A beginner's guide to "is equal to" vs "is equivalent to".

[bobn2]

So, "Total Noise" simply represents the average SNR for corresponding portions of the photo.

So why is sensor size a factor? In good light the SNR is so high that total noise is not an issue.

What you're saying is that isn't an issue for you, which is just fine, but clearly there are people for whom it is an issue. It might be that you believe their standards are unnecessarily high, but I think everyone is at liberty to decide for themselves in the end.

Sometimes, until one runs the numbers, these things are not obvious. For instance, suppose that a photographer really does insist on high levels of image quality that offer around 50MP resolution (and as a note here, if your idea that 'total noise' is 'not an issue' was universal, then no-one would be buying these large sensor cameras). Here are 3 camera/lens combination that all seem aimed at more or less the sam kind of outcome.

1. Fujifilm GFX50R with Fujifilm 100-200/5.6. $4,500 camera, $2,000 lens - $6,500 total

2. Canon EOS 5Dsr with Canon 70-200/4, $3,900 camera, $650 lens - $4,550 total

3. Nikon D850 with Nikon 70-200/4, $3,300, $1,400 lens - $4,700 total.

The relative crop factor of the Nikon and Canon with respect to the FUJI is 1.27.

So, it's relatively uncontroversial to quote the angle of view as an 'equivalent' focal length, so we can see that the 70-200 lenses on the Canon and Nikon are 'equivalent' to a 90-250 lens on the Fujifilm. So far, so good, the Canon and Nikon have a zoom range advantage over the Fuji at both ends.

But what about image noise, what is achievable? Given that the sensors on these cameras have the same efficiency to within a percent or two, the noise will depend on the 'total light' that we can project onto the sensor. So, to quantify that we can think in terms of 'equivalent f-number', to see how much light those lenses will project. This is convenient, because f-number is the generally used metric for how much light a lens projects, but unfortunately breaks down when you compare across formats. So now we find that the f/4 of the Nikon and Canon is projecting the same amount of light on the sensor as would an f/5 lens on the Fuji. So, in terms of the light that the lens can project, advantage once again to the Nikon and Canon. That also gives them the advantage in terms of SNR at any given shutter speed. The final issue is absolute SNR. The Fuji has a base ISO of 100. Again, most people think of how much light is collected in an exposure in terms of ISO, but like f-number, when used for that purpose, it doesn't translate across formats, so we use an 'equivalent ISO'. The base ISO of the Canon (100) results in capture of the same amount of light as would an ISO of 160 on the Fuji, so clearly the Fuji can accept more light (about 2/3 stop) and thus produce a less noisy image, if not shutter speed constrained. On the other hand, the Nikon has a base ISO of 64, which captures as much light as would an ISO of 100 on the Fujifilm, so in that respect the two are quits.

Without the use of these equivalence calculations, the simple assumption would be that the Fuji, with the bigger sensor, would give a better SNR but, at least with these lenses (and in fact with most lens match ups across the range) this isn't the case.

Of course, there are many other characteristics that this comparison doesn't capture, but that is true of any simple numerical comparison. At least it does prevent the making of incorrect assumptions based on a simplistic view of the effects of sensor size.

--
Ride easy, William.
Bob

 

[stevo23]

Sure, people can express mathematical equations to explain something.

It doesn't mean other people have to think it's that terribly important to the outcome of their photography work.

Sure. You don't need to understand gravity to enjoy (and be good at) skydiving. But you wouldn't argue against gravity would you? And you wouldn't say that someone computing how much time you have before you hit the ground in the event of a chute malfunction is full of it, would you?

www.dpreview.com/forums/post/62048291

Oh man -- you posted that link? That didn't end well for you.

You don't understand the f/ratio of a lens...

In fact I do. You link and quote *anything* I've said that you feel supports that bogus assertion, and I'll link and quote where I've showed you to be mistaken about that. Deal?

Sure.... www.dpreview.com/forums/post/62129091

Huh? I don't see GB even contributing to that thread. Are you seriously posting links to links to that same thread where you were repeatedly thrashed? You're not planning to crash and burn on purpose are you?

Some people feel that because they can [repeatedly] block a punch with their face, often by their own hand, that they've won the fight.

Good analogy.

 

[stevo23]

You're wasting your effort. Mr. Llama doesn't believe in a spherical earth...eh...equivalency.

Oh please.

So anybody that doesn't swallow hook line and sinker all of the nonsense the self

appointed experts spout in these forums must be a flat-earther...???

What does any of that have to do with the subject at hand? This is about equivalence. You don't "believe in" equivalence?

I'm saying that not believing in equivalence, which can be tested and proven, is not very dignified. It's like being a flat earther - foolishness in the face of proof.

What is your definition of equivalence that can be tested and proved. Is your proof mathematical or intuitive?

So you're saying you don't buy it, is that it? I seem to recall that you scoff at these explanations.

I don't know what it is that you think I'm not buying.

You do think equivalency works?

According to Richard Butler, "Equivalence, at its most simple, is a way of comparing different formats (sensor sizes) on a common basis".

Of course that works. It is just a way of comparing different formats on a common basis.

How some of these comparisons are used to draw certain conclusions is what is debatable.

Specifically which is debatable?

Why total noise in an image, which the equivalency advocates

So you're not an equivalency advocate - that much is now clear.

keep emphasizing, is a more useful metric than the signal-to-noise ratio?

But you're not actually debating it, you're just questioning it's importance. What specifically is wrong in GB's original post?

 

[Airolo]

I don't need two lenses to be equal. *Equivalence* is a good rough term.

I just accept that any two *equivalent* FL lenses (for different formats, of course) do have a common set of shooting scenarios and each one has it own set, which the other is not capable of.

And while the larger aperture advantage of larger formats is mentioned frequently, the shorter Minimum Focus advantage of the smaller formats is not.

Not even in the DPR's article on equivalence. And GB's essay as well.

Peter

 

[bobn2]

Sorry but there is just one definition of Equivalence relation:

A <=> B when and only when A=> B & B => A. Which is not the case with *equivalent* lenses because of different focus ranges.

Well, if you insist on there being 'only one definition of an equivalence relation' then you have defeated your own argument, because the most widely accepted definition of 'equivalence relation' isn't that. It is: "a binary relation that is reflexive, symmetric and transitive. "

In fact, the relation 'Equivalence' as discussed here is binary, reflexive, symmetric and transitive, so is indeed an equivalence relation.

What I think you're actually saying is that you have your own definition of an 'equivalence relation' and you think that it should be compulsory for everyone else to adopt it.

Your point about different focus ranges is not relevant to the discussion, because in the 'Equivalence' relation, A and B in your terms are photographs, not lenses. The issue of focus ranges between lenses only defines the range of settings (in this case focus) over which equivalent photographs are available (although so far as I'm aware, the 'Equivalence' relation doesn't say anything about whether the equivalent photos need both to be in focus - GB might correct me).

And excuse me, but to compare usually rather minor differences in Minimum Focus Distances of real 70-200mm lenses from different manufacturers with say 50cm to 95cm is not very reasonable.

I'm not getting what you're saying here. Are you saying, for instance, that if one manufacturer's 70-200/ has a minimum focus distance of 50cm whilst another has a minimum focus distance of 95cm it would be unreasonable to use the word 'equivalent' in relation to those two lenses?

There are some people who prefer Lumix 42.5/1.7 over the mentioned PL lens because of 31cm MFD. If not important for you, good. But it doesn't mean for all

So?

Equivalence theory is sound.

Yes, it is.

 

[bobn2]

I don't need two lenses to be equal. *Equivalence* is a good rough term.

It's not a rough term, it's a precise term, and it doesn't mean 'equal'.

I just accept that any two *equivalent* FL lenses (for different formats, of course) do have a common set of shooting scenarios and each one has it own set, which the other is not capable of.

And while the larger aperture advantage of larger formats is mentioned frequently, the shorter Minimum Focus advantage of the smaller formats is not.

Not even in the DPR's article on equivalence. And GB's essay as well.

No disagreements there.

 

[stevo23]

What is your definition of equivalence that can be tested and proved. Is your proof mathematical or intuitive?

So you're saying you don't buy it, is that it? I seem to recall that you scoff at these explanations.

I don't know what it is that you think I'm not buying.

You do think equivalency works?

According to Richard Butler, "Equivalence, at its most simple, is a way of comparing different formats (sensor sizes) on a common basis".

Of course that works. It is just a way of comparing different formats on a common basis.

How some of these comparisons are used to draw certain conclusions is what is debatable.

Specifically which is debatable?

Why total noise in an image, which the equivalency advocates keep emphasizing, is a more useful metric than the signal-to-noise.

"Total Noise" is simply about comparing the noise over the same proportion of the photo, rather than one pixel on one photo to one pixel on another photo when the pixels cover different portions of the photo.

So, "Total Noise" simply represents the average SNR for corresponding portions of the photo.

So why is sensor size a factor?

At base ISO, larger sensors can typically record more light for a given DOF by using a longer exposure time before oversaturating. However, at higher ISO settings, the noise advantage doesn't come from the larger sensor, but the wider aperture that lenses designed for larger sensors typically (but not always) offer, so it's the lens, not the sensor.

However, in this situation, the less noisy photo for the larger sensor system *necessarily* requires a more shallow DOF, since a wider aperture not only projects more light onto the sensor for a given scene and exposure time, but also results in a more shallow DOF for a given perspective and framing, which may be desirable, undesirable, or neither here nor there, depending on the scene and aesthetics of the viewer.

In good light the SNR is so high that total noise is not an issue.

It's less of an issue, to be sure, but not necessarily a non-issue. It depends on many factors, not the least of which is the scene, processing, display size, display medium, viewing conditions, and the aesthetics of the viewer.

That said, in my personal opinion, at base ISO, even a smart phone is more than "good enough" for the vast majority of situations for the vast majority of people, with regards to IQ. Indeed, I have a 16x20 inch photo from a smartphone hanging on my wall, and if I told people who aren't into photography that it was from a $3000 camera, I don't think anyone would question me.

On the other hand, if I had a photo of the exact same scene take with a D850 and displayed at the same size, I'm pretty sure the difference would be obvious, but not so sure that people would care about the difference. But if I were trying to sell the photo, and both versions were for sale at the same price, I'm thinking I'd not sell many of the smartphone photos compared to the D850 photo. The smaller the size of the photo, however, the less the difference would matter.

I am not arguing that there is no benefit for having a larger sensor just as since the dawn of photography everybody knew that the larger format film was always an advantage.

There are downsides to larger sensor systems, too. Furthermore, if you would typically be using the larger sensor system at the same DOFs and exposure times you'd be using with a smaller sensor system, then the advantages of the larger sensor system pretty much vanish.

I said what was debatable about what so many of the equivalence evangelists keep touting about total noise in the image being so important.

They simply mean the noisiness of the photo as a whole, as opposed to per-pixel noise for photos made with different numbers of pixels.

I know all the premises about the same scene from the same position, etc, but I am not going to get into responding to your 1000 word essay to lead me down a rat hole.

The deeper the understanding you want, the more involved the explanation. The more complete of an answer you want to cover more scenarios, the more involved the explanation. Would you tell a teacher or professor, "I am not going to get into reading a thousand pages to lead me down a rat hole" on *any* subject?

It's fine if you don't want to know all the details

No. I just don't need to know YOUR details.

Like he said, have some class. Don't want to read it all? Don't scoff because you're impatient.

 

[Michael Fryd]

First of all, I agree with the principles of equivalence. It's sound and correct when applied to individual images.

But to call two lenses with different FLs equivalent is oversimplification.

While risking of being called overly pedantic, I've to say that there is one condition, which makes no lenses truly equivalent is missing from OP, your essay and the DPR article on equivalence.

The lenses do not have same shooting ranges - the shorter one is able to focus closer.

In reality lenses between formats can be *equivalent* just over a common part of their shooting envelopes. And (usually) none of these envelopes is superset of the other - they have just some overlap and some unique parts

Petr

PS. I'm fine with calling two lenses *equivalent*, but I think that our essay and DPR article should mention the difference in shooting envelopes.

P.

There are two issues here, first what are we calling "equivalent", and what is the context of that equivalence?

Generally, we are not discussing whether or not two lenses are equivalent, we are discussing whether the lenses are capable of an equivalent setting. For instance a 50mm f/1.8 prime lens on a 2x crop body is not an "equivalent lens" to a 70-200 f/2.8 zoom lens on a full frame. However, if the 50mm is set to f/4, and the full frame is set to 100mm at f/8, we say the settings are equivalent.

On this forum, the context for equivalent lens settings is typically same angle of view, same aperture diameter, and same subject distance. If two lenses can meet those specs for a particular shot then (in this context) they are at equivalent settings.

Note: If you are uncomfortable with the use of "same aperture diameter" then substitute "same depth of field".

 

[Airolo]

Excuse me, but it starting to be ridiculous.

I don't deny equivalence theory, I use the underlying principles for a long time.

I don't insist on any my definition of Equivalence. I just provided evidence, that the current use of term equivalence when applied to lenses is not fully correct.

The Equivalence relation is indeed reflective, symetric and transitive, witch is expressed by the expression I provided.

The problem is that *equivalence* relation as described by OP is NOT symetric when applied to lenses. Domains of two *equivalent* lenses are not the same. There are some shots you can produce with one lens but not equivalently with the other. And usually vice versa.

So once more, I can live with equivalence theory as it is. I just wanted to note one aspect which is often missing. To help OP and DPR to make it more correct.

Regards

Peter

 

[Airolo]

Michael, thanks for your reply.

I wanted to address just the case when people talk about *equivalent* lenses. And they do - OP in his essay and DPR in the article.

I don't deny equivalence theory, I just wanted to make it more correct when applied to lenses.

Regards.

Peter

 

[stevo23]

Compare that to lenses with the same F-stop. They can have widely different sizes, weights, DOF, and resolution. Saying a F/2.8 lens on a cell phone has any resemblance to F/2.8 on a crop, medium format, or a telescope is a joke. The only thing they have that is similar is exposure--which has nothing to do with photographic quality or properties.

The difference between a cell phone and a FF camera is the 50 times difference in sensor area. It collects 50 times more light with the same exposure of the same scene....

But if you stop down your larger camera, you will find that at the same angle of view, and same depth of field, the larger sensor collects the same total number of photons.

For instance, the iPhone 6 has a crop factor of 7.21, focal length of 4.15mm and a f/2.2 maximum aperture.

A full frame body with a 30mm lens at f/16 will have the same depth of field and collect the same light per unit area as that iPhone.

The sensor on the full frame has about 50 times the area of the iPhone sensor. However, at f/16, it gets about 1/50 the light per unit area as the iPhone sensor.

Both the iPhone with a 4.15mm at f/2.2 and the full frame with a 30mm at f/16 have an aperture diameter of about 1.9 mm.

The advantage of the full frame, is that you can open up wider that a 1.9mm aperture diameter. With the iPhone, that 1.9mm is your maximum aperture.

It really does boil down to aperture diameter and angle of view.

Where are you fitting noise into that?

If the shutter speeds are the same, the iPhone at f/2.2 will have about the same shot noise as the full frame at f/16. (Both sensors are seeing the same total amount of light).

Obviously, the full frame will need to be set to a higher ISO in order to get the same image lightness as the iPhone photo.

That's backward from how I would do it. I would set the same ISO - I'm always wanting to compare at the same ISO. Changing ISO negates the point of the comparison for me. But as long as it's know that ISO is different between the two, the same story is told.

I don't have a comparison of the sensor technologies, so I won't comment on how other factors contribute to the noise.

I think we typically assume "similar generation" but these two sensors may or may not be very similar in technology generation.

 

[Michael Fryd]

...

If the shutter speeds are the same, the iPhone at f/2.2 will have about the same shot noise as the full frame at f/16. (Both sensors are seeing the same total amount of light).

Obviously, the full frame will need to be set to a higher ISO in order to get the same image lightness as the iPhone photo.

That's backward from how I would do it. I would set the same ISO - I'm always wanting to compare at the same ISO. Changing ISO negates the point of the comparison for me. But as long as it's know that ISO is different between the two, the same story is told.

ISO is an implementation detail. In terms of the final image, what's the advantage of comparing different sensor sizes at the same ISO? A small sensor at ISO 800 isn't going to act anything like a large sensor at ISO 800. More importantly, if you are shooting at the same depth of field, you probably will have the larger sensor at a higher ISO than the smaller sensor.

When we shot film, we didn't compare different focal lengths at the same aperture diameter, we compared them at the aperture diameter relative to the focal length. When your'e shooting film, the light per unit area on the film is an important implementation detail.

With digital, and the option of Auto-ISO, there is far less of a need to start with ISO. I think you're much better off starting with total light captured (light per unit area times the sensor size).

...

 

[J A C S]

Excuse me, but it starting to be ridiculous.

I don't deny equivalence theory, I use the underlying principles for a long time.

I don't insist on any my definition of Equivalence. I just provided evidence, that the current use of term equivalence when applied to lenses is not fully correct.

The Equivalence relation is indeed reflective, symetric and transitive, witch is expressed by the expression I provided.

The problem is that *equivalence* relation as described by OP is NOT symetric when applied to lenses. Domains of two *equivalent* lenses are not the same. There are some shots you can produce with one lens but not equivalently with the other. And usually vice versa.

This is also true if you keep the format fixed. A macro lens can take a shot a non-macro cannot.

Also, even at non-macro distances, lenses of different designs provide somewhat different DOF on the same camera. It has been measured (by Zeiss, for example).

Next, DOF depends on the position on the frame. A good fast, say 50/1.2 FF lens would provide much shallower DOF at 50/2.4 than an equivalent m43 25/1.2 lens away from the center because the latter would vignette much more and both would produce blur very different from disks away from the center; so even the premise of DOF would not hold.

Next, field curvature would change the DOF across the frame. The derivation of DOF does not really apply for complex lenses, etc. The blur even in the center varies depending on the lens design (and it is a big factor for some), so the DOF definition becomes questionable.

All that does not invalidate equivalence. It just says that we have to be aware of the limitations of our models but that applies to everything.

 

[Michael Fryd]

Michael, thanks for your reply.

I wanted to address just the case when people talk about *equivalent* lenses. And they do - OP in his essay and DPR in the article.

I don't deny equivalence theory, I just wanted to make it more correct when applied to lenses.

Regards.

Peter

I guess we read the OP differently. When he says:

"For example, a lens at 85mm f/2.8 on Canon APS-C (crop factor = 1.6) will be *equivalent to* (as opposed to "equal to") a lens at 136mm f/4.5 on FF."

I thought it was implied that the equivalence was in the context of angle of view and Depth of Field (aperture diameter).

I didn't think he was commenting on lens size, weight, minimum focus distance, lens mount, build quality. pincushion distortion, cost, warranty, or any of the other factors to be considered when selecting a lens.

 

[stevo23]

...

If the shutter speeds are the same, the iPhone at f/2.2 will have about the same shot noise as the full frame at f/16. (Both sensors are seeing the same total amount of light).

Obviously, the full frame will need to be set to a higher ISO in order to get the same image lightness as the iPhone photo.

That's backward from how I would do it. I would set the same ISO - I'm always wanting to compare at the same ISO. Changing ISO negates the point of the comparison for me. But as long as it's know that ISO is different between the two, the same story is told.

ISO is an implementation detail.

?

In terms of the final image, what's the advantage of comparing different sensor sizes at the same ISO?

That's part of the equivalency discussion I would think.

A small sensor at ISO 800 isn't going to act anything like a large sensor at ISO 800.

Exactly. You're trying to equalize them, I'm trying to see how they're different.

 

[Airolo]

Thanks J A C S.

I do agree with all of your points. I just wanted to make visible one aspect, which is often missing. Non-equal shooting envelopes.

That's why I put - strictly speaking...

Regards.

Peter

 

[Airolo]

We read it slightly different. I agree with you, that there are some other independent parameters you mentioned, which is reasonable to left aside.

But the MFD is dependent on reall FL and to some extent to max aperture. It's true even for macro lenses.

So when you choose a lens with given FL and max aperture, you more or less set your shooting envelope. Which is different from the shooting envelope of a lens with different FL.

That's why I put- Strictly speaking...

Regards.

Peter

 

[Lee Jay]

"Total Noise" is simply about comparing the noise over the same proportion of the photo, rather than one pixel on one photo to one pixel on another photo when the pixels cover different portions of the photo.

So, "Total Noise" simply represents the average SNR for corresponding portions of the photo.

So why is sensor size a factor?

Because, at the same f-stop, a larger sensor captures more light, and since SNR=sqrt(light captured), more light = higher SNR.

In good light the SNR is so high that total noise is not an issue.

And in bad light, it is an issue.

 

[Michael Fryd]

...

If the shutter speeds are the same, the iPhone at f/2.2 will have about the same shot noise as the full frame at f/16. (Both sensors are seeing the same total amount of light).

Obviously, the full frame will need to be set to a higher ISO in order to get the same image lightness as the iPhone photo.

That's backward from how I would do it. I would set the same ISO - I'm always wanting to compare at the same ISO. Changing ISO negates the point of the comparison for me. But as long as it's know that ISO is different between the two, the same story is told.

ISO is an implementation detail.

?

Imagine we are using a pinhole camera. We can hold a 2X sensor 50mm from the "lens", or a FF sensor 100mm from the lens. In both cases, the sensors are getting the same photons. The only difference is that they are spread more thinly over a wider area with the larger sensor that's further from the lens.

Assuming they were both high quality sensors and we had a reasonable amount of light, you would be hard pressed to see a difference in the images from either sensor. If it doesn't make a difference in the final print, I call it an "implementation detail". It's not information we need to know in order to create an image.

My position is that the size and distance from the lens (i.e. focal length) are implementation details.

If you wanted to shoot 2X crop at 50mm, f/2, 1/500 ISO 400, and I substituted FF 100mm, f/4, 1/500 ISO 100, you wouldn't be able to see a difference in the final print. From my perspective the import details are 24° angle of view, 25mm aperture diameter, 1/500 shutter. The sensor size/focal length/f-stop we used to get that angle of view and aperture diameter are, in my opinion, "Implementation Details".

In terms of the final image, what's the advantage of comparing different sensor sizes at the same ISO?

That's part of the equivalency discussion I would think.

A small sensor at ISO 800 isn't going to act anything like a large sensor at ISO 800.

Exactly. You're trying to equalize them, I'm trying to see how they're different.

Do you test at the same actual focal lengths or do you equalize the focal length?

Do you test at the same f/stop or same aperture diameter?

If you are equalizing any of these, why wouldn't you equalize ISO as well?

 

[Airolo]

I think, I was more precise in my reply to Michael Fryd up the thread.

Subject is Yes.

I don't want to pollute the discussion to much, so if you're interested...

Regards.

P.