Circle of Confusion

Prepared 2006-06-28 by Bill Claff

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.

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.

_{}

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_{}.

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.

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

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.

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)

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.

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.