Discussion of 'Equivalence'

[simsurace]

No.  It is simply pointing out that "nearly identical" means nothing.  It is like the often heard inanity that humans and chimpanzees are nearly identical because they have 98% (or whatever the number is) of their DNA the same.  But 98% is only "nearly identical" to 100% if you already think chimpanzees and humans are "nearly identical", and if you think humans and chimpanzees are quite different 98% is quite different to 100%.

In science, one common meaning of "equivalent" would be "the same within measurement error".  In the case of lens focal length, eg, the measurement error is +/- 3%, so the precision of DoF calculations cannot be greater than that, so DoF within +/- 3% must be "equivalent".  In everyday photography, some measurements have much larger error - subject distance, eg.  You may blithely say "suppose I am 3m away", but that distance is, in practice, rarely measured at all, let alone to +/- 3%.  And measurement errors accumulate: if the focal length is measured to +/- 3%, and the distance is measured to +/- 3%, the DoF is accurate to +/- 18% - because DoF is proportional to the square of distance and inversely proportional to the square of focal length. (This is leaving aside the issue of significant figures, and the indefensible practice of calculating DoF to four or five significant figures when the CoC has one).

So, is your definition of "equivalent", "within measurement error"?  If so, you will know what all the errors are, so you can tell us. 

Another common meaning of "equivalent" would be that the difference is less than is practically important.  My professional expertise is in medicine, so I will use a medical example.  In the pre-antibiotic era, 3% of previously healthy people with pneumococcal pneumonia died.  With penicillin, none die.  So you have to treat 100/3 = 33 previously healthy people with pneumonia with penicillin to prevent one death.  This is called the "number needed to treat", or NNT.  Everyone can set the NNT they care about for themselves, but most ordinary people regard numbers over a few hundred as not worth bothering about, even for serious outcomes.  So if you have to give 500 people treatment A compared to treatment B to prevent one day off work, the two treatments are "equivalent" (there are plenty of commonly used treatments with NNT to prevent one death in the thousands - 5000 for taking a statin if you are a woman under 50 with no known heart disease, eg).  Thresholds of worthwhileness come in here, in photography as they do in medicine - the difference being that you can't get a D5 on the NHS, like you can a statin, so there is no need for a public concensus. 

So, what is your definition of a meaningful difference, and, if you want everyone else to accept it, where did it come from?  (If it is for your personal use only, you can choose whatever you like - but it would help if you said what it was).  I  am not saying that 70mm and 75mm are not different: I am asking why, if 50 and 54mm are not different, how come 70mm and 75mm are? 

I think for the uncertainties in measurement similar ideas apply as for the uncertainty in the comparison of images. Both just increase the space of parameters that is equivalent to a certain parameter.

To give you an example, I will consider again the calculation of blur circles that I showed earlier. We saw that equivalent blur circles imply that the ratio of focal length and aperture number (also known as the absolute aperture) is the same (where it is understood that the format is appropriate to achieve the same angle of view as the focal length changes -- this is important, I hope that this does not cause too much confusion, otherwise go back to Reply #184 to read the full example).

I plotted the focal length and aperture number that will produce the same size of the blur circle as 24mm and f/5.6 on DX.

In the upper left, it is assumed that the focal length and aperture number on DX are known exactly (i.e. f=24.00000000...) and that blur circle diameter can be measured to arbitrary precision. Unsurprisingly, the equivalent parameters lie on a line. Anything that is ever so slightly besides the line would not be equivalent. This definition of 'equivalent' would not be very practical, but note that even in this case, there are equivalent parameters that are different from f=24mm, N=5.6!

In the remaining three plots, I introduce uncertainties for the focal length and aperture number and/or the blur circles. The parameters that are equivalent now occupy an entire wedge of the space, so hitting the wedge would be easier. I took 3% measurement errors as an example. The wedge would get wider as the error increases.

[David H. Hartman]

Could some less confusing terms be use rather than EDR and PDR. It seems to me it would be less confusing to reserve "dynamic range" for the image sensors performance. Obviously cropping half a frame out of a D810's image doesn't change the D810's performance but enlarging that half frame to various common sizes may cause a practical difference in image quality.

For old timers like myself a Tri-X negative's characteristics don't change if you crop half a frame out of a full frame but if you print that half frame to 8x10 or worse 11x14 the image quality really takes a dive. Fortunately cropping half a frame out of a 36MP full frame isn't so costly nor is shooting with today's DX cameras.

I fear to suggest a different term for PDR but perhaps there should be one, one that won't confuse a fool devil's advocate like me.

Dave Hartman

[Les Olson]

I think for the uncertainties in measurement similar ideas apply as for the uncertainty in the comparison of images. Both just increase the space of parameters that is equivalent to a certain parameter.

Yes, but the question to be asked in relation to "equivalence" is, what is the relationship between the uncertainty in the size of the blur circles caused by measurement error, and the uncertainty in DoF when you add subject distance with its measurement error, to the difference between DX and FX calculated, as it always is, on the assumption of no measurement error?  My rough calculation suggests that answer is that the difference between DX and FX is within the uncertainty caused by measurement error, taking +/- 3% as a plausible value for each of the errors.  Subject distance is the key measurement, because an error of +/- 3% in a subject distance of 3m would be plausible for an averagely careful person using an ordinary domestic tape measure with the camera on a tripod, but if the distance is guesstimated, or the camera is hand-held, the error would be much bigger. 

[simsurace]

Yes, but the question to be asked in relation to "equivalence" is, what is the relationship between the uncertainty in the size of the blur circles caused by measurement error, and the uncertainty in DoF when you add subject distance with its measurement error, to the difference between DX and FX calculated, as it always is, on the assumption of no measurement error?  My rough calculation suggests that answer is that the difference between DX and FX is within the uncertainty caused by measurement error, taking +/- 3% as a plausible value for each of the errors.  Subject distance is the key measurement, because an error of +/- 3% in a subject distance of 3m would be plausible for an averagely careful person using an ordinary domestic tape measure with the camera on a tripod, but if the distance is guesstimated, or the camera is hand-held, the error would be much bigger.

Just to clarify: my calculation in Reply #184 and my plots in Reply #225 are for blur circles of point light sources at infinity. They do not apply for DOF calculations.

My intuition is that it is possible that the difference in DOF between DX and FX is masked by uncertainties when subject distance is not carefully controlled, if the errors are propagated in a very nonlinear way. I would be interested to see your calculation.

Right now I don't fully understand where you want to go with this, but I will follow you to see where we end up.

[Les Olson]

Right now I don't fully understand where you want to go with this, but I will follow you to see where we end up.

Another way to put the point is to imagine a group of photographers with DX cameras, and an equally skilled group with FX cameras. They set out, each on her own, to make photographs of a range of subjects - portraits, buildings, landscapes, nature, etc - almost like real life.  The aim is to have the same framing and perspective on DX as for the corresponding image on FX.  Aperture is up to each photographer.  Because measurement error is inescapable, they rarely succeed in achieving exactly the same framing and perspective for the corresponding DX and FX images.  Because both groups are equally skilled,  the images are equally good, just slightly different.  Because the difference in DoF between FX and DX for identical framing, perspective and aperture is less than the variability of DoF arising from measurement error, some of the DX images have less DoF than some of the corresponding FX images, some have more, and some have the same.   

Of course, if there were a person whose sins were so grievous that their penance was to examine all the images, and the photographers were trying hard to achieve the same framing and perspective, the penitent sinner would, over time, see more FX images with shallower DoF than the corresponding DX image than vice versa.  How many more will depend on two factors.  One is the size of the measurement errors - and as anyone knows who has done anything depending on accurate measurement, whether it is scientific research or carpentry, measurement errors are more common and larger than most people expect. 

The other, far more important, factor, is the nature of the images.  DX with 16mm at f/8 and FX with 24mm at f/8 both have infinite DoF, and near limits of a couple of meters, so the penitent sinner will see no difference for the landscapes or the buildings.  And there will be no discernible difference for full length portraits using a plain backdrop.  Or - although here I am straying beyond the narrow issue of equivalence and into the realm of - gasp - photography - for portraits where the photographer wants the background in focus, as in Annie Liebowitz' brilliant portraits of Queen Elizabeth, where she is referencing the royal portraits by Gainsborough and Thomas Lawrence rather than blindly re-using photographic cliches like shallow DoF.  Or where the photographer, noting that the finest royal portrait of the past 150 years (Lucien Freud's 2001 portrait of Queen Elizabeth) is referencing photographic style - it is the size of a photograph and uses a tight, photographic crop - decides to get a bit closer and/or crop a bit tighter and not have a background at all. 

[simsurace]

Another way to put the point is to imagine a group of photographers with DX cameras, and an equally skilled group with FX cameras. They set out, each on her own, to make photographs of a range of subjects - portraits, buildings, landscapes, nature, etc - almost like real life.  The aim is to have the same framing and perspective on DX as for the corresponding image on FX.  Aperture is up to each photographer.  Because measurement error is inescapable, they rarely succeed in achieving exactly the same framing and perspective for the corresponding DX and FX images.  Because both groups are equally skilled,  the images are equally good, just slightly different.  Because the difference in DoF between FX and DX for identical framing, perspective and aperture is less than the variability of DoF arising from measurement error, some of the DX images have less DoF than some of the corresponding FX images, some have more, and some have the same.   

Well, whether the two groups will get different images will depend on (among other things) what lenses they carry and at what apertures they use them. But what is the point of this? You are introducing a whole bunch of external variables in order to make a point about how the concept of equivalence does not matter. I understand your argument, but I don't think it is a criticism of equivalence. Equivalence is a concept that is based on optics; to see optical effects clearly it is best to work in a controlled environment. First of all, it is about how to set the parameters to get pictures that look the same on different formats, and as a corollary, how to set the parameters if you want them to differ in certain ways. It is not (as you seem to imply with your example) about some claim of FX getting systematically lower DOF when you look at a random sample of photographs*. And if you don't look closely or introduce a bunch of uncertainties and other variables, you might get pictures that look more or less the same despite not setting the parameters exactly right.

*Especially if the aperture is chosen randomly as well. What does random mean in this context?

[Les Olson]

Well, whether the two groups will get different images will depend on (among other things) what lenses they carry and at what apertures they use them. But what is the point of this? You are introducing a whole bunch of external variables in order to make a point about how the concept of equivalence does not matter. I understand your argument, but I don't think it is a criticism of equivalence. Equivalence is a concept that is based on optics; to see optical effects clearly it is best to work in a controlled environment. First of all, it is about how to set the parameters to get pictures that look the same on different formats, and as a corollary, how to set the parameters if you want them to differ in certain ways. It is not (as you seem to imply with your example) about some claim of FX getting systematically lower DOF when you look at a random sample of photographs*. And if you don't look closely or introduce a bunch of uncertainties and other variables, you might get pictures that look more or less the same despite not setting the parameters exactly right.

*Especially if the aperture is chosen randomly as well. What does random mean in this context?

You have not made the claim that FX is about systematically less DoF (for the same perspective and framing), but that claim is made: "Larger format, longer lenses for the same perspective and framing, more background blurring" (http://nikongear.net/revival/index.php/topic,5942.15.html).  And right of the bat Joseph James says "This essay is about relating different systems [...] Equivalence relates the visual properties of photos from different formats [...] the advantage of a larger sensor system over a smaller sensor system is that the larger sensor system will generally have lenses that have wider aperture (entrance pupil) diameters for a AOV (diagonal angle of view) than smaller sensor systems, which allows for more shallow DOFs". 

I did not propose comparing a random sample of photographs, I proposed comparing a random sample of photographs that were intended to be equivalent. 

If it is true that equivalence is intended to tell people "how to set the parameters to get pictures that look the same on different formats, and as a corollary, how to set the parameters if you want them to differ in certain ways", it is an abject failure.  The reason is simple: I learn nothing from being told how to change the parameters to take the same photographs, or photographs with some desired difference, on different systems, if I do not already know the parameter settings on at least one system.  What is helpful is to say "DoF is directly proportional to F-number (double the F number = double the DoF); DoF is directly proportional to the square of subject distance (double the subject distance = four times the DoF); DoF is inversely proportional to the square of focal length (double the focal length = one quarter the DoF)". 

And "setting the parameters exactly right" takes us back to circularity: the only reason to do that is to make it possible to make statements about "equivalence".

[simsurace]

You have not made the claim that FX is about systematically less DoF (for the same perspective and framing), but that claim is made: "Larger format, longer lenses for the same perspective and framing, more background blurring" (http://nikongear.net/revival/index.php/topic,5942.15.html).  And right of the bat Joseph James says "This essay is about relating different systems [...] Equivalence relates the visual properties of photos from different formats [...] the advantage of a larger sensor system over a smaller sensor system is that the larger sensor system will generally have lenses that have wider aperture (entrance pupil) diameters for a AOV (diagonal angle of view) than smaller sensor systems, which allows for more shallow DOFs". 

I do not want to convince anyone that they should think in a certain way or adopt a certain conceptual framework. But I believe that when criticising something, one should at least not take things out of their proper context.

The reality is that different formats are more or less well developed in terms of lens lineup. There is a scarcity of fast primes for DX. To give an example, I have a 50/1.2 from Nikon. This lens is fairly special because of the fast aperture and rendering wide open. I do not find an equivalent lens for DX, which would have to be a 35/0.8 or thereabouts. If I conclude that FX is superior (for me) than DX because there is no 35/0.8 lens, it is not a conclusion that can be deduced from equivalence. The latter only told me that I have to look for a 35/0.8. The closest thing I can get is one of various 35/1.4 lenses. But despite all measurement errors that occur in casual photography, I will still not be convinced that the 50/1.2 on FX and a 35/1.4 on DX give the same rendering wide open (not only because of DoF, but this is the easiest aspect to compare). In a parallel universe, it could be different, with hundreds of native lenses for DX and almost none for FX.

Now some people are telling me that I should just accept each format for what it is and not try to do the same things on both formats. Well, fine, but that doesn't change the fact that certain lenses don't have equivalents in other formats, and if I decide that I want to do certain things with my photography, that rules out certain formats for me. To draw that conclusion, I either need to do a lot of experiments, or have some knowledge of optics (the relevant parts of which are neatly summarized in equivalence). By the way, short of owning every possible format there is, I sometimes have to do stuff on a format which is not ideal, and knowledge of optics helps in that situation as well.

The statement by Dave is along a similar vein. The problem that I see, is that we often see statements like these without all the proper qualifications, but this is part of informal language/conversations. It can be confusing for people starting out in photography. But if you have to make a decision whether to purchase an FX or DX camera, you have to take lens selection into consideration, and a big part of that is maximum aperture.

If it is true that equivalence is intended to tell people "how to set the parameters to get pictures that look the same on different formats, and as a corollary, how to set the parameters if you want them to differ in certain ways", it is an abject failure.  The reason is simple: I learn nothing from being told how to change the parameters to take the same photographs, or photographs with some desired difference, on different systems, if I do not already know the parameter settings on at least one system.

Well, if you have a general formula for taking a photograph, you might as well program a computer to go out and take pictures.
The reason you are giving for the failure is not about the intended purpose.

What is helpful is to say "DoF is directly proportional to F-number (double the F number = double the DoF); DoF is directly proportional to the square of subject distance (double the subject distance = four times the DoF); DoF is inversely proportional to the square of focal length (double the focal length = one quarter the DoF)". 

That is exactly what equivalence is doing. However, if you open a book (or the Wikipedia article) on DoF, you will see that there are a dozen ways to rearrange the formulae depending on what is fixed and what is variable. The average person that takes up photography is not very comfortable in manipulating formulae like this. In addition, DoF is confusing and trips up many people, even experienced ones. Equivalence makes a certain decision what is to be held fixed when going from one format to the other, and gives the optical relationships that remain to be specified.

[David H. Hartman]

DoF is based on the limitations of human vision and the expectations of the viewer. If all the factors are accounted for it should be possible to calculate DoF nicely for one viewer. For 2? For 3? For many? The odds can go down quickly as DoF isn't a hard fact.

If graphs and charts and articles don't help with understanding background blurring practical experience with 35mm, 6x6 and larger should give the photographer an intuative feeling of what to expect. No experience with film? Then CX, DX and FX or similar formats should do it.

If one has a fast zoom try a little systematic observation. Choose a perspective then zoom to frame. While zooming observe the background blurring. A clipboard and pen will make people watching think one really knows what they are doing. Paper on the clipboard isn't required.

If perspective is the concern a distance of say 1.9m to 2.1m should be close enough if 2.0 was desired. The distance need not be measured. One can see perspective with their eyes.

Instead of using a prime and "zoom with your feet" one might try setting the perspective with their feet and then zooming to frame the subject. The exact focal length when focused and zoomed isn't important, only that the subject is framed as one likes.

One can select a point of view without lifting the camera to their face. I'm sure I look like a fool bobbing around but it's much easier to select one's point of view by eye than to hold a monrail view camera to my face. I'm sure my Karden Colour 45s taught me to select my point of view by eye.

I posted charts and grafts hoping they might be helpful. I'm not sure if they were or not.

Dave Hartman

Caution: using a zoom with a variable maximum aperture may cause vertigo. I read it in a magazine.

[Les Olson]

To give an example, I have a 50/1.2 from Nikon. This lens is fairly special because of the fast aperture and rendering wide open. I do not find an equivalent lens for DX, which would have to be a 35/0.8 or thereabouts. If I conclude that FX is superior (for me) than DX because there is no 35/0.8 lens, it is not a conclusion that can be deduced from equivalence. The latter only told me that I have to look for a 35/0.8.

Then it mislead you, because you do not have to look for a 35/0.8: you can use the 50/1.2 perfectly well on DX, and the properties you value - rendering and large aperture - will be the same; other properties will be better - vignetting, eg.  Of course, you will have to adapt a little: you can keep the same perspective and DoF and have a tighter crop, or you can have the same framing and move a little further away and therefore have a few centimetres more DoF.  Why is that a problem?  You could even innovate, and use the lens differently. 

Of course, if, for reasons of your own, you particularly like the framing, perspective etc you get with your current subject on FX with a 50mm lens at f/1.2, then you will want FX. As I said in the other thread, you can get any framing, perspective, DoF etc you want with DX, unless your requirements are highly specialised.  An example of highly specialised requirements is wanting exactly what you get on FX with 50mm at f/1.2.  That is fine, and that is a reason you, individually, need FX.  But the fact that you are perfectly happy using the 50/1.2 on FX is also a reason buying a DX camera would never cross your mind: so far from showing why "equivalence" is important, it shows why it is pointless.   

Of course lens selection plays a role in deciding whether DX or FX is a better fit.  If you use wide angle focal lengths at large apertures a lot then you are a natural FX user (people complain because Nikon has not provided wide DX primes, but because DX and FX have the same registration distance the complexity and size of wide lenses scales nearer actual focal length than equivalent focal length, so a 14mm DX prime would be very hard to make competitive with the 20/1.8 FX).  Conversely, if you use 500mm a lot you are a natural DX user.   

[simsurace]

Then it mislead you, because you do not have to look for a 35/0.8: you can use the 50/1.2 perfectly well on DX, and the properties you value - rendering and large aperture - will be the same; other properties will be better - vignetting, eg.  Of course, you will have to adapt a little: you can keep the same perspective and DoF and have a tighter crop, or you can have the same framing and move a little further away and therefore have a few centimetres more DoF.  Why is that a problem?  You could even innovate, and use the lens differently.

The character of the 50/1.2 changes completely when mounted on FX vs. DX. I'm not saying it is not useful on DX, I would even say it is very useful, but for completely different purposes. Moving a few steps backwards does not do much to breach that abysmal difference. Cropping a significant chunk from the image circle has profound implications on rendering on temperamental lenses like the 50/1.2. It does not do much on technically perfect, more clinical lenses. Moreover, the difference between the AOV of a normal lens and a short tele is also not small by any means.

buying a DX camera would never cross your mind

This is certainly false. I have owned DX cameras and I will consider them again in the future. I have used the 50/1.2 on DX and it was marvellous, much less temperamental, I wish there were something equivalent for FX .

so far from showing why "equivalence" is important, it shows why it is pointless.   

I'm afraid that I cannot give better explanations; we just have to agree to disagree.

[simsurace]

DoF is based on the limitations of human vision and the expectations of the viewer. If all the factors are accounted for it should be possible to calculate DoF nicely for one viewer. For 2? For 3? For many? The odds can go down quickly as DoF isn't a hard fact.

The perceived absolute amount of DoF is maybe dependent on the viewer. But different viewers would probably agree about changes in DoF, or about which of two images has more or less DoF. For background blur it's even easier to notice changes. Do you agree?

[Andrea B.]

Here is a legitimate question for which the optics of Equivalence can be useful.

I have the Nikkor 200-500 f/5.6E ED VR lens for taking photos of birds.
^{(Disclaimer: I am not a "bird photographer". I simply like to make a little field record of some of the birds I see when out walking or hiking. :-) )}

What are the trade-offs between using this zoomer on my 36MP D810 versus my 24MP D500 versus my 24MP D750??

Leave aside for the moment, the question of which camera body can provide better tracking, auto-focus or frames/second. As it turns out, I've actually successfully used the 200-500/5.6 VR on all 3 cameras for bird shooting. Which is not to say that I don't have my preferences. I do. But I want to better understand what I might be losing/gaining amongst these three camera choices.

For now, I'll leave this as an exercise for the interested reader. (My math books used to use that phrase!) Meanwhile, I'll try to work out an answer and post it later.

Edit:  I had to remove a double post.

[JohnMM]

Of course, if, for reasons of your own, you particularly like the framing, perspective etc you get with your current subject on FX with a 50mm lens at f/1.2, then you will want FX.

I like the "look" but I can't afford FX. I want to know if I can get the same "look" with DX. How do I find out?

[Les Olson]

I like the "look" but I can't afford FX. I want to know if I can get the same "look" with DX. How do I find out?

A new D610 costs $1500.  B&H has a D610 in near new condition for $1000, exactly $100 more than a new D7200 at its currently discounted price.  A 50mm f/1.2 is $700 and a 50mm f/1.4D is $300.  You can afford the extra shaving of lens speed but not the FX body?  Well, OK, if you say so. 

However, if you want to know if you can get "the look" on DX I recommend a website (http://www.naturfotograf.com/lens_norm.html), where it says: "There is an endearing slight softness (bokeh) when the lens is deployed on a D1/D2-series camera and shot wide open, but the image even at f/1.2 has plenty of detail. Stopped down in the range f/2.8-f/5.6, image contrast is enhanced, sharpness is very good to excellent, and veiling flare has gone entirely. Quality deteriorates rapidly as expected with the lens stopped down beyond f/8. [...] Image contrast even at f/1.2 is higher on the D3, so pictures come across crisper and appearing sharper with this camera. Focusing the lens on a D3 was easy." 

The 50/1.2 is unique.  There is no 24/1.2.  There is no 85/1.2.  It will not be the same lens on FX as on DX: on FX it is a "normal" lens - ie, focal length is close to the image diameter - and on DX it is a - loose portrait lens?  Personally, I would have more use for a loose portrait lens with f/1.2 than a normal lens with f/1.2, but to each his own.  I wouldn't pay $400 to use an f/1.2 lens for the contrast, so I think the answer is "Yes if you buy the lens".  However, you are free to disagree - in which case you had better grab that D610 while you can.  Either way, that was easier than reading God knows how many pages of bumph about "equivalence", wasn't it?