Circle of confusion

In photography, the circle of confusion is used to determine the depth of field, the part of an image that is acceptably sharp.

The smallest such spot that a lens can produce is often referred to as the circle of least confusion.

Two important uses of this term and concept need to be distinguished: For describing the largest blur spot that is indistinguishable from a point.

Defocused object points are imaged as blur spots rather than points; the greater the distance an object is from the plane of focus, the greater the size of the blur spot.

Such a blur spot has the same shape as the lens aperture, but for simplicity, is usually treated as if it were circular.

The common criterion for "acceptable sharpness" in the final image (e.g., print, projection screen, or electronic display) is that the blur spot be indistinguishable from a point.

In idealized ray optics, where rays are assumed to converge to a point when perfectly focused, the shape of a defocus blur spot from a lens with a circular aperture is a hard-edged circle of light.

In photography, the circle of confusion diameter limit (CoC limit or CoC criterion) is often defined as the largest blur spot that will still be perceived by the human eye as a point, when viewed on a final image from a standard viewing distance.

With this definition, the CoC limit in the original image (the image on the film or electronic sensor) can be set based on several factors: The common values for CoC limit may not be applicable if reproduction or viewing conditions differ significantly from those assumed in determining those values.

If the original image will be given greater enlargement, or viewed at a closer distance, then a smaller CoC will be required.

All three factors above are accommodated with this formula: For example, to support a final-image resolution equivalent to 5 lp/mm for a 25 cm viewing distance when the anticipated viewing distance is 50 cm and the anticipated enlargement is 8: Since the final-image size is not usually known at the time of taking a photograph, it is common to assume a standard size such as 25 cm width, along with a conventional final-image CoC of 0.2 mm, which is 1/1250 of the image width.

Kodak recommended 2 minutes of arc (the Snellen criterion of 30 cycles/degree for normal vision) for critical viewing, yielding a CoC of about f/1720, where f is the lens focal length.

This criterion evidently assumed that a final image would be viewed at perspective-correct distance (i.e., the angle of view would be the same as that of the original image): However, images seldom are viewed at the so-called 'correct' distance; the viewer usually does not know the focal length of the taking lens, and the "correct" distance may be uncomfortably short or long.

Depth of field formulas derived from geometrical optics imply that any arbitrary DoF can be achieved by using a sufficiently small CoC.

Using a smaller CoC requires increasing the lens f-number to achieve the same DoF, and if the lens is stopped down sufficiently far, the reduction in defocus blur is offset by the increased blur from diffraction.

Because the f-number and CoC occur only as the product Nc, an increase in one is equivalent to a corresponding decrease in the other.

For example, if it is known that a lens DoF scale is based on a CoC of 0.035 mm, and the actual conditions require a CoC of 0.025 mm, the CoC must be decreased by a factor of 0.035 / 0.025 = 1.4; this can be accomplished by increasing the f-number determined from the DoF scale by the same factor, or about 1 stop, so the lens can simply be closed down 1 stop from the value indicated on the scale.

It depends only on these distances and the aperture diameter A, via similar triangles, independent of the lens focal length:

More generally, this approach leads to an exact paraxial result for all optical systems if A is the entrance pupil diameter, the subject distances are measured from the entrance pupil, and the magnification is known:

This we may consider as the nearest approach to a simple focus, and term the circle of least confusion.The Society for the Diffusion of Useful Knowledge (1832, p. 11) applied it to third-order aberrations: This spherical aberration produces an indistinctness of vision, by spreading out every mathematical point of the object into a small spot in its picture; which spots, by mixing with each other, confuse the whole.

The diameter of this circle of confusion, at the focus of the central rays F, over which every point is spread, will be L K (fig.

); and when the aperture of the reflector is moderate it equals the cube of the aperture, divided by the square of the radius (...): this circle is called the aberration of latitude.Circle-of-confusion calculations: An early precursor to depth of field calculations is the TH (1866, p. 138) calculation of a circle-of-confusion diameter from a subject distance, for a lens focused at infinity; this article was pointed out by von Rohr (1899).

He finally observes "long-focus lenses have usually a larger aperture than short ones, and on this account have less depth of focus" [his italic emphasis].

"On the Use of Diaphragms or Stops" of unknown date), says: Thus every point in an object out of focus is represented in the picture by a disc, or circle of confusion, the size of which is proportionate to the aperture in relation to the focus of the lens employed.

If a point in the object is 1/100 of an inch out of focus, it will be represented by a circle of confusion measuring but 1/100 part of the aperture of the lens.This latter statement is clearly incorrect, or misstated, being off by a factor of focal distance (focal length).

He goes on: and when the circles of confusion are sufficiently small the eye fails to see them as such; they are then seen as points only, and the picture appears sharp.

This choice of CoC limit remains (for a large print) the most widely used even today.

207–08) takes a similar approach based on a visual acuity of one minute of arc, and chooses a circle of confusion of 0.025 cm for viewing at 40–50 cm, essentially making the same factor-of-two error in metric units.

For example, Wall (1889, p. 92) says: To find how quickly a shutter must act to take an object in motion that there may be a circle of confusion less than 1/100 in.

in diameter, divide the distance of the object by 100 times the focus of the lens, and divide the rapidity of motion of object in inches per second by the results, when you have the longest duration of exposure in fraction of a second.