How to understand the definition of dynamic range?

Question

From "The ISO Definition of the Dynamic Range of a Digital Still Camera", I have the following formula:

$$ \mathrm{DR} = \frac{1.4L_\mathrm{sat}G_\mathrm{i}}{N_\mathrm{D}} $$

where

• \$\mathrm{DR}\$ is the dynamic range;
• \$L_\mathrm{sat}\$ is the luminance at the “margin level”，70.1% of the maximum recordable（saturation）luminance（yes，\$L_\mathrm{sat}\$ is a peculiar symbol for that);
• \$G_\mathrm{i}\$ is the incremental gain at the reference black luminance;
• \$N_\mathrm{D}\$ is the measured digital noise at the reference black luminance

But \$L_\mathrm{sat}\$ is 100/140 (1/1.4) of the actual saturation luminance (which we call here \$L_\mathrm{clip}\$).

Thus we can recast equation 2 as:

$$ \mathrm{DR} = \frac{L_\mathrm{clip}G_\mathrm{i}}{N_\mathrm{D}} $$

I cannot understand this \$L_\mathrm{clip}\$ and \$L_\mathrm{sat}\$ and the relationship between them.

Answer 1

I have found the following interesting reference "The ISO Definition of the Dynamic Range of a Digital Still Camera" which, at pages 6, clarifies your formula.

It is based on the assumption that a useful signal is one for which the Signal-to-Noise ratio is >= 1.

This luminance level is your denominator \$N_\mathrm{D}\$. The numerator is "the maximum luminance that receives a unique coded representation (the “saturation” luminance)", \$L_\mathrm{clip}\$ in your notation. But digital luminance is not equal to luminance (The famous "gamma correction"). This slope factor is the \$G_\mathrm{i}\$ (the incremental gain).

\$L_\mathrm{sat}\$ and \$L_\mathrm{clip}\$ are directly related: if one is known, the other follows by simply multiplying (or dividing) by a factor. It is a matter of definition.

The 1.4 factor (roughly the inverse of 70%, as you note) provides a sort of a buffer (the paper calls it "the well-known so-called 'half stop margin' against overexposure").

The ratio of:

• (numerator) Gamma corrected maximum luminance
• (denominator) lowest luminance level where the SNR is >= 1

gives the output, the dynamic range of the sensor. It is (as it must be) based on a series of convention: but if you apply them consistently to various sensor you can numerically describe them according to this metric.