Depth of field and that pesky circle of confusion

[Doug Kerr]

Depth of field is a man-made outlook, not an optical phenomenon.

Focus and misfocus

With the camera focus set in a certain way, then a point on an object at the "plane" of perfect object focus (of course it may actually be a curved surface) will, setting aside lens aberrations, result in a point on the image. We may say that such an object point is perfectly focused by the camera.

Object points not in that surface (simplistically, at other distances from the lens) will not result in a point in the image. Rather, they will result in a finite-sized blur figure. We can consider this to be a circular blob of light with a clean boundary and thus a certain diameter. This blur figure is the manifestation of imperfect focus of the object point that spawned it.

The blur figure is often called a circle of confusion.

Depth of field

Consider a situation in which the camera is focused at a certain distance. Object points not at that distance will be subject to misfocus.

We now ask the question, "for what range of object distances, embracing less than and greater than that perfect focus distance, will the degree of misfocus not exceed a limit that we will establish". We may speak of that degree of misfocus as the "limit of acceptable misfocus", or "the limit of misfocus that can be considered negligible", or in other terms (not necessarily synonyms), but it is any case a limit, one we have established. That range of object distance is called the depth of field. (There are several ways it can be stated.)

What metric might we use to indicate "degree of misfocus"? We might adopt the amount of decline in the modulation transfer function at some spatial frequency as that metric. But most commonly, in this area of work, we use the diameter of the circle of confusion as that metric of misfocus - the greater that is, the greater the misfocus. And we normally state that as it occurs in the image itself - that is, on the film or digital camera sensor, since the basic optical equations we will soon encounter "work there".

Thus, expressing a limit on the degree of misfocus becomes establishing a limit on the diameter of the circle of confusion - what I call (curiously enough) the circle of confusion diameter limit (COCDL).

Now as many of you know, it is most common in this work to call that value the "circle of confusion". That's like calling the nominal diameter of the largest wheel that can be used on a car the "wheel". It is not good notation.

Once we have chosen this limit, straightforward optical calculations, using the distance at which the camera is focused, the f-number of the lens, the focal length of the lens, and our choice of a circle of confusion diameter limit (COCDL) as inputs, will tell us the least and greatest subject distance for which the amount of blurring from misfocus will not be greater than our limit. That range (expressed in any of several ways) is the depth of field for the conditions specified. When it is stated as a single number, it is the difference between least and greatest distance.

CHOOSING A COCDL

Now, how might we choose our limit? In reality, this should depend on many particulars of the individual photographic mission. How will the image be delivered, how will it be viewed, how will misfocus impact the viewer's experience, and so forth. No simple answer comes forth.

But, in the interest of giving "some basic idea" of depth of field for general use, a number of rather arbitrary approaches are often used.

Human visual acuity

One says that our limit is intended to reflect the degree of misfocus that will be seen as "bothersome" by the viewer. Perhaps we will start with some widely-used measures of human acuity, such as that the eye can resolve two objects separated in angle by and angle of about 1/2800 radian in "viewing space". Perhaps we will conclude that a circle of confusion whose (angular) diameter is twice that would be a manifestation of "bothersome" blurring from misfocus. That diameter would be equivalent to 1/1400 radian in "viewing space".

But our calculation asks for a limit in terms of a diameter (in mm, perhaps) at the focal plane, not an angle as the image is viewed. So we must make some assumption as to how the image is to be viewed. This might relate directly to the expected use of the image, or it might just be chosen arbitrarily so we can get an answer without knowing exactly what we are doing.

In the latter vein, it is common to arbitrarily think of viewing the "delivered" image (print, screen display, etc.) from a distance equal to the diagonal dimension of the image. Suppose we have an image with an aspect ratio of 3:2, and that we put it to a size of 12" x 8". Then we would assume viewing of that "print" from a distance of 14.4 inches. Is that reasonable? Reasonable how? Might someone look at a 12" x 8" image from a distance of 14.4"? Sure, why not. Is that "typical"? Don't ask.

The, using the 1/1400 radian guideline, the circle of confusion (as "enlarged" onto our print) would be 0.0103" in diameter (0.261 mm).

But of course, we must use a diameter limit as it pertains to the film or sensor to enter into our calculations. Lets assume a "full-frame 35-mm" format size - 36 x 24 mm (with a diagonal of 43.3 mm).

Well, by proportions, the corresponding diameter there will also be 1/1400 of the diagonal dimension, or 0.031 mm.

And that is in fact one of the widely used values for COCDL for the full-frame 35-mm format size.

Camera resolution

Another approach says, "not knowing exactly how the image will be used, we should set as our misfocus limit any misfocus that would cause a noticeable deterioration of the camera resolution". This might cause us to choose a COCDL that is, for example, twice the camera's pixel pitch.

Is that reasonable? Well, if this outlook rings your bell, sure.

Consider for example the Canon EOS 1Ds Mark II dSLR, with a pixel pitch of 0.0064 mm. Then, on the basis above, we might choose for use in connection with this camera a COCDL of 0.0128 mm. That is about 2.4 times smaller than the value mentioned above as being widely used. What does that mean?

Well, it means we will consider as "acceptable blurring" blurring only about 2.4 times less that under that other outlook. Why? Because we decided than some notion of a viewer looking at the final product from some arbitrary distance is meaningless, and we wish to plan our work to avoid any misfocus for the "critical elements" of the scene that would significantly diminish the camera's resolution potential.

Now this can lead to some peculiar happenings. Suppose a photographer has an EOS 1Ds and considers getting a 1Ds Mark III. In connection with some project, he makes a depth of field calculation contemplating his 1Ds (using the camera resolution premise for choosing a COCDL), and then just for kicks, does it again for the 1Ds Mark III. He gets a smaller depth of field for the same shot setup!

Does that mean that the blurring would be worse for an object at a certain distance (not at the perfect focus distance) with the 1Ds III than with the 1Ds? No, of course not.

What it means is that with the 1Ds III, we are expecting higher resolution of the photographic result, and if we expect the degree of focus to keep pace with that, we will need to confine our critical object features to a smaller range of distance (to decrease the blurring that occurs).

#

Best regards,

Doug

[Cem_Usakligil]

Hi Doug,

An apt and useful article, as usual. Thanks for that.

Originally Posted by Doug Kerr

....
Camera resolution

Another approach says, "not knowing exactly how the image will be used, we should set as our misfocus limit any misfocus that would cause a noticeable deterioration of the camera resolution". This might cause us to choose a COCDL that is, for example, twice the camera's pixel pitch.

Is that reasonable? Well, if this outlook rings your bell, sure.

Consider for example the Canon EOS 1Ds Mark II dSLR, with a pixel pitch of 0.0064 mm. Then, on the basis above, we might choose for use in connection with this camera a COCDL of 0.0128 mm. That is about 2.4 times smaller than the value mentioned above as being widely used. What does that mean?

Well, it means we will consider as "acceptable blurring" blurring only about 2.4 times less that under that other outlook. Why? Because we decided than some notion of a viewer looking at the final product from some arbitrary distance is meaningless, and we wish to plan our work to avoid any misfocus for the "critical elements" of the scene that would significantly diminish the camera's resolution potential.

Now this can lead to some peculiar happenings. Suppose a photographer has an EOS 1Ds and considers getting a 1Ds Mark III. In connection with some project, he makes a depth of field calculation contemplating his 1Ds (using the camera resolution premise for choosing a COCDL), and then just for kicks, does it again for the 1Ds Mark III. He gets a smaller depth of field for the same shot setup!

Does that mean that the blurring would be worse for an object at a certain distance (not at the perfect focus distance) with the 1Ds III than with the 1Ds? No, of course not.

What it means is that with the 1Ds III, we are expecting higher resolution of the photographic result, and if we expect the degree of focus to keep pace with that, we will need to confine our critical object features to a smaller range of distance (to decrease the blurring that occurs).

Just a quick correction if I may. The pixel (sensel) pitch of Canon EOS 1Ds Mark II dSLR is not 0.0064 mm but 0.0071 mm. The newer Canon EOS 1Ds Mark III dSLR's pixel (sensel) pitch is indeed 0.0064 mm. But this does not impact your message, which is clear and correct.

My personal choice of COCDL is to use 1.5 times the pixel (sensel) pitch (*) but this is a semi-arbitrary decision which works for the worst case scenario. As you've rightfully stated, the blurring would not be worse or better for an object at a certain distance by one's choice of COCDL. What it helps me achieve is that if I want to have an object to be in perfect focus at pixel level for a specific camera sensor (notwithstanding the limitations of the lens used), I then calculate the DoF range using this extremely conservative COCDL value (which is 1.5 times the pixel pitch).

(*) Coincidentally, this is the formula Bart uses to calculate the maximum COCDL which defines when the system becomes diffraction limited due to a certain aperture chosen. This is when the diffraction pattern (the so-called airy disk) at a certain aperture has a diameter which is larger than the maximum COCDL.

[Bart_van_der_Wolf]

Originally Posted by Cem_Usakligil

Hi Doug,

An apt and useful article, as usual. Thanks for that.


Just a quick correction if I may. The pixel (sensel) pitch of Canon EOS 1Ds Mark II dSLR is not 0.0064 mm but 0.0071 mm. The newer Canon EOS 1Ds Mark III dSLR's pixel (sensel) pitch is indeed 0.0064 mm. But this does not impact your message, which is clear and correct.

Correct, I assumed it to be a typo because Doug does mention the 1Ds3 later.

My personal choice of COCDL is to use 1.5 times the pixel (sensel) pitch (*) but this is a semi-arbitrary decision which works for the worst case scenario. As you've rightfully stated, the blurring would not be worse or better for an object at a certain distance by one's choice of COCDL. What it helps me achieve is that if I want to have an object to be in perfect focus at pixel level for a specific camera sensor (notwithstanding the limitations of the lens used), I then calculate the DoF range using this extremely conservative COCDL value (which is 1.5 times the pixel pitch).

Indeed. Reality doesn't change when we use a different criterion. It's just a tool to decide whether we consider the sharpness to be sharp enough for the purpose we have in mind. With large format output in mind we need to use a stricter level of tolerance than for web publishing. Of course when we want to keep our options open then the stricter tolerance is what we want to use (better safe than sorry).

(*) Coincidentally, this is the formula Bart uses to calculate the maximum COCDL which defines when the system becomes diffraction limited due to a certain aperture chosen. This is when the diffraction pattern (the so-called airy disk) at a certain aperture has a diameter which is larger than the maximum COCDL.

In my experience with a number of different cameras (Bayer CFA + anti-aliasing filter) the 1.5x sensel pitch criterion marks the onset of visible microdetail losses caused by diffraction at the pixel level. Again, because diffraction is only determined by the F-number (and wavelengh), diffraction is of a fixed size at a given F-number and doesn't care about our criteria or sensel sizes. However, when the pixels become small enough to resolve the airy disc (or the airy disk large enough for the sensels), then the per pixel resolution will suffer (which impacts the output magnification potential without introduction of more blurry pixels per unit length).

Having said that, it's also not meant as a restriction against using smaller apertures than that criterion would suggest. As we stop down, and the diffraction blur pattern diameter grows, we will start to subsample the diffraction pattern more and more accurately. That can be exploited with subsequent deconvolution sharpening. Because the shape of the point spread function (PSF) of the Airy disk can be calculated/approximated, we can also make a good estimate of the required PSF to use for deconvolution, so some resolution can be restored.

Confused yet?

Cheers,
Bart

[Asher Kelman]

Originally Posted by Bart_van_der_Wolf

Having said that, it's also not meant as a restriction against using smaller apertures than that criterion would suggest. As we stop down, and the diffraction blur pattern diameter grows, we will start to subsample the diffraction pattern more and more accurately. That can be exploited with subsequent deconvolution sharpening. Because the shape of the point spread function (PSF) of the Airy disk can be calculated/approximated, we can also make a good estimate of the required PSF to use for deconvolution, so some resolution can be restored.

Confused yet?

Not at all, Bart! Thanks for adding this last hopeful addition and in fact I expected you would as I started to read you post. After all, I know you are keeping up to date with deconvolution at the edge of diffraction limits.

Asher

[Doug Kerr]

Hi, Cem, Bart,

Originally Posted by Bart_van_der_Wolf

Correct, I assumed it to be a typo because Doug does mention the 1Ds3 later.

Indeed. Mea culpa.

Thanks for the catch, guys.

Best regards,

Doug