What is the optimal estimator?

Fred Marshall <fmarshallxremove_the_x@acm.org> wrote:

(snip)
> Since you mentioned pixels, which are usually only positive-valued, I'd 
> understand better if you were to tell us if the noise is zero-mean 
> before the sampling/digitizing limits are imposed. I suspect that's the 
> case but it's not clear.

(snip)
> DC value at 128 before sampling / quantization.
> Noise zero mean with variance of 90 is added to the DC value.
> This means that the noisy data is both positive and negative at times.

If the noise is that big, then there might not be much image left.

Maybe I don't understand CCD sensor quite well enough to say,
but as you increase the ISO value, as many cameras allow, you
increase the noise level in the image.  (Well, decrease the signal,
and then adjust the gain.)  

Some of the noise is in the quantization of electrons in the 
pixel transistor.  The transistor is charged to a known value,
some charge leaves as photocurrent, and the result is measured.
For small changes, the statistical distribution of electrons left
is noise, but the noise depends on the signal.  There can't be
a negative number, either.

Also, as I understand it, there is a ROM giving the properties
(such as gain) for each pixel, such that they can be normalized
to allow for variations in fabrication for each one.  That might
mean that the A/D converter has a wider range than the result,
such that clipping to 0 and 255 is done after normalization.

> Now sample and quantize between 0 and 255.
> The negative-going noise is limited at 0.
> The positive-going noise is limited at 255.
> The mean of the data before sampling is 128.
> Since it's 128 then the mean after sampling is apparently also 128.

As with audio signals, you usually don't want to clip.  
Since you can't get darker than no light, maybe it doesn't matter
on that end, but it is best not to reach 255.  (Though for very
white parts, maybe it doesn't matter so much.)

> But, if the data were something different, say 64 and the noise is as 
> above, then after sampling there will be more sample points at 0 
> than at 255.  The mean before sampling will be 64.  The mean 
> after sampling will be something else I do believe.

> Does this match with your situation?

-- glen

On Dec 7, 3:41=A0pm, Fred Marshall <fmarshallxremove_th...@acm.org>
wrote:
> On 12/6/2011 3:18 PM, zqchen wrote:
>
>
>
> > Is there any problem in my question?
>
> Since you mentioned pixels, which are usually only positive-valued, I'd
> understand better if you were to tell us if the noise is zero-mean
> before the sampling/digitizing limits are imposed. I suspect that's the
> case but it's not clear.
>
> Example:
>
> DC value at 128 before sampling / quantization.
> Noise zero mean with variance of 90 is added to the DC value.
> This means that the noisy data is both positive and negative at times.
>
> Now sample and quantize between 0 and 255.
> The negative-going noise is limited at 0.
> The positive-going noise is limited at 255.
> The mean of the data before sampling is 128.
> Since it's 128 then the mean after sampling is apparently also 128.
>
> But, if the data were something different, say 64 and the noise is as
> above, then after sampling there will be more sample points at 0 than at
> 255. =A0The mean before sampling will be 64. =A0The mean after sampling w=
ill
> be something else I do believe.
>
> Does this match with your situation?
>
> Fred

Yes, it's what I imagined an image could be formed. I'm not sure
whether it reflects the fact of physics or not. English is not my
native language, so sorry for the ambiguity.

On 7 Des, 12:22, zqchen <zhiqun.c...@gmail.com> wrote:
> On Dec 7, 3:41=A0pm, Fred Marshall <fmarshallxremove_th...@acm.org>
> wrote:
>
>
>
>
>
> > On 12/6/2011 3:18 PM, zqchen wrote:
>
> > > Is there any problem in my question?
>
> > Since you mentioned pixels, which are usually only positive-valued, I'd
> > understand better if you were to tell us if the noise is zero-mean
> > before the sampling/digitizing limits are imposed. I suspect that's the
> > case but it's not clear.
>
> > Example:
>
> > DC value at 128 before sampling / quantization.
> > Noise zero mean with variance of 90 is added to the DC value.
> > This means that the noisy data is both positive and negative at times.
>
> > Now sample and quantize between 0 and 255.
> > The negative-going noise is limited at 0.
> > The positive-going noise is limited at 255.
> > The mean of the data before sampling is 128.
> > Since it's 128 then the mean after sampling is apparently also 128.
>
> > But, if the data were something different, say 64 and the noise is as
> > above, then after sampling there will be more sample points at 0 than a=
t
> > 255. =A0The mean before sampling will be 64. =A0The mean after sampling=
 will
> > be something else I do believe.
>
> > Does this match with your situation?
>
> > Fred
>
> Yes, it's what I imagined an image could be formed. I'm not sure
> whether it reflects the fact of physics or not. English is not my
> native language, so sorry for the ambiguity.

The ambiguity is not in the language, but in the
topic of your stated question.

The estimators you ask for are textbook material
if you find the simple Gaussian model acceptable.
Since you *don't* want to use the estimators for
the Gaussian case, we naturally assume that there
is something about the stated problem that prevents
you from using the Gaussian model. Most of the
discussion has been about what that 'something'
might be, and why it would be important.

Rune

On 12/7/2011 3:22 AM, zqchen wrote:
> On Dec 7, 3:41 pm, Fred Marshall<fmarshallxremove_th...@acm.org>
> wrote:
>> On 12/6/2011 3:18 PM, zqchen wrote:
>>
>>
>>
>>> Is there any problem in my question?
>>
>> Since you mentioned pixels, which are usually only positive-valued, I'd
>> understand better if you were to tell us if the noise is zero-mean
>> before the sampling/digitizing limits are imposed. I suspect that's the
>> case but it's not clear.
>>
>> Example:
>>
>> DC value at 128 before sampling / quantization.
>> Noise zero mean with variance of 90 is added to the DC value.
>> This means that the noisy data is both positive and negative at times.
>>
>> Now sample and quantize between 0 and 255.
>> The negative-going noise is limited at 0.
>> The positive-going noise is limited at 255.
>> The mean of the data before sampling is 128.
>> Since it's 128 then the mean after sampling is apparently also 128.
>>
>> But, if the data were something different, say 64 and the noise is as
>> above, then after sampling there will be more sample points at 0 than at
>> 255.  The mean before sampling will be 64.  The mean after sampling will
>> be something else I do believe.
>>
>> Does this match with your situation?
>>
>> Fred
>
> Yes, it's what I imagined an image could be formed. I'm not sure
> whether it reflects the fact of physics or not. English is not my
> native language, so sorry for the ambiguity.

zqchen,

OK.  So, I'll paraphrase that back.  I'm not sure that I presented it 
all that well:

- There is an image sensor whose output is noiseless (for the sake of 
analysis).  All of the output is magnitude information, thus positive.
- Zero-mean Gaussian noise is added to the output.  The resulting 
composite can have negative values as a result.
- the composite signal S + N is sampled and quantized from 0 to 255.

The resulting quantized output value distribution:
- peaks at a value closest to the image sensor output (and because the 
noise is zero-mean).
- has "spikes" at 0 and 255 due to the tails of the noise distribution.
- the lower the sensor output value, the higher the "spike" at zero and 
vice-versa.

I hope this helps clarify .. if I got it right.

One of the nagging problems I've had with this is that the time frame 
hasn't been stated.  This raises some issues in a practical sense:

- can the image data change during the analysis period?  If not then the 
notion of lowpass filters, etc. works better but the potential of the 
spikes at 0 and 255 will perturb the output.  But, as someone mentioned, 
that would be a *really* noisy S + N !!  So maybe the end spikes are 
only worthy of analytical mention and of no practical importance.

Fred