PROOF that the Larger the Camera Format the Shallower the Depth of Field!

It is a sad fact that today some algorithm ensures the populist proliferation of gross misinformation on the internet. In an age of misinformation being spread like wildfire over the internet, readers are asked to critically evaluate the information they have been presented with:

1. Is the source of information credible?
2. Is it mechanistically plausible (in photography, on the basis of the science of optics)?
3. If objections have been raised, have these objections been adequately dealt with?

Sadly, the chances are that the internet will ignore credible sources of information along with science and unquestioningly listen to Suckerberg-type algorithm driven popularisations of egregious falsehoods instead. But despite the chances that all reasoned arguments and scientifically credible evidence will be ignored, proof that the claims made in this YouTube segment are utter nonsense will be shown to the rare thinkers amongst readers:

This flies in the face of what Canon state on their website:

Left: 35mm format
Right: APS-C format
Canon state that the larger the format, the shallower the depth of field
(images: Canon Japan)

So which is correct, the statement by the largest camera company in the world endorsing the conventional view that the larger the camera format, the shallower the depth of field (DOF); or some vlogger's statement that camera format has no impact on DOF? I put it to the reader that the optical engineers at Canon know more about what they are talking about than some random vlogger.

No mathematical proof to accompany Granger's arguments in support of his assertions, that fly in the face of conventional optics, are provided. One commentator on another site even asserts:

This is probably one of the most argued about topics in digital photography. For as long as we can remember people have been arguing that if you will use a camera with a larger sensor (with the same lens and same settings) you will get more depth of field (DOF) or basically better separation between your subject and its background. 

Well, in this video from 2012, photographer Matt Granger (aka ThatNikonGuy) set out to disprove this claim once and for all. He used the Canon 1D-X with a [35mm small format] sensor and the tiny EOS-M (with its APS-C sensor) along with the same 70-200mm F/2.8 lens in exactly the same settings to shoot an object with a clear second object in the background.

Interestingly from the reaction to the video – lots of people continue to claim that DOF is in one way or another dependent on the sensor size… 

I often encounter recent things on the internet perpetuating Granger's piece of misinformation ad infinitum to this day despite heroic but futile attempts by more sensible educators like Tony Northrup to debunk him. While the foolish might like to furiously argue that the earth is flat, as with the question as to whether the earth is round or flat, the science of optics on this subject is similarly well settled.

To understand where Granger has erred, it is necessary for the reader to grasp the factors that reduce the DOF:

1. Distance from the subject
The closer the camera is to the subject, the shallower the DOF. This is why the DOF becomes very shallow shooting at macro photography distances. In macro photography, the DOF becomes so shallow that it becomes desirable to focus stack to win back DOF.

2. The aperture size
The larger the aperture size, the shallower the DOF. At any equivalent field of view, the larger the aperture, the shallower the DOF. Shooting with the same lens, on the same camera, at the same distance from the same subject, you get shallower DOF at f/1.2 than you get at f/22.

3. Format size
The larger the digital sensor or film format, the shallower the DOF becomes due to the optical impact on hyperfocal length and circle of confusion. This is why a phone camera gives you less background blur than a medium format camera. Reducing the sensor size doesn't "just crop" the view like cropping a photo in Photoshop, as Granger falsely asserts, for it profoundly changes the optics at a fundamental level to impact upon the hyperfocal length and circle of confusion.

4. The effective focal length/field of view
The longer the focal length, the shallower the DOF becomes. Conversely, wider angle lenses have more DOF than telephoto lenses. This is why you get little background blur shooting with ultra-wide lenses, whereas long telephoto lenses give you a lot of background blur. To isolate the effect of format size on DOF from the effect of focal length/field of view, the 35mm format (henceforth "35mmF") equivalent focal length must be calculated by correcting the absolute focal length by the crop factor so that you get equivalence in field of view. That way you are comparing like with like i.e. two lenses with equivalent fields of view. You can't compare the effect of format size on DOF when one camera has a wide angle lens and the other a telephoto lens. For example, a 25mm focal length M4/3 format lens (crop factor of 2×) gives the same field of view as a 50mm focal length lens in 35mmF (25 × 2 = 50), or a 63mm 4433 medium format lens. An M4/3 format 25mm lens gives a "normal" field of view, whereas on a medium format camera a 25mm lens is a wide angle lens. Even if you keep the absolute focal length the same (eg 25mm), the longer the effective focal length in 35mmF equivalent terms, the shallower the DOF becomes. That is why at an f/2 aperture, shooting at a distance of 2m, an M4/3 format 25mm lens has a shallower DOF (0.38m DOF) than a 25mm lens (absolute focal length) on a 35mmF camera (0.79m DOF), or a 25mm lens on a medium format camera (1.4m DOF ). Increasing the effective focal length (adjusted to 35mmF equivalent terms) has a much stronger impact on reducing DOF than increasing format size (or even increasing the aperture size). These fundamental changes are predicted by the standard mathematical formulae for the calculation of the depth of field:

Circle of confusion (C), focal length (f), F number (N), and distance to subject (u)
Notice that variables u and f are squared giving them more leverage. 

The statement that a 25mm M4/3 format lens has more DOF than a 35mmF 25mm lens or a medium format 25mm lens (all at f/2, 2m) and that therefore smaller formats give you shallower DOF at any one absolute focal length is thus "correct", but it is an empty redundancy that merely states that the wider the field of view, the greater the DOF (i.e. you get less bokeh with wide angle lenses). It says nothing useful about the impact of format size on the DOF on its own, independent of the distance to subject, effective focal length, and f stop.

Readers are asked to independently corroborate that 1, 2, and 4 are correct by experimenting for themselves. Readers are also asked to confirm 3 for themselves after reading the rest of this article.

Based upon 1-4, you now know that to make the DOF extremely shallow you should get as close to the subject as possible, shooting with the largest maximum aperture possible, at the longest effective focal length possible, with the largest format camera possible—but that increasing format size reduces the effective focal length i.e. larger format lenses have less reach. Of these factors, increasing the effective focal length and getting closer to the subject have a stronger impact on reducing the DOF. You can also understand how you can get a lot of background blur shooting extremely close up at macro distances even with an ultra-wide angle lens, but you might not get that much background blur if you focus on a distant subject with a super telephoto lens. You should now understand why you can't compare the amount of background blur you get with a 70mm M4/3 lens with that from a 70mm (4433 format) medium format lens if you want to isolate the impact that camera format alone has on DOF, because that would be to effectively compare a 55mm lens to a 140mm lens (in 35mmF equivalent terms). That would be no better than comparing two camera formats focusing on a subject at macro distances with one camera, but from miles away on the other. Or for that matter comparing two lenses set at different f stops. On the Pentax Q10 camera (crop factor of 5.53×), a 70mm lens would be the 35mmF equivalent of a 387mm telephoto lens, whereas a 70mm lens on a 4433 medium format camera would be the 35mmF equivalent of a 55mm "normal" lens.

To isolate the impact of just increasing camera format size on the DOF independent of whether the lens is a wide angle or telephoto lens, the following conditions must be kept constant:

1. The distance to the subject must be kept the same for both cameras
2. The apertures must be kept at the same f stop eg f/2.8 for both lenses
3. The focal lengths of both lenses must give the same equivalent field of view (or the same effective focal length converted into 35mmF terms)**

**It is standard to convert to 35mmF focal length equivalence as a convenient point of comparison by multiplying the absolute focal length by the crop factor to get the 35mmF equivalent focal length/field of view. For APS-C sensors made by Sony multiply by the crop factor of 1.52× and for Canon APS-C sensors multiply by 1.6. For 4433 medium format sensors multiply by 0.79. For M4/3 sensors multiply by 2.

Unless the three variables of distance to subject, aperture, and effective focal length/field of view are controlled for, they act as confounding variables. The experiment ends up looking more at the influence of these confounding variables than just the effect of format size in isolation on DOF. After all, the question that is being asked is what the effect of changing format size is on DOF, and NOT whether telephoto lenses have a shallower DOF than wide angle lenses, or whether increasing aperture size reduces DOF, or whether moving closer to the subject reduces DOF.

Can you spot where the Granger YouTube methodology goes wrong? He fails to fulfil condition number 3 before comparing the effect of format on DOF. He needed to have corrected for the impact of sensor size on field of view and effective focal length. Granger takes the same Canon 70-200mm lens and puts it on a 35mmF camera before switching the lens onto a Canon APS-C format camera with the zoom lens set at the same absolute focal length—but they are at different effective focal lengths converted to 35mmF terms. The test results end up being contaminated by the confounding effect of the impact of changes in effective focal length/field of view on DOF (where the longer the effective focal length, the shallower the DOF). With scientific experiments, methodology is everything.

If you shoot (the f stop and distance to subject being kept constant) with a Canon 70mm focal length (35mmF) lens and place it on a Canon APS-C camera shooting again at 70mm focal length, the field of view you now get on the Canon APS-C camera (crop factor of 1.6) is the equivalent of what you would see shooting in 35mmF with a 112mm focal length lens. (70 × 1.6 = 112mm). Now, remember: the longer the focal length, the shallower the DOF. So at any given absolute focal length (e.g. 70mm) what you gain in DOF from shooting with a camera with a smaller format is compensated for by the increase in effective focal length reducing the DOF back down again. Thus shooting at a 70mm absolute focal length, increasing the camera format size predictably fails to cause the DOF to become shallower and indeed the DOF should paradoxically increase, since increasing the effective focal length has more of an impact on reducing DOF than increasing format size:

Canon 5D series 35mmF body with a 70mm lens at f/2.8:
Subject distance 2 m

Depth of field
Near limit 1.94 m
Far limit 2.07 m
Total 0.13 m

In front of subject 0.06 m (48%)
Behind subject 0.07 m (52%)

Hyperfocal distance 57.8 m
Circle of confusion 0.03 mm 

Canon M series APS-C body with a 70mm lens at f/2.8:
Subject distance 2 m

Depth of field
Near limit 1.96 m
Far limit 2.04 m
Total 0.08 m

In front of subject 0.04 m (49%)
Behind subject 0.04 m (51%)

Hyperfocal distance 91.2 m
Circle of confusion 0.019 mm

Calculations using: https://dofmaster.com/dofjs.html 

A better updated DOF calculator can be found here (though it does not give values for the circle of confusion): https://www.photopills.com/calculators/dof

Equations used for the calculations are provided:
https://dofmaster.com/equations.html 

Further information on the DOF equations can be found on Wikipedia:
https://en.wikipedia.org/wiki/Depth_of_field

The sensor format change (keeping the lens and f stop the same) induces a change in the hyperfocal distance and the circle of confusion. In the Granger example, he switches the same lens, first mounted on a 35mmF body, onto an APS-C format body, while maintaining the same absolute focal length, causing the total DOF to become shallower rather than stay the same as he alleges. Granger wrongly claims that the DOF has not changed, but in fact, it has—the DOF on his set-up with the APS-C camera has become shallower than on the 35mmF camera! The reason for this is that increasing effective focal length has a much greater impact in reducing the DOF than increasing camera format size. Had he more accurately measured the DOF, he would have erroneously claimed on the basis of his methodologically flawed set-up that the smaller the camera format, the shallower the DOF.

We could exaggerate Granger's error a little by comparing the DOF from a 23mm f/4 medium format wide angle lens with an M4/3 format 300mm f/4 lens (keeping distance to the subject at 3m and f stop constant at f/4):

Olympus Micro Four Thirds Camera mounted with 300mm f/4 lens set at f/4

Subject distance 3 m

Depth of field
Near limit 2.99 m
Far limit 3.01 m
Total 0.01 m

In front of subject 0.01 m (50%)
Behind subject 0.01 m (50%)

Hyperfocal distance 1500.3 m
Circle of confusion 0.015 mm

Fujifilm on GFX series 4433 medium format camera mounted with 23mm f/4 lens at f/4  

Subject distance 3 m

Depth of field
Near limit 1.41 m
Far limit Infinity
Total Infinite

In front of subject 1.6 m
Behind subject Infinite

Hyperfocal distance 2.67 m
Circle of confusion 0.05 mm

You could argue based on this one set up that M4/3 format cameras have a vastly shallower DOF than medium format cameras. The M4/3 camera and lens have a DOF of only 0.01m or 1cm, whereas the medium format camera-lens combination has an infinite DOF. The problem is that we are not looking at the impact on DOF of increasing format size, but the impact on DOF of effective focal length/field of view i.e. it merely proves that telephoto lenses have shallower DOF than wide angle lenses.

One can hear Matt Granger complaining that he did not compare a medium format 23mm wide angle lens with an M4/3 format 300mm telephoto lens—after all, he kept the absolute focal length the same between the formats. Surely what he did would suffice to keep the playing field level for the two formats? Keeping the absolute focal length at 70mm for both formats (in our hypothetical example, this being 4433 medium format vs M4/3 format) under comparison wouldn't help Granger one bit, because he would still be comparing the difference in DOF between a 55mm effective 35mmF equivalent focal length lens and a 140mm effective 35mmF equivalent focal length lens:

Fuji GFX medium format camera with 70mm lens set at f/4:
Subject distance 3 m

Depth of field
Near limit 2.68 m
Far limit 3.41 m
Total 0.73 m

In front of subject 0.32 m (44%)
Behind subject 0.41 m (56%)

Hyperfocal distance 24.6 m
Circle of confusion 0.05 mm

Olympus M4/3 camera with 70mm lens set at f/4:
Subject distance 3 m

Depth of field
Near limit 2.9 m
Far limit 3.11 m
Total 0.22 m

In front of subject 0.1 m (48%)
Behind subject 0.11 m (52%)

Hyperfocal distance 81.7 m
Circle of confusion 0.015 mm

Once again, Granger might have erroneously concluded that the smaller the camera format, the shallower the DOF. In this example, shooting with a 70mm lens, the medium format set up has over three times the DOF than the M4/3 setup. The problem is that this set up is so overwhelmed by the effect of changes in 35mmF equivalent relative focal lengths on DOF, that you cannot draw any conclusions based on this on the impact of sensor format on DOF in isolation from other confounding variables i.e. the only conclusion that can be drawn from the Granger YouTube set up is that the longer the effective 35mmF equivalent focal length, the shallower the DOF i.e. it merely proves that telephoto lenses have shallower DOF than wide angle lenses.

There is a way to keep the comparison between formats fair, eliminating the confounding effect of the impact of focal length on DOF, and that is to compare two lenses with the same relative focal lengths adjusted to their 35mmF equivalent fields of view. What Granger should have done is to compare a 35mmF camera (eg 5D or 1D series) mounted with a 70-200mm zoom lens set at 112mm f/2.8, and then compared this with the results on a Canon APS-C body (eg Canon 7D or M series) shooting at 70mm f/2.8:

Canon 5D series body with a 70-200mm lens at 111mm* f/2.8:
Subject distance 2 m

Depth of field
Near limit 1.97 m
Far limit 2.03 m
Total 0.05 m

In front of subject 0.03 m (49%)
Behind subject 0.03 m (51%)

Hyperfocal distance 145.3 m
Circle of confusion 0.03 mm

* the calculator only permits the choice of 111mm or 114mm. The value of 111mm has been chosen

Canon M series body with a 70-200mm lens at 70mm f/2.8:
Subject distance  2 m

Depth of field
Near limit  1.96 m
Far limit  2.04 m
Total  0.08 m

In front of subject  0.04 m (49%)
Behind subject  0.04 m (51%)

Hyperfocal distance  91.2 m
Circle of confusion  0.019 mm

Once the field has been levelled by adjusting for effective focal length according to the crop factor, the total DOF on the larger format sensor camera can be shown to be shallower (5cm on the 5D vs 8cm on the 7D). The difference in the DOF is predicted by the crop factor (0.05 × 1.6 = 0.075 or 0.08 once rounded to 2 decimal places).

I have replicated Canon's results by running tests shots of charts at equivalent fields of view across different format sizes on two lenses of the roughly same field of view but in different formats:

1. 4433 Medium Format (43.8×32.9mm) vs Sony-made APS-C Format (23.5mm×15.6mm)
2. Focus point in both cameras set to the centre focus point
3. Focus on centre "X" of the three  X X X figures inside an "O" on the chart to the front
4. Blue and yellow square positioned at the centre-top edge of the image
5. Distance to subject kept constant
6. F stop for both lenses kept at f/2.0
7. Both formats mounted with the 35mmF equivalent of an 85mm lens.

The field of view is slightly different due to differences in sensor aspect ratio.

1. Fujifilm GFX50s with GF 110mm f/2 lens (35mmF equivalent of an 87mm focal length) at f/2 (ISO 1250):

2. Fujifilm X-Pro2 with 56mm f/1.2 APD lens (35mmF equivalent of 85mm focal length) shooting at f/2 (ISO 1250):

You can see that both in front of, and behind the centre "X", the figures on the chart go out of focus earlier with the larger format camera. For example, look at the "+" signs on the chart in the front, and they go out of focus sooner with the larger sensor.

Here is a telling crop of a key part of the back chart:

Left: GFX50s with 110mm f/2 lens at f/2.
Right: X-Pro2 with 56mm f/1.2 APD lens at f/2

The results corroborate those demonstrated by Canon:

Left: 35mm format
Right: APS-C format
Images: Canon Japan

Some will point out caveats in my experiment:

1. The two lenses are not exactly equivalent focal lengths, 87mm vs 85mm in 35mmF terms
2. The sensor aspect ratios are not the same
3. The 56mm f/1.2 APD lens has the unfair advantage of having an apodisation filter in it which makes the bokeh slightly more pronounced
4. Even if both cameras are set to ISO 1250, the APS-C camera will have lost more contrast at this modestly high ISO stop than the medium format camera, which may make the out of focus areas look less contrasty and more blurry.  This will again give the APS-C camera an unfair advantage.

One important deciding factor as to which of the two fields test results (mine vs Granger's) are valid is that of scientific theoretical plausibility. It is possible to use the physics of optics to mathematically calculate the predicted DOF d based on sensor format, distance to subject, and focal length. Once again, we will use this calculator: https://dofmaster.com/dofjs.html. Readers are encouraged to run their own tests.

So for the Sony-fabricated APS-C sensor used on the Fujifilm X-Pro series, shooting at 1m from subject, with a 56mm absolute focal length lens (35mmF equivalent of an 85mm lens):

Subject distance 1 m

Depth of field
Near limit 0.99 m
Far limit 1.01 m
Total 0.02 m

In front of subject 0.01 m (49%)
Behind subject 0.01 m (51%)

Hyperfocal distance 78.5 m
Circle of confusion 0.02 mm

For the 4433 medium format sensor used on the Fujifilm GFX50s and Pentax 645D, shooting at 1m from subject, with a 110mm focal length lens (35mmF equivalent of an 87mm lens):

Subject distance 1 m

Depth of field
Near limit 0.99 m
Far limit 1.01 m
Total 0.01 m

In front of subject 0.01 m (50%)
Behind subject 0.01 m (50%)

Hyperfocal distance 121.1 m
Circle of confusion 0.05 mm

The predicted total DOF for the medium format camera is 0.01m (1cm) compared to 0.02m (2cm) for the APS-C camera. This is what you expect, given that the crop factor for the larger format here is about half that of the smaller format (crop factor of 0.79 vs 1.52). If you halve the crop factor, you halve the total DOF.

What you see based on these predictive calculations is reflected in what you see in the two images of the charts taken on the GFX50s vs X-Pro2. You can also enter a focal length of 106mm for the medium format lens instead of 110mm in the DOF calculator to see if might impact on the results, but it makes no meaningful difference at all to the calculations rounding to two decimal places. The mathematics of the theory of optics and the empirical findings in practice are in agreement with each other. The science of optics wins over mere blogger assertion.

Conclusion: the larger the camera sensor format, the shallower the DOF—but only if the three variables are kept constant of f stop, distance to subject, and effective focal length/field of view. The Canon optical engineers are correct, Granger is wrong.


[EDIT: some follow-up thoughts on additional experimental setups]