Noise spectrum

Discussion in 'Digital Photography' started by tsingh, Mar 8, 2005.

  1. tsingh

    tsingh Guest

    Many years ago, when I was a graduate student, I had read that noise
    was concentrated in the higher frequencies. Is this true of
    contemporary sensors?
    Also, is the noise in a pixel (fairly) independent of the noise in
    other pixels?

    Now consider the following (brute force) scheme for producing low noise
    images of a given resolution: oversample the image with a higher
    resolution sensor, filter out the resulting higher frequencies and
    scale down. If noise is independent across pixels, this should yield a
    lower noise image than a lower resolution sensor would. How far can you
    take this process, asymptotically?

    As a concrete example, consider two 2 megapixel images - one from a 2M
    sensor and another from a 4M sensor of the same size. Downsize the 4M
    image to 2M using a decent algorithm like Lanczos. Which image will
    have lower noise?

    tsingh, Mar 8, 2005
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  2. Unfortunately, you can't win this way. The reasons are:
    the 2M sensor would have larger pixels, thus gather
    more photons (in proportional to the area). The noise
    from modern digital cameras is photon noise limited
    at high signal levels and sensor read noise limited
    at low levels. In your example, the signal-to-noise
    of the 4M sensor would be half that of the 2M sensor
    in each pixel. In that case, there is no theoretical
    difference. But the read noise is relatively constant
    for the two sensors (typical values are 10 to 20 electrons),
    so the read noise is a higher proportion of the signal on
    the 4M sensor, thus overall the 4M sensor with the smaller
    pixels will be noisier, even if you averaged 2x2 pixels.

    Examples with real digital cameras are shown at:

    Roger N. Clark (change username to rnclark), Mar 9, 2005
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  3. tsingh

    tsingh Guest

    Ignore the read noise for the sake of argument.
    If you average 2x2 pixels, you'd be comparing a 2M with an 8M image,
    and not a 4M image. Let us stick with 8M as it is more convenient.

    Sure, the 8M will be noisier, but what fraction of that noise will be
    filtered out when downsizing? A good downsizing scheme will filter
    differently than averaging 2x2 pixels. If noise is high-frequency
    heavy, then the downsized image should be cleaner.

    tsingh, Mar 9, 2005
  4. You need to factor in:

    - the 8MP pixels will be smaller, and the readout circuitry may occupy a
    greater fraction of the available area, making capture less efficient.

    - how the eye responds to noise of different spatial frequencies at
    different viewing distances.

    - what the minimum resolveable contrast (or modulation) is for the eye at
    a given spatial frequency.

    I suspect that people who view prints close up may say: the 8MP is noiser,
    but that people who view at "normal" viewing distances (arm's length for a
    10 x 8 inch print) may say: the 8MP looks sharper. Between 2MP and
    resampled 8MP - that would be good test to try!

    David J Taylor, Mar 9, 2005
  5. SNIP
    That depends on the noise spectrum, but for argument sake let's assume
    a White noise power spectrum.
    I don't see why it should be high-frequency heavy, the Nyquist
    frequency will still be the limit for the highest frequency. I made
    the following empirical evaluation, using the Imatest program. A test
    object (slanted edge for SFR/MTF measurement) without noise was
    Gaussian blurred in Photoshop with 0.5 radius. That was to model a
    resolution limited linear gamma image. I added a Uniform (white) noise
    of 4%. That was down-sampled with ImageMagick's Sinc filtered resize
    routine to 50% of each dimension.

    This is the result for the three images:
    Image 1 at the top: Original resolution limited image has perfect S/N,
    Image 2 in the middle: Noise added results in flat noise spectrum,
    Image 3 at the bottom: Down sampled image still has a relatively flat
    spectrum, although a bit high frequency attenuated due to
    anti-aliasing Sinc prefilter, but the S/N is 2.4x as good which is
    even better than the theoretical 2x for 4 samples due to change in
    noise spectrum and MTF.

    A better model would also include Poisson distributed Photon shot
    noise, but it would make it more difficult to see the single effect of
    downsampling because it varies with luminance as well.

    Bart van der Wolf, Mar 9, 2005
  6. There are LOTS of sources of noise. Many are independent of frequency.
    In a scanned sensor the time spectrum has an effect. In a mosaiced
    sensor, however, like a CCD, the temporal makeup of the noise is
    unimportant. Each pixel is essentially a seperate sample. So in terms
    of spatial frequency the noise content is indeed a high frequency
    component. This is true for things like electronic noise.

    If we consider things like flare and ghosts a 'noise', these will,
    however, have lower spatial frequency values.
    Don Stauffer in Minneapolis, Mar 9, 2005
  7. tsingh

    tsingh Guest

    With a white noise spectrum, there should be no difference between the
    two cases.
    I don't know why it should be, but I was under the impression that it
    was. That was many years ago, I don't know if it is still true.
    Thanks for the detailed study. The two cases should be identical.
    Perhaps sharpening the down sampled image to compensate for he
    pre-filter will also reduce the S/N down to 2.

    This also raises a separate question - will the 8M down sampled image
    have higher SNR than the 2M image because the camera's antialiasing
    filter won't suppress the lower frequencies need to generate the 2M
    tsingh, Mar 10, 2005
  8. [A complimentary Cc of this posting was sent to

    Probably you were looking for the noise in log-frequency scale; with
    (creative) area-preserving log-scaling you may get an impression of

    E.g., there is 2x as much white noise power between 10KHz and 20KHz
    than between 5KHz and 10KHz (taking aural analogy); but these
    intervals are the same in log-scale.

    Hope this helps,
    Ilya Zakharevich, Mar 10, 2005
  9. In theory that might be the case, but the downsampling process makes
    assumptions and suffers from limitations such as quantization errors
    and non-infinite size. There are also things like the Gibbs phenomenon
    (ringing) that will amplify and attenuate certain frequencies near the
    Nyquist limit. That's why I determined the results empirically.

    Yes, it is common to sharpen after resampling in digital imaging, but
    it will also increase the aliasing potential. That will manifest
    itself as jagged lines and the finest noise will become more visible
    because the aliases are imaged as larger noise structure. Therefore a
    very small radius USM, e.g. amount 300 radius 0.3, and a threshold
    suited for the image content is commonly applied, preferably with an
    edge only mask.
    The sensor's dark current and readout noise is added independent of
    the photon signal, which is the only signal that's modified by the
    optical AA-filter. That's what I also did with my test. The image was
    pre-blurred first, the noise was then added.

    However, there are also other considerations like dynamic range, that
    will be worse for the smaller sensor elements. Also, smaller elements
    require better lenses to deliver actual resolution. Down-sampling does
    significantly improve the MTF of the image and the S/N ratio, but the
    2MP image loses resolution versus the 8MP image for same size output,
    because it requires more magnification.

    In my example the resolution improved between 35-65% (depending on the
    criterion), so magnifying that downsampled image by 2 again, would
    reduce the resolution to less of what it was. If the final image needs
    no magnification, then downsampling only brings quality benefits.

    Bart van der Wolf, Mar 10, 2005
  10. I think in these discussions, when we use the term spectrum, we need to
    clarify whether we are talking spatial spectrum or temporal ones.
    Ordinarily in speaking of electronic noise, we are referring to temperal
    spectrum. However, when we view a still photograph, it is the spatial
    spectrum, or spatial frequencies, we view.

    In a scanning sensor, like a vidicon or mirror scanned IR sensors, there
    IS a relationship between temporal and spatial spectra. In a mosaic
    sensor, however, there is usually no relationship.
    Don Stauffer in Minneapolis, Mar 10, 2005
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