Aperture - Equivalence or Equivalency?

Lets start with an well-understood(?) equiv scenario, two sensors, one (BS) having twice the linear dimensions of the smaller one (ss):
same number of pixels, same aspect ratio, lenses with same AoV (i.e. equiv focal length), same shutter speed, same f-number (i.e. same exposure(??)), both sensors isoless (i.e. we'll count photons per pixel)

So bigger sensor lens aperture (BA) is twice the diameter (and four times the area) of the smaller one (sa).

Focused at inf the BS will capture 4 times as much photons (NPI) per pixel as the ss (npi). So far so good, right? Thus
NPI/npi = 4

Now we want to take an frame filling equiv image of an object half the linear size of the ss, i.e. magnification 2:1 for the ss lens.
That should reduce the number of photons for the ss shot to
npi/(1+2)^2 = npi/9

To get an equiv image on the BS we need magnification raised to 4:1 for the BS lens.
That should reduce the number of photons for the BS shot to
NPI/(1+4)^2 = NPI/25

As we know the aperture sizes didn't change between shots the ss will get
npi/9 = (NPI/4)/9 = NPI/36

Ratio of photons (NP_BS/np_sm)at selected object size (same FoV?)
(NPI/25)/(NPI/36) = 36/25 = 1.44

Why doesn't the BS get four times the photons of the ss anymore?

If we had started with "equiv aperture at inf" (i.e. f_number_BS = 2*f_number_ss) and hence
NPI = npi
we'd instead come out at
(NPI/25)/(NPI/9) = 9/25 = 0.36
Almost three times more photons for the small sensor than for the big one for an equiv image of that tiny object??

I have no idea what your scenario is, the way you develop it. Why can't you just state distances, pupils, for the two systems, as a one-moment-in time comparison instead of developing your scenario(s?) over time. How many scenarios are in your post? You use language that implies you are changing things mid-post in a long chronology. If there are multiple scenarios, it would be easier to follow along if you gave the full parameters for each one, instead of building on previous ones. We don't need your earlier scenarios to evaluate the later ones, anyway. Building only contributes to confusion.

I don't know where you're getting "9" and "25" and all these other strange numbers you have injected into your scenario. Perhaps you are taking things that would be cancelled out in some model and highlighting them as if they didn't cancel and needed yet to be applied? Why do you start out with "infinity"? Infinity has no place in total light calculations with one exception; the disk blur area of a point at infinity, relative to subject size, when the subject is in focus, is proportional to the rate of subject light projected by the lens. Everything you shoot is at a finite distance, and the available rate of light from an object is inversely proportional to distance squared, with the same pupil size.. You might be mixing and matching from different models in your head.

With the same shutter speed, in the same illumination, with the same subject, the total subject light projected by a theoretically perfect lens is proportional in a very simple way, to the angular area of the pupil, as seen by the subject. For 0.36x as many subject photons, from a pupil with 4x the area and the same shutter speed, you'd have to be shooting at different distances; the square root of 4/0.36 (11.111), or 3.333, which would be the ratio of distances. I don't recall you mentioning that huge difference in distance, or a difference in shutter speed, so otherwise this huge difference in distance is the only way to effect 0.36x the total subject light with the larger system.

This is why I sound like a broken record and often say that you need to check all your other photographic math against the etendue model of subject distance and pupil size. There are so many mistakes that people can make in judgment, with other models, which would not be made with etendue, because etendue only deals with standalone reality, and not with abstractions that need exact proper context to be relevant. The fact is, unless your subject is so out of focus that its OOF blur gets cropped, you get the same total light from it as if it is perfectly focused. See how simple the pupil-driven model is? If the subject is a ping-pong ball, and multiple systems are photographing it at multiple distances, with multiple pupil sizes, then the angular area of the pupil, as seen by the ping-pong ball, determines how many photons from the ping-pong ball are projected by the lens per unit of time. Only if the sensor is too small to capture all of the subject, either focused or blurred, will any further losses occur from the basic geometry. Of course, as we all know, lenses fall short of ideals, and they lose some of that projected light, and so can photosites at very low f-numbers, but the starting point for evaluation is very, very simple, and the optical imperfections affect all models, anyway, although using t-stops instead of f-stops ("t-pupils", anyone?) is truer to total light.

This is why Occam invented the razor. Pupil sizes and distances are all we need to know to (roughly) solve the total subject light ratios, for the same subject in the same light.

That formula is widely used for macro photography. To keep things simple I assume a simple optical design, e.g. a symmetrical double gauss, at pupil ratio 1:1.

I can't be of any help with the math, but I did some practical experiments with my own gear. I tested with three different formats. The testing wasn't as rigorous as I would like, but the results consistently showed that the larger sensor system requires a higher ISO when working in the macro realm with the same f-number, shutter speed and framing to produce the 'same' image. For example, I found that a full frame system working at 1:1 magnification required two stops higher ISO than an APS-C system producing the 'same' image with an equivalent focal length at the same f-number and shutter speed. Does that fit the math? I don't know.

The reasons behind why that happens must be due to the basic fact that an image of a small object made on a small sensor requires less magnification to fill the frame than it does with a larger sensor, and to the differences in extension of the lenses to get the required magnifications.

I guess this is all working in the way it's supposed to work, but it's a phenomenon that isn't normally mentioned when discussing different formats and equivalence, so it surprised me. Turns out that the simple equivalence 'rules' that I'm familiar with are only technically applicable at infinity focus, and they have to be expanded to accurately describe what to expect at closer distances.

I stated distance "infinity" at the start, scenario #1.
Equivalence|equivalency except for aperture where I started with same f-number for both sensor sizes.
(If you prefer full equivalence|equivalency kindly refer to the text after the horizontal ruler in the OP, call it scenario #3, which starts with equivalent aperture at infinity distance as only difference to scenario #1)

Then I change distance to get a really small object to fill the entire image frame, hence for the small sensor I changed magnification from 0 (i.e. infinity distance) to 2:1 (if I'm not making an error then for a thin lens that might be 3*f_small_sensor_lens distance between object and the small sensor).

To get the "same" (i.e. "equivalent" as I understand that word) image from the bigger sensor needs twice the magnification, hence 4:1 (5*f_big_sensor_lens distance is needed for that).

To keep things simple I neither care about DOF nor perspective, so let's assume the really small object is 2D, e.g. a detail of a print or something).

Please see my reply to your other post.

I think i somewhat understand equivalence|equivalency at large distances, hence i start at infinity to establish kind of a "ground truth".

I'm somehow afraid equivalence|equivalency breaks at closer distances, or at least it seems to get much more complicated than what i've read to date on that matter.
Will mull over your message later.

On the contrary, you cannot assume that at all. The apparent size of the aperture from the point of view of the sensor is extremely important. It describes why f/8 with a 20mm lens captures the same total light as f/8 with a 500mm lens. The apertures look exactly the same size to the sensor, and the reason why they do is easy to understand: The one in the 500mm lens has a much longer apparent distance from the sensor. BTW, that also helps describe why diffraction will be the same in both cases.

Sorry, I didn't notice anything about close focus when I read your original post. Unqualified, I usually assume photography where subject distance is large compared to lens and camera dimensions.

At very close distances, the pupil size starts to plummet, so my basic idea can still be valid if the actual pupil is measured, but of course, then number written on the lens may be invalid for calculating pupil size.

This article by The Online Photographer seems to hit the nail on the head regarding many discussions of equivalent aperture. 😀

It doesn't, though. I'm gonna be quoting about half of it to show why:

This will probably annoy a few people, but I'm sorry, I think it's true, even if it's not true of you. I personally think the equivalent aperture fallacy mainly exists because it relates to status and prestige. It's usefulness for many (not all) of those who promulgate it is in asserting that one can achieve shallower DoF with a "full-frame" (FF) camera than others can using smaller-sensor cameras even if the latter have the same speed or slightly faster lenses. ("My FF ƒ/1.4 lens is better than your Micro 4/3 ƒ/1.2 lens.") At root it's a way of showing off a claimant's ownership of expensive equipment, and there's not really much more to it than that.

Um, no. Not even remotely true. I'm not saying that there aren't people that think that way, but in every single Equivalence argument I've been in (and, for the record, it's a number much larger than 1), that was not anywhere in the discussion. A situation related to that was in the discussion, however: cost. Why spend so much for an f/1.2 lens on mFT when it's equivalent to an f/2.4 lens on FF? Well, that's a valid question, much like asking why spend 4x as much for a FF f/2.8 Zeiss lens than an f/1.4 Canon lens, both to be used on FF. There's more to a lens than aperture, and that is worth discussing.

Three assertions in support of this contention: First, I've observed (over many years of observing) that EA is almost always asserted (not always, so don't take offense, please) as an argument against smaller-than-FF sensors, and of the superiority of FF sensors and fast lenses. Second, people seldom point out that you can get even shallower DoF with larger-than-FF formats. The reason for the latter is probably because shallow DoF isn't actually the point. Showing that one's camera is cooler and mo bettah and more he-man than gnarly liddle-sensor cameras and baby zooms is the point. Third are all those people who shoot wide open all the time even when they shouldn't, getting important areas of the image (like the dog's nose) out of focus even when more DoF would be better for the picture.

Again, no. No one ever asserted that more shallow DOF is always (or even usually) better. What was asserted was that the option to use a more shallow DOF was good to have, even if shallow DOF, per se, was not desirable, since it also allows for less noisy photos. For example, it's nice to have the choice of using f/2.8 1/800 ISO 1600 over f/5.6 1/800 ISO 6400 (let's say for shooting a flying bird in lower light) even given that the deeper DOF might be preferable, because the lower noise matters more than the wingtips and tail being within the DOF.

All of the EA adherents' assumptions also depend on the proposition that where depth-of-field is concerned, shallower is better. This is exactly contrary to how most photographers have felt over most of the history of photography. It's perfectly easy to turn that around and say something like: 4/3 sensors are more ideal because it's easier to get more DoF.

Again, no. What was said is that there is no advantage to mFT because it forces a deeper DOF. All the mFT "aficionados" were constantly harping about how deeper DOF is "always" better (then why do they open up the aperture in low light, if true?), all the time being willfully ignorant of how DOF, noise, and motion blur all go hand-in-hand in lower light.

More to the point is when I addressed this very issue on DPR:

www.dpreview.com/forums/post/53803757

Note: I was defending those claiming that shallow DOF was used only as a "cover" for "lazy composition". So, no, no one was saying shallow DOF is always better -- people were defending the reason shallow DOF as an option was nice to have.

So, the article you linked not only does not "hit the nail on the head" -- it scored a direct hit on the hand of the person using the nail. The author shares the same mistaken attitude of the mFT forum moderator who imposes his twisted view on those simply correcting misrepresentations and out-and-out lies as being "the bad guys" in the whole drama, all the while never once (and I do mean "never once") going after the willfully ignorant and/or those who intentionally misrepresent what is being said.

Exposure time is a key parameter in Equivalence:

www.josephjamesphotography.com/equivalence/#equivalence

Equivalent photos are photos of a given scene that have the:

1) Same Perspective
2) Same Framing
3) Same Exposure Time
4) Same DOF / Diffraction / Total Amount of Light Projected on the Sensor
5) Same Lightness
6) Same Display Dimensions

As a corollary, Equivalent lenses are lenses that produce Equivalent photos on the format they are used on which means they will have the same AOV (angle of view) and the same aperture diameter. The following rules of thumb, which are a consequence of the above definition, are also helpful to understand:

1) For a given exposure, more light is projected on a larger sensor.
2) For a given scene, DOF, and exposure time, the same amount of light is projected on all sensors, regardless of size.

Thus, the only way for a larger sensor to collect more light is to use a more shallow DOF or longer exposure time.

As for who cares about Equivalence:

www.josephjamesphotography.com/equivalence/#purpose

A common criticism of Equivalence is that some people say that it does nothing to help them to take better pictures, but this represents a misunderstanding of what Equivalence is all about. Equivalence is simply a framework by which six visual properties -- perspective, framing, DOF / diffraction / total amount of light projected on the sensor, exposure time (motion blur), lightness, and display size -- relate between different formats. It is not an "instruction manual" for how to take a photo, it is not an argument that "FF is best", it does not say that "bigger is always better". In a word or two, it simply explains why the mantra "f/2 = f/2" is no more or less true, or useful, than saying "50mm = 50mm".

If one system can take a photo that another system cannot, and that results in a "better" photo, then, of course, we would do so. For example, if low noise meant more than a deeper DOF in a scene where motion blur were a factor, then we would compare both systems wide open with the same shutter speed, as that would maximize the amount of light falling on the sensor and thus minimize the noise. Equivalence tells us, however, that this would necessarily result in a more shallow DOF for the system using a wider aperture, and thus most likely result in softer corners. So, we surely would not criticize the larger sensor system for having softer corners on the basis of a choice the photographer made.

The point of photography is making photos. As such, one doesn't choose the particular system to get photos which are equivalent to another system. A person chooses a particular system for the best balance of the factors that matter to the them, such as price, size, weight, IQ, DOF range, AF, build, etc.. By understanding which settings on which system create equivalent images, these factors can be more evenly assessed to choose the system that provides the optimum balance of the needs and wants of a particular photographer.