The Mathematics of Depth of Field / Part Two: Crop Factor, Magnification and the 1/3 Myth

This is a continuation of the article I wrote a few years ago, “The Mathematics of Depth of Field.”

In the earlier article, I presented some equations and plotted some results from them. I concluded with a series of “rules” about depth of field (DOF). There were several conditions for using the rules:

• The distance to the subject is significantly larger than the focal length of the lens (i.e., don’t use the equations for macro work).
• Only DOF is considered. There are other factors that can affect sharpness, including diffraction, lens quality and stability, and sensor/film resolution.
• We’re not dealing with lenses that can shift and tilt.

The four rules I originally presented describe how DOF is affected by changing one variable while keeping the others the same:

1. As distance to the subject increases, DOF increases.
2. As f stop increases, DOF increases.
3. As focal length increases, DOF decreases.
4. As sensor/film size increases, DOF decreases.

The main focus of this article is to provide more detail on the fourth rule, sensor/film size, by looking at how “crop factor” affects DOF. I’ll also look at how magnification—expressed in terms of crop factor, focal length and distance to the subject—influences DOF. The last issue I’ll briefly discuss is the 1/3 rule (what I call the 1/3 myth) that describes the proportions of DOF in front of and behind the point of focus.

Crop Factor

For the purposes of this article, I will compare 35 mm film (or a full frame DSLR, e.g., Canon 5D) to a DSLR with a crop factor of 1.6x (e.g., Canon 40D). A full 35 mm film frame is 36 mm by 24 mm. Since the 1.6x sensor is smaller, those dimensions are divided by 1.6, which gives 22.5 mm by 15 mm (or something close to that; many 1.6x cameras are only approximately 1.6x). Basically, the effect is cropping the middle out of a 35 mm slide—hence the term “crop factor.”

To understand how crop factor affects DOF, we need to revisit the idea of “circle of confusion.” As a reminder, circle of confusion defines what it means to be in focus. Only points in the scene that are actually at the plane of focus are perfectly in focus; all other points are fuzzy circles that get fuzzier the farther away they are from the plane of focus. The circle of confusion is the size below which the fuzzy circles will still look to the viewer as if they are sharp points, and, therefore, in focus.

In making the comparisons, I’m going to use the same criteria I did before: to be considered in focus, the size of the circle of confusion must be smaller than 0.2 mm on an 8×10 inch print (cropped from the image’s original 2:3 aspect ratio which would produce an 8×12 print). This means the circle of confusion that I use in the equations is 0.0236 mm for the 35 mm slide and 0.01475 mm for the 1.6x DSLR.

In the previous article, I compared sensor/film sizes in only one way; I wanted to compare cameras capturing the same images (other than DOF). The only way two cameras can capture the same image is if they are in the same location. As location changes, the perspective changes. So what I did before was use different lenses on the two cameras—the camera with the smaller sensor (or film format) needed a shorter lens to capture the same image. For example, a full frame camera with a 160 mm lens and a 1.6x DSLR with a 100 mm lens, at the same place, will capture the same image. Except for differences in DOF, all the elements in the images will look the same relative to one another. Given this assumption, I presented the fourth rule: As sensor/film size increases, DOF decreases. Consider this the first way of comparing two cameras with different crop factors. But it’s not the only way we could compare them. For example, we could keep the same location and focal length, or we could keep only the same focal length and move locations. If we make these different comparisons, the fourth rule won’t be true.

Let’s first consider keeping the same location and focal length. Frequently, nature photographers (particularly when working with birds or animals) want as much magnification as possible. If a photographer buys a 1.6x DSLR, he is not going to trade in a 600 mm lens for a 375 mm lens so he can get the same images that were photographed with a full frame camera. Instead, he is going to keep the 600 mm and work as close to the subject as before.

What does that do to DOF? I’ve seen people say that the same lens at the same f-stop should give exactly the same DOF—the optics don’t care about the sensor/film size. On the sensor/film plane, that is perfectly true. What this overlooks is that to get the same print size, the 1.6x DSLR image has to be blown up more, since it starts from a sensor with a smaller physical size. This makes any softness more apparent. Figure 1 shows this.

Figure 1

The full frame camera has more DOF. In fact, it has 1.6x more DOF, since it needs to be enlarged that much less to get the same print size.

Now let’s consider keeping the same focal length and moving locations. Perhaps the photographer with a 600 mm lens decides to stand back a bit further to put less stress on the subject. With a 1.6x DSLR, the photographer can stand 1.6x further away to get the subject the same size as with the full frame camera. (The perspective will be different, but when dealing with subjects like birds, often the background is out of focus, so the different perspective won’t be apparent.) The effect of this on DOF is illustrated in Figure 2.

Figure 2

The “Full Frame” line in Figure 2 is the same as the one in Figure 1. But in this case, the 1.6x DSLR is being used at a distance that is 1.6x more than what’s shown on the scale to get the same subject magnification. This time, even though the two cameras have the same magnification, the 1.6x DSLR has more DOF. It’s about 1.6 times more.

So now we have three ways of comparing the two cameras. Starting with a restatement of the original fourth rule, the rules related to crop factor are:

1. If we shoot from the same location, but use a shorter lens on the 1.6x DSLR, we will have more DOF (about 1.6x more than the full frame).
2. If we shoot from the same location with the same lens, we will have less DOF on the 1.6x DSLR (the full frame will have 1.6x more).
3. If we shoot with the same lens, but further back with the 1.6x DSLR to get the same subject size, we will have more DOF (1.6x more than the full frame).

In cases 1 and 3 where the subject magnification (i.e., the size of the subject on the final print) was the same, the 1.6x DSLR has more DOF than full frame. In case 2 when the magnification was greater with the 1.6x DSLR, the DOF was less.

Magnification

The principal equation on which both this article and the previous one are based uses these four variables to calculate DOF: focal length, distance to the subject, aperture and circle of confusion. Many people consider that depth of field is a function of just two variables: magnification and aperture. If you have only one film or sensor format, for DOF purposes the use of magnification is essentially correct, though not 100% accurate. Where it breaks down is in considering crop factors.

Given the same assumptions built into the DOF equations, as listed at the top of this article, magnification at the sensor/film plane is given by:

Using the same notations as in the previous article (and adding M):

• M is the magnification
• L is the focal length
• D is the distance to the subject (it must be substantially larger than L)

However, for this article I’m not interested in magnification at the sensor/film plane. I have brought up the question of crop factors because I am interested in the magnification on a print. On a print, total magnification is influenced by the magnification at the sensor (i.e., distance to the subject and focal length), and also by the relative sizes of the print and sensor (or film frame). Instead of overly complicating the equations, I will introduce a “relative” magnification, which can be used to compare examples without determining their actual magnification:

Where:

• CF is the crop factor

Figure 3 below shows the effect of relative magnification on DOF as each of the three variables (focal length, distance to the subject, and crop factor) is changed separately. The starting point is a 600 mm lens, 6 m (19.685 ft) from the subject, in a full frame camera with a circle of confusion of 0.0236 mm. This comparison is not affected by aperture, so it will be kept constant at f/8. If we move closer to the subject with this set up, we will have more relative magnification. If we use a longer lens, we will have more relative magnification. If we use a cropped camera we will have more magnification.

Figure 3

In the figure, the curves for changing magnification by distance or focal length are almost (but not exactly) on top of each other. Therefore, if you’re only thinking about one film or sensor format, it is okay to say that DOF depends only on magnification and aperture, since it doesn’t matter how you change the magnification. Note that it is not a linear relationship—a 50% increase in magnification does not result in a 50% decrease in DOF.

However, when you want to compare the DOF of different formats, considering the crop factor very much does matter. Changing relative magnification by changing the sensor/film format has less of an effect on DOF than changing the magnification at the sensor/film plane.

In summary:

1. Greater magnification (on the final print) will result in less DOF.
2. Increases in magnification from increasing the focal length or getting closer to the subject will give essentially the same reduction in DOF.
3. Increases in print magnification from increasing the crop factor will have less of an effect on DOF than increases from focal length or distance.

The 1/3 Myth

Common photographic wisdom holds that 1/3 of the DOF is in front of the point that is focused on, and 2/3 of the DOF is behind it. This is a myth, but like many myths, it is based on a kernel of truth.

Figure 4 shows the proportion of the total DOF which is in front of the subject (or point of perfect focus) as compared to the focal point distance divided by the hyperfocal distance. (Recall that the hyperfocal distance is the point of focus at which everything from half that distance to infinity is in focus.)

Figure 4

If you focus on a point that is 1/3 of the hyperfocal distance, which is shown by the red dashed line in the figure, then you will get the 1/3 of the DOF in front of the subject and 2/3 behind it. This is the origin of the myth. However, as you get closer or farther away from that point at 1/3 of the hyperfocal distance, the DOF proportions change.

Focusing farther out, the figure shows that the proportion of DOF in front of the focal point drops until it becomes effectively zero at the hyperfocal distance. This doesn’t mean that there is nothing in focus in front of the focal point, since the definition of hyperfocal distance tells us otherwise. Rather, there is an infinite amount in focus behind that point, and so the ratio of the amount in front divided by infinity behind is zero.

As the focal point approaches the minimum focus distance of the lens, the proportion approaches 1/2, not the mythical 1/3. Note that the equations being used weren’t intended for macro, and I wouldn’t trust the results on the extreme left side of the graph. If you’re shooting close objects—but not so close as to be considered macro—you can consider half your DOF to be in front of the point you focus on.

In summary:

1. The 1/3 “rule” is really only true when the lens is focused at one third of the hyperfocal distance.
2. Close to the camera, the DOF is split with half in front and half behind the point of focus.
3. As the point of focus is moved further from the camera, the proportion of DOF that is behind the point of focus increases.
4. At or beyond the hyperfocal distance, everything past the point of focus to infinity will be within the DOF.