- Apples to Apples-


... comparing cameras and lenses the right way ...







Preamble


If you want to buy a car for mainly cruising at a certain speed, it would be logical to compare cars when driving at this speed, in order to see which one is the most comfortable or economic or whatever. The same thing goes for camera systems (body + lenses). If you want to produce a certain image (composition, print size) in a certain circumstance (available light), you would typically want to compare systems which are set up to produce the same output (given the same input). We will see that if the sensor (or film) sizes of the systems are different, this generally requires that the cameras should be compared at different iso levels, and the lenses at different F-numbers. The persistant bad habit of comparing cameras with different sensor size at same iso levels, and their lenses at same F-numbers, leads to misconceptions and wrong conclusions, as I have whitnessed many times in real life and in discussions on the internet. In January 2006 I was probably one of the very first on the internet to describe the framework of what later became generally known as equivalence. Despite (or because of) the many efforts and arguments, the leading publishers on photography today still use the same iso comparison schemes, and this for reasons which are obvious to me but on which I will not elaborate. I will limit myself to the same observations I made in 2006, as they still kind of speak for themselves, as do the results from a more recent test :


System A
System B




Introduction


When an experienced (or at least educated) photographer takes a picture, he or she will try to obtain an optimal result via the control of composition and image quality. Especially when shooting handheld in difficult lighting conditions, this often means finding a specific compromise between magnification, shutter speed, aperture, and sensitivity. We will see that in a wide range of conditions, the settings that are needed to obtain a certain desired result, depend on the sensor size. We will develop and discuss a mathematical model describing how settings on different cameras relate to eachother. This is not only useful for people like me who actually own different types of cameras, but also for people with experience in one format and considering the switch to a different format. Or for people interested in how things work.





Requirements


We will tackle image composition and image quality parameters one by one, the question being what settings make 2 shots as identical as possible.




1. Same perspective

The pictures should obviously show the subject(s) from the same spot and angle.




2. Same framing

The pictures should show the same amount of subject and / or surroundings. We will suppose the framing is determined at the moment of recording, and not by cropping afterwards.




3. Matching depth of field

We will also suppose that the output display size and viewing distance are predetermined by the user, irrespective of sensor size or film format.




4. Same motion blur

Especially important for moving subjects.




5. Same brightness

This point doesn't really require going into details.




6. Same noise or grain

Indeed, sometimes noise or grain can be part of the 'composition'. We will see that this point will automatically match as a result of the previous conditions, and as such does not add an extra constraint.





Equivalent settings


We will now see how the requirements translate into specific camera or lens settings




1. Same perspective

The optical center of the lens has to be in the same spot (no, not the sensor or the tripod mount, but the difference is usually small) and the lens axis should be aligned.




2. Same framing

Technically speaking, this means that the image projected on the larger sensor has to have a higher optical magnification than the image projected on the smaller sensor :

m2 = k ⋅ m1




3. Matching depth of field

This translates into the need to have the near and far in-focus planes situated at the same distances from the lens center :

s'2 = s'1

s"2 = s"1


In the upcoming formulas, we will also use the fact that the circle of confusion limit on the sensor necessarily scales with sensor size :

c2 = k ⋅ c1




= Intermezzo =

So far, many parameters have been introduced, none of which is a direct camera (or lens) setting. We will now introduce the following formulas valid for thin lenses and derived in this seperate tutorial :

v = m/s

s = 2s's" / (s"+s')

a = c/m ⋅ ( s" + s' ) / ( s" - s')


Using these formulas, the conditions above translate into :

s2 = s1

v2 = k ⋅ v1

a2 = a1

Which means that the requirements 2 and 3 can be met by selecting the same (in focus) subject distance, the same aperture diameter, and an image distance proportional to the format. The focal length is then the result of 1/f = 1/s + 1/v (thin lens formula). In case the subject distance is large compared to the focal length, the subject-to-image distance is close to the subject distance ( d ≈ s ) and the image distance is close to the focal length ( v ≈ f ). And because a = f/N is supposed to be equal, the 'F-number' N should now be proportional to the format. The 3 equations above can be replaced by the following ones, which are at last expressed in terms of direct camera or lens settings :

d2 ≈ d1

f2 ≈ k ⋅ f1

N2 ≈ k ⋅ N1




4. Same shutter speed

Now this is a direct camera setting, easy.

SS2 = SS1




5. Same brightness

The shutter speeds match. The aperture diameters are equal. This means the same amount of light is projected on to the sensor. For the image of the larger sensor to be as bright, it has to be amplified by a factor equal to the sensor area ratio, by turning up the iso. On a more local scale, one can say that the amplification needs to be proportional to the square of the magnification (as light is spread out over the 2-dimensional sensor surface). And we already knew that for same subject distances, magnification is proportional to the sensor size. Either way, we find :

ISO2 = k² ⋅ ISO1




6. Same noise or grain

The most important source of noise, photon noise, is induced by a lack of light. In a first order approximation, equalling noise boils down to having the same amount of light to hit the sensor, which is already a result of previous requirements. Note that if the sensors have the same amount of pixels, the corresponding pixels will receive the same amount of light, and one can expect the same per pixel noise too.





Definition


We will call settings equivalent if they meet the following conditions :

d2 ≈ d1

f2 ≈ k ⋅ f1

N2 ≈ k ⋅ N1

SS2 ≈ SS1

ISO2 ≈ k² ⋅ ISO1


The formulas are for distance between camera and in-focus plane (d), focal length (f), F-number (N), shutter speed (SS) and sensitivity (ISO). The formulas are such that the crop factor (k) is larger than unity in case sensor2 is larger than sensor1. The value I typically use for the crop factor is the square root of the sensor area ratio, but that's a different discussion. One can also use the following strickter definition for equivalent settings, allowing a truly exact match in case of thin lenses :

s2 = s1

v2 = k ⋅ v1 or m2 = k ⋅ m1

a2 = a1

SS2 = SS1

ISO2 = k² ⋅ ISO1


This time, the formulas are for lens to in-focus distance (s), lens to sensor distance (v), aperture diameter (a), and again shutter speed (SS) and sensitivity (ISO). But do note that aperture diameters and iso values can typically only be selected in steps, and perfect matching is not allways possible in practice even if one would like to.





Example


Here's an example of equivalent settings between 3 setups ('Full Frame', '1.4x crop' and '2x crop') :

Full Frame @ iso 400, 70mm f/2.8 1/250 sec

1.4x crop @ iso 200, 50mm f/2 1/250 sec

2x crop @ iso 100, 35mm f/1.4 1/250 sec






FAQ


Frequently asked questions.




1. Equivalent settings, equal results ?

Using equivalent settings does not guarantee equal results, it is merely a model for levelling the field when comparing or using cameras. If one wants equal results, one needs extra conditions. To give an example : the lens has to be sharper for a cropped camera to bring the same information to the sensor (as the pixels of the sensor are closer together). But as lens sharpness is not a camera or lens setting, we left it out of the definition. And there are other things like colour rendition, chromatic aberration, distortion, AA-filter design, sensor efficiency, ...




2. And what about equal settings ?

If you compare cameras and lenses at equal settings, you will often just prove the obvious : at equal sensitivity, the camera with the larger sensor has better noise behavior, and at equal aperture, the lenses for crop are sharper wide open but diffraction kicks in earlier when stopped down. This will nearly allways be the case. But what matters (well, for many people that is : those who often shoot handheld), is if the cropped lens is still sharper after the other lens is stopped down to achieve the same depth of field ; and if the larger sensor is still less noisy after it has been bumped up to compensate for the stopping down of the lens.




3. Ok but what system should I choose then ?

Everything seems to equal out by using equivalent settings, does it then really not matter which system one uses ? Well, it does. On a large format system, it is normally always possible to select equivalent settings to a setting on the smaller format system. But the inverse is typically not true. Especially low iso settings on large format sytems do not find their equivalent on smaller format sytems. In this case comparing base iso's does make sense in order to see the potential in high image quality. The same is typically true for large aperture diameter lenses : the larger the format, the more choices one typically has as a user.




4. What about diffraction, dynamic range and colour rendition ?

Colour rendition is quite sensor specific, but diffraction equals under the same conditions as DOF, and dynamic range reduces with amplification and thus also tends to match for equivalent settings.





Myths


Common misunderstandings.




1. Each and every image I take will be better with a larger format camera

This most common misunderstanding comes from the fact that when comparing for same iso, and at same F-number, the noise of the large format will be noticebly lower. The reason why this is so is that the large camera setup uses a larger aperture diameter and thus is allowed to capture more light. If you allow the smaller format camera to capture the same amount of light, by using equivalent settings, the noise levels will decrease to roughly the same level. This priciple can be used in the field too (if at least the equivalent setting is available on the smaller format camera).




2. Large sensors are better for low light shooting

Large sensors are not better in low light. But large aperture diameters are (if you can live with shallow DOF that is). The fact that larger apertures typically require larger lens mounts and thus camera bodies, which on their turn allow accomodating larger sensors, does mean that there is a correlation between availability of large aperture diameter lenses, and sensor size.




3. Smaller formats provide more DOF

Not true : if you stop down the lens on the system with the larger format, you will obtain the same DOF ; this is nearly allways possible.




4. f/2.8 = f/2.8 = f/2.8

See above : equal settings do provide equal exposure, in case sensitivities and shutter speeds are equal, but give images with a different effect.




5. Larger format systems are heavier

Not true. Now the sensor might be heavier, but optical viewfinders will be less heavy for a same magnification. The weight of the lenses typically depends on the aperture diameter, which, for same DOF possibilities, is the same regardless of the format.




6. Larger format systems require better lenses

Again, not true. One can actually argue that the smaller format system requires lenses to be sharper in order to project the same information onto a smaller area.




7. Small format lenses are sharper wide open.

Well, they might be. But the equivalent setting on a large format lens is typically not the most wide open setting, so before comparing one might allow the large format lens to stop down.





More Examples


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