Depth of Field, Part II: The Math

In Part I we talked, in a fair amount of detail, about what DOF is and how it is determined. If you are happy with what you learned in Part I and do not feel the need to know the math behind all of the different factors in DOF, then you are free to go out and start confidently manipulating DOF while taking photos. However, if you are having difficulty pressing the “I believe” button and want to see the “concept” or “theory” of DOF become a cold, hard fact with mathematics, then keep on reading.

Circle of Confusion

Were you confused by the first foray in to the topic of Circle of Confusion (COC) in Part I? Well, the COC is a confusing, yet important part of how DOF is calculated. As we mentioned in Part I, there are three things that factor into the COC:

1. Viewing distance
2. Enlargement
3. Visual acuity

Here, in Part II, we will dive more deeply into the elements of COC so we can enter the numbers into a formula and then use this to figure out hyperfocal distance and then use that figure to calculate DOF mathematically.

Viewing distance is measured in centimeters for the purposes of our equation.

Enlargement is measured as a factor of magnification. In the example below, an 8 x 10" print is 7x larger than a 24 x 36mm sensor or frame of 35mm film.

Visual acuity is measured by line pairs per millimeter (lp/mm). At your optician’s office, your eyes are measured and the acuity is assigned a number; 20/20 being “normal” vision. This means that at 20' you can see clearly what a “normal” eye can see clearly at 20'. If you need to be at 20' to see what a normal eye sees clearly at 40', you have 20/40 vision. For the purposes of COC, a line pair comprises alternating black and white lines of equal size. You may have seen these lines on certain eye charts. Acuity is measured by determining how many line pairs a person can see at a designated distance. “Normal” vision is measured at 5 lp/mm.

Now let’s apply the COC factors into a simplified equation to mathematically resolve COC so we can use it in our DOF calculations.

The math behind COC
 

This COC value represents the maximum blur spot diameter, measured at the image plane, which looks to be in focus. A spot with a diameter smaller than this COC value will appear as a point of light and, therefore, in focus in the image. Spots with a greater diameter will appear blurry to the observer.

For simplicity, camera and lens manufacturers use a standard COC. The standard value varies between manufacturers, but it is generally around 0.03mm for full-frame cameras. The example above illustrates the variables used to formulate the COC.  

Non-Symmetry of DOF

Are you ready for another wrinkle in your DOF world? DOF is not symmetrical. This means that the area of acceptable focus does not have the same linear distance before and after the focal plane. This is because the light from closer objects converges at a greater distance aft of the image plane than the distance that the light from farther objects converges prior to the image plane.

At relatively close distances, the DOF is nearly symmetrical, with about half of the focus area existing before the focus plane and half appearing after. The farther the focal plane moves from the image plane, the larger the shift in symmetry favoring the area beyond the focal plane. Eventually, the lens focuses at the infinity point and the DOF is at its maximum dissymmetry, with the vast majority of the focused area being beyond the plane of focus to infinity. This distance is known as the “hyperfocal distance” and leads us to our next section.

Hyperfocal Distance

Hyperfocal distance is defined as the distance, when the lens is focused at infinity, where objects from half of this distance to infinity will be in focus for a particular lens. Alternatively, hyperfocal distance may refer to the closest distance that a lens can be focused for a given aperture while objects at a distance (infinity) will remain sharp. The hyperfocal distance is variable and a function of the aperture, focal length, and aforementioned COC.

34' is the hyperfocal distance. If the lens is set at f/8 and focused at 34' (or at the infinity mark if it appears before 34'), everything from 17' to infinity should be in focus.

Remember that your solution will be presented in millimeters, as your lens focal length is most likely measured in millimeters. Using that formula, you will see that the smaller you make the lens aperture, the closer to the lens the hyperfocal distance becomes.

Back in the old days, lenses used to have hyperfocal distance markings on the lens barrels and/or near the focus rings. This is rarely seen in today’s autofocus world, but there are many smartphone applications and websites that will crunch the numbers for you so that you can determine your lens’s hyperfocal distance for a given aperture. I will discuss these calculators later.

The practical application of hyperfocal distance is that, in the above example, you can set your 50mm f/1.8 lens to f/8 and turn your focus dial (if marked) to 34' or ∞ and everything in your image from 17' to the horizon and beyond should be in acceptable focus.

Why is this important? Hyperfocal distance also featured in—you guessed it—the calculations used to compute DOF.

Calculating DOF

It is now time, finally, to calculate DOF. In review, the four factors that determine DOF are:

Above, we used the standard COC and then added the lens focal length and aperture to calculate the hyperfocal distance. Now, we can add the fourth factor, subject-to-lens distance, into an equation to figure out our DOF. As we now know, DOF is a linear range before and aft of the focal plane, and we also know that the DOF range is not symmetrical on each side of the focal plane.

Our next voyage into the mathematical realm is to figure the DOF near point (remember we are working in millimeters, so don’t forget to convert):