[3dem] [ccpem] on FSC curve (A can of worms...)

[Alexis Rohou]

Hi Marin,

My two cents:

(1) SNR = CCC/(1-CCC).
It is also my understanding that equating actual (as opposed to 
estimated) SNR to CCC or FSC from a single experiment is incorrect. 
Strictly speaking, it should be made clear that the left-hand-side is an 
estimate only, ideally with some mention of what the error on that 
estimate might be.

(2) Information content from SNR
I believe the correct interpretation of your thought experiment requires 
the Shannon-Hartley theorem 
(https://en.wikipedia.org/wiki/Shannon%E2%80%93Hartley_theorem), which 
relates SNR to information rate (bits per seconds or, in our case, bits 
per pixel). My understanding is that upon binning, the information 
content per pixel goes up. Of course, we also lose a lot of pixels and 
so the total information content in the image goes down. Thus your 
paradox is gone.

(3) N
Yes, in some cases, N is not so large. This can occur with high 
symmetry, small objects in large reconstruction volumes etc. That should 
mean that, for example, icosahedral and C1 cases should be treated 
differently if we want a resolution criterion that is robust.
The reason this isn't done today is that in many cases the effect due to 
N is believed to be marginal. Arguably this is not a good reason to not 
do the right thing - just because something is only marginally wrong 
doesn't mean we shouldn't fix it.

(4) As SNR approaches 0.0, CCC oscillates around 0.0
Yes, in that case, the B&R estimator will give negative SNR estimates 
~50% of the time. I'm OK with that, you're not.

(5-6) Signal-noise cross terms in Bershad & Rockmore
Of course, signal and noise are not orthogonal, and the cross-terms are 
not zero. It's just that the expectation value of these cross terms is zero.
I think B&R did the right thing. In Eqn (2) they are describing the 
expectation value of the multiplication of the two noisy data streams. 
Therefore, when they derive Eqn (6) from it, they are justified in not 
having cross terms. Then Eqn (7), their estimator, doesn't have the 
cross-terms in it, but that's because it only need be correct in the 
ensemble (unbiased), not for a single experiment. B&R's Eqns (14) and 
(15), which give the error of their estimator, must have the cross-terms 
- this is reflected in the prominent role of N in those expressions.

To summarise: B&R are correct; a fixed criterion is not always a good 
tool for trustworthy estimates of resolution; in many cases this isn't a 
significant problem, but we might as well improve on what we have.

All the best,
Alexis


-- 
Alexis Rohou

Research Specialist
Grigorieff Lab
http://grigoriefflab.janelia.org
Tel. +1 571 209 4000 x3485