Some confusion about the circle

I thought it was about time to give my periodic plea for more care in our use of terminology in connection with the infamous circle of confusion.

To set the stage, note that there are three things we from time to time refer to in this area:

1. Circle of confusion. This is the "blur figure" that is produced on the focal plane from a point on the object when focus is imperfect. It is not a number or quantity, but a two-dimensional physical phenomenon - a "thing".

2. Diameter of (a) circle of confusion. This is the diameter that some real (or hypothetical) circle of confusion, created in some imaging situation, has. We do not choose it - it results from the parameters of the imaging situation.

3. Circle of confusion diameter limit (my term). This is a criterion we choose as a parameter of the calculation of depth of field. It is the largest diameter we allow the circle of confusion to have and still consider blurring to be acceptable (that is, focus to be acceptable).

It is of course common to refer to the quantity in (3) above as "the circle of confusion" (often abbreviated "COC").

The problem with doing this is that it is then almost impossible to speak of item (1) or (2), as we often need to do, without -- confusion.

Thus I recommend using the term "circle of confusion diameter limit" (which we may abbreviate "COCDL") for the parameter in item (3) above, reserving the term "circle of confusion" for item (1) and "circle of confusion diameter" for item (2).

Hi Doug,

Come on, give us a photo or drawing, units of measurement, etc.

Best wishes,

Ray

Hi, Ray,

Ray West said:

Hi Doug,

Come on, give us a photo or drawing, units of measurement, etc.

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Sorry to be so (unaccustomedly) succinct- I didn't want to miss breakfast!

Best regards,

Doug

The conventional term that I've seen for #3 is "acceptable circle of confusion." My lens books are packed away at the moment, but if you Google it as a phrase, you'll turn up references.

Hi, David,

David A. Goldfarb said:

The conventional term that I've seen for #3 is "acceptable circle of confusion." My lens books are packed away at the moment, but if you Google it as a phrase, you'll turn up references.

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That is certainly a good phrase. I had seen it, but didn't realize that is was so widely cited. (I just did a little scan and found many citations.)

I may wish to consider recommending it rather than my own coinage. And of course there would be a tidy abbreviation: "ACOC".

Of course, by rights, the word "diameter" should be in the name (as it often is in actual usage), but there seems to be little hazard by using the shorter form.

Thanks for bringing this to my attention.

Best regards,

Doug

Well, in that case . . .

David Goldfarb pointed out here this morning that the phrases "diameter of the acceptable circle of confusion" and "acceptable circle of confusion diameter" are widely used for the parameter under discussion in textbooks and technical papers. I had of course seen those expressions but I was unaware of the breadth of their use. This is certainly an attractive terminology.

That being the case, I have reversed course. I will adopt that phrase (certainly the short form "acceptable circle of confusion" will be free from any hazard of misunderstanding in almost any context) for my own writing, and I now encourage others to do so. I will adopt (and I suggest) the abbreviation "ACOC" for it.

Thanks, David, for helping with this.

Best regards,

Doug

Hi Doug,

I guess it's now your lunch time ;-)

The reason I asked for drawings, was that if the circle is blurred, how do you measure its diameter? Is it a linear blur, or some other pattern? If you are doing things for your own tests, then you can make sure you are consistent, but it becomes difficult, if folk are using their own definition of diameter.

Something along the lines of 'little fleas, ad infinitum', somehow comes to mind.

Best wishes,

Ray

ps I can often create confusion, circular or otherwise.

Hi, Ray,

Ray West said:

Hi Doug,

I guess it's now your lunch time ;-)

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Indeed.

The reason I asked for drawings, was that if the circle is blurred, how do you measure its diameter?

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No, the circle isn't blurred. It is the turning of points on the object into circles on the image that makes the image blurred. In theory (and neglecting diffraction), the circle of confusion has a well-defined edge. It is created by a cone of rays striking the focal plane.

If focus were perfect, the rays in the cone would all converge at a point in the image. If the focus error is such that the rays would converge behind the focal plane, the cone still has a non-zero diameter where it strikes the focal plane.

If the focus error is such that the rays converge in front of the focal plane, then as the rays progress beyond that point, they form an "expanding" cone again, which also has a non-zero diameter when it strikes the focal plane.

I'll post some figures in a little while that shows how that works. Gotta wash off the car wheels now - they are disgusting and Carla has a social visit to make tomorrow!

Best regards,

Doug

Consider widely spaced points of light.

The lens focuses the light to an image plane but has to go through a tiny aperture.

Due to diffraction (imagine the light as waves being "scraped" by the edge of the aperture), each point of light in focus will yield, not a point but a cone of higher light intensity and in addition, concentric circles of light, like expanding ripples in still water after dropping in a stone. Each successive ring of light around the central cone will peak in intensity and then fall to a minimum before rising somewhat to form the next larger diameter ring, but with lower light intensity than the one before. There will surely be no defined edge only a decision of how many circles to include.

Hi, Asher,

Asher Kelman said:

Consider widely spaced points of light.There will surely be no defined edge only a decision of how many circles to include.

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Indeed, which is why I explicitly stipulated to "excluding diffraction" when I said that the circle of confusion has a well-defined edge.

And of course, when we consider diffusion, the image of a point, "perfectly focused", is not a point either, but an Airy disk. Do we consider that to also be a circle of confusion of non-zero size (in the sense of depth of field reckoning)?

But note that if we (in a case in which diffraction is consequential) decide to include the effects of diffraction on our definition of the diameter of the circle of confusion. we would be unable to adopt a very small acceptable circle of confusion (ACOC) and thus our computed result for depth of field, for a very small aperture, would be larger than otherwise.

There is a way to rationalize that: there is no sense choosing an aperture so small as to give a certain, desired circle of confusion diameter (COCD) for an object at a certain displacement from the perfect focal position if the circle of confusion per se were "swamped" by the effects of diffraction for that aperture.

But then, we would need to establish a criterion for how large would the (theoretical, sharp-edged) COC need to be, compared to the diameter of the Airy circle (given as one of the conventional metrics, such as to the first null) before we would consider blurring due to imperfect focus to have an impact on the image quality.

An interesting problem.

In fact, we need to keep these two phenomena distinct, nevertheless including both (as well as that of various aberrations) in our analysis of the overall image result.

In another thread it was suggested that these various phenomena combine on a "root-sum-square" basis (often called "addition in quadrature", although I wish that metaphor were not used in that setting). But in fact the distribution across the COC is not likely Gaussian, so that isn't necessarily - or even likely - how it would work.

In fact, if the edge of the joint blur figure (the combined effect of a non-zero COCD and the Airy circle from diffraction) were defined by sweeping an Airy circle around the circumference of the COC proper, I think that the "diameter" of that figure (to the "first null") would be the simple sum of the COCD and the "diameter" of the Airy circle.

In fact, the actual radial profile of luminance would not be what I have described (owing to the contributions of the Airy circles whose centers were not on the periphery of the COC proper). I'm sure that result is well known, just not to me!

Best regards,

Doug

Doug_Kerr said:

In another thread it was suggested that these various phenomena combine on a "root-sum-square" basis (often called "addition in quadrature", although I wish that metaphor were not used in that setting). But in fact the distribution across the COC is not likely Gaussian, so that isn't necessarily - or even likely - how it would work.

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I'm sorry. I was using the former until recently, but in some contexts I was in people were using the latter, so I adapted quickly, as the compact nature of "in quadrature" was appealing.

Neither really is important, however. The only significance of adding gaussian distributions is that they have that simple relationship, but really, all noise (or dispersion) distributions add non-linearly. None team up positive with positive, and negative with negative (relative to the mean) by rule, so there is both constructive and destructive additions going on. If you make a 50% gray square in PS, and add 5% linear monochromatic noise, and do so repeatedly, the histogram goes from a rectangle to a virtually gaussian distribution after a few steps, with intensities the same as if the distribution were Gaussian:

7.84 -> 11.04 -> 13.47 -> 15.62

IOW, the distribution does not seem to be particularly important. Nothing needs to have a Gaussian distribution to add in a root-sum-square manner. I would suspect that most if not all symmetrical distributions would result in a Gaussian distribution after adding a number of them in equal intensity.

So, anyone prepared to show me some circles and show me how to measure the diameter? Surely, there is an image somewhere, with these confusing circles. Then, we can discuss 'what is acceptable' I guess.

Best wishes,

Ray

PS funny, we can't discuss art, since understandably nobody agrees on definitions, and that is complicated - but a circle???

Hi, John,

John Sheehy said:

I'm sorry. I was using the former until recently, but in some contexts I was in people were using the latter, so I adapted quickly, as the compact nature of "in quadrature" was appealing.

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I understand, and the terminology is widely used.

I think it was probably adopted as a parallel to, or an offshoot of, the well-accepted use of "orthogonal" to mean things far beyond the original geometrical meaning.

Neither really is important, however. The only significance of adding gaussian distributions is that they have that simple relationship, but really, all noise (or dispersion) distributions add non-linearly. None team up positive with positive, and negative with negative (relative to the mean) by rule, so there is both constructive and destructive additions going on. If you make a 50% gray square in PS, and add 5% linear monochromatic noise, and do so repeatedly, the histogram goes from a rectangle to a virtually gaussian distribution after a few steps, with intensities the same as if the distribution were Gaussian:

Click to expand...

Note that the histogram of a uniform-luminance rectangular image is not conceptually a rectangle but an "impulse" (which of course looks rectangular - one unit wide - owing to the discrete nature not only of teh data but of a histogram itself).

Thus adding a number of random variables to that is no different than adding a number of random variables to zero (that is, just adding them), in this case variables that are probably Gaussian to begin with.

[/quote]IOW, the distribution does not seem to be particularly important. Nothing needs to have a Gaussian distribution to add in a root-sum-square manner. I would suspect that most if not all symmetrical distributions would result in a Gaussian distribution after adding a number of them in equal intensity.[/QUOTE]

But in any case, I suspect that if we take a circular figure (regardless of how generated) and then allow it to be impacted by diffraction (where the Airy circle "diameter" is on the same general order as the diameter of the base figure), the RSS summation doesn't well represent the "diameter" of the resulting figure.

But I might be wrong. I'll try and play with that when I get a chance.
I'm sure this matter is well treated in formal statistics, but my feeble background there doesn't bring it to hand!

Thanks for your input.

Ray West said:

So, anyone prepared to show me some circles and show me how to measure the diameter? Surely, there is an image somewhere, with these confusing circles. Then, we can discuss 'what is acceptable' I guess.

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Just look at the OOF highlights of a point lightsource, and you'll also notice that it's perimeter is not always a perfect circle.

The diameter is a function of the diameter (or shape) and position of the exit pupil of the lens, the focal plane position, and the distance from that focal plane. The intensity distribution across the 'circle' (or rather the intersection of the focal plane and a cone (of sorts) of light) depends on the aperture and optics.

http://www.vanwalree.com/optics/dof.html
http://www.cambridgeincolour.com/tutorials/depth-of-field.htm
http://en.wikipedia.org/wiki/Circle_of_confusion
http://www.largeformatphotography.info/articles/DoFinDepth.pdf

Bart

Hi Bart,

Thanks for the links. But in a quick read through, I can see no picture showing such circles in a photo, nor means of measuring the same, (plenty of discussion as to estimating/calculating, theory, but nothing in practical terms such as 'this is a circle with a diameter of x units', or 'here is a photo, look at all the circles'. Is it something that in practice can be isolated from other effects?

Plus, it seems undecided if it is circles we are really talking about. I thought it was only circles that had a single diameter.

Now, it is a simple concept, I think, maybe, perhaps, possibly, but it is made difficult. I expect, but probably will not attempt it, that I could find conflict between any of the links you have given. There must be, since a single concise explanation would suffice. Already we are in the thick of it wrt diffraction.

I know we are trying to get there, in this thread, I am try to start it from a basis that no assumptions are made, because I expect there is some disagreements within the very basics of the whole of our concepts. Well, where is the starting point?

So, I think a definition of the circle, if it is a circle should be given. If it is not a circle, then give it some other name - area/region/cloud of confusion. If it's not a circle, it can not have a diameter. If you want to build a house from a deck of cards, fine, but let us at least agree on the very basics, use the same deck of cards. Assume nothing. Just because someone else has given it the wrong name, has it wrong, we don't have to, in our little world.

Maybe it's just me, but this is getting to be one of 'the what is art' type of discussions, so I think it may be best if I clear off.

Best wishes,

Ray

Ray West said:

Hi Bart,

Thanks for the links. But in a quick read through, I can see no picture showing such circles in a photo, nor means of measuring the same, (plenty of discussion as to estimating/calculating, theory, but nothing in practical terms such as 'this is a circle with a diameter of x units', or 'here is a photo, look at all the circles'.

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As I said, look at OOF point lightsources. For images of COCs, look at e.g. figures 6 and 9 of the following reference:
http://www.luminous-landscape.com/essays/bokeh.shtml

Plus, it seems undecided if it is circles we/you are really talking about. I thought it was only circles that had a single diameter.

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The reference above also demonstrates that the circles (or rather disks, the perimeter modeled as a circle) needn't be circles by definition. The circle perimeter only applies to a defocused image of a point lightsource (our subject could be regarded as point lightsources of varying luminance across the field of view), seen through a circular aperture, on the optical axis. When closing down the aperture, the 'roundness' of the disk takes on the shape of a octagon or other shape, depending on the aperture blades and optical correction.

As Van Walree states in the link to his explanation,
"Circle of confusion
To calculate the depth of field, one needs a sharpness criterion. This criterion is taken as the so-called circle of confusion (COC). A COC value corresponds to the blur spot diameter, measured on the film/sensor, of an unsharply imaged point in object space. In DOF calculations it is customary to use the designation COC for the largest permissible circle of confusion. The blur disk diameter is zero for points in the plane of sharp focus and progressively grows as we move forward or backward from this plane in object space. However, as long as the blur disk is smaller than the acceptable COC it is considered sufficiently sharp and part of the DOF range.
"

He later continues:
"A circle?
The designation circle of confusion is widely used, but it is rarely correct in the strict sense of the word. For the blur spot will be a circle only in the image center, and that only for a lens used at full aperture. Towards the image periphery, at full aperture, the shape is more like a cat's eye. When a lens is stopped down the blur takes on the shape of the diaphragm opening. A lens with a six-sided diaphragm, for example, comes with a hexagon of confusion. Fortunately, the depth of field does not noticeably depend upon the shape of the aperture. Whether the aperture is triangular, hexagonal, octagonal, ... the depth of field will not really depart from that of a truly circular aperture at the same F-number, although one may expect an influence on the rendering of out-of-focus areas.
"

Bart