Re: Noise levels as a function of pixel size

Ilya Zakharevich wrote:

[A complimentary Cc of this posting was NOT [per weedlist] sent to
Kennedy McEwen <rkm@xxxxxxxxxxxxxxxxxxxx>], who wrote in article <eu4tEzFbXLqDFw2h@xxxxxxxxxxxxxxxxxxxx>:

Photographically it doesn't make any sense at all to reduce the pixel below about 3um pitch. 


There is a physical limit, but it is much more than an order of
magnitude below the size you claim.

The wavelength of the spacial cut-off frequency of a lens with
incidence angle phi is

  lambda/(2*sin(phi)*n)

here lambda is wavelength, n is the refraction coefficient at the
focus plane.  The sensor which will collect ALL the information
available in the focal plane should have the step 1/2 of this.  This gives

  lambda/(4*sin(phi)*n)

for the limit sensor pitch.  I do not know n for silicon; however, if
there is an air gap between the last optical element and the sensor,
then n should be replaced by 1.  Anyway, assume n=1.6 for silicon
(this is closer to one for SiO2).  For lambda=0.55 micron, and phi
close to 90 degrees, one gets the absolute limit for pitch of 0.086
microns.

Even with phi about 30 degrees (f/1 lens), and n=1 one gets
0.275microns.  This is not "unthought of", but most probably still
outside of practical range of lenses.  What is definitely easy to do
today is a cheap about-f/2 diffraction-limited lens (in about 2/3''
formfactor); it requires 0.55micron pitch.

Your errors are that

  a) you consider Airy disk which has very little to do with
     intensively post-processed digital photography (as opposed to
     Nyquist limit of the cut-off frequency), and

  b) take estimates for useful f-numbers from larger format lenses.
     Since a price of a lens element is about 4th or 5th degree of its
     size, much better lenses become economically feasible for smaller
     formfactors.

     (As a limit example, having little to do with photography: in 1mm
     focal lens, one can get diffraction-limited sin(phi)=0.93 very
     cheap today - I estimate the price of pennies with mass
     production.)

Hope this helps,
Ilya


There are several factors that are being ignored in this discussion.

1) To do color photography, you need to focus the visible spectrum
onto the pixel.  That means from about 0.38 to 0.7 microns.

2) You can't use pixel sizes less than the wavelength of light
(0.7 microns for color photography).  The equations for diffraction
spot size you have been using are for conditions resulting in larger
than the wavelength of light for the diffraction disk.  When you
approach the wavelength of light (this means above the wavelength),
more complex equations will apply.

2a) As pixel size becomes smaller, other problems arise in the
sensor, including leakage of electrons into adjacent wells.

3) You state that you can get these fast lenses with great performance
in the small form factor.  This may be true for monochromatic
wavelengths, but fast f/ratio lenses in the small form factor
notoriously have poor chromatic aberration.  You state that small
format cameras have great resolution, but I would argue that, and
when you consider other aberrations, like chromatic, image quality
is poor compared to good lenses in larger formats that don't need
to be as fast.

4) Realistic quantum efficiencies can be found from references on my
web pages.  Even though I do not derive quantum efficiencies, you still
find the space to take shots at my work yet you have never given
a reference that show any of it is wrong.  Quantum efficiencies
are summarized here, for example, see Figure 1:
http://huhepl.harvard.edu/~LSST/general/Janesick_paper_2003.pdf

For the discussion on detection and signal-to-noise ratios, here is one
I co-authored that has some relevance, using a least squares algorithm:

Swayze, G.A., Clark, R.N., Goetz, F.H., Chrien, T.G., and Gorelick,
N.S., 2003, Effects of spectrometer band pass, sampling, and
signal-to-noise ratio on spectral identification using the
Tetracorder algorithm: Journal of Geophysical Research (Planets),
vol. 108(E9), 5105, doi: 10.1029/2002JE001975, 30 p.

This paper analyzes detections in imaging data using receiver-operator-
characteristic (ROC) curves.  The relevant result to this discussion
is shown in Figure 21, which concludes a 50% error of correct
detection per pixel at a signal-to-noise ratio of 5.  With multiple
pixels, you can detect lower signal-to-noise targets.

Concerning a read noise of 3 electrons, and a signal to noise of
9, the Poisson error in the signal is square root 9 = 3, thus the
combined noise in the signal is 4.2, not 3 as previously discussed.
Thus the signal-to-noise ratio is only 2.1.

There has been an implied discussion in this thread that smaller cameras
can match the resolution (in pixels per picture height) a larger
camera.  The absurdity of this concept is well known by all those
photographers who have moved to larger formats to get better image
quality.  4x5 images have stunning detail compared to 35mm, and
35mm format will always be better than cell phones with 2 mm sensors.

Roger

.

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