Noise spectrum

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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?

Thanks,
Tripurari
 
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tsingh@cnds.jhu.edu wrote:

> 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?
>
> Thanks,
> Tripurari
>

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:
http://www.clarkvision.com/imagedetail/does.pixel.size.matter

Roger
 
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Roger N. Clark (change username to rnclark) wrote:
> tsingh@cnds.jhu.edu wrote:
>
> > 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?
> >
> > Thanks,
> > Tripurari
> >
>
> 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.

Ignore the read noise for the sake of argument.

> 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.

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.

Regards,
Tripurari


> Examples with real digital cameras are shown at:
> http://www.clarkvision.com/imagedetail/does.pixel.size.matter
>
> Roger
 
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tsingh@cnds.jhu.edu wrote:
[]
> 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.

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
 
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<tsingh@cnds.jhu.edu> wrote in message
news:1110338452.987131.239290@f14g2000cwb.googlegroups.com...
SNIP
> 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?

That depends on the noise spectrum, but for argument sake let's assume
a White noise power spectrum.

> A good downsizing scheme will filter differently than averaging
> 2x2 pixels. If noise is high-frequency heavy, then the
> downsized image should be cleaner.

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:
<http://www.xs4all.nl/~bvdwolf/temp/Noise_before_and_after_Downsampling.png>
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
 
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tsingh@cnds.jhu.edu wrote:

> 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?
>
> Thanks,
> Tripurari
>
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.
 
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Bart van der Wolf wrote:
> <tsingh@cnds.jhu.edu> wrote in message
> news:1110338452.987131.239290@f14g2000cwb.googlegroups.com...
> SNIP
> > 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?
>
> That depends on the noise spectrum, but for argument sake let's
assume
> a White noise power spectrum.

With a white noise spectrum, there should be no difference between the
two cases.

> > A good downsizing scheme will filter differently than averaging
> > 2x2 pixels. If noise is high-frequency heavy, then the
> > downsized image should be cleaner.
>
> I don't see why it should be high-frequency heavy, the Nyquist
> frequency will still be the limit for the highest frequency.

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.

> 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:
>
<http://www.xs4all.nl/~bvdwolf/temp/Noise_before_and_after_Downsampling.png>
> 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.

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
image?

> 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

-Tripurari
 
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tsingh@cnds.jhu.edu wrote:

> Bart van der Wolf wrote:
>
>><tsingh@cnds.jhu.edu> wrote in message
>>news:1110338452.987131.239290@f14g2000cwb.googlegroups.com...
>>SNIP
>>
>>>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?
>>
>>That depends on the noise spectrum, but for argument sake let's
>
> assume
>
>>a White noise power spectrum.
>
>
> With a white noise spectrum, there should be no difference between the
> two cases.
>
>
>>>A good downsizing scheme will filter differently than averaging
>>>2x2 pixels. If noise is high-frequency heavy, then the
>>>downsized image should be cleaner.
>>
>>I don't see why it should be high-frequency heavy, the Nyquist
>>frequency will still be the limit for the highest frequency.
>
>
> 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.
>
>
>>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:
>>
>
> <http://www.xs4all.nl/~bvdwolf/temp/Noise_before_and_after_Downsampling.png>
>
>>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.
>
>
> 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
> image?
>
>
>>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
>
>
> -Tripurari
>
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.
 
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[A complimentary Cc of this posting was sent to

<tsingh@cnds.jhu.edu>], who wrote in article <1110434818.182043.48340@z14g2000cwz.googlegroups.com>:
> > > 2x2 pixels. If noise is high-frequency heavy, then the
> > > downsized image should be cleaner.
> >
> > I don't see why it should be high-frequency heavy, the Nyquist
> > frequency will still be the limit for the highest frequency.
>
> 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.

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

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
 
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<tsingh@cnds.jhu.edu> wrote in message
news:1110434818.182043.48340@z14g2000cwz.googlegroups.com...
>
> Bart van der Wolf wrote:
>> <tsingh@cnds.jhu.edu> wrote in message
>> news:1110338452.987131.239290@f14g2000cwb.googlegroups.com...
SNIP
> With a white noise spectrum, there should be no difference
> between the two cases.

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.

SNIP
> 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.

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.

> 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 image?

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
 

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