Originally posted by Ian Stuart Forsyth If we first look at your 12mp compared to 7.5mp that is not a 1.6 crop factor it is 1.26crop factor, you have to look at it in 2 dimensions, For example 36mp camera cropped by a factor of 1.5 is not 36 / 1.5 as that would give you a a final pixel count of 24mp
we have to take into account that 36mp has both height and width so 1.5 x 1.5 = 2.25 so 36mp / 2.25 = 16mp
Now lets go back to the 12mp compared to 7.5mp 12 / 7.5mp= 1.6 √2= 1.265 crop factor
So a 12mp camera with a 300m will have an additional crop ability of 1.265 giving you an image that is 7.5mp with a FOV that is 380mm and not the 480mm FOV when using that 1.6
Now with a FF 36mp camera if you are cropping down to a 24mp image you are only cropping by 36/24=1.5√2=1.22 and not below the level of an apsc
So a FF36mp cropped to 24mp will give you a crop factor of 1.22 with a 600mm lens you would have a FOV 730mm
A 24mp cropped camera with a 1.5 crop factor will give you a 24mp image with the FOV 900mm
You have to also remember because you are enlarging the 24mp 1.5 cropped image more to be viewed as the same size as the larger 24mp 1.22 cropped image there will be some resolution loss due to this enlarging
A very simple way to look at it is if you want to double your lenses reach (600mm-1200mm) you need to quadruple you pixel count by place 4 times more pixels into the same area
With a 36mp camera I seldom go below a 1.2 crop and that is only for shooting raw for buffer and frame rate
Based on this set up.
And these two images
I'm estimating in the real world, the magnification of the K-3 and the exact same set up an position, the real world advantage is about 36%. Someone cooking a bunch of numbers means nothing to me. This is what's real.
SO, to get the same image I could be 36% further away with APS-c, a real benefit for birding. As for the total noise thing and worse noise on APS-c. Ya slightly worse. But not meaningfully worse. These images haven't been run through Topaz Denoise. It's a safe bet that once it has, the noise will be the same. The K-1 however will never recover the missing detail. And there's not enough difference to make any kind of case anyway, unless you were making the case that there is virtually no benefit based solely on total light.
Any attempts to describe this difference have to come up with the "right answer" or they shouldn't be entertained.
With all due respect to the above, the difference is significant any way you look at it. The numbers are just numbers. The proof is in the pudding. The biggest problem with the math, as demonstrated here is, you make one little mistake and you create falsehood. Without confirmation in the real world it's useless.
This is why a notion is just a notion until supported by empirical data. Then it may become a hypothesis and eventually a theory. What I'v posted represents real world usage, not some convoluted math worked out to try and prove some kind of a point that makes no sense in the real world.
But then I'm not using arbitrary crops that I would never use in real life to try and prove a point. That's the thing about mathematical proofs, you can demonstrate something that might be mathematically true, that is practically irrelevant. But you still need to get your numbers right, confirming with real world shooting, hopefully, before you post the information.
The measured lw/ph of a k-5 is ~2100. The measured lw/ph of a k-3 is ~2700. Nothing you can do with the K-1 images cropped will make up for the extra 600 lw/ph in the FF mode. That 600 lw/ph is just gone, no matter how you crop it. And 600 is significant, unlike many 100 lw/ph differences.
The crop factor, APS_c to FF is 1.5, no matter how you stack it. 600mm FF is the same field of view as 400mm on APS-c, and there is more pixels on the subject (and therefore more resolution) where the pixels are smaller on APS-c, which is the true until you get to really expensive 51 or higher MP FFs.