# How to understand the definition of dynamic range?

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.

• This question has strayed off-topic for this site. It's no longer about photography as such; it has become a question belonging in the physics, math or electronics SE sites, so I have voted to close.
– user2719
Jul 31, 2012 at 12:21

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.