circle of confusion to an accept-
able number is commonly called the depth of field and is a function of both the focus distance
and the aperture, as shown diagrammatically by many lens markings (Figure 2.20). Since this
depth of field depends on the aperture diameter d, we also have to know how this varies with
the commonly displayed f-number, which is usually denoted as f/# or N and is defined as

f
f/#= N = , (2.97)
d
where the focal length f and the aperture diameter d are measured in the same unit (say,
millimeters).
The usual way to write the f-number is to replace the # in f/# with the actual number,
i.e., f/1.4,f/2,f/2.8,...,f/22. (Alternatively, we can say N =1.4, etc.) An easy way to
interpret these numbers is to notice that dividing the focal length by the f-number gives us the
diameter d, so these are just formulas for the aperture diameter. 8

7
If the aperture is not completely circular, e.g., if it is caused by a hexagonal diaphragm, it is sometimes possible
to see this effect in the actual blur function (Levin, Fergus, Durand et al. 2007; Joshi, Szeliski, and Kriegman 2008)
or in the “glints” that are seen when shooting into the sun.
8 This also explains why, with zoom lenses, the f-number varies with the current zoom (focal length) setting.