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Firstly, the term "crop sensor" is a misnomer, and I only use it in the above title to explain the misunderstanding and promotional agenda.
The implication is that 35mm is the holy grail of digital formats to which we all must aspire, and smaller formats are mere crops of the "real deal".
APS-C and 4/3 or no more crops of 35mm than 35mm is a crop of medium format. Each is its own format. Yes we can compare the relative strengths and weaknesses, but bigger isn't necessarily better; just different.
Origins of the Term
The term is valid if you attach a 35mm format lens to an APS-C format body - as people would perhaps have done having bought their first DSLR and making use of their 35mm film lenses. Here, the smaller sensor is only using a cropped, central portion of the image available from the larger format lens, and the field of view will be narrowed by the ratio of the "crop factor".
If you use an APS-C lens on an APS-C body, there's no crop!
There's no crop on any format using its own lens.
One of my tuition clients believed that shooting with an APS-C format camera and lens was inferior because the image isn't the "full picture"; it's cropped.
How powerful the disinformation is!
This term is used to compare other formats to 35mm, which can be useful. However, a better term in most applications is "35mm equivalent ratio", which dispenses with derogatory connotation.
APS-C has a crop factor, or 35mm equivalent ratio, of 1.5 (1.6 Canon), and 4/3 or M4/3 has a crop factor of 2. You would multiply the actual focal lengths of APS-C or M4/3 lenses by these values to determine the equivalent 35mm focal length/ field of view. (And as you will see, ratios between various formats can be useful for other approximate calculations too).
Full Frame 35mm Properties Compared
Ok, I'm going to keep this as simple as possible with approximate / average values for illustration.
Yes, full frame 35mm does have the potential to yield better image quality than APS-C and M4/3. But, together with the relative properties of the format, how much better is it, and under what circumstances?
If you like to stand a few inches away from metre-wide prints, you will generally see more fine detail with 35mm. This is because the image from the physically larger sensor doesn't have to be scaled up as much as that of a smaller format.
Low Light ISO Performance and Image Quality
You will generally get better image quality with 35mm compared to smaller formats (mostly noticeably in low light), which can be measured in terms of ISO.
How much? Just over 1 stop (1 EV) on average when compared to APS-C, and 2 stops when compared to M4/3.
What does that mean? Well, purely in terms of noise / image quality, (assuming a base ISO of 100) it would be like shooting at ISO 50 on APS-C, and ISO 25 on M4/3 - all approximations, of course.
Less Depth of Field - or More Bokeh
Smaller formats have more depth of field (DOF) at the same f/ stop value, and for full frame 35mm to match the DOF of the smaller formats, you would need to stop down (narrow) the aperture by just over 1 stop compared to APS-C and 2 stops compared to M4/3.
Again, we see the same figures.
So in practice, f/11 FF on 35mm, f/8 on APS-C and f/5.6 on M4/3 will all give the same approximate depth of field.
The rub is that when using FF 35mm with an equivalent aperture in terms of depth of field, you will need a slower shutter speed or higher ISO to maintain the same exposure.
How much? Yes, you've guessed it: Just over a stop compared to APS-C, and 2 stops compared to M4/3.
If you choose the higher ISO, you're giving back the ISO performance and image quality you were suppose to gain with 35mm.
So why are we seeing the same figures again and again?
These figures are all related to the same format ratio: The 35mm equivalent ratio, or crop factor.
Remember: Aperture's f/ stops are logarithmic; each integer is 1.41 times or divided by its neighbour (double or half the light):
So if we take f/11 on 35mm and divide it by the ratio or crop factor of APS-C, we get f/7.3 - just over 1 stop.
And if we take f/11 on 35mm and divide it by the ratio or crop factor of M4/3, we get f/5.5 - 2 stops (roughly).
This is, however, only a starting point for assessing ISO performance and potential Image quality. Technique and sensor generation also play a roll.
Another factor affecting ISO performance and image quality is the camera's sensor. Sensor technology has improved considerably over the years, and newer sensors significantly outperform older ones. So if you have an old 35mm full frame camera, a new APS-C or M4/3 outfit could out-perform it.
The maximum available ISO can be a guide; if your camera only goes up to ISO1600, a newer model of any of the above formats would give you better low light image quality.
(Lenses also play a part in image quality too, of course, but this article is about formats.)
If you mainly value low light performance, wide apertures and bokeh, then 35mm full frame has the advantage (like-for-like sensor generation), and you now know roughly how much. Would the extra bulk and weight be too restrictive? And is the price prohibitive? That's your decision.
But if you shoot landscape, architecture, and genres requiring lots of depth of field, the trade off with 35mm is either in ISO - at the same value as you had gained - or in shutter speed, perhaps requiring the use of a tripod.
Incidentally, if you shoot RAW and practice Exposure Optimisation, you can often gain a stop or two in ISO/ image quality with any format. This technique is more practical with live view cameras than with DSLRs though.
For me in particular, I value image content more than ultra-fine detail, and a lightweight compact system camera helps me optimise the quality of my images and the opportunity of taking them.
It depends what genres of photography you mainly pursue.
The main feature of full frame 35mm is that it has less depth of field and more bokeh than smaller formats, which some will appreciate; but it won't be an advantage to everyone.