Photon noise: is it white?

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

  1. 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
    #1
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  2. Ilya Zakharevich

    Stewy Guest

    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
    #2
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  3. 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
    #3
  4. 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
    #4
  5. Ilya Zakharevich

    acl 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).


    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
    #5
  6. 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
    #6
  7. Ilya Zakharevich

    acl Guest

    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
    #7
  8. Ilya Zakharevich

    Paul Rubin Guest

    (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
    #8
  9. Ilya Zakharevich

    Paul Rubin Guest

    "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
    #9
  10. 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
    #10
  11. 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.
    >>

    >
    >"Traditional" where?


    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
    #11
  12. 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
    #12
  13. Ilya Zakharevich

    acl Guest

    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.
    >>>

    >>"Traditional" where?

    >
    > 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
    #13
  14. Ilya Zakharevich

    Marvin 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 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
    #14
  15. Ilya Zakharevich

    Marvin 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.
    >


    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
    #15
  16. [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
    #16
  17. Ilya Zakharevich

    minnesotti 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 also have a question: what is the diffrerence between "white noise",
    "pink noise" and "grey noise" ?
    minnesotti, Oct 22, 2006
    #17
  18. David J Taylor, Oct 22, 2006
    #18
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