# Photon noise: is it white?

Discussion in 'Digital Photography' started by Ilya Zakharevich, Oct 21, 2006.

1. ### Ilya ZakharevichGuest

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

Ilya Zakharevich, Oct 21, 2006

2. ### StewyGuest

In article <ehbpki\$2duu\$>,
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).
>
> 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?

Stewy, Oct 21, 2006

3. ### Kevin McMurtrieGuest

In article <ehbpki\$2duu\$>,
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).
>
> 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.

Kevin McMurtrie, Oct 21, 2006
4. ### Randy BerbaumGuest

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.

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

Randy Berbaum, Oct 21, 2006
5. ### aclGuest

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.

acl, Oct 21, 2006
6. ### Philip HomburgGuest

In article <ehbpki\$2duu\$>,
Ilya Zakharevich <> 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

Philip Homburg, Oct 21, 2006
7. ### aclGuest

Philip Homburg wrote:
> In article <ehbpki\$2duu\$>,
> Ilya Zakharevich <> 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.

acl, Oct 21, 2006
8. ### Paul RubinGuest

(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, Oct 21, 2006
9. ### Paul RubinGuest

"acl" <> 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.

Paul Rubin, Oct 21, 2006
10. ### Philip HomburgGuest

In article <>,
Paul Rubin <http://> wrote:
> (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

Philip Homburg, Oct 21, 2006
11. ### Philip HomburgGuest

In article <>,
acl <> wrote:
>Philip Homburg wrote:
>> In article <ehbpki\$2duu\$>,
>> Ilya Zakharevich <> 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.

In practice there will be a correlation. Most images have relatively
little energy around Nyquist. So, neighboring sensor elements will have
'similar' values, which in turn results in similar photon noise.

Of course, in a Bayer pattern sensor, you have to consider neighboring
elements of the same color.

>> 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.
>>

>

I don't know where you see sources of white noise. I see them mostly in
electronics, and not much in other areas.

>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.

Fortunately, optics tends to be linear.

>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.

So, to get back on track, is 'f(x)' white or not?

--
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

Philip Homburg, Oct 21, 2006
12. ### Roger N. Clark (change username to rnclark)Guest

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).
>
> 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

Start here
http://www.imatest.com/docs/noise.html

then study Poission statistics.

Roger

Roger N. Clark (change username to rnclark), Oct 21, 2006
13. ### aclGuest

Philip Homburg wrote:
> In article <>,
> acl <> wrote:
>>What he is arguing is that there is a correlation in the noise detected
>>by neighbouring sensor elements. He is wrong.

>
> In practice there will be a correlation. Most images have relatively
> little energy around Nyquist. So, neighboring sensor elements will have
> 'similar' values, which in turn results in similar photon noise.

Well yes, but that is because of the characteristics of the image
(neighbouring areas have similar intensities). This is not what he is
arguing, as far as I can tell. If I take what he says literally, he is
arguing that there is correlation because of the PSP, ie, because of the
wave-like nature of light (ie mostly because of diffraction, in this
case). Well yes there is. This is why there's an image formed. So yes,
as you say, this will result in in neighbouring areas to be correlated,
but this is trivial. That's why in my other post I specified that I
include things like diffraction etc in the definition of "image".

>
> Of course, in a Bayer pattern sensor, you have to consider neighboring
> elements of the same color.
>
>>>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.
>>>

>
> I don't know where you see sources of white noise. I see them mostly in
> electronics, and not much in other areas.

Maybe you mostly look at electronics. In, for instance, physics, almost
nothing else is used except white noise. And when coloured noise
appears, it may be eliminated in favour of twice the number of equations
and white noise only; this is what people usually do to study such
systems. Also in theoretical biology, etc.

Think, for example, of the statistics of particles arriving on a surface
randomly (eg deposited from above) and the morphology of the resulting
configuration. There is a subsubfield of physics dealing with this,
motivated by molecular beam epitaxy, and which usually aims to derive
and study stochastic differential equations describing the growth of the
resulting pile of particles. There, there is intrinsic noise (same
reasons as for photons), and the necessary stochastic equations involve
white noise. This is derived, as opposed to postulated (but this was
correctly done only very recently).

Anyway, there is a very large body of literature in physics dealing with
processes with white noise. There is no question about that.

>
>>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.

>
> Fortunately, optics tends to be linear.
>
>>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.

>
> So, to get back on track, is 'f(x)' white or not?

I had said it's normally distributed. White noise=noise with power in
all frequencies, so yes, it's white, uncorrelated noise. On second
thought, though, maybe it's better to use s(x)+sqrt(s) f(x), so that the
average of the square of the noise is proportional to the signal itself.
f is still white and uncorrelated here.

But this is hand waving. I'm sure it will give the right answers,
though, and that it can be derived as follows: We know the photon
arrival times are governed by Poisson stats. Given the exposure time and
the intensity of the signal, we can easily find the magnitude of the
standard deviation of the noise for each site. It will be something like
what I wrote above, I suppose. But I am drowning in work at the moment
and cannot actually check details.

By the way, to avoid confusion, what people normally mean with
Poisson-distributed "shot noise" for photons is a Poisson process in
time. It's different than the s(x) I have above, which models the fact
that noise is present and uncorrelated across pixels. Well, except
through the dependence on s.

acl, Oct 21, 2006
14. ### MarvinGuest

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 noise arises mainly in the sensor, so it has nothing to
do with the lens. Technically, it is described as "white",
meaning that the noise pulses are randomly distributed in
time. The confusion comes from having two meanings for
"white", depending on the context.

Marvin, Oct 21, 2006
15. ### MarvinGuest

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.
>

A correction to my posting a few minutes ago. I said that
white noise is random with time. That is true for photon
counting detectors. For the kind of detectors in digicams,
it is more significant that the noise pulses are randomly
distributed in amplitude. In some cases, the pulses become
systematically less frequent as the amplitude increases, and
it may be called 1/f noise, meaning that the number of
pulses per unit time is inversely proportional to the amplitude.

Marvin, Oct 21, 2006
16. ### Ilya ZakharevichGuest

[Answer] Photon noise: is it white? - YES

[A complimentary Cc of this posting was NOT [per weedlist] sent to
Ilya Zakharevich
<>], who wrote in article <ehbpki\$2duu\$>:
> 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.

Thanks to everybody how answered; however, even the people who
actually understand what is a "white noise" and "correlation" could
only make hand-waving arguments; AFAIU, nobody could claim a precise
answer... Hand-wave can even I; the problem was that I could wave in
two different ways. ;-)

Anyway, I made the calculation, and I can sleep well again: the noise
is indeed white, as I was supposing all the time. The calculation
attached.

Thanks again,
Ilya

=======================================================

Consider an image of an evenly lighted while surface created by a lens.
Is the photon noise in the image white? In other words, is there a
correlation between photon count in the (neighboring) pixels?

It turns out that photon noise is indeed white, notwithstanding the
correlation between sensels when individual photons hit the sensor.

To simplify the situation so that analysis is handy, consider a sensor
with two sensels, A and B; model the evenly lighted white source by two
point sources, sA and sB. Assume sA is focused on sensel A, sB is
focused on sensel B. Assume that photons from sA hit sensel A with
probability (1+p)/2 and hit sensel B with probability (1-p)/2; likewise
for sB - with A and B exchanged.

What we need to do is to calculate the distribution of the sum S of
readings of sensels A and B, and the difference D between these readings.
If the photon noise is white, these distributions should be the similar.
S is obvious; I do not discuss it here.

It turns out that it makes sense to break D into two parts, D = D0 + D';
here each photon from sA increases D0 by p, each photon from sB decreases
D0 by p. Then expectation of D' is 0, the sigma^2 contributed by
each incoming photon is Var(D') = 1 - p^2.

Assume that sA emits N + Na photons, sB emits N + Nb photons, with
N being the expectation, and Na, Nb variations (of order of magnitude
sqrt(N)). Then D0 is p*(Na - Nb), and, for given values of Na and Nb,
Var(D') = (2N + Na + Nb)(1 - p^2).

The key step is that although D0 and D' are not independent, this
interdependence does not contribute into Var(D). Indeed,

Var(D) = Var(D0) + Var(D') + 2 Expectation(D0 * D').

To show that the last term vanishes, it is enough to show that
the conditional distribution of D' for the given value of D0 (i.e.,
of Na - Nb) has expectation 0; however, it has conditional expectation
0 even for fixed values of both Na and Nb.

Finally,
Var(D0) = p^2 * Var(Na - Nb) = p^2 * 2N,
Var(D') = (1 - p^2)(2N + Expectation(Na + Nb)) = (1 - p^2) * 2N.

Thus Var(D) is 2N, as expected for the white noise.

Ilya Zakharevich, Oct 22, 2006
17. ### minnesottiGuest

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 also have a question: what is the diffrerence between "white noise",
"pink noise" and "grey noise" ?

minnesotti, Oct 22, 2006
18. ### David J TaylorGuest

David J Taylor, Oct 22, 2006