The useful bit depth

[John Sheehy]

Here is the clipped sigma 5 histogram. It does appear quite different.
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Yes, but it doesn't demonstrate what I was talking about.  The unclipped histogram shows populations for -1, 0, and 1 that are very close for the sigma=5.0, but are radically different for sigma=0.5.  The 5.0 allows for simple interpolation; IOW, if the difference between P and Zc were 10,000, and the population of value 1 was 5000, you'd know that the blackpoint was about 1 ADU off.  This is where my first, mistated criterion comes from, now that I think of it.

[ejmartin]

That seems reasonable, and is in agreement with the result obtained by assuming that the clipped distribution is half-Gaussian.
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BTW, while I correctly gave the contraction in width for a half-gaussian vs a full gaussian as ~1.66, the actual state of affairs is of course a half gaussian on the right plus a delta function at zero containing the weight of the negative values that have been clipped to zero.  For the latter case the factor is ~1.71.  

But the problem is that one cannot know precisely where the clipping has been done, and the noise spectrum is not a pure gaussian anyway.  So I have some doubt as to whether applying such correction factors will give a reliable result.  For instance in the D300 the signal is clipped well to the right of the mean of the noise distribution (over 90% of the raw values are zero up to about ISO 800).  This is why methods using near-black but unclipped noise histograms extrapolated to zero give the most accurate results; no assumptions are being made.

[ejmartin]

Here are ISO 400 histograms of noise vs lower exposure for a D200; green channel data of a Kodak IT8 chart underexposed by three stops.  These are histograms of the grayscale patches along the bottom of the chart. The growing spike at zero as the mean level is reduced is the aggregate weight of the histogram to the left of zero being lumped into a delta function *at* zero by the clipping of black:



The darkest patch (22) approaches the endpoint, which is roughly a half gaussian of unaffected raw values with positive read noise, plus a delta function incorporating all the pixels whose negative raw value has been set to zero by clipping.

From this camera I was getting about 14 electrons of read noise at ISO 100, somewhat in between John's figure and Roger's.

[bjanes]

But the problem is that one cannot know precisely where the clipping has been done, and the noise spectrum is not a pure gaussian anyway.  So I have some doubt as to whether applying such correction factors will give a reliable result.  For instance in the D300 the signal is clipped well to the right of the mean of the noise distribution (over 90% of the raw values are zero up to about ISO 800).  This is why methods using near-black but unclipped noise histograms extrapolated to zero give the most accurate results; no assumptions are being made.
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Emil,

Thanks for the update. Certainly you have convinced me of the validity of your method. However, some will remain unconvinced.

Bill