The two faces of "depth of field"

[Part 1]

Introduction

We tend to use the term "depth of field" to refer to two quite different, although related, issues:

• Depth of field [DoF]
• Out-of-focus blur performance (OFB)

With regard to depth of field, we ask this question:

With regard to out-of-focus blur performance, we ask this question:

In connection with each of these matters, we may be interested in the implications of the use of different sensor sizes.

Depth of field

In calculating the depth of field to be experiences in any given camera setup, we must deal with the question "how do we quantify 'amount of blurring' " and how do we decide what is 'acceptable' "?

We normally use as the numerical indicator of amount of blurring the diameter of the circle of confusion, the "blur figure" created from a point on the object in the case of imperfect focus, on the focal plane. We impose our view of how much blur is "acceptable" by establish a limit on this diameter, called the circle of confusion diameter limit (COCDL).

But how might we choose what value of COCDL to use? The are two popular underlying approaches:

Approach A: Visual perception

Here we say that acceptable blurring is blurring that is not significantly perceptible to the viewer of the image. Of course that is a best an arbitrary judgement, but the important fact is that this must be expressed expressed in terms of the angular diameter of the circle of confusion as measured from the eye.

But to calculate depth of field, we must adopt a COCDL, which is measured as a distance on the focal plane. To relate the two, we must adopt (probably arbitrarily) some ratio between the size of the viewed image and the size of the image on the focal plane, plus a viewing distance.

The need for this (often) arbitrary assumption of the viewing context is often cited as a reason not to adopt traditional values of COCDL (such as "1/1400 of the diagonal size of the image on the focal plane").

Approach B: Degradation of resolution

Here, we define acceptable blurring as that which does not significantly degrade the resolution of the camera. Again, what diameter of the circle of confusion, compare to the resultion of the camera "in perfect focus", constitutes "significant degradation" is rather arbitrary, but a practical interpretation is to use for COCDL some small multiple (maybe 1x, maybe 2x) of the sensel pitch.

What is the advantage of using this approach over Approach A? Well, one advantage is that we don't have to think about the viewing environment.

In most cases, depth of field calculated using a COCDL developed under this approach will be less than when using a COCDL developed under Approach A.

The irony

Of course the irony here is that regardless of the approach we take with respect to the COCDL to be used for reckoning the depth of field, or the COCDL value we adopt, the image will look identical for any given setup of subjects. The camera of course has no idea what COCDL we would use when calculating depth of field, or how we got it. Depth of field is a creature of our outlook on what a given optical setup does, not an innate property of the optical result.

Effect of sensor size

We often ask, "if I go from a sensor size of 27 mm (diagonal) to 43 mm, what will that do to the calculated depth of field, 'all other factors remaining equal' "?

Well it is that latter qualification than must be carefully thought out. A reasonable prescription might include these:

• Focal length that gives a comparable field of view
• Same focus distance

But we also need to make some decisions about the COCDL to be used as an input to the calculation.

If we are "Approach A" believers (this is a largely ideological issue), then a reasonable choice would be:

A: Use a COCDL that is a consistent fraction of the sensor dimensions.

But suppose we believe in Approach B? Well, we might do various things. Let's say we will not want to consider the actual resolutions of two cameras being compared, but just want a "generic" result.

Then we might:

B1: Assume that the two cameras have the same resultion in terms of cycles per mm, or
B2: Assume that the two cameras have the same resultion in terms of cycles per image height

Oh. boy!

Well, lets see what happens. We will start with a camera with a 27 mm sensor ("APS-C"), with the following setup:

• Focal length: 50 mm
• Focus distance: 10 m
• Aperture: f/4.0

We will start with a COCDL of 1/1400 of the sensor diagonal size (0.019 mm).

The calculated depth of field (double-sided) would be 6.66 m.

Now we consider the use of a camera with a 43 mm sensor size ("35-mm full-frame"), with the:"same setup" (as defined above). Now, depending on the approach we take regarding COCDL, the calculated depth of field would be:

A: 4.03 m [0.61 times the original value]
B1: 2.41 m [0.36 times the original value]
B2: 4.03 m [0.61 times the original value]

In Part 2, we will do a similar exercise with regard to out-of-focus blur performance.

Time now for breakfast.

[to be continued]

[Part 2]

Out-of-focus blur performance

Recall that with regard to out-of-focus blur performance, we ask this question:

To deal with this, we need not establish any arbitrary parameter of the calculations (as we do with COCDL in the case of depth of field). The answer comes out of the calculations as the diameter (on the focal plane) of the circle of confusion formed from a point on the "foreground" or "background (out-of-focus) object.

But we do have to decide just how to interpret that diameter (especially if we are making comparisons between two different cameras with different sensor sizes).

But here there is only one reasonable premise for that assessment: the diameter of the circle of confusion compared to the image size (perhaps the image diagonal dimension). That will be consistent for any size of displayed image, and independent of viewing distance.

Now, lets do an exercise. Again, we will start with the assumption of a 27 mm (diagonal) sensor, and this "setup":

• Focal length: 50 mm
• Focus distance: 10 m
• Aperture: f/4.0

Then, let's consider a background object at a distance of 30 m.

A point on that will be imaged as a circle of confusion of diameter 0.042 mm. That is 1/643 of the image diagonal.

Now we will move to a 43 mm sensor (with a focal length that preserves the field of view). Now, a point on that same background object will be imaged as a circle of confusion of diameter 0.107 mm. That is 1/402 of the image diagonal - a "fatter" blur figure in terms of image size, by about 1.6 x (in fact, just about the ratio of sensor sizes).

Best regards,

Doug

Doug Kerr said:

The answer comes out of the calculations as the diameter (on the focal plane) of the circle of confusion formed from a point on the "foreground" or "background (out-of-focus) object.

Click to expand...

You suppose that the image of a punctual light source outside the locus of focus is a disc. For real world lenses, this is rarely the case. Therefore your brilliant demonstration and all the similar demonstrations and formulas that one usually finds about depth of field and out of focus blur are based on false premises.

Hi, Jerome,

Jerome Marot said:

You suppose that the image of a punctual light source outside the locus of focus is a disc. For real world lenses, this is rarely the case.

Click to expand...

Yes of course. I just used the familiar simplification to help get the concept across.

Therefore your brilliant demonstration and all the similar demonstrations and formulas that one usually finds about depth of field and out of focus blur are based on false premises.

Click to expand...

I'll write that down so I don't misquote it later.

D

Doug Kerr said:

Yes of course. I just used the familiar simplification to help get the concept across.

Click to expand...

Now that you got the concept across, I can't wait for a study of the subject without the simplification.

For your convenience, I type "bokeh" in google image search and found the following example of real lens rendition of an out of focus point source:

6 blades hexagonal diaphragm, with an aspherical element:


effect of vignetting and, apparently, overcorrection of spherical aberration


aspherical lenses

Hi, Jerome,

Jerome Marot said:

Now that you got the concept across, I can't wait for a study of the subject without the simplification.

Click to expand...

Me too. I look forward to your paper. It looks as if you are off to a good start.

Note that my discussion was solely as to the "size" of the blur figure (and that certainly in a simplistic way, assuming that the figure has a sharply-defined, circular outline, and based on idealized lens behavior - as we do for so many optical theory topics here), not the many other aspects of bokeh with which we are concerned.

It will be interesting to learn how we can predict, even approximately, the "size" of the blur figure taking into account the many aspects of real lenses you mention.

Best regards,

Doug

Oh, come one, Doug! You were off to a good start. You are the one who posted about "Out-of-focus blur performance" first!

Hi, Jerome,

Jerome Marot said:

Oh, come one, Doug! You were off to a good start. You are the one who posted about "Out-of-focus blur performance" first!

Click to expand...

Indeed, and I did what I came to do, and I stand by my findings and observations, which I hope might be useful to some.

My basic objective was to present a practical way in which we might estimate the impact on one property of the blur figure (its "diameter") of a change in sensor size.

I don't know what further results, useful to the members, I would be in a position to look into. I'm not, for example, really prepared to take actual images of point sources with various lenses under various setups and refocus situations and, for example, show the distribution of luminance across the blur figure. Nor would I know what I might say about such findings.

And I don't really think I am in a position to be able to predict the luminance distribution in the face of, for example, quantifiable spherical aberration. Not do I know what the members might make of such an analysis.

And I don't have a lot of enthusiasm for developing a metric analogous to diameter but applicable to a blur figure whose outline was not essentially circular.

A lot has been written, for example, about the "visual impact" of various luminance profiles for the blur figure and which ones are often considered desirable insofar as the "quality" of the bokeh is concerned. I have nothing to add to that.

What kind of understandings do you think would be valuable to the members as a next step in this investigation? Perhaps there is something I could undertake.

Thanks for your help with this.

Best regards,

Doug

I would not have participated to this thread if you had not written "Out-of-focus blur performance". The formulas you cite are only useful when the objective is to keep blur minimal. They allow to compute the aperture necessary so that an object of a given extension gives a blur figure below a value where everything appears to be sharp. That they do very well.

Using them to assert what happens outside of the depth of field zone is not useful, it is actually misleading since different blur functions will render the subject more or less recognisable or "blurred" for a given distance.

You are writing that lot has been written about the "visual impact" of various luminance profiles for the blur figure. I suppose that you know that there is considerable research at present on this particular field. One of the applications of this research is, for example, the Lytro lightfield camera, a device which can essentially synthesise any kind of blur function one wants. Is this research what you were referring to?

Hi, Jerome,

Jerome Marot said:

Using [the equations] to assert what happens outside of the depth of field zone is not useful, it is actually misleading since different blur functions will render the subject more or less recognisable or "blurred" for a given distance.

Click to expand...

I did not attempt to discuss how "recognizable" a blurred out-of-focus object was. My discussion was specifically of the "diameter" of the blur figure from an out-of-focus point source. Perhaps characterizing that as a principal aspect of "blur performance" was improvident.

I proceeded on the assumption that the classical equations for spot size give a reasonable approximation of the "diameter" of the blur figure for substantially out-of-focus situations. Perhaps I am gravely wrong in that.

Do you have reference to any articles of illustrations that would show the degree to which that may be in error?

Perhaps tonight I will shoot some stars, substantially out-of-focus, and see what I can make of their images.

Of course if the blur figure does not exhibit circular symmetry, the concept of a "diameter" does not actually apply. It is for that reason that there is no f-number for a lens with a substantially-non-circular entrance pupil (although we often reference such). (They do have a T-stop value.)

You are writing that lot has been written about the "visual impact" of various luminance profiles for the blur figure. I suppose that you know that there is considerable research at present on this particular field. One of the applications of this research is, for example, the Lytro lightfield camera, a device which can essentially synthesise any kind of blur function one wants. Is this research what you were referring to?

Click to expand...

No. I was referring to the numerous articles on various photographic sites that discuss different luminance profiles across the blur figure (including different shapes of the nominal "boundary") that may occur and comment on their impact on the "quality" of the resulting bokeh.

Best regards,

Doug

Yes, I sure have trouble recognizing the background objects here:


Even here:


Or here:


But here I sort of can:


I wonder if that means the bokeh work was successful or unsuccessful.

I'm not sure that when we intentionally seek to blur the background (the context of my note), we necessarily expect the background objects to be "recognizable" (or "unrecognizable").

Best regards,

Doug

Allow me to give you some examples.

Hi, Jerome,

Jerome Marot said:

Allow me to give you some examples.

Click to expand...

Mostly nice work.

What is your point? That in these the images the blur figures from points on the objects may not have the size predicted by the geometric blur formula?

Or is is just that there is more to "bokeh planning" than concern with the diameter of images of out-of-focus point sources.

No kidding.

Best regards,

Doug

The point was to give examples of backgrounds which have been blurred to emphasise the subject but are still recognisable so as to give context.

Hi, Jerome,

Jerome Marot said:

The point was to give examples of backgrounds which have been blurred to emphasise the subject but are still recognisable so as to give context.

Click to expand...

Good examples. An important genre. We sometimes get it "unavoidably".

Do you believe that a useful metric of the "degree of blurring" in such cases is the actual "diameter" of the blur figure created at the focal plane for each point on the object?


Or do you feel that other factors (probably the only one is the distribution of illuminance across the blur figure, which recognizes the "shape" of the "boundary") are of such impact on the overall visual appearance in real cases that the simple "diameter" metric may be of little value in characterizing or predicting the appearance?

Best regards,

Doug

Hi, Jerome,

This is one of the articles I referred to:

http://www.bobatkins.com/photography/technical/bokeh.html

Best regards,

Doug

Doug Kerr said:

Do you believe that a useful metric of the "degree of blurring" in such cases is the actual "diameter" of the blur figure created at the focal plane for each point on the object?

If so, do you think that the "geometric blur" calculation gives a good estimate of that metric?​

Or do you feel that other factors (probably the only one is the distribution of illuminance across the blur figure, which recognizes the "shape" of the "boundary") are of such impact on the overall visual appearance in real cases that the simple "diameter" metric may be of little value in characterizing or predicting the appearance?

Click to expand...

Keep in mind that the example I gave with a heart shaped aperture is just illustrative. The blur figure in common lenses is different. Most importantly it varies across the field, so even if you could define a diameter, the diameter varies.

As to your question: the "diameter" of the blur figure is not always a useful metric. Sure, it is useful when we compare a given lens at one aperture to the same lens stopped down. It is less useful when we compare two different lenses at the same aperture.

To recapitulate:

1. In my original report (part 2), I show that, using what I consider to be a reasonable premise for comparison, the calculated "diameter" of the blur circle from an out-of-focus point source, measured with respect to the image size, will remain approximately constant as we vary sensor size. By calculated, I mean using the traditional "geometric" equation for defocus circle of confusion diameter.

2. We were reminded that the visual nature of an out-of-focus background or foreground area was not wholly a creature of the diameter of the point object blur circle. (I had not discussed the matter of the visual nature of an out-of-focus background or foreground area, but it's certainly a valid point.)

3. It was aptly pointed out that with a real lens, aberrations (most prominently spherical aberration) would result in a blur figure whose "diameter" would be greater than that calculated as mentioned in item 1.

4. It was aptly pointed out that the discrepancy between the "calculated" and "actual" blur figure might well be greater for off-axis regions.

5. I had no feeling as to the typical degree of this discrepancy. Might we likely encounter a blur circle whose diameter is 5% greater than the calculated value? Might we encounter a blur circle whose actual diameter is 40% greater than the calculated value? I solicited inputs (perhaps from ray tracing, perhaps from actual image tests, whatever) that would give some idea of this. I await the appearance of any such.

Thanks to all for your help with this.

Best regards,

Doug

I can quickly offer this test, which I made a few years ago:


The lights are more or less punctual, although not as much as distant stars. The bat-like figures on the top right corner is what the light points become when imaged by the lens (the main aberration being coma here). I would suppose that the "diameter" does not decrease when out of focus.

This is one of my worst lenses for coma. On lens tests, this would be considered a relatively poor lens full open (although coma does not show that much on resolution charts), but this was standard for a fast 28mm at the time. I keep the lens because I like the rendering in some cases actually. The full resolution is here if you need it.

Hi, Jerome,

Jerome Marot said:

I can quickly offer this test, which I made a few years ago:

Minolta__28_f2.0​

The lights are more or less punctual, although not as much as distant stars. The bat-like figures on the top right corner is what the light points become when imaged by the lens (the main aberration being coma here). I would suppose that the "diameter" does not decrease when out of focus.

This is one of my worst lenses for coma. On lens tests, this would be considered a relatively poor lens full open (although coma does not show that much on resolution charts), but this was standard for a fast 28mm at the time. I keep the lens because I like the rendering in some cases actually. The full resolution is here if you need it.

Click to expand...

That's very nice. Thank so much.

Indeed, from a customary terminology standpoint, the cause of the large blur figures off-axis is typically coma rather than spherical aberration (although of course actually "coma" is the general case of "spherical aberration", or, alternatively, spherical aberration is the degenerate case of coma or, alternatively, coma comprises two phenomena, spherical aberration and inconsistent lateral magnification across the field).

Thanks again for the nice example.

Best regards,

Doug

Then, can you explain why coma takes this peculiar shape in that lens with two tails that extend perpendicular to a line crossing the center of the picture? Note that this is not unique to this lens and is more or less the general case.

Hi, Jerome,

Jerome Marot said:

Then, can you explain why coma takes this peculiar shape in that lens with two tails that extend perpendicular to a line crossing the center of the picture? Note that this is not unique to this lens and is more or less the general case.

Click to expand...

I'm not sure what that means. The concept of inconsistency of lateral magnification (the "new ill" that we have in coma that is not a factor for on-axis points) would not seem to lead to such a figure.

So I need to ponder that some.

Perhaps this reflects some interaction of astigmatism with the other phenomena.

Thanks

Best regards,

Doug

Jerome,

do you see that nice little arrow to the right?
Looks like coma plays a role here too.



Best regards,
Michael

Hi, Jerome,

Interesting mention of "bat-like coma" on dpr:


This was in a discussion of the [make?] 42.5/1.2 and Nikkor 58/1.4:

http://www.dpreview.com/forums/thread/3605182

Interesting.

Best regards,

Doug

Hi, Jerome,

In the literature, I find reference to both sagittal and meridional coma. I believe these refer to the radial and circumferential aspects of spreading of the spot in an off-axis situation (not necessarily on that order - I always get very confused about that).

Evidently, "classically" (for a simple lens with spherical surface(s) ) there is an expected ratio of the two (3:1), leading to the familiar "teardrop-shaped" spot from which the name coma comes. But that does not always obtain.

I think what happens is that the component "circles" can be ellipses with their major axes either radial or circumferential, even though the variation in magnification from zone to zone always places them along a radial line. If the ellipses are quite eccentric with major axis circumferential, then we can get the kind of circumferentially-oriented "wings-shaped" spot seen so clearly in your recent example image.


In this article:

http://www.nikkor.com/en/story/0049/

in section IV, there is a discussion of the resulting shape of the spot under different situations of correction of coma. It seems to embrace both the classical "radially-oriented teardrop" and the circumferentially-oriented "wings-shaped" spot.

But the detailed discussion, as is so often the case, is a bit confusing. Among other things, while "sagittal" and "meridonal" coma are probably most aptly applied to two orthogonal aspects of the "spreading" process, in some cases Nikon uses the terms to mean a spot whose major direction is radial and circumferential (not necessarily in that order) (maybe).

Nevertheless, I think we know know that the insects in your image are not of an unexpected species.

************

At the moment I am hoping for thin enough cloud cover tonight that I can do some more star tests. On the agenda are:

• Testing with the focus distance of the lens set to a known distance by preliminary focusing onto a target at that distance.

• Use of a shorter exposure time (in part by escalation of the ISO sensitivity) to minimize the effect of motion spreading due to the Earth's rotation. This may require a compromise in exposure result, but I think that is not of any consequence to the major interest here (the spot seems to have a fairly rapid fall-off).

• Testing with the point object (perhaps Alpha Orionis) both near the center of the field and also near a corner.

But a look out the window is not optimistic as to the odds for the cumulus vs. Alpha Orionis struggle.

Best regards,

Doug