Re: [APML] CMY Filters and Signal-to-Noise

[Aplanatic@aol.com]

Chuck wrote:

> I appreciate you picking up the ball and running through this
>  analysis. Let me see if I understand this correctly. It looks
>  like the CMY method will produce lower S/N than the RGB method
>  because noise from the other parts of the spectrum work to
>  reduce the S/N. A red image taken with a red filter contains
>  no noise from the green or blue channels. Although RGB exposures
>  need to be longer, the S/N will be higher.

Yes, that's how I see it, but there are some further issues.  
See:

http://astroccd.com/terre/buil/us/cmy/cmy.htm

To summarize this article, Buil introduces the following "Signal" as 
the figure of merit:

S=R+G+B

and then shows (correctly, I believe) that the signal-to-noise ratio of 
this metric is

S/N = 2/sqrt(3)  S/N(native)

with the CMY process, where S/N(native) is the signal-to-noise ratio 
of the same metric for the RGB process.  In other words, S/N is larger 
for CMY than for RGB for this particular definition of "Signal"

As I have shown, each individual reconstructed color channel has a 
lower S/N in the CMY process, so how could the "combined" S/N 
be better?  Well, if you work out the noise equation for this metric, 
(R+G+B), you will find that the cross products have a negative 
correlation.  For example, when we square (R+G+B) we will have 
terms like R*B which will expand to (M^2 - Y^2 - C^2) and you can 
now see the negative noise correlation directly.  When all the terms 
are added together I get exactly the S/N reported by Buil in his 
more direct approach.

What does the metric S=R+G+B mean?  It is the total, unfiltered 
signal, i.e., the luminance.  So, it would appear that the CMY process 
causes a small increase in the signal-to-noise ratio for the luminance 
information at the expense of a small decrease in the signal-to-noise 
ratio for the chrominance information.  

In some cases I see this as an advantage and in other cases as a 
disadvantage.  Clearly, if the object is primarily an H-alpha emission 
region, then the loss of S/N in the red channel is certainly not attractive.  
On the other hand, the increased luminance S/N probably helps for 
images of galaxies.

In any case, the effect is rather small, at most plus or minus about 
15% in SNR.  This small effect is probably overshadowed by the 
color response function of the filters and their peak transmission 
characteristics, which seem to favor a CMY filter set, especially 
considering the difficulty and expense in procuring a good RGB set 
that adequately covers the OIII line and has high passband 
transmission.  

The above analysis is based upon a linear detector.  As noted before, 
film mixes photons of different colors and thereby may suffer an 
additional loss when used with CMY filters, especially, I think, at 
the very lowest light levels.  I have been trying to analyze this situation 
in more detail but so far the nonlinear noise calculation has me 
stumped.

Dave Rowe.

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