Maths behind full frame (FF from now on) vs APS-C (APS from now on) vs small sensors. I used to use film SLR several years ago, now having used APS-C DSLR for some years, wondering what exactly is the difference, and is it worth it?
FF has 1.5*1.5=2.25 times the cell size compared to APS (more with Canon's smaller APS, but let's use the standard crop ratio of 1.5 for now), so it gathers 2.25 times the light, right?
Well, kind of...
A camera's light input simply depends on:
1. Field of View (FoV) - from how big cone are photons received from - double the area, double the photons
2. Absolute aperture area, which the photons have to pass - again, double the area, double the photons
3. Optics quality
Optic transmission rate and reflection differences are some percents, very small when we are talking about steps, each 50% less or 100% more light, thus we can pretty safely forget the last line.
Points 1 and 2 can be combined to the relative aperture, the "f-number", or 1/f. These are extremely convenient for comparing lenses of different focal lengths as long as the sensor size/crop ratio stays the same. As a single number without knowing crop ratio and focal length the f-number is mostly suitable for marketing cell phone cameras.
Example:
Let's say we have a lens with 50 mm focal length and f-number of 2.4. From this we can calculate the aperture diameter 50/2.4=20.8 mm and area Pi*r^2=3.1416*(50/2.4)^2=341 mm^2. With an APS sensor this lens would give FoV diameter of 32.2 degrees and for a FF sensor 46.8 degrees. FoV width of FF is apparently about (we're forcing the FoV cone to a rectangle photo, so have to accept some approximation) 1.5 times that of APS, so area (width*height) would obviously be 2.25 times, thus the full frame would be collecting light from 2.25 times the area. Myth confirmed? Not that simple, follow on.
Usually when photographing, I would choose the area that will fit my photo, instead of choosing a focal length and being happy with what happens to fit the frame (experience has wired my brain to think directly in focal length, of course, but that only works with the format(s) I am familiar with). If I wanted the 32.2 degree diagonal FoV (about 1 meter wide at 2 meters), I would choose the above 50 mm lens for APS or a 75 mm lens for FF.
Now we have two lens-sensor combinations with the same FoV - point 1, let's then find how to match apertures, point 2. For raw light throughput, we are interested in the absolute aperture size, so can just divide focal length by the f-number and match the aperture diameters. 50 mm/2.4=20.8 mm, 75 mm/20.8 mm=3.6, so the f-number required by the full frame setup for the same light input would be 3.6 (which happens to be 2.4*1.5, seen that multiplier somewhere before?).
Let's check Depths of Field (DoF) for the above sets from the on-line DoF Master.- Camera=Canon 5D for example, Focal length=75, f-stop=3.6, distance = (say) 200 cm -> DoF total=14.7 cm
- Camera=Nikon D300 for example, Focal length=50, f-stop=2.4, distance = (same) 200 cm -> DoF total=14.9 cm
Hmm.. Both happen to have a DoF of 15cm, when field of view and light input are the same, interesting. If I switch the 75 mm lens to the same f2.4, I will naturally get more light in, but my already small DoF will be reduced to less than 10 cm. So, full frame can grab more light with the same FoV and f-number, but by the price of reduced DoF, exactly like stepping our APS-C lens aperture value from 2.4 to 1.6.
So, is the full frame equal to APS-C with a bit faster lens? No. First, if taking a photo of a given scenery with APS-C with a 50 mm lens and full frame with a 75 mm lens, you will not get the same results, proportions of objects in the photos will be different. Take two equally tall people for example, place them close enough you can fit them in a photo, but so that one is a couple of meters farther from the camera than the other. Take a photo of them with 18 mm and 200 mm focal lengths, note that in the 200 mm photo they look pretty much equally sized and in the 18 mm one the closer guy will seem much bigger. This same effect you will notice, when comparing those seemingly equal full frame and APS-C configurations; to make APS-C look like full frame 50 mm lens, you need to use a 50 mm lens, and get used to cropping.
Second, pixels in a 36x24 mm sensor are generally bigger than in a 24x16 mm APS-C sensor, which should make them less noisy, but that is not necessarily the case, and a tightly packed full frame sensor might actually have smaller pixels.
If you look at sensor measurements (in http://www.dxomark.com for example), you will find that full frame cameras have about 2-3 times the low-ISO performance of APS-C cameras. Is this because of bigger pixels, the aperture advantage, or what? This difference seems more often than not to be suspiciously close to the 1.5^2=2.25x difference, which would hint that most of the difference was due to the aperture advantage (lens with the same FoV would have bigger absolute aperture), which would sacrifice DoF, which would often be a challenge even for APS-C in low light.
ISO, unlike the physical parameters shutter speed and aperture (absolute or relative), is just an arbitrary number given to an amplification coefficient, to match parameters between cameras. In a given environment, two cameras with equal f-number, shutter speed and ISO number should give pretty much similarly bright image, and as f-number and shutter speed are physical measurable values, ISO number is the parameter that has to be adjusted to make a camera behave like any other camera. ISO number is not amplification level, just a mapping to suitable amplification levels in a given camera.
Lets try mathematical approach to a real world example.
I want to take a photo of my daughter playing in a rather dark environment, rather close, DoF of something like 10 cm and shutter speed at least 1/50.
With my APS-C camera, I'll pick my 30mm 1.4 lens, and place the lens 80 cm away, giving a nice 64 cm wide Field of View at that distance. For the desired DoF, aperture value of 4 would give 11 cm, plenty. Let's say my camera wants to use ISO 6400 for this, fine.
The big question: Would I get faster shutter speed or less amplification=less noise with a full frame in this situation? Let's see.
In the full frame camera we will need a 45 mm lens for the same FoV, for the desired DoF we would select aperture value of 5.6 to get DoF of 10.2 cm - DofMaster, or most cameras, don't let me select 4*1.5=6, which would match the same DoF.
How about ISO? The combination of shutter speed, aperture value and ISO should give more or less the same brightness image with any camera, including these two. ISO is just a number, so we cannot directly tell which ISO is "better", but we can relate to gain, or amplification, to get an idea about noise. Let's assume our setup would cause 12 billion photons reflected from the target hit our APC-C sensor in the 1/50=20 milliseconds of shutter time, and that our sensor has 12 megapixels, so 1000 photons would hit each pixel area, and that would cause, say, 50% intensity (red, green, blue, depending on the pixel) with the amplification my camera uses for ISO 6400, lets say this amplification is 90x. To compare ISO between the cameras, we will keep shutter speed and aperture value equal, and figure out the gain needed in the full frame camera. Shutter open time is again 20 ms, but with the 2.25x larger FoV we are collecting photons from 2.25 times the area, so are actually receiving 27 billion photons, and if we keep pixel count equal, we will now receive 2250 photons per pixel. With the amplification above we'd overdrive the sensor, so keep the same 1000 photons per pixel for 50% intensity, we need amplification of 90/2.25=40x, which this camera should use for ISO 6400. As we are now using less amplification, but the signal level is the same, we get about 3dB better signal to noise ratio with the same ISO. Full frame now proven to be better? Read on.
ISO matching has to be done using equal shutter speed and aperture value, in which case the full frame shines. But, when trying to get a decent photo of my daughter, I don't really care about aperture value - I care about FoV to get the framing I want, and just enough DoF to get both eyes and hair in focus. Above we calculated the suitable lens and aperture setting for our full frame camera, how about ISO? The above calculated ISO and the related amplification used the matching focal length and aperture value or 30 mm/4, not the 45 mm/5.6 one we want. Absolute aperture doesn't change, but FoV area is reduced by the famous 2.25 coefficient, which we would compensate by cranking up ISO to 14400 (probably 12800 in the real camera), thus increasing amplification to 90x, matching the amplification and noise level used in the APS-C camera.
Hmm.. Mathematically looks like full frame is no more sensitive than APS-C, when using the same FoV and DoF, and the 3dB SNR difference and 2.25x high ISO performance are just an illusion caused by the shallower DoF of the bigger sensor. Seems against public belief, but rather logical really; also APS-C SLR shines in high ISO compared to compacts, but by the price of reduced DoF, bigger sensor just allows trading DoF for sensitivity in dark. Also, overhead of wiring etc is very small in APS, so I don't get where over 2x sensitivity differences would come from.
Note that even if FF doesn't actually have better sensitivity, so no benefit in close quarters, it does allow more than one step effective aperture speedup; a 70-200/2.8 would behave like a stellar 47-133/1.9 lens in an APS-C, and be sharp from 2.9 meters to infinity (hyperfocal) at 70 mm, pretty nice for weddings or sports fields.
Findings:
- FF does receive 2.25x the photons with the same lens as APS.
- If the lens is changed to 1.5x focal length to match FoV, and f-number stays the same, it will still receive 2.25x the light. It should, as f-number doesn't change.
- With above lenses, ISO would stay the same as in APS, but FF needs only 44% of the amplification for that same ISO, thus also getting about 3 dB better Signal to Noise Ratio (read: lower noise level)
- With above lens/f-number, Depth of Field is reduced, just the same as adjusting APS aperture to 2.25x the size=a bit more than one step of f-number.
- To match the DoF, aperture value of the FF needs to be increased by one step and some.
- To compensate above change, also ISO needs to be increased by one step and some, thus using the same amplification and getting the same Signal to Noise Ratio as APS.
- Now our lenses are matching in FoV, DoF and brightness: We get the same amount of stuff in the photo, as much of that photo is in focus, and the same general brightness.
- Thus, for the "same" photo, FF needs to use about double the ISO, but the amplification level and noise margin behind that is actually equal to what the APS would have.
- 2.25x of the "high/low light ISO performance" advantage of FF is due to sensor size and not sensitivity - to compare sensitivity (which doesn't sacrifice DoF), multiply "high/low light ISO performance" by crop ratio^2. Let's try comparing Canon 650D vs 5D MkII with this multiplier: 650D low-light ISO 722 and 5D MkII 1815 (values from DxOMark), multiplying 722*1.6*1.6=1848, pretty close to 5D, looks like the sensors have pretty much equal technology behind. 650D scoring a bit better might be due to 5D having more pixels dividing those precious photons, thus needing a bit more amplification. Do the same test with Nikon APS D5200 vs. FF D800, you'll get 2889 vs 2853, see a pattern?
- When having a fast enough lens and going hyperfocal, or otherwise able to use the shallower DoF, this 2.25x size advantage is fully realized, hard to beat a 70-200/2.8 lens in the APS world.
- Another major benefit of FF is increased perception of depth, FF would look more "3D". For the same depth effect, APS needs to be moved closer to match the FF, not sure if this is a real benefit or something you are/get used to.
- If you swap your APS with 17-50/2.8 to FF with 24-70/4.0, you will be using 12800 instead of 6400, but the amplification and noise level is actually the same. For about $/€ 2500 extra price for FF you will get a bigger viewfinder and greater increased perception of depth, probably also some more sharpness, not necessarily better low-light sensitivity. If you swapped from a 2.8 lens to another 2.8 lens instead of the 4.0, you will get improved low-light performance, but with lower DoF, which won't matter if you do hyperfocal, which would more often than not be the case when shooting various events.
- Conclusion from the previous one would be, if you mostly do non-studio inside photography, a sensitive APS sensor will give you better low light performance than a mediocre FF sensor; multiply crop-sensor low-light ISO values by crop ratio^2 (2.56 for Canon, 2.25 for most others). For wedding, sports etc event photography, where your targets are generally far enough to not need to worry that much about DoF, get the FF, after making sure you can also afford a good 70-200/2.8 or so. In studio, where you'll always use a tripod anyway, and have full control of everything, FF or even a medium format camera would serve you well.
Some things to dig in further:
- Effect of pixel count/density, here we used the same conservative 12Mpix for both APS and FF.
- Effect of Circle of Confusion (CoC), as that is used for determining DoF, will need to figure out how exactly does CoC function.
- Perception of depth, is the difference only what you are used to, or is FF more "real"? 36x24 mm is just as arbitrary size as 24x16 mm, why would it be any more "real", than APS?
Any errors? Probably, please tell me. I'll do a bit clean up later and add figures.
BTW, me being skeptic, not believing without fully understanding, doesn't mean I wouldn't love full frame cameras:)
References
* http://www.bobatkins.com/photography/technical/field_of_view.html
* http://dofmaster.com/dofjs.html
* http://www.dxomark.com/index.php/About/In-depth-measurements/Measurements/ISO-sensitivity
* http://www.dxomark.com/index.php/Cameras/Compare-Camera-Sensors
