Circle of Confusion
Prepared 2006-06-28 by Bill Claff

Circle of Confusion

The Circle of Confusion () is the diameter of a circle on film or sensor that serves as the upper threshold for acceptable focus.
Note that the term focus is chosen carefully and is not to be confused with sharpness.
An image can be in perfect focus and not appear sharp due to reasons unrelated to .

* is an important photographic concept that is used in a number of areas, most notably as part of the calculation of Depth Of Field (DOF).

The treatment of  in this article is somewhat unusual but is consistent with write-ups and results you may have seen elsewhere.

The value of  depends on three factors:

1.      The visual acuity of the human eye.

2.      The viewing distance to the print or projected image.

3.      The enlargement factor that was used to produce the print or projected image.

For the balance of this article I will use the term print rather than the more wordy print or projected image.
However, it is understood that the same principles apply to projected images as to prints.

Visual Acuity of the Human Eye

As with any human characteristic visual acuity varies from person to person.
For the purposes of this article I will use  to denote this quantity and a value of 3.75 minutes of arc.

Viewing Distance

As viewing distance () increases, the cone this is formed by the visual acuity of the human eye expands.
Let’s call this quantity the Circle of Acuity ().
By simple trigonometry:

Note that for the value of  we are using that  (for small angles .)
I’ll use the value of .0011, which is within 1% of the “exact” value for computational simplicity.

At a “typical” hand-held distance of 200mm (nearly 8 inches), and.

Enlargement Factor

Usually some amount of enlargement is required to form a print from film or a digital sensor.
I have chosen the term enlargement and avoided the term magnification because magnification has a very specific meaning in the context optics and close-up photography.
Let’s call enlargement  and by similar triangles:

In the “trivial” case of an 8x10 inch large format negative and an 8x10 inch print,  and .

So at a “typical” hand-held viewing distance of 200mm, .
This agrees with the commonly cited value for 8x10 inch film.


Geometry

The following diagram may help in understanding the formulas given above and the examples that follow:


Common Values for CoC

Let’s continue to assume that .
More typical film and sensor sizes are 35mm film and the DX (APS-C) sensor.
35mm film is 36x24mm and the DX sensor is (nominally) 23.4x15.6mm.

Let’s assume enlargements to a print that is 10 inches (254mm) long.

For 35mm film  and .
And for the DX sensor  and .
This agrees with commonly cited values.

However, let’s now assume enlargements to a print that is 8 inches (203.2mm) wide.

Now for 35mm film  and .
And for the DX sensor and
This is the case that is diagramed above. Values similar to these are also commonly cited.

It’s obvious from the formulas, and clear from the above examples that how you crop to perform your enlargement affects the  value.

Less Typical Values for CoC

Example 1: Hi-speed crop mode on a Nikon D2X

Hi-speed crop produces a 3216x2136 pixel image from the center of the 4288x2848 pixel DX sensor.
To achieve the same size print this requires multiplying the full DX frame  by ;
and the common DX  values computed above would drop from between .017mm and .020mm to .013mm and .015mm respectively.

(The factor for cropping a D50/D70 size image out of a D200 image is nearly identical at 1.29)

Example 2: Exhibition viewing

What  is required if we intend to display 20x30 inch (508x762mm) uncropped prints from a DX sensor to be viewed at a distance of 24 inches?

, , and

Note that the result is the same for 10x15 inch prints cropped from 50% of the DX sensor.

Conclusion

Although, knowing the amount of cropping, enlarging, and eventually viewing distance can be problematic; calculating the value of the Circle of Confusion () is straightforward and intuitive.

Having a reasonable estimate of the required  is indispensable in choosing an aperture for desired Depth Of Field (DOF), and estimating exposure times at avoid camera shake.