DOF versus lens length and sensor size

[DaveW]

"In practice, lens aberrations change this form a little bit by either rolling off the edges (producing a soft-edged blur) or brightening the edges (producing a bright-ringed blur)."

That was the reason I gave the Rockwell link previously as it shows the difference in blur circles due to lens aberrations just under the plant picture:-

http://www.kenrockwell.com/tech/bokeh.htm

"The blur circles that I'm talking about are essentially images of the aperture. (You see already that the term "circle" is a perversion, since most apertures are obviously polygonal!) "

I suppose when the concept of blur circles was first introduced apertures were truly circular in that Waterhouse stops were used rather than variable aperture blades?:-

http://en.wikipedia.org/wiki/Waterhouse_stop

DaveW

[Harold Gough]

Unfortunately, the term "circle of confusion" is often taken to mean "the maximum permissible circle of confusion", which is not what I intended.

Just to clarify, I have come across this previously in terms of maximum circle (disc) size acceptable on film for the required resolution. The context was principally that of hyperfocal distances/focusing.

Harold

[rjlittlefield]

you get more in focus behind the subject plane than in front.

This becomes less the case as you go up the magnification scale. Intuitively, this should disappear at 1:1 but that seems too simple to be the case.

Here's an intuition that works: It becomes less the case all the way up the magnification scale, and disappears only at infinite magnification.

--Rik

[ChrisR]

So does the Lefkowitz DOF formula (as in the spreadsheets) give the same asymmetry at greater distances? I thought not, something I must find out. What I haven't worked out if where the asymmetry becomes important
I'm not sure if this will be clarification, or just laboring the point:
What might be happening, is that with the smaller camera, habitually used with a wider angle of view lens at close distances, the background quickly becomes non-macro where the asymmetric effect is significant. ANd on top is the point made by Dave, about the resolution at which we normally view things. A print a metre wide , would have a C of C =1mm, but if you're looking at it from say three paces back, can you see the difference between a 1mm and a 2mm circle? If not, what does that do to the perceived DOF?

We've discussed before this concept that "you get more stuff looking sharp when you use a smaller camera".
The outcome of those earlier discussions was (and remains) pretty straightforward

What we're now talking about in this thread is a different situation. We're imaging the same in-focus field of view from different distances

I'm not so sure.
What's being talked about is what happens when you use the camera, without thinking about maths. I believe that if you gave someone two cameras to go round a wood with taking pics of acorns on oak trees, and then asked another person to pick out the prints with the greater dof, a little camera would win. The mathematics is interesting, revealing, surprising, yes, but not everything.

If one were a camera phone and the other a 10x8, the 10x 8 would have a tripod. "The outcome is pretty straightforward" if you eliminate such differences from the maths.

[rjlittlefield]

I see that ChrisR and I have been typing at the same time. I'll go ahead and submit this post, and then respond separately to the points raised in his last posting.

(It's 1:30 am and I'm not sure I follow 100% of the above, yet)

One "error" I haven't got round to checking on a spreadheet is a comparison with the "other" standard dof formula pair which show you get more in focus behind the subject plane than in front. The background in wide-angle shots is more likely to be non-macro of course, where the difference becomes larger.

I tried a couple of mods to the sheet to see what angle of view you would get with lenses you might actually choose, where the focal length is in proportion to sensor size. (I arbitrarily used 1.5x) It shows how much wider the small-sensor camera view angle becomes. That with Rik's posts above, and a contribution from the "error" mentioned, combine to show the effect I'm familiar with - you get more stuff looking sharp when you use a smaller camera.
It seems that Mathematical analysis of a subset of the factors, and tests designed to demonstrate specific relationships, don't necessarily tell the whole story.

The last line here is the one I added to Rik's SH, formula shown below it.

To clarify...

The spreadsheet that ChrisR has used for illustration here is not the one that I linked earlier in this thread (HERE).

Instead it is derived from a different spreadsheet that I pointed him to on Sept.6, when he and I were discussing by PM the relationship between DOF, aperture settings, and sensor size.

One key relationship is simple: at same FOV and same DOF, the effective f-number is proportional to sensor size.

However, "simple" does not necessarily mean "obvious" or "believable".

So I prepared that spreadsheet to help get the point across.

The computation in that case is not done from first principles and does not compute blur circle diameters. Rather, it is based on the standard DOF formula that appears in Lefkowitz et.al., that DOF=2fc(m+1)/(m*m). The inputs are FOV, sensor widths, focal lengths, and the f-number for one lens. The f-number for the second lens is computed such that the standard DOF formula using that f-number give the same value as for the first lens and sensor. Then from the two f-numbers and another standard formula, the spreadsheet computes effective f-numbers for both setups. It also computes the ratios of sensor sizes and effective f-numbers (not shown in ChrisR's screenshot).

In playing with that spreadsheet, the observation is that no matter how you mess with the sensor sizes, FOV, and focal lengths, in the end the ratio of effective f-numbers turns out to be exactly the same as the ratio of sensor sizes. Of course that's the point I was trying to make, but having it fall out of a simple computation from accepted formulas adds some weight to its believability.

In addition, the spreadsheet automatically produces the second f-number needed to get the same DOF. The relationship between those two nominal f-numbers is decidedly not simple, so having them computed by spreadsheet provides nice guidance for correlating theoretical results with physical experiments.

What ChrisR has done is to extend that earlier spreadsheet to also compute the angle of view. That's a valuable addition, and I've added it to my copy as well.

If you want to play with that spreadsheet, you can download it HERE. I've plugged in ChrisR's numbers, so what you should see when the spreadsheet loads up is this:
.

I've included here the other ratios that get computed. Notice that in cells E14 and E15, the ratios are equal, sensor sizes versus effective f-numbers. Cell E16 shows the ratio of ISO ratings that would be required to have the same exposure time. This ratio is equal to the square of the effective f-numbers and sensor sizes, 1:36 in this case. If the smaller sensor were set for ISO 100, then the larger sensor would have to be set to ISO 3600. Do this, and both sensors end up counting the same numbers of photons, hence producing the same level of noise in the same pixel counts.

One last interesting question is this: What lens is needed to get the same wide-angle view on a 36 mm sensor that the 9 mm lens produces on the 6 mm sensor? The answer is focal length 31.5 mm. Certainly one could fit such a short lens on a camera with 36 mm sensor, but it would probably require some special fittings in order to focus properly.

So the case that ChrisR has defined makes a nice illustration of what's written at "Sensor size: how does it matter?":

the smaller sensor naturally comes with a shorter lens, which allows to easily get in closer, which gives more of a "wide-angle macro" appearance and also works better with auto-focus.

And in addition, we now understand that if you take advantage of that wide-angle setup, distant backgrounds will look sharper than in a narrow-angle setup that has the same FOV at sharpest focus. Progress is made...

--Rik

[rjlittlefield]

SO does the Lefkowitz DOF formula (as in the spreadsheets) give the same assymetry at greater distances? I thought not, something I must find out. What I haven't worked out if where the asymmetry becomes important.

No, the Lefkowitz DOF formula does not have asymmetry. It is essentially a limiting form for high magnifications and shallow DOF. The exact equations do show asymmetry even under those conditions, but for macro work the asymmetry becomes too small to matter and the Lefkowitz equation is used for convenience. BTW, there are two spreadsheets, as explained in my previous post. The later one can be used to explore asymmetry; the earlier one cannot.

What's being talked about is what happens when you use the camera, without thinking about maths. I believe that if you gave someone two cameras to go round a wood with taking pics of acorns on oak trees, and then asked another person to pick out the prints with the greater dof, a little camera would win.

I agree about the effect, but I think it is important to consider why the effect occurs. There are two reasons, very different from each other.

The first reason is that most people will pick settings that make it happen. Standard advices like "f/8 and be there", and "use the smallest ISO the camera provides" will naturally lead to setting a larger entrance pupil on the larger camera. The larger entrance pupil produces less DOF. The larger camera can be made to produce the very same pictures as the smaller, but the aperture and ISO settings needed to make that happen may be non-intuitive.

The second reason is the one that we've been talking about in this thread, that the smaller camera with its shorter lens can retain a wide-angle view for smaller subjects. In this case the larger camera can not be made to produce the very same pictures as the smaller, given commonly available lenses.

I think it is important to distinguish between these two reasons.

For a wide range of applications, the larger camera can be made to give exactly the same DOF and background appearance as the smaller one. The issue is primarily a matter of education about the required settings, so the user can choose which way to go when using the larger camera.

For a much narrower range of applications, the smaller camera has intrinsic advantages due to its ability to shoot wide-angle views of smaller subjects. I've written about this from a standpoint of perspective, but until working through this thread I had not thought about its effect on the appearance of far background. I will pay more attention to this issue in the future.

--Rik

[elf]

I think the spreadsheet shows that DOF and diffraction effects are essentially the same for different sensors and focal length lens given the constraints, but how would the images be different?

I would expect the following to be dependent on the sensor, CPU, or pixel dimensions:

• Resolved details
  Noise
  Color depth
  Dynamic range

[rjlittlefield]

I would expect the following to be dependent on the sensor, CPU, or pixel dimensions:

• Resolved details
  Noise
  Color depth
  Dynamic range

They are, in ways that may or may not matter depending on the application.

Noise, color depth, and dynamic range are all related to the maximum number of photons that can be counted in a pixel. Larger sensors have the potential to give less noise and greater dynamic range & color depth. To exploit that potential, you have to give them more light, which typically means raising the illumination level (brighter flash), increasing the exposure time, or using a larger aperture (with attendant changes in DOF and diffraction blur). If you give them only the same illumination level through the same aperture for the same time, then both large and small sensors capture the same number of photons and as a result experience the same noise levels due to sampling uncertainty, which is the primary cause of noise in modern sensors. With flash lighting or static subjects, larger sensors can definitely give less noisy images.

Resolved details depend primarily on pixel count and the quality of optical image. If DOF is not a limiting factor, then larger sensors win here because they allow using longer lenses with larger apertures. This effect can be huge at very low magnifications, for example shooting a distant landscape. The ability of a larger sensor to capture more detail diminishes with increasing magnification, say shooting through a microscope, and it disappears altogether if one adjusts the aperture to maximize DOF at some specified resolution that is within reach of the sensor.

There is a weak interaction between noise level and the ability to capture fine detail at low contrast. Here again larger sensors have the advantage, but the effect is not important for most applications.

--Rik

[ChrisR]

I made a slip on the previous page - note repeated here as the spreadsheet's copied over -
Edit:
I've just noticed a slip. I meant to enter 54mm for the FL of the lens on the 36mm sensor, to match the 9 for the 6mm. This yields a 19 degree fov instead of 14. Shows the same effect as described, but to a lesser extent.
--

That ("Sept 6")is a useful spreadsheet, and I've altered it quite a lot. For example to allow insertion of different CofCs, to suit different outputs.

Another dof calculator would be an interesting addition.

[PaulFurman]

So if I'm following this, there aren't any 'rules' that change at macro distances. Things like the amount of dof in front vs behind are the result of an external factor like exceeding the designed working distance so the complex optics don't function the same. Would any of these effects occur at moderate closeup distances using a large format 8x10 camera?

[rjlittlefield]

So if I'm following this, there aren't any 'rules' that change at macro distances.

It depends on what you mean by 'rules'.

The physics of DOF are the same at all magnifications and all distances. As a result, the "exact" DOF equations are also the same.

The wrinkle is that a lot of the standard "rules" are only approximations, and those approximations do change at macro distances.

For macro, it is a good approximation that you get equal DOF in front and behind the focus plane; for shooting scenics, you get vastly more behind.

Things like the amount of dof in front vs behind are the result of an external factor like exceeding the designed working distance so the complex optics don't function the same.

No, these DOF effects are all in the geometry, nothing to do with lens design. They would appear even with a perfect thin lens.

Would any of these effects occur at moderate closeup distances using a large format 8x10 camera?

Yes, they surely would. If you set up the camera to photograph a landscape, you'll get vastly more DOF behind focus than in front. Extend the bellows to focus closer, and the ratio shrinks. As you continue to extend the bellows, the ratio of front/back DOF gets closer and closer to 1. It never quite reaches 1, but it gets close enough "for all practical purposes" at fairly low magnifications. That's the place where people stop using the exact equations and switch over to the simplified equations such as appear in Lefkowitz.

--Rik

[PaulFurman]

Would any of these effects occur at moderate closeup distances using a large format 8x10 camera?

Yes, they surely would. If you set up the camera to photograph a landscape, you'll get vastly more DOF behind focus than in front. Extend the bellows to focus closer, and the ratio shrinks. As you continue to extend the bellows, the ratio of front/back DOF gets closer and closer to 1. It never quite reaches 1, but it gets close enough "for all practical purposes" at fairly low magnifications. That's the place where people stop using the exact equations and switch over to the simplified equations such as appear in Lefkowitz.

OK, thanks. The math is over my head but I can accept that it's a question of geometry and it helps clarify knowing that scaling up like that works the same.

Visiting space aliens with bodies a mile tall would have to struggle to see us through their huge cameras.