In early analyses of ETTR, it was theorized that the benefits were derived by pushing up the exposure towards sensor saturation where more levels would be available in the digital capture inasmuch as half the levels are in the brightest f/stop of the image at saturation. Later analysis (such as that of Emil Martinec
) showed that the advantage was in a higher signal:noise rather then the number of levels. It makes no sense to quantize the data in finer steps than the noise at that level. At higher levels of exposure, noise in a digital capture is largely photon noise which follows a Poisson distribution where the noise is the square root of the number of photons collected. Camera data numbers (ADUs) are related to the number of photo electrons by the gain, which is the number of photons/ADU. For a sensor with a full well of 50,000 electrons, the gain (assuming that amplification is such that the full range of the ADC (analog to digital converter) is used at the full well of the sensor) would be 3.05 e-/ADU for 14 bit quantization and 12.21 e-/ADU for 12 bit.
For a highlight captured with this sensor with 40,000 electrons, the noise would be 200 e-. With this sensor one level would be 3 e- with 14 bit quantization and 12 e- for 12 bits, and it is apparent that the finer quantization is wasted. This is the basis for compressed NEF raw files used by Nikon, which records 2753 levels for 14 bits and 689 levels for 12 bit (according to Emil) with no visual loss, visually lossless in Nikon terminology. The number of levels that can be distinguished by human perception is described by the Weber-Fechner law (see Norman Koren
for details). According to this law, visual perception is sensitive to relative differences of illumination of about 1%, so 1 f/stop would be about 70 levels (1.01 ^ 70 = 2.0). Accordingly, only 70 levels would be needed for the brightest f/stop, but more levels would be advisable to allow for subsequent editing. These considerations underlie the DXO measurement of tonal range
, which is the effective number of gray levels the system can produce. For example, the tonal range of the Nikon D800 is far less than 14 bits (readers can look this up, the DXO site is not responding at the time of this writing).
The benefit of a higher bit level is in the shadows where a change in signal from 1 to 2 ADUs would represent an increase of 100%, whereas a change from 14,000 to 14,001 ADUs is insignificant.
Discussion as to the practical effect of these theoretical considerations is welcome.