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Circle of Confusion formula

Zak Ray

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I get the concept behind CoC, I'm just trying to wrap my head around the formula. I've seen the simple approach ("d/1500") but I want to actually plug in different values for screen size. My understanding of the full formula is this:

CoC = (Visual Acuity * Viewing Distance) / (Display Diagonal / Camera Sensor Diagonal)


I know the "standard" CoC is 0.001". I assume that's for Super35 1.78, which is a diagonal of 27.54mm. 

I don't fully understand the visual acuity part, but from what I've read it should be around 1/1000 for normal vision?

So with that information, I can reverse engineer the formula to be:

(1/1000 * 52') / (57' / 27.54mm) = 0.001" (obviously after converting all the units)

57' seemed like a reasonable diagonal for a theater screen, and 52' was the required viewing distance to get to an answer of 0.001".


Is this anywhere near correct?

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A circle of confusion was used as a base for calculation in lens design. You’re transferring it to viewing conditions in a cinema theatre. That’s what is not correct.

My main objection, however, is that you omit the medium. Purely abstract values are quite senseless. The eye must be included, correct, and what it sees. There is always a medium, an image carrier, if you want so, film or a pixel screen. Changes everything.

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Thanks Simon. My understanding was that besides lens design, it's also necessary for DOF calculations, and in order to select the right value, one has to consider the size of and distance to the final screen. For example, this post...

...gets at what I'm trying to achieve; specific CoC values for different presentations. I want to know how those values are calculated. 

I'm not quite sure I understand what you mean by the medium affecting the result.

Edited by Zak Ray
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Perhaps now, if I tell you that a depth of field calculus is hollow as well without connection to the real world. The very base is the resolution or better resolving power of the eyes. CoC are so-to-say a cheap way around the problem or a shortcut to make an optician’s life easier. S/he just puts a thou diameter in his reckoning, done. Imagine an executive officer with an optical company to rule that lenses be calculated with this or that CoC. Relieves work a lot and there you have the reason why lenses with very high resolving power such as used in circuits photoprinting are expensive.

Cine camera lenses are expensive due to fancy mechanics, chauvinism, and a market of shrinking size.

To make myself fully understandable, the medium is the worldly termination of theory and calculation. A processed film holds the images as a silver wool suspended in gelatine or as dye clouds in several layers of gelatine. Object edges have been degraded to image patterns. A sensor breaks them down to a raster, the well-known stairs instead of continuous lines. You see, circles of confusion have no value on this plain.

Another example is the most subjective judging one makes on the image seen on a ground glass. Light is scattered into a multitude of small shining spots at various angles. When you observe an image on a smooth glass plate you have breathed upon you have refractions in tiny water droplets. Not much comes close to the vision we have by our rods and cones. The retina is the gauge.

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The issue with Circle of Confusion is that it entirely rests on some vague moment when a point of light going out-of-focus looks too large to look like a point anymore, based on estimates of average viewing size/distance, etc.  Many people think that the current standard is not critical enough for modern digital cinemas, plus not everyone will be sitting at some theoretical perfect distance from the screen. Since you are dealing with perceptions of sharpness and depth of field, these calculations should only be used as a rough guide, or for comparing one lens against another -- i.e. relative to each other.

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Thanks David, I agree they should only be used as guidelines, but broadly speaking, I would think the CoC for theatrical projection vs. a home television would be noticeably different; for most people, the movie screen fills up a larger amount of the field of vision, and so the need for precision is greater. Or are you saying even then the difference is negligible? 

Simon, I agree that human vision has to come into the equation, which is why the "visual acuity" parameter is part of the formula. My (admittedly shaky) understanding of why visual acuity = 1/1000 is derived from here. I could be totally off base here, but thats why I'm asking for the formula 🙂

Edited by Zak Ray
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I don’t know any formula that takes visual acuity into account, and honestly I’m not sure what the point would be anyway. Would you use “average” acuity or 20/20 vision? 

As others have pointed out, depth of field is a very hazy, subjective metric and attempting to apply rigorous mathematical formulae to work it out is a bit like trying to measure happiness to several decimal places.

There are lots of different CoC values people and companies have used to create depth of field charts or apps over the years. Factors like the film/sensor diagonal, the viewing magnification, the viewing distance, the pixel pitch, the viewer’s eyesight, the quality of the presentation, the type of lenses used, even the specific shot and how it’s lit can all affect whether something appears sharp or not, and still be argued over by two different viewers. 

This is a great old thread where David discusses the frustration of dealing with ACs who rely on depth of field calculations rather than their eyes:


As for using different CoC values for content made for TV or small screens as opposed to cinema screens, 15 years ago Doug Hart (author of “The Camera Assistant“) used to use 1/700” (0.035mm) for 35mm cinema shows and 1/500” (0.05mm) for 35mm TV. Others, including the AC manual and many lens manufacturers used the more traditional and tighter toleranced 1/1000“ (0.025mm) for 35mm in general. If you apply the quick diagonal/1500 calculation, cine 35mm works out to be even tighter, at 0.021mm. These days, with higher resolution cameras and lenses, and much higher resolution TVs, I imagine a case could be put forward to move on from the old CoC standards and tighten the tolerance for what looks “out of focus“ even further, to maybe 1/1500“ (0.017mm) for S35 sized sensors. I think apps like pCam default to around this value for modern cameras (maybe someone can confirm).

But in the end, it’s all just a loose guide, something to help with certain shots or sequences, definitely not a prescription. 


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Thanks Dom, that's very helpful. The reason I keep pressing for a formula is because I'm working on a DOF calculator and I thought it'd be useful to allow users to plug in their own numbers. But I appreciate the futility of trying to narrow it down to multiple decimal places, fair enough.

It's good to know that there's some talk about updating the standard for modern cameras. Does anyone else have an opinion on what that "updated standard" might be?

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1 hour ago, Dom Jaeger said:

I think apps like pCam default to around this value for modern cameras (maybe someone can confirm).

PCam appears to use 0.021 for S35 1.78.

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