http://en.wikipedia.org/wiki/Crop_factor [accessed 02/10/2010]
http://en.wikipedia.org/wiki/Charge-coupled_device [accessed 02/10/2010]
http://www.december.com/john/photography/cropfactor.html [accessed 02/10/2010]
to try and ascertain a good formula for calculating the crop factor between a full frame sensor and an APS-C sensor (used by 550D / 7D etc.).
The dimensions of a FULL FRAME sensor is 36mm x 24mm with a aspect of 3:2. Dimensions of an APS-C sensor is 22.7mm x 15.1mm, again with a 3:2 aspect.
Square root of (36^2 + 24^2) DIVIDED BY the square root of (22.3^2 + 14.9^2) = 1.6 approx.
This explains why the DoF on camcorders with such a small image sensor (6mm diagonal, 4.8mm x 3.6mm in the case of a 1/3" CCD), are so deep.
The crop factor of this 1/3" CCD sensor would be around x 7. Now, the ratio between each successive f stop is ROOT 2 or 1.414. Comparing a FULL FRAME f stop to an APS-C 1.6 crop shows that the difference in effective f number is around 1.13 fstops (i.e. 1.414 x 1.6). By this what I mean is that f4 on a FULL FRAME camera equates to f2.5 on a 1.6 crop. Now f2.8 would be ONE FSTOP wider, (i.e. 4 / 1.414) so f2.5 is slightly wide than this. Therefore, returning back to the previous example of a 1/3" CCD sensor with a crop factor of x7, how many effective f stops wider would the lens need to be to give the same DoF? Well, 7 / 1.414 = 4.95. Therefore, approx. 5 fstops wider! A 50mm lens with an f8 aperture on a FULL FRAME camera would therefore require 7mm lens with an f1.4 aperture to give approx. same DoF. If we took the same 50mm lens with an aperture of f2.8, we would need an aperture of f0.5???
The 1/3" example above doesn't have the same aspect ratio however, so the calculations will probably differ slightly :)
[Addition]: What about the difference between a x1.6 cropped sensor and 1/3" CCD? Well, as mentioned above, the dimensions of an APS-C sensor is 22.3mm x 14.9mm (diagonal of 26.7mm) and the dimensions of a 1/3" sensor is 4.8mm x 3.6mm with a diagonal of 6mm. Therefore, the crop factor between the APS-C and 1/3" is 26.8 / 6 = 4.55.
It would therefore seem that the effective change in f stops between these two sensors is 4.55 / 1.414 = 3.22 f stops. What this means is that f/8 on an APS-C camera would be the equivalent of a little over f2.8 on the 1/3". Also, to obtain the same field of view on the 1/3" camera, one would need to use a lens 4.55 times smaller (50mm lens, 35mm equivalent = 80mm on APS-C camera and 364mm on 1/3" sensor). So, to give same field of view, I would need to set the lens on the 1/3" camera to 80 / 4.55 = 18mm approx.
A test is going to be carried out to establish whether this is the case, by comparing both the field of view and Depth of Field between a 1/3" video camera and a Canon 550d (APS-C sensor). Watch this space!..... will place a demonstration video on Vimeo.
Here is an interesting article which shows some example shots taken with two different sensor sizes and it sort of agrees with what is being said here:
http://www.janrik.net/DOFpostings/PM1/Depth_of_Field_Versus_Sensor_Size.html [ accessed 14/11/2010].
The crop factor between the two cameras is x3.07. To obtain the same DoF on both sensors, one needed to be set to f/4 and the other f/13. This is very interesting as this represents a little over 3 f stops difference (i.e. f/5.6, f/8, f/11 and a bit more to f/13). The focal length must also have been set to 3.07 times shorter too, although it doesn't stage this in the article. My theory that that the change in f number is equal to the crop factor / divided by 1.414 doesn't therefore ring true here, so it will be interesting to see what my test demonstrates.
[EDIT - 10/01/2010] - JUST FOUND THIS AWESOME FIELD OF VIEW CALCULATOR / DEMONSTRATION TOOL:
http://www.abelcine.com/fov/
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