Velocity Reviews > Photon noise: is it white?

# Photon noise: is it white?

Ilya Zakharevich
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Posts: n/a

 10-21-2006
For a long time I lived under impression that photon noise IS white.
Now I thought about it more, and I'm not absolutely sure. I can make
two "English language" arguments, and by one of them the noise is
white, by another its spectrum is (related to) the MTF of the lens.

The problem is that a photon emitted by the source hits several
sensels SIMULTANEOUSLY (distributed by PSP). At most one of the
sensels will register the photon, so THERE IS a correlation between
readings of different sensels. The question is whether this
correlation survives when one considers not one photon, but a Poisson
distribution of photons (possibly emitted by different sources).

The ultimate solution is to consider it as an honest
quantum-mechanical system, and just calculate the answer. I'm too
lazy to do it right now; maybe somebody already KNOWS the answer?

Thanks,
Ilya

Stewy
Guest
Posts: n/a

 10-21-2006
In article <ehbpki\$2duu\$(E-Mail Removed)>,
Ilya Zakharevich <(E-Mail Removed)> wrote:

> For a long time I lived under impression that photon noise IS white.
> Now I thought about it more, and I'm not absolutely sure. I can make
> two "English language" arguments, and by one of them the noise is
> white, by another its spectrum is (related to) the MTF of the lens.
>
> The problem is that a photon emitted by the source hits several
> sensels SIMULTANEOUSLY (distributed by PSP). At most one of the
> sensels will register the photon, so THERE IS a correlation between
> readings of different sensels. The question is whether this
> correlation survives when one considers not one photon, but a Poisson
> distribution of photons (possibly emitted by different sources).
>
> The ultimate solution is to consider it as an honest
> quantum-mechanical system, and just calculate the answer. I'm too
> lazy to do it right now; maybe somebody already KNOWS the answer?
>

Search me mate. Have you tried the star trek engineers?

Kevin McMurtrie
Guest
Posts: n/a

 10-21-2006
In article <ehbpki\$2duu\$(E-Mail Removed)>,
Ilya Zakharevich <(E-Mail Removed)> wrote:

> For a long time I lived under impression that photon noise IS white.
> Now I thought about it more, and I'm not absolutely sure. I can make
> two "English language" arguments, and by one of them the noise is
> white, by another its spectrum is (related to) the MTF of the lens.
>
> The problem is that a photon emitted by the source hits several
> sensels SIMULTANEOUSLY (distributed by PSP). At most one of the
> sensels will register the photon, so THERE IS a correlation between
> readings of different sensels. The question is whether this
> correlation survives when one considers not one photon, but a Poisson
> distribution of photons (possibly emitted by different sources).
>
> The ultimate solution is to consider it as an honest
> quantum-mechanical system, and just calculate the answer. I'm too
> lazy to do it right now; maybe somebody already KNOWS the answer?
>
> Thanks,
> Ilya

That doesn't make any sense. The noise is what eventually assembles
itself into the image over time. The intermediate stages are, well,
noise; random photons that converge towards averages.

Randy Berbaum
Guest
Posts: n/a

 10-21-2006
Ilya Zakharevich <(E-Mail Removed)> wrote:
: For a long time I lived under impression that photon noise IS white.
: Now I thought about it more, and I'm not absolutely sure. I can make
: two "English language" arguments, and by one of them the noise is
: white, by another its spectrum is (related to) the MTF of the lens.

You may be confusing "white noise" which is an electronic signal that
includes many (or all) frequencies with "white" noise which is an
unintended photonic signature that has the characteristics of a white
color.

Noise in the parlance of digital cameras (which is what we talk about
here) is an unexpected or unwanted signal from a light sensitive element
that does not correctly reflect the intensity or color characteristics of
the incoming light from which an image is to be made. In this case "Noise"
can be reflected in an inconsistancy that can be any color, even black
where white is expected.

This noise can come from many sources, some more likely than others.
External electromagnetic waves can cause incorrect signals to be
registered by many kinds of electronic sensors and this includes sensor
chips in a camera. Another source can be improper reflections in the light
path that project light in the wrong place (tho this is normally not
specifically called "noise"). Since this reflected light can be of any
color, it wouldn't be just white. Another source of noise is in the
electronic quantifying and amplification of the signals from the sensors.
if there is some slight variation from sensor to sensor in an array and
these differences are multiplied (as in electronically brightening a dark
image or collecting charges over an extreemly long exposure), slight
differences can drastically multiply. Since each individual sensor element
is reading a single color, the variations can be of many different color.
And if these variations are even slightly influencing several elements the
resulting color can be almost any color reproducable by the camera
sensors.

There are many other possible sources of noise in cameras. Some cameras
may be able to compensate or correct some types of noise better than
others. But there is no camera that can be absolutely immune to all forms
of noise. The best we can hope for is that the visual results of noise can
be minimized by the time we are viewing the finished product.

So tho many engineers may refer to "white noise" it may or may not have
anything to do with a specific color in the visual light spectrum.

Randy

==========
Randy Berbaum
Champaign, IL

acl
Guest
Posts: n/a

 10-21-2006
Ilya Zakharevich wrote:
> For a long time I lived under impression that photon noise IS white.
> Now I thought about it more, and I'm not absolutely sure. I can make
> two "English language" arguments, and by one of them the noise is
> white, by another its spectrum is (related to) the MTF of the lens.
>
> The problem is that a photon emitted by the source hits several
> sensels SIMULTANEOUSLY (distributed by PSP). At most one of the
> sensels will register the photon, so THERE IS a correlation between
> readings of different sensels. The question is whether this
> correlation survives when one considers not one photon, but a Poisson
> distribution of photons (possibly emitted by different sources).

If what you mean is that the photon's wavepacket will, just before
"detection", have a nonzero amplitude over several sensels, then here
is the mistake: The only correlation that will be caused will be
detectable if I shoot a single photon and detect it at some sensel.
Then, I know with certainty it has not been detected anywhere else. It
implies absolutely nothing about correlations between different photons
or different detection events. Why? Because of the Hilbert space
structure of QM, which means that the state vector of many incoming
photons is a linear superposition of the state vectors of individual
photons, and the fact that the evolution equation for this state vector
is linear. This is also true for a random distribution of incoming
photons.

If, instead, you mean that a photons emitted from a given point has a
nonzero prob of being detected over several sensels, due to the PSP,
then you're right and there is a correlation, but it is the trivial
one: all the photons must eventually form an image, so all emitted from
that source will end up within some area on the sensor/film (otherwise,
no image would be formed). I include the effects of difraction etc in
the definition of image.

Maybe you mean something else but I can't interpret your words any
other meaningful way.

> The ultimate solution is to consider it as an honest
> quantum-mechanical system, and just calculate the answer. I'm too
> lazy to do it right now; maybe somebody already KNOWS the answer?

Most anybody who knows quantum mechanics, I'd have thought.

Philip Homburg
Guest
Posts: n/a

 10-21-2006
In article <ehbpki\$2duu\$(E-Mail Removed)>,
Ilya Zakharevich <(E-Mail Removed)> wrote:
>The ultimate solution is to consider it as an honest
>quantum-mechanical system, and just calculate the answer. I'm too
>lazy to do it right now; maybe somebody already KNOWS the answer?

I think it is in the end very simple: the number of photons that are
recorded by a sensor element has an uncertainty of the square root of
that number.

So, if you take many pictures of exactly the same (constant) subject and then
plot the recorded values for one sensor element, you get a Poisson
distribution.

The thing is that traditional white noise is to a large extent independent of
the signal. However, photon noise is part of the signal. Wherever your
signal goes (diffraction, colored filters, etc) photon noise comes with it.

--
That was it. Done. The faulty Monk was turned out into the desert where it
could believe what it liked, including the idea that it had been hard done
by. It was allowed to keep its horse, since horses were so cheap to make.
-- Douglas Adams in Dirk Gently's Holistic Detective Agency

acl
Guest
Posts: n/a

 10-21-2006
Philip Homburg wrote:
> In article <ehbpki\$2duu\$(E-Mail Removed)>,
> Ilya Zakharevich <(E-Mail Removed)> wrote:
> >The ultimate solution is to consider it as an honest
> >quantum-mechanical system, and just calculate the answer. I'm too
> >lazy to do it right now; maybe somebody already KNOWS the answer?

>
> I think it is in the end very simple: the number of photons that are
> recorded by a sensor element has an uncertainty of the square root of
> that number.

What he is arguing is that there is a correlation in the noise detected
by neighbouring sensor elements. He is wrong.

>
> So, if you take many pictures of exactly the same (constant) subject and then
> plot the recorded values for one sensor element, you get a Poisson
> distribution.
>
> The thing is that traditional white noise is to a large extent independent of
> the signal. However, photon noise is part of the signal. Wherever your
> signal goes (diffraction, colored filters, etc) photon noise comes with it.
>

"Traditional" where? There are whole books discussing how noise comes
into the discussion and the distinction between external (due to the
environment, independent of the signal) and internal (what we have eg
here, part and parcel of the "signal") noise. For example, van Kampen,
"Stochastic processes in physics and chemistry", discusses this
distinction (internal vs external) and also gives a way to
systematically incorporate internal noise in the mathematical
description of a system. This has been further extended recently in
several papers by other people. So, to get back to the topic, white
noise does not necessarily refer to external noise, which is what you
seem to be saying.

What you are saying (that the noise goes with the signal) could be
modelled phenomenologically (ie in a hand-waving way) by writing, for
the signal, s(x)+f(x), where x is a 2-d vector, s is the signal and f
is the noise, and specifying a zero-mean normal distribution for f (ie
it is white). This will work because, at this level of approximation,
nothing nonlinear ever happens to the signal (notwithstanding
complicated signal processing). If there was nonlinear processing, we'd
need all the moments of f.

Anyway, I think I discussed this with you some time in the past (or
maybe it wasn't you). And this is going seriously off topic, so I'll
stop.

Paul Rubin
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Posts: n/a

 10-21-2006
http://www.velocityreviews.com/forums/(E-Mail Removed) (Philip Homburg) writes:
> So, if you take many pictures of exactly the same (constant) subject and then
> plot the recorded values for one sensor element, you get a Poisson
> distribution.

Would the noise spectrum then be the same as the Fourier transform of
the Poisson distribution? Since the Poisson distribution approximates
the normal distribution, should it be about equal to its own Fourier
transform? In that case the noise spectrum would be centered at some
particular frequency that depended on the rate of incoming photons.
But when this rate is very low, maybe the approximation fails.

Paul Rubin
Guest
Posts: n/a

 10-21-2006
"acl" <(E-Mail Removed)> writes:
> Anyway, I think I discussed this with you some time in the past (or
> maybe it wasn't you). And this is going seriously off topic, so I'll stop.

The parts of this discussion that I can understand, I find
interesting, and certainly more on-topic than the endless political
diatribes. If you're not bored yet I don't see much reason to stop.

Philip Homburg
Guest
Posts: n/a

 10-21-2006
In article <(E-Mail Removed)>,
Paul Rubin <http://(E-Mail Removed)> wrote:
>(E-Mail Removed) (Philip Homburg) writes:
>> So, if you take many pictures of exactly the same (constant) subject and then
>> plot the recorded values for one sensor element, you get a Poisson
>> distribution.

>
>Would the noise spectrum then be the same as the Fourier transform of
>the Poisson distribution? Since the Poisson distribution approximates
>the normal distribution, should it be about equal to its own Fourier
>transform? In that case the noise spectrum would be centered at some
>particular frequency that depended on the rate of incoming photons.
>But when this rate is very low, maybe the approximation fails.

Remember that in this case, the noise is function of the signal. So if you
want to do a spectrum analysis, you probably want to limit yourself you
uniformly colored areas.

Each value in a sensor element is an indepent (randomized) selection from a
Poisson distribution.

So my guess is that a 2D Fourier transform will show a lot of energy around
the Nuquist frequency.

You can't just use the Fourier transform of the Poisson distribution,
you have to use random selections that follow the Poisson distribution.

--
That was it. Done. The faulty Monk was turned out into the desert where it
could believe what it liked, including the idea that it had been hard done
by. It was allowed to keep its horse, since horses were so cheap to make.
-- Douglas Adams in Dirk Gently's Holistic Detective Agency