*An Analytical Comparison
of the Signal to Noise Ratios Between CMY
and RGB
Imaging Techniques*

*By Ed Grafton*

**One of the most significant obstacles,
which must be overcome in imaging astronomical objects, is the inherent
faintness of these objects. From the early days of photographic film to
present day CCDs, great strides have been made in capturing and rendering
these faint objects. With the advent of CCD technology an astro-imager
was no longer plagued by photographic film reciprocity failure and finally
had a linear detector to capture these objects. The most efficient way
(in terms of detector speed) at present to capture these faint objects
in color is to take three filtered images and combine them into a single
color image. The traditional methodology of producing color astronomical
images is to take an image through a red, green, and blue filter and combine
the three into a color image. This process was first developed in the 19th
century and applied to film emulsions to produce the first color photographs.**

**In 1998 Al kelly, Richard
Berry, Chuck Shaw and Ed Grafton began to experiment with using cyan,
magenta and yellow filters to produce color astronomical images in stead
of using the traditional red, green, and blue methodology. Since the cyan
magenta and yellow are each comprised of two colors each (and thereby passing
twice as much data per unit exposure time), it seemed as if this methodology
would be more efficient that the traditional red, green, blue (RGB) methodology.
The charts below show the typical band pass of characteristics of RGB and
CMY filter sets.**

**The Bandpass Characteristics
of RGB and
CMY
Filters**

**In order to construct
an image from the CMY data set, the images must first be converted in to
an RGB data set. The relationship between RGB and CMY is:**

**R
= M + Y
- C**

**G
= C + Y
- M **

**B
= C + M
- Y **

**The derivation of the RGBs from
CMY data has the effect of degrading the S/N ratio of the RGBs derived
from CMY filters. The reason is that noise adds as the root sum of squares
so the RGB derivatives adds sqrt(3) to the noise. Since the signal is twice
as large the S/N is degraded by a factor of sqrt(2/3). The S/N of an RGB
derived from a CMY data set has 81% the S/N ratio of RGBs from RGB filters.
So the CMY RGBs are noisier than the RGBs from RGB filters.**

**In practice many if not most
imagers are producing images that are LRGB or WCMY. So what does this mean
to the S/N of a LRGB compared to a WCMY image?**

**Suppose we wanted to bring the
CMY RGBs up to the S/N level of the RGBs from RGB filters. Let the R, G
and B filter Signal=S, then the C, M and Y filter signal=2S and an IR blocked
"L" or "W" image would have a signal of 3S. Let the
noise for each image = N.**

**1) The RGB S/N is = S/sqrt(S)
**

**2) The RGB S/N for CMY derived
RGBs is = 2S/(Sqrt(sqrt(2S)^2 + sqrt(2S)^2 + sqrt(2S)^2)) = 2S/Sqrt(6S).
If the exposure of the CMYs were increased by 50% then: **

**3) Then equation 2 becomes 3S/(sqrt(sqrt(3S)^2
+ sqrt(3S)^2 + sqrt(3S)^2)) = 3S/sqrt(9S) = S/sqrt(s) = to equation 1**

**The math shows that you would
need 50% longer exposures for the CMYs to get to the same S/N level as
the RGBs from RGB filters.**

**Now lets construct the luminance
(W) from these 50% longer CMY images (as I typically do) and compare them
to the luminance constructed from the RGB filters and also do the same
for the RGBs**

**4) The W for the 50% longer CMY
exposures = 3s+3s+3S/(sqrt(sqrt(3S)^2 + sqrt(3S)^2 + sqrt(3S)^2))= 9S/sqrt(9S)
**

**5) The L for the RGBs is = 3S/sqrt(3S)
**

**6) Dividing eq.4 by equation
5 = 1.73 = 173%**

**The math shows that the SNR of
the constructed W from the CMYs is 173% better than the L constructed from
the RGB filters. So with a 50% longer exposure for the CMY you will get
173% better luminance S/N and equal S/N on the chrominance.**

**Now instead of equalizing the
chrominance, lets equalize the luminance by taking longer RGB exposures.**

**Since the RGB signal = S and
CMY signal = 2S:**

**The math shows that you would
need to take 100% longer exposures through the RGBs to construct an L that
has the same S/N of the CMY constructed W image, but the RGBs (do some
more math) would have a chrominance S/N that is 173% better than the CMY
chrominance derived RGBs.**

**Now lets go back to where we
had equal chrominance S/N (this is where we had to have 50% longer CMYs)
and a W S/N that was 173% better than the RGBs. In order to bring the LRGB
L image up to the S/N of the W (173% better S/N) from the 50% longer CMYs,
we will take a no filtered L image and add it to the RGBs to construct
a L that has the same S/N as the W. This means that the chrominance and
the luminance will then have the same S/N for each method.**

**7) From equation 4 the W image
has a S/N of 9S/sqrt(9S). **

**8) The L (R+G+B+L) image would
have a S/N = S+S+S+3XS/(sqrt(sqrt(S)^2 + sqrt(S)^2 + sqrt(S)^2 + sqrt(3XS)^2)).
Where X is the multiplication factor applied to the no filter image so
the R+G+B+L constructed luminance has the same S/N as eq.7 (i.e. the S/N
of LRGB Luminance = to the S/N of the WCMY Luminance). Now if X=2 then:
**

**9) The S/N of the LRGB luminance
is S+S+S+6S/(sqrt(9S) = 9S/sqrt(9S) which is equal to the W luminance in
equation 7.**

**So now we have a WCMY from 3
images and a LRGB from 4 images with equal Luminance and chrominance. The
exposure time units for the LRGB was 1+1+1 and 2(the X component in equation
8) for the no filtered image and is equal to 5 time units.**

**The WCMY exposure time was (1+1+1)
times 1.5 (the 50% increase in exposure time that was necessary to bring
the S/N of the CMY derived RGBs up to the chrominance of S/N of the RGBs
from RGB filters) = 4.5 time units.**

**The Bottom-Line (all things being
equal which they rarely are!):**

**A WCMY with an equal S/N in both
the chrominance and luminance to the S/N of a LRGB will require 3 images
that are a total duration of 90% (4.5/5) as much as the total duration
of the 4 images needed by the LRGB.**

**How does this work in real life?
Below are two images, with equal exposure times, of the Globular Cluster
M3.**

**M3
was imaged with equal exposures using CMY
filters and RGB
filters. The CMY
image shows fainter stars and a greater expanse of the globular M3. Since
each of the CMY
color filters pass two colors of R,
G, and B;
the CMY
image reaches to a fainter magnitude than the RGB
image with equal exposures. Taken with a C14 @ f/7. The RGB
is 5min. R, 10min.
G, 15 min. B.
The CMY
image is 10min each C,
M and Y. **

**............Ed Grafton**

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