Dynamic Range and Bit Depth

My camera has an engineering Dynamic Range of 14 stops, how many bits do I need to encode that DR?  Well, to encode the whole Dynamic Range 1 bit will suffice.  The reason is simple, dynamic range is only concerned with the extremes, not with tones in between:

    \[ DR = \frac{Maximum Signal}{Minimum Signal} \]

So in theory we only need 1 bit to encode it: zero for minimum signal and one for maximum signal, like so

14 Bit DR

This image is comprised of four squares, two at maximum white and two at deepest black, made up of pixels encoded at 1 bit depth.  Think of it as a fax: 0 for black, 1 for white.  What is its DR?  The answer depends on the physical characteristics of the capture/display medium, not on the bit depth of the information.

If it were displayed on a typical 24″ consumer monitor, the DR of the resulting image would be independent of the 1-bit encoding depth and entirely due to the physical characteristics of the monitor.  The ratio of the brightest to the darkest luminance  such monitors are currently able to produce in cd/m^2 is typically about 250:1 (often referred to as static ‘contrast ratio’), or 8 stops of DR (log_2(250)).

If it were printed on paper the DR of the resulting image would be independent of the 1-bit encoding depth of the information and entirely due to the physical characteristics of the paper and the ink.  The ratio of brightest to darkest reflected luminance in cd/m^2 that can be produced by photographic papers and inks is typically about 100:1, or about 7 stops of DR (log_2(100)).

If the image had been captured by a modern full frame Digital Still Camera,  chances are maximum signal at full scale would have been around 80,000 photoelectrons (e-) and minimum acceptable signal around 5e-, for  an eDR of about 14 stops (log_2(80k/5)).   However, the information to describe this image could have been encoded at 1-bit depth.

Black and white newspaper images are created by more or less densely spaced black dots (ones and zeros), as are grayscale inkjet images, a process known as halftoning.  1-bit Audio ADC/DACs produce dynamic ranges above 100dBs.

All of this to say that Dynamic Range and bit depth are not directly related.  The question then becomes: How many bits do I need to properly encode all tones in the scene? Next.