A tedious essay on Depth-of-Field:

Now and again, there are questions about depth-of-field and hyperfocal distance. These questions and their answers are often fact often lore.

 

As you all know, depth-of-field is that zone of acceptable sharpness before and after the point focused upon. This zone expands and contracts based on subject distance, aperture (f-number setting), subject contrast, observer distance to displayed image, magnification used to make the displayed image, acuity of eyesight of the observer, illumination level of displayed image and perhaps a few more variables.

 

I have had spent a long time studying this topic and I sometimes chuckle when I read – span of depth of field is 4 feet 6 ¼ inches to 7 feet 5 ¾ inches etc. How can these computations be so exact? Answer they can’t.

 

The basis of it all – An image is comprised of tiny fragments of information. At the lens level, the smallest fraction that can convey inelegance is the so called circle of confusion.

 

The vista we are to image can be thought of a being covered by a googolplex of tiny points of light. Each can be a reflector or a radiator of image forming rays. When these rays traverse the lens, the shape of the glass and the density of lens material refract (bend backwards) these rays. In other words they change their direction of travel.

 

These rays emerge from the rear of the lens tracing out a cone of light. These cones converge forming an apex. If not intercepted by film or digital sensor, they fan out.

 

A tack sharp image materializes when we adjust focus and cause the apex of these cones to just kiss the surface of the sensitized material. If we succeed, a point source on the vista will be replicated as a point on the light-sensitive surface. If we miss focus, even slightly, the image points will be enlarged. If it’s too large we will declare the image to be blurry, too small is OK.

 

If we scrutinize the camera image using a magnifier, we will see these image points as fuzzy circles with indistinct boundaries. Additionally, they are juxtaposing. Nevertheless, we are taking about the smallest possible fraction of an image that is able to convey intelligence. We name them “circles of confusion”.

 

We declare an image to be tack sharp if these circles are super tiny, so tiny that they appear as dimensionless points. The question becomes, what is the maximum size they can be for us to proclaim the image is tack sharp? The answer is a variable dimension based on conditions.

 

The factors are: viewing distance, image contrast, subject matter, image brightness, and the acuity of vision of the observer.

 

Also consider, a pin-hole camera sports a super tiny aperture, no lens required to form an image. Interesting, the pin-hole camera projects an image that displays unlimited depth-of-field. The key problem, the image is super dim; the image is only marginally shape. We enlarger the pin-hole to make the image brighter we succeed but sharpness is deterred. We substitute a lens for the pin-hole we get image brightness, and improved sharpness but we lose infinite depth-of-field.

 

The basis for much of this stuff:

 

A person with good vision is able to close focus at or about 10 inches (250mm). Tests reveal that the circles of confusion will appear as dimensionless points when viewed from a distance of about 1000 times their diameter. For a 250mm viewing distance, this will be a diameter of 1/100 of an inch (0.025mm). Now such close viewing is not common: We normally examine pictorial content from our comfortable reading distance, commonly 500mm (20 inches). For this viewing distance, the circle size relaxes to become 0.5mm (1/50 inch).

 

A circle size of 0.5mm is OK for viewing finished pictorials; however we must now pay attention to circle size at the image plane inside the camera. Since we use miniature cameras like 35mm size or even smaller, the images they make are too small to be viewed without being magnified (enlarged). When figuring out depth-of-field tables and charts, we must take the degree of magnification needed into account.

 

The venerable 35mm frame size is 24mm height by 36mm length. Typically depth-of-field calculations are based on what it takes to enlarge this image to make an 8x10 inch displayed image. To enlarge this frame an make an 8x10 requires about 8 ½ X magnification.

 

Best we err on the side of caution and round this magnification up to 10X. For this degree of magnification the circle size becomes 0.5mm ÷ 10 = 0.05mm. Using this circle size we can enlarger and make a 9 ½ x14 inch displays.

 

The DX format cameras (APS-C aka Advanced Photo System – Classic format) measures 16mm height by 24mm length. These require 15X magnification to do the same task. To accommodate this the circle at the image plane of the camera must be readjusted. Thus --- 0.5mm ÷ 15 = 0. 33mm.

 

There you have it. Now we know what size circle of confusion will be needed to make an 8x10 from FX = 0.05mm --- from the DX 0.033mm.

 

Now my goal is to work out a simple way to help you construct depth-of-field data. Normally this is the stuff of algebraic lens formula math.

 

I think the easiest approach is to figure out the hyperfocal distance first; This is actually easy math. Once the hyperfocal distance is known, you can continue using simple math to find the near and far limits of the span of depth-of-field.

 

When focusing a landscape vista, it is practical to set focus distance equal to the hyperfocal distance. This procedure maximizes the span of depth-of-field. When the focus distance is set to the hyperfocal distance, the span of depth-of-field is infinity ∞ (as far as the eye can see) to the near point of acceptable focus which is hyperfocal distance divided by 2. As an example you calculate the hyperfocal distance and find it is 15 feet. Setting your focus distance to 15 feet yields a span of depth-of-field infinity ∞ to 7 ½ feet.

 

Finding the hyperfocal distance is easy! We multiply the working diameter of the camera’s lens (aperture diameter in millimeters) by 1000. Why 1000? The diameter of the aperture (f-number) viewed from 1000 diameters distance will be seen as a super tiny dimensionless point. Thus 1000x the dimeter works nicely to derive the hyperfocal distance. This value might not be stringent enough for giant enlargements. These are often based on 1/2000 of the dimeter of the iris. Kodak often used 1/1750 and Leica uses 1/1500 for critical work. Depth-of-field calculators often use different sizes. The choice is a guesstimate based on the magnification required to make the display image and the distance the observer is from this image.

 

 

OK let’s use the tried and true 1/1000 of the diameter of the iris.

 

We mount a 100mm lens and set the f-number to f/22. The working diameter is 100 ÷ 22 = 0.45mm (OK to round). Now 0.45 X 1000 = 450mm. This is the hyperfocal distance Now 450mm = 4.5meters = 15 feet. Set the focus on 15 feet, depth of field carries from ½ the hyperfocal to infinity ∞ = 7 ½ feet to ∞ = 2.25meters to ∞. Simple stuff! This is the stuff of the Brownie point-and–shoot no focusing required.

 

For another example, let’s reset the 100mm to f/11.

 

The hyperfocal distance is 100 ÷ 11 X 1000 = 9,091mm = 30 feet. Span is 15 feet to ∞ = 4.6meters to ∞.

 

Now let’s work out the depth-of-field span for a 100mm set to f/11 with the subject distance of 7 feet. All distance units must be the same, so 7 feet = 2,134mm

 

We will use the 2,134mm hyperfocal in both the following questions.

 

N = Near limit

 

F = Far limit

 

H = Hyperfocal

 

D = Distance focused upon

 

Formula near limit (H X D) ÷ (H + D)

 

Formula far limit (H X D) ÷ (H - D)

 

 

 

Now find near limit

 

N = (H X D) ÷ (H + D)

 

N = (9091 X 2134) ÷ (9091 + 2134

 

N = 19400194 ÷ 11225

 

N = 1728mm = 1.73 meters = 5.7 feet.

 

 

Now find far limit

 

 

F = (H X D) ÷ (H - D)

 

F = (9091 X 2124) ÷ (9091 – 2124)

 

F= 19400194 ÷ 6967

 

F = 2785mm = 2.78 meters = 9.1 feet.

 

 

Set a 100mm to f/11 Set focus to 7 feet

 

Span of Depth-of-field is

 

Near 5.7 feet

 

Far 9.1 feet.

 

 

This tedious example sets the maximum size of the circle of confusion at 1/1000 of the focal length. For more critical applications, like making giant displays, enlargements, 1/2000 is often substituted.

 

 

The basis for the rest of the stuff. A person with normal vision, can resolve lines that subtend an angle of 1/3000 the distance. This is a small angle. A wagon wheel 3 feet in diameter viewed in bright sunlight from 9,000 feet (1.7 miles). However photographs have lowered contrast and are viewed under reduced illumination. For this reason and angle of 3.4 minutes of arc i.e. a circle viewed from 1/000 times is diameter. This equivalent to 1/100 inch viewed from 10 inches or 1/50 of an inch from 20 inches. These are the distances whereby a disk appears as a dimensionless point. This is the criteria for 1/1000 of the diameter which is generally accepted for these applications.

Alan,

 

Thank you for this succinct, illuminating (no pun intended) essay on Depth of Field. Like so many of us, I have read in book after book of what the hyperfocal distance is, and how to (approximately) calculate it, but very few of these either agree or use the same references. You have summed it all up beautifully - and I even managed to follow the Math !

 

Thank you

Not tedious in the least. I think this may be the clearest explanation I have read, and I have read a bunch. Having the knowledge is one thing; being able to communicate it and teach is quite another. You clearly have both. Thanks for taking the time to post this.

Ah, the old quarter-millimeter blur circle. I think it was Leitz or Zeiss that first calculated that size?

As usual, useful and well-done. Thanks.

Thanks for the nice explanation.

 

It is usual on prime lenses, less often on zooms, to put DoF scales next to the

focus numbering, such that one can easily estimate a good DoF.

 

As you note, we don't know how the lens manufacturer came up with

the scale, and (at least I) tend to believe it more than I should.

 

I do notice no mention of diffraction.

 

In the case of a pinhole camera, as the hole gets smaller, diffraction effects get larger.

There is an optimal pinhole size that minimizes the image spot size. Tradition is

to make pinhole cameras with that size, maybe also because it tends to be close

to the size of actual pins.

 

With a perfect lens, diffraction means that even at exact focal distance,

we still get a circle. For nearby focal distance, the circle will get larger, but not

so fast as it would without diffraction. Even without considering visual acuity and

viewing distance, there is already a circle size, though likely much smaller

than you note.

 

Also, real lenses have some aberrations, most often spherical and chromatic

to start with. Spherical means that we get a larger size spot than diffraction, even

at best focus. Chromatic means that different colors focus at a different distance,

so we get colored spots. Including these, we get an even larger focus range than

just considering diffraction.

 

If you add Gaussian distributions, the size of the result is the square root of the

sum of the squares of those being added. (I used to know how to show that.)

 

If the lens at perfect focus gives a spot size d, and we defocus such that the

above calculation gives a spot size of d, then the result is about d*sqrt(2),

so not so much larger. From this, the DoF that give a spot sqrt(2) times the

perfect focus spot might be considered the diffraction and aberration

limited DoF. I don't know that any lens manufacturers use this.

Discussing diffraction in lens systems reminds me of stories about astronomers

in the early days (or not so early) of telescope astronomy. For many years,

astronomers had no good measurement of the actual distances to the moon, planets,

or stars. They could compute relative distances, which Kepler used to come up with

his laws, but no absolute distances.

 

Even more, they had no thought that stars might be as far away as they are.

(I suspect most people today don't know how far away they are.)

 

Because of diffraction, stars would appear as a disk when viewed through even

the best telescope. (Even larger with the lenses of the day.) This made it

difficult to believe that stars are so much farther away from us than planets.